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# The affine Springer fiber – sheaf correspondence

Eugene Gorsky  Department of Mathematics, UC Davis<EMAIL_ADDRESS>,  Oscar Kivinen  Institute of Mathematics, École polytechnique fédérale de
Lausanne<EMAIL_ADDRESS>and  Alexei Oblomkov  Department of
Mathematics, UMass Amherst<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

Given a semisimple element in the loop Lie algebra of a reductive group, we
construct a quasi-coherent sheaf on a partial resolution of the trigonometric
commuting variety of the Langlands dual group. The construction uses affine
Springer theory and can be thought of as an incarnation of 3d mirror symmetry.
For the group $GL_{n}$, the corresponding partial resolution is
$\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}})$. We also
consider a quantization of this construction for homogeneous elements.

###### Contents

  1. 1 Introduction
    1. 1.1 3d Mirror Symmetry
    2. 1.2 Conjectures
    3. 1.3 Relation to conjectures stemming from knot theory
    4. 1.4 Outline
      1. 1.4.1 Outline of the argument
      2. 1.4.2 Outline of the paper
  2. 2 Affine Springer fibers
    1. 2.1 Definitions
    2. 2.2 The Springer action
    3. 2.3 Extended symmetry
    4. 2.4 The lattice action
    5. 2.5 Equivariant versions, endoscopy
  3. 3 The commuting variety
    1. 3.1 The commuting scheme
    2. 3.2 Partial resolutions
    3. 3.3 The universal centralizer
    4. 3.4 Explicit antisymmetric polynomials
  4. 4 Trigonometric Cherednik algebra
    1. 4.1 Definitions
    2. 4.2 Shift isomorphism
    3. 4.3 ${\mathbb{Z}}$-algebras
    4. 4.4 ${\mathbb{Z}}$-algebras from Cherednik algebras
    5. 4.5 ${\mathbb{Z}}$-algebra for $GL_{n}$
  5. 5 Coulomb branches and ${\mathbb{Z}}$-algebras
    1. 5.1 Coulomb branches
    2. 5.2 A category of line defects
    3. 5.3 ${\mathbb{Z}}$-algebras and the flavor deformation
    4. 5.4 ${\mathbb{Z}}$-algebras in the abelian case
  6. 6 Coulomb branches and ${\mathbb{Z}}$-algebras in the adjoint case
    1. 6.1 From Coulomb branch to Cherednik algebra
    2. 6.2 Localization of the spherical algebra in the adjoint case
    3. 6.3 Factorization of bimodules
    4. 6.4 The geometric ${\mathbb{Z}}$-algebra for the adjoint representation
    5. 6.5 A flag ${\mathbb{Z}}$-algebra
      1. 6.5.1 Bimodules
  7. 7 Generalized affine Springer theory
    1. 7.1 Generalized affine Springer fibers
    2. 7.2 Springer action from the Coulomb perspective
    3. 7.3 The adjoint case
    4. 7.4 Action on representations
  8. 8 Finite generation and examples
    1. 8.1 Finite generation conjecture
    2. 8.2 Examples in type A
    3. 8.3 Beyond type A

##  1\. Introduction

Let $\mathbf{k}=\overline{\mathbf{k}}$ be an algebraically closed field of
characteristic zero or large enough positive characteristic. We fix a
connected reductive group $G/\mathbf{k}$ and a maximal torus $T\subset G$. We
let
$\operatorname{Lie}(G)=:\mathfrak{g}\supset\mathfrak{t}:=\operatorname{Lie}(T)$,
and denote by $\check{G}$ the Langlands dual group over ${\mathbb{C}}$ (or
$\overline{{\mathbb{Q}}}_{\ell}$).

In this paper we explain how one can naturally associate to an affine Springer
fiber for $G$, or rather to a conjugacy class of the loop Lie algebra
$\mathfrak{g}(\\!(t)\\!)$, a quasi-coherent sheaf on a partial resolution of
the commuting variety (henceforth PRCV) associated to
$\check{G}\times\check{\mathfrak{g}}$. We prove that the sheaves constructed
this way are coherent in a number of cases and conjecture they are coherent in
general. The sheaf on the PRCV remembers homological invariants of the affine
Springer fiber, and we expect that our construction provides a unified
perspective on the affine Springer fibers and their various functorial
properties.

Our first main character is a partial resolution of the trigonometric version
of the commuting variety for $\check{G}$, which we denote by
$\widetilde{\mathfrak{C}}_{\check{G}}=\widetilde{T^{*}T^{\vee}/W}$. It is in
general a singular Poisson variety which we conjecture to locally agree with
those constructed in [53, 1, 66]. There is a natural map
$\widetilde{\mathfrak{C}}_{\check{G}}\to\operatorname{\mathrm{Proj}}\bigoplus_{d\geq
0}A^{d}_{\check{G}}$ where $A_{\check{G}}\subset{\mathbb{C}}[T^{*}T^{\vee}]$
is the subspace of antisymmetric polynomials with respect to the diagonal Weyl
group action, and $A^{d}_{\check{G}}=(A_{\check{G}})^{d}$. For $G=GL_{n}$ this
is an isomorphism, so that we recover the Hilbert scheme of points on
${\mathbb{C}}^{*}\times{\mathbb{C}}$ [64]. A more detailed construction is
given in Section 3.

Let $\operatorname{Gr}_{G}$ be the affine Grassmannian of $G$. On the level of
$\mathbf{k}$-points this is $G(\mathcal{K})/G(\mathcal{O})$, where
$\mathcal{O}=\mathbf{k}\llbracket t\rrbracket$ and
$\mathcal{K}=\mathbf{k}(\\!(t)\\!)=\mathbf{k}\llbracket t\rrbracket[t^{-1}]$.
Our second main character is the affine Springer fiber
${\mathrm{Sp}}_{\gamma}\subset\operatorname{Gr}_{G}$,
$\gamma\in\mathfrak{g}(\mathcal{K})$, defined as the fixed locus of the vector
field $\gamma$. More precisely, let $\gamma\in\mathfrak{g}(\mathcal{K})$ be a
compact semisimple element. On the level of $R$-points

${\mathrm{Sp}}_{\gamma}(R):=\left\\{gG({R\llbracket
t\rrbracket})|\operatorname{Ad}(g^{-1})\gamma\in\operatorname{Lie}(G)({R\llbracket
t\rrbracket})\right\\}\subset\operatorname{Gr}_{G}(R).$

Under these assumptions, ${\mathrm{Sp}}_{\gamma}$ is a nonempty ind-scheme
over $\mathbf{k}$. If $\gamma$ is also regular, ${\mathrm{Sp}}_{\gamma}$ is
finite-dimensional and locally of finite type. We will only be interested in
the étale or singular cohomologies of the ${\mathrm{Sp}}_{\gamma}$, so will be
writing ${\mathrm{Sp}}_{\gamma}$ for the $\mathbf{k}$-points
${\mathrm{Sp}}_{\gamma}(\mathbf{k})$. If $\mathbf{k}={\mathbb{C}}$ we will use
the analytic topology and if $\mathbf{k}=\overline{{\mathbb{F}}}_{q}$ we will
use the étale topology. Our main result is the following.

######  Theorem 1.1.

Let $\gamma\in\mathfrak{g}(\mathcal{K})$ be a semisimple element and
$K_{\gamma}=\operatorname{\mathrm{Stab}}_{G(\mathcal{K})}$ its centralizer.
Then for every compact (in the $t-$adic sense) subgroup $L_{\gamma}\subseteq
K_{\gamma}$, there exists a quasi-coherent sheaf
$\mathcal{F}_{\gamma}\in\mathrm{QCoh}_{{\mathbb{G}}_{m}}(\widetilde{\mathfrak{C}}_{\check{G}})$
such that:

  * (1)

$\mathcal{F}_{t\gamma}=\mathcal{L}\otimes\mathcal{F}_{\gamma}$ where
$\mathcal{L}=\mathcal{O}(1)$ is the Serre twisting sheaf coming from the Proj-
construction of $\widetilde{\mathfrak{C}}_{\check{G}}$.

  * (2)

There exists an integer $M$ such that for $m>M$ we have

$H^{0}(\mathcal{F}_{t^{m}\gamma},\widetilde{\mathfrak{C}}_{\check{G}})=H_{*}^{L_{\gamma}}({\mathrm{Sp}}_{t^{m}\gamma})$

Moreover, the homological grading on the affine Springer fiber side can be
recovered from the torus action dilating the cotangent fibers on the coherent
sheaf side.

We conjecture below that $\mathcal{F}_{\gamma}$ is actually coherent (see
Conjecture 1.7) and prove it in some cases.

######  Remark 1.2.

Note that the quasicoherent sheaf $\mathcal{F}_{\gamma}$ only depends on the
reduced structure of ${\mathrm{Sp}}_{\gamma}$.

######  Remark 1.3.

Here and in the rest of the paper, $H_{*}(-)$ denotes Borel-Moore homology,
defined as $H_{*}(X):=H^{-*}(X,\omega_{X})$. The Borel-Moore homology of an
ind(-proper) variety $X=\varinjlim X_{i}$ will be defined as
$H_{*}(X)=\varinjlim H_{*}(X_{i})$.

It is natural to wonder what kind of sheaves $\mathcal{F}_{\gamma}$ Theorem
1.1 yields. For $G=GL_{n}$ and $\gamma$ homogeneous, we have the following.

######  Theorem 1.4 (Proposition 8.10).

If $G=GL_{n}$ and $\gamma$ homogeneous of slope $\frac{kn+1}{n}$ then
$\mathcal{F}_{\gamma}$ agrees with the restriction of $\mathcal{O}(k)$ to the
punctual Hilbert scheme at $(1,0)$.

######  Theorem 1.5.

For $G=GL_{n}$ and $\gamma$ of integral slope $k$, the sheaf
$\mathcal{F}_{\gamma}$ agrees with $\mathcal{P}\otimes\mathcal{O}(k)$, where
$\mathcal{P}$ is the Procesi bundle restricted to
$\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}})$.

######  Remark 1.6.

For arbitrary $G$ and homogeneous $\gamma$ of integral slope, we also get an
explicit description of $\mathcal{F}_{\gamma}$, see Theorem 8.12.

We also prove a noncommutative version of the above results. The ring of
functions on $\widetilde{\mathfrak{C}}_{\check{G}}$ admits a deformation, or
quantization known as the spherical trigonometric or graded Cherednik algebra
(or graded DAHA). By the work of Yun [88, 89], this algebra acts in homology
of ${\mathrm{Sp}}_{\gamma}$.

The sheaf $\mathcal{L}=\mathcal{O}(1)$ and its powers are quantized to
bimodules between two trigonometric Cherednik algebras with different values
of quantization parameters. These algebras and modules are assembled to a
large ${\mathbb{Z}}$-algebra, and one of the main results of the paper
(Theorem 7.9) associates a module over this ${\mathbb{Z}}$-algebra to a
collection of affine Springer fibers
$\\{{\mathrm{Sp}}_{\gamma},{\mathrm{Sp}}_{t\gamma},{\mathrm{Sp}}_{t^{2}\gamma},\ldots\\}$.
Roughly speaking, considering the bimodules between different algebras allows
one to move between different affine Springer fibers. For $G=GL_{n}$, the
${\mathbb{Z}}$-algebra is similar to the one considered by Gordon and Stafford
for rational Cherednik algebras of $GL_{n}$ in [35, 36].

The main tool we use is a novel construction of ${\mathbb{Z}}$-algebras
related to Coulomb branches of $3d\ \mathcal{N}=4$ theories. These (but not
the ${\mathbb{Z}}$-algebras we construct) were defined by Braverman,
Finkelberg, Nakajima [9, 10] and further explored by Webster [85]. This
algebra is defined as the convolution algebra in the equivariant Borel-Moore
homology of certain spaced related to the affine Grassmannian of $G$. This
construction admits a natural quantization by considering additional
equivariant parameters, and one can study both commutative and quantized
versions. We use the machinery of Coulomb branches to achieve the following
goals:

  * •

We realize the full graded DAHA as the Coulomb branch algebra associated to
the affine flag variety and construct its Dunkl-Cherednik embedding to
$\hbar$-difference operators on the Lie algebra of the torus $T^{\vee}$.

  * •

We give explicit formulas for the Coulomb branch ${\mathbb{Z}}$-algebra in
difference presentation.

  * •

We prove the shift isomorphisms for the spherical/anti-spherical subalgebras
of the graded DAHA, in the difference-operator representation. This allows us
to define the shift bimodules and ${\mathbb{Z}}$-algebras associated to graded
DAHA in all types.

  * •

In the commutative version, the Coulomb branch ${\mathbb{Z}}$-algebra is
equivalent to a graded algebra which we identify explicitly. This allows us to
define $\widetilde{\mathfrak{C}}_{\check{G}}$ using
$\operatorname{\mathrm{Proj}}$ construction.

  * •

Finally, we prove that a collection of affine Springer fibers
$\\{{\mathrm{Sp}}_{\gamma},{\mathrm{Sp}}_{t\gamma},{\mathrm{Sp}}_{t^{2}\gamma},\ldots\\}$
yields a module over the Coulomb branch ${\mathbb{Z}}$-algebra. This is done
similarly to “BFN Springer theory” developed by Hilburn, Kamnitzer and Weekes
[42], and Garner and the second author [26].

We give a more detailed outline of the results and arguments in Section 1.4.
We also comment on various conjectures and connections to physics of “3d
Mirror Symmetry” and link homology (for $G=GL_{n}$).

We note that the technology of Coulomb branches works in far greater
generality than the ”adjoint matter” case studied in this paper. The
associated ${\mathbb{Z}}$-algebras and their modules can be defined and
studied as well [50]. It is likely that their representation theory would be
very interesting as well.

###  1.1. 3d Mirror Symmetry

The construction can be thought of as a part of the $3d$ mirror symmetry of
$3d\;\mathcal{N}=4$ gauge theories [45]. In particular, it is known that the
Higgs branch of the mirror of the $(G,\operatorname{Ad})$-theory is a partial
resolution of $T^{*}T^{\vee}/W$. In physics terms, our construction starts
with a ”boundary condition” for the category of line operators in the A-twist
of the $(G,\operatorname{Ad})$-theory (skyscraper sheaf on the stack of
conjugacy classes in the loop Lie algebra) and produces another line operator
in the $B$-twist of the dual theory (a (quasi-)coherent sheaf on the Higgs
branch) [21]. While the present construction is far from giving any sort of
categorical equivalence (even definitions of the categories involved is
delicate), it gives an explicit ”mirror map” for some of the boundary
conditions and exchanges the algebras of local operators on a resolved Higgs
branch and on a resolved Coulomb branch. We hope this construction gives a
starting point for rigorous constructions of 3d mirror symmetry for line
operators. The fact that these categories have putative definitions in terms
of vertex operator algebras [20] is an interesting topic for further
investigations.

###  1.2. Conjectures

The main construction of the paper produces a ${\mathbb{C}}^{*}$-equivariant
quasi-coherent sheaf

$\mathcal{F}_{\gamma}\in\text{QCoh}_{{\mathbb{C}}^{*}}(\mathfrak{C}_{\check{G}})$

for $\gamma\in\mathfrak{g}(\mathcal{K})$. Quasi-coherence of the sheaf follows
directly from our construction, but we suspect that a stronger statement is in
fact true:

######  Conjecture 1.7.

For any regular semisimple $\gamma\in\mathfrak{g}(\mathcal{K})$ the sheaf
$\mathcal{F}_{\gamma}$ is coherent:

$\mathcal{F}_{\gamma}\in\operatorname{Coh}_{{\mathbb{C}}^{*}}(\mathfrak{C}_{\check{G}}).$

The coherence conjecture already has interesting numerical corollaries. It is
known that $H^{*}({\mathrm{Sp}}_{\gamma})$ is finite-dimensional if $\gamma$
is elliptic and $G$ is simply connected. Thus the conjecture above implies an
estimate on the growth of the dimensions of these cohomology spaces, as we
multiply $\gamma$ by $t^{m}$.

######  Conjecture 1.8.

For any elliptic regular semisimple $\gamma\in\mathfrak{g}(\mathcal{K})$ with
$G$ being simple and simply connected there are $c_{i}\in{\mathbb{Q}}$ and
$M\in{\mathbb{Z}}$ such that

$\dim(H^{*}({\mathrm{Sp}}_{t^{m}\gamma})\\!)=\sum_{i=0}^{r}c_{i}m^{i},\quad
m>M,\quad r=\mathrm{rank}(\mathfrak{g}).$

In the case of homogeneous elliptic $\gamma$ it was shown in [83] that the
conjecture is true, the left-hand side being given by variants of rational
Coxeter-Catalan numbers, in particular the $c_{i}$ can be explicitly
computed.111For ${\widetilde{\mathrm{Sp}}}_{t^{m}\gamma}$ i.e. the version in
affine flags, the result is easier to state and simply says $c_{i}=0$, $i<r$
Many low-rank examples are treated in a lot of detail in [72]. More
complicated non-homogeneous elliptic cases for $G=SL_{n}$ were studied in
[75].

###  1.3. Relation to conjectures stemming from knot theory

The case of $G=GL_{n},SL_{n}$ is of special interest because of the
applications to knot theory [33, 31, 32, 71, 70, 68]. In particular, the
characteristic polynomial of $\gamma\in\mathfrak{gl}_{n}(\mathcal{K})$ defines
an germ of planar curve singularity and the link of this singularity is the
closure of the braid (conjugacy class) $\beta(\gamma)\in\mathfrak{B}r_{n}$.
When $G=SL_{n}$, the conjecture [70, Conjecture 2 and Proposition 4] predicts
an isomorphism between the (reduced) triply graded homology of $\beta(\gamma)$
and $H^{*}({\mathrm{Sp}}_{\gamma})$ enhanced with the perverse filtration [61,
60, 62]. Notice that these papers use cohomology whereas the present work uses
BM homology, but this distinction is immaterial for these numerical
comparisons of multiply graded finite dimensional vector spaces.

In this paper, we enrich the algebro-geometric side of the above conjectures
by considering an infinite family of affine Springer fibers
$\\{{\mathrm{Sp}}_{\gamma},{\mathrm{Sp}}_{t\gamma},{\mathrm{Sp}}_{t^{2}\gamma},\ldots\\}$.
It is easy to see that multiplication of $\gamma$ by $t^{m}$ corresponds to
the multiplication of the braid $\beta(\gamma)$ by $\operatorname{FT}^{m}$,
the $m$-th power of the full twist braid $\operatorname{FT}$. Since
$\operatorname{FT}$ is central in the braid group, the conjugacy class of
$\beta(\gamma)\cdot\operatorname{FT}^{m}$ is determined by the conjugacy class
of $\beta(\gamma)$.

By construction of triply graded homology $\operatorname{\mathrm{HHH}}(-)$,
there are natural multiplication maps

(1)
$\operatorname{\mathrm{HHH}}(\beta(\gamma))\otimes\operatorname{\mathrm{HHH}}(\operatorname{FT}^{m})\to\operatorname{\mathrm{HHH}}(\beta(\gamma)\cdot\operatorname{FT}^{m}),\quad\operatorname{\mathrm{HHH}}(\operatorname{FT}^{m})\otimes\operatorname{\mathrm{HHH}}(\operatorname{FT}^{m^{\prime}})\to\operatorname{\mathrm{HHH}}(\operatorname{FT}^{m+m^{\prime}}),$

and hence
$\bigoplus_{m}\operatorname{\mathrm{HHH}}(\beta(\gamma)\cdot\operatorname{FT}^{m})$
has a structure of a graded module over the graded algebra
$\bigoplus_{m}\operatorname{\mathrm{HHH}}(\operatorname{FT}^{m})$. The latter
graded algebra, as conjectured in [31] and proved in [32], is closely related
to the homogeneous coordinate ring of
$\operatorname{Hilb}^{n}({\mathbb{C}}^{2})$, and to the
${\mathbb{Z}}$-algebras appearing in this paper. In other words, in this paper
we establish a precise analogue of multiplication maps (1) on the affine
Springer side by means of geometric representation theory.

In a series of papers [68, 69] the third named author and Rozansky took a
different approach to knot invariants and defined a
${\mathbb{C}}^{*}\times{\mathbb{C}}^{*}$-equivariant complex of coherent
sheaves

$\mathcal{G}_{\beta}\in
D^{b}_{{\mathbb{C}}^{*}\times{\mathbb{C}}^{*}}(\text{Coh}(\operatorname{Hilb}_{n}({\mathbb{C}}^{2})))$

such that the hypercohomology $H^{*}(\mathcal{G}_{\beta})$ is isomorphic as a
bigraded vector space to the ”lowest row” of the triply-graded homology of
$\beta$.

To connect these constructions with the present one, note that the natural
inclusion map
$i_{{\mathbb{C}}^{*}}:{\mathbb{C}}^{*}\times{\mathbb{C}}\to{\mathbb{C}}^{2}$,
$i_{{\mathbb{C}}^{*}}(x,y)=(x-1,y)$ induces an inclusion
$i_{{\mathbb{C}}^{*}}:\operatorname{Hilb}^{n}({\mathbb{C}}^{*}\times{\mathbb{C}})\to\operatorname{Hilb}^{n}({\mathbb{C}}^{2})$
which identifies the punctual Hilbert schemes at $(1,0)$ and $(0,0)$.

######  Conjecture 1.9.

For any regular semisimple $\gamma\in\mathfrak{gl}_{n}\llbracket t\rrbracket$
there is an isomorphism of ${\mathbb{C}}^{*}$-equivariant sheaves

$\mathcal{F}_{\gamma}\simeq
i^{*}_{{\mathbb{C}}^{*}}(\mathcal{G}_{\beta}),\quad\beta=\beta(\gamma).$

Let us point out that the results of Maulik [58], the aforementioned [60, 61,
62], and the results of the third author with Rozansky in [69] can be combined
to show that the conjecture is true on the level of Euler characteristics:

$\chi({\mathrm{Sp}}_{\gamma})=\chi_{{\mathbb{C}}^{*}}(i^{*}_{{\mathbb{C}}^{*}}(\mathcal{F}_{\beta})),\quad\beta=\beta(\gamma),$

for $\gamma\in\mathfrak{sl}_{n}\llbracket t\rrbracket$ elliptic regular
semisimple.

In particular, we derive an Euler characteristics version of the weak
coherence conjecture:

######  Corollary 1.10.

For any elliptic regular semisimple $\gamma\in\mathfrak{sl}_{n}\llbracket
t\rrbracket$ there are $c_{i}\in{\mathbb{Q}}$ and $M\in{\mathbb{Z}}$ such that

$\chi(H^{*}({\mathrm{Sp}}_{t^{m}\gamma}))=\sum_{i=0}^{n-1}c_{i}m^{i},\quad
m>M.$

Finally, note that Maulik’s result actually keeps track of the Euler
characteristics of the Hilbert schemes of points on the germ defined by
$\gamma$ and hence the perverse filtration [60, 61]. In particular, for
elliptic $\gamma$ we may also conjecture that there exists a Springer-
theoretic construction of a sheaf
$\operatorname{gr}^{P}\mathcal{F}_{\gamma}\in\text{Coh}_{{\mathbb{C}}^{\times}\times{\mathbb{C}}^{\times}}(\operatorname{Hilb}^{n}({\mathbb{C}}^{2}))$
which agrees with $\mathcal{F}_{\beta}$
${\mathbb{C}}^{\times}\times{\mathbb{C}}^{\times}$-equivariantly.

It is interesting to consider the extension of the conjecture to other than
elliptic elements. While the homology in this case is always infinite-
dimensional, one has promising results for for instance the lattice quotients
${\mathrm{Sp}}_{\gamma}$. In the unramified case, the paper [16] proves that
the multivariate generating function (over root valuations) of the weight
polynomials is rational. On the other hand, the second author has computed in
[51] the equivariant cohomology of the affine Springer fibers in the
unramified case and related them to the Procesi bundle on the Hilbert scheme
of points.

###  1.4. Outline

####  1.4.1. Outline of the argument

The key ingredients of the construction are 1) the technology developed by
Braverman, Finkelberg and Nakajima [9] on the affine Springer side (Topology)
and 2) noncommutative geometry methods akin to the work by [35] on the Hilbert
scheme side (Algebraic Geometry). The theory of the 3) Double affine Hecke
algebras (Algebra) links these two theories together. Our work provides a
dictionary between objects in the three theories. A part of this dictionary is
as follows222For simplicity of introduction we discuss the type $A$ case, for
other types see section 8:

$\begin{array}[]{|c|c|c|}\hline\cr\text{Topology}&\text{Algebra}&\text{Algebraic
Geometry}\\\
\hline\cr{}_{i}\mathcal{A}_{i}:=H_{*}^{{\widetilde{G}}_{\mathcal{O}}\rtimes{\mathbb{C}}^{\times}}({}_{i}\mathcal{R}_{i})&\mathbf{e}\mathbb{H}_{c+i\hbar,\hbar}\mathbf{e}&{\mathbb{C}}[T^{*}T^{\vee}/W]\\\
{}_{i-1}\mathcal{A}_{i}:=H_{*}^{{\widetilde{G}}_{\mathcal{O}}\rtimes{\mathbb{C}}^{\times}}({}_{i-1}\mathcal{R}_{i})&{}_{i-1}\mathcal{B}_{i}=\mathbf{e}_{-}\mathbb{H}_{c+i\hbar,\hbar}\mathbf{e}&\mathcal{O}(1)\\\
{}_{i}\widetilde{\mathcal{A}}_{i}:=H_{*}^{{\widetilde{\mathbf{I}}}_{\mathcal{O}}\rtimes{\mathbb{C}}^{\times}}({}_{i}\widetilde{\mathcal{R}}_{i})&\mathbb{H}_{c+i\hbar,\hbar}&H^{0}\left(\widetilde{T^{*}T^{\vee}/W},\mathcal{P}\right)\\\
{}_{i-1}\widetilde{\mathcal{A}}_{i}:=H_{*}^{{\widetilde{\mathbf{I}}}_{\mathcal{O}}\rtimes{\mathbb{C}}^{\times}}({}_{i-1}\widetilde{\mathcal{R}}_{i})&{}_{i-1}\widetilde{\mathcal{B}}_{i}&H^{0}\left(\widetilde{T^{*}T^{\vee}/W},\mathcal{P}\otimes\mathcal{O}(1)\right)\\\
H_{*}^{{\mathbb{C}}^{\times}}({\widetilde{\mathrm{Sp}}}_{\frac{nk+1}{n}})&\mathbb{H}_{c+k\hbar,\hbar}\curvearrowright
L_{\frac{nk+1}{n}}({\mathbb{C}})&\mathcal{O}_{\pi^{-1}(1,0)}(k)\\\
\hline\cr\end{array}$

In the Algebra column of the table we have the algebraic objects related to
the representation theory of the graded double affine Hecke algebra
$\mathbb{H}_{c,\hbar}$ defined in Definition 4.1. This algebra is also known
under the names trigonometric or degenerate DAHA. The algebra
$\mathbb{H}_{c,\hbar}$ contains the finite Weyl group $W$ and
$\mathbf{e},\mathbf{e}_{-}\in W$ are the projectors for the trivial and sign
representations. We define an explicit representation of these algebras using
difference operators and use it to prove the following:

######  Theorem 1.11.

(Theorem 4.12) The spherical subalgebra
$\mathbf{e}\mathbb{H}_{c,\hbar}\mathbf{e}$ is isomorphic to the anti-spherical
subalgebra $\mathbf{e}_{-}\mathbb{H}_{c-\hbar,\hbar}\mathbf{e}_{-}$ with
shifted parameter.

Similar shift isomorphisms are well known in the theory of rational Cherednik
algebras and for the Dunkl differential-difference representation [4, 43, 74],
but it appears to be new for the difference representation of trigonometric
DAHA.

Thus $\mathbf{e}_{-}\mathbb{H}_{c+i\hbar,\hbar}\mathbf{e}$ naturally has left
$\mathbf{e}\mathbb{H}_{c+(i-1)\hbar,\hbar}\mathbf{e}$ and right
$\mathbf{e}\mathbb{H}_{c+i\hbar,\hbar}\mathbf{e}$ action and we set:

${}_{i}\mathcal{B}_{i}=\mathbf{e}\mathbb{H}_{c+i\hbar,\hbar}\mathbf{e},\quad{}_{i}\mathcal{B}_{i+1}=\mathbf{e}_{-}\mathbb{H}_{c+i\hbar,\hbar}\mathbf{e},\quad{}_{i}\mathcal{B}_{j+1}={}_{i}\mathcal{B}_{j}\otimes_{{}_{j}\mathcal{B}_{j}}{}_{j}\mathcal{B}_{j+1}.$

The direct sum ${}_{\bullet}\mathcal{B}_{\bullet}=\bigoplus_{i\leq
j}{}_{i}\mathcal{B}_{j}$ is an example of a ${\mathbb{Z}}$-algebra, introduced
by Polischuk and Bondal [8] and studied in a setting relevant to us by Gordon
and Stafford [35, 36, 76].

We now explain how the above mentioned structures exist in the affine Springer
theory, which corresponds to the Topology column. The key geometric object is
an ind-scheme ${}_{i}\widetilde{\mathcal{R}}_{j}$, a variant of ”the space of
triples $\mathcal{R}$” central to the work of Braverman, Finkelberg and
Nakajima on Coulomb branches:

${}_{j}\widetilde{\mathcal{R}}_{i}=\left\\{(g,v)\in G(\mathcal{K})\times
t^{i}\operatorname{Lie}(\mathbf{I})|g\cdot v\in
t^{j}\operatorname{Lie}(\mathbf{I})\right\\}/\mathbf{I},$

where $\mathbf{I}$ is the Iwahori subalgebra. On the level of sets, the
quotient space $\mathbf{I}\backslash{}_{0}\widetilde{\mathcal{R}}_{0}$ is the
quotient $\widetilde{St}/G(\mathcal{K})$ of the affine Steinberg space
$\widetilde{St}$ as also explained in the introduction to [10].

It was explained by Varagnolo and Vasserot [82] that the equivariant homology
group of the affine Steinberg variety
$H_{*}^{{\widetilde{\mathbf{I}}}\rtimes{\mathbb{C}}^{*}_{\mathrm{rot}}}({}_{0}\widetilde{\mathcal{R}}_{0})$
is isomorphic to $\mathbb{H}_{c,\hbar}$, here the parameter $c$ depends on our
choice of the equivariant structure with respect to the loop rotation group
${\mathbb{C}}^{*}_{\mathrm{rot}}$. Their work however uses localization
techniques which we are able to avoid, thereby providing an isomorphism over
the full parameter space, see Theorem 6.1.

Similarly, one can define the affine Grassmannian version
${}_{i}\mathcal{R}_{j}$ of the above spaces. Since the fibers of the
projection ${}_{i}\widetilde{\mathcal{R}}_{j}\to{}_{i}\mathcal{R}_{j}$ are
classical Springer fibers, we have a geometric model for the spherical algebra
$\mathbf{e}\mathbb{H}(c)\mathbf{e}=H_{*}^{G(\mathcal{O})\ltimes{\mathbb{C}}^{*}_{\mathrm{rot}}}({}_{0}\mathcal{R}_{0})$.
Thus, it is natural to define

${}_{i}\mathcal{A}_{j}=H_{*}^{G(\mathcal{O})\ltimes{\mathbb{C}}^{*}_{\mathrm{rot}}}({}_{i}\mathcal{R}_{j}),\quad{}_{\bullet}\mathcal{A}_{\bullet}=\bigoplus_{i\leq
j}{}_{i}\mathcal{A}_{j}.$

As explained in [11, 21, 85], there is a natural convolution product

$H_{*}^{G(\mathcal{O})\ltimes{\mathbb{C}}^{*}_{\mathrm{rot}}}({}_{i}\mathcal{R}_{j})\otimes
H_{*}^{G(\mathcal{O})\ltimes{\mathbb{C}}^{*}_{\mathrm{rot}}}({}_{j}\mathcal{R}_{k})\to
H_{*}^{G(\mathcal{O})\ltimes{\mathbb{C}}^{*}_{\mathrm{rot}}}({}_{i}\mathcal{R}_{k}).$

By associativity the convolution descends to give bilinear product maps

${}_{i}\mathcal{A}_{j}\bigotimes_{{}_{j}\mathcal{A}_{j}}{}_{j}\mathcal{A}_{k}\to{}_{i}\mathcal{A}_{k}.$

One of our main results partially identifies the Coulomb branch
${\mathbb{Z}}$-algebra ${}_{\bullet}\mathcal{A}_{\bullet}$ in terms of the
algebraic ${\mathbb{Z}}$-algebra ${}_{\bullet}\mathcal{B}_{\bullet}$.

######  Theorem 1.12.

The Coulomb branch ${\mathbb{Z}}$-algebra ${}_{\bullet}\mathcal{A}_{\bullet}$
satisfies the following properties:

  * (a)

For all $i$ the algebras ${}_{i}\mathcal{A}_{i}$ and ${}_{i}\mathcal{B}_{i}$
are isomorphic.

  * (b)

For all $i$ the bimodules ${}_{i}\mathcal{A}_{i+1}$ and
${}_{i}\mathcal{B}_{i+1}$ are isomorphic.

  * (c)

For $G=GL_{n}$, the bimodules ${}_{i}\mathcal{A}_{j}$ and
${}_{i}\mathcal{B}_{j}$ are isomorphic for all $i\leq j$. Moreover, the
${\mathbb{Z}}$-algebras ${}_{\bullet}\mathcal{A}_{\bullet}$ and
${}_{\bullet}\mathcal{B}_{\bullet}$ are isomorphic.

We prove part (a) as Theorem 6.1, part (b) as Theorem 6.5 and part (c) as
Theorem 6.21. In Proposition 6.10 we also provide an explicit basis for the
associated graded of ${}_{\bullet}\mathcal{A}_{\bullet}$ with respect to
Bruhat filtration in all types, see in particular Theorem 6.8 for $G=GL_{n}$.

The main difficulty in proving part (c) is that
${}_{\bullet}\mathcal{B}_{\bullet}$ is generated by the “degree one bimodules”
${}_{i}\mathcal{B}_{i+1}$ by definition, while this is not clear at all for
${}_{\bullet}\mathcal{A}_{\bullet}$. For $G=GL_{n}$, we resolve this
difficulty by a careful combinatorial analysis of the basis in Theorem 6.8,
and using deep results of Gordon and Stafford on ${\mathbb{Z}}$-algebras for
rational Cherednik algebras [35, 36].

Next, we turn to the Algebraic Geometry column of the table. In the
commutative limit $c=\hbar=0$ the ${\mathbb{Z}}$-algebra
${}_{\bullet}\mathcal{A}^{\hbar=0}_{\bullet}$ becomes a graded commutative
algebra, as ${}_{i}\mathcal{A}^{\hbar=0}_{j}$ only depends on the difference
$d=j-i$. For $d=0$ the algebra ${}_{0}\mathcal{A}^{\hbar=0}_{0}$ can be
identified with the algebra of symmetric polynomials on $T^{*}T^{\vee}$, or,
equivalently, the algebra of functions on $T^{*}T^{\vee}/W$. For $d=1$ the
module ${}_{0}\mathcal{A}^{\hbar=0}_{1}$ can be identified with the space of
antisymmetric polynomials on $T^{*}T^{\vee}$. We can then define an algebraic
variety

(2)
$\widetilde{\mathfrak{C}}_{\check{G}}:=\operatorname{\mathrm{Proj}}\bigoplus_{d=0}^{\infty}{}_{0}\mathcal{A}^{\hbar=0}_{d}$

which is a partial resolution of
$\operatorname{Spec}{}_{0}\mathcal{A}^{\hbar=0}_{0}=T^{*}T^{\vee}/W$. In other
words, we identify the above graded algebra with the homogeneous graded ring
of $\widetilde{\mathfrak{C}}_{\check{G}}$. Our next main result identifies
this graded algebra explicitly.

######  Theorem 1.13.

(Theorem 3.8) Let $\epsilon$ be the sign representation of ${\mathbb{C}}[W]$
and $\mathbf{e}_{d}$ be the idempotent corresponding to $\epsilon^{\otimes
d}$. We have

${}_{0}\mathcal{A}^{\hbar}_{d}\simeq\mathbf{e}_{d}\bigcap_{\alpha\in\Phi^{+}}\langle
1-\alpha^{\vee},y_{\alpha}\rangle^{d}$

where in the right hand side we have an intersection of ideals in
${\mathbb{C}}[T^{*}T^{\vee}]$. The isomorphism agrees with the convolution
structure on the left hand side and the multiplication on the right hand side.

By the work of Haiman [40], for $G=GL_{n}$ this implies the following:

######  Theorem 1.14.

For $G=GL_{n}$, we get

$\operatorname{\mathrm{Proj}}\bigoplus_{d=0}^{\infty}{}_{0}\mathcal{A}^{\hbar=0}_{d}=\operatorname{Hilb}^{n}({\mathbb{C}}^{*}\times{\mathbb{C}}),$

in particular, it is a smooth resolution of
$T^{*}T^{\vee}/W\simeq({\mathbb{C}}^{*}\times{\mathbb{C}})^{n}/S_{n}$.

######  Remark 1.15.

A different proof of Theorem 1.14 essentially follows from [11, Theorem 3.10],
which identifies a resolution of the Coulomb branch for
$(GL_{n},\mathfrak{gl}_{n}\oplus{\mathbb{C}}^{n})$, constructed using flavor
symmetry, with $\operatorname{Hilb}^{n}({\mathbb{C}}^{2})$. The same proof in
loc. cit. applied to $(GL_{n},\mathfrak{gl}_{n})$ yields Theorem 1.14.

Another proof can be extracted from more general results of Nakajima and
Takayama [65, 80] which identify the resolved Coulomb branches of quiver gauge
theories of affine type A with certain Cherkis bow varieties [19], and certain
moduli spaces of parabolic sheaves on $\mathbb{P}^{1}\times\mathbb{P}^{1}$.
See, in particular, [11, Theorem 4.9] for more context and details.

Next, we study graded modules over all of the above ${\mathbb{Z}}$-algebras
and graded algebras. The convolution structure of spaces
${}_{i}\widetilde{\mathcal{R}}_{j}$ allows us to define a correspondence
between the affine Springer fibers in the affine flag variety
$\widetilde{{\mathrm{Sp}}}_{t^{i}\gamma}$,
$\widetilde{{\mathrm{Sp}}}_{t^{j}\gamma}\subset G(\mathcal{K})/\widetilde{I}$.
Similarly, we prove in Theorem 7.9 that $\bigoplus_{j\geq
0}H_{*}({\mathrm{Sp}}_{t^{j}\gamma})$ is a module over ${\mathbb{Z}}$-algebra
${}_{\bullet}\mathcal{A}_{\bullet}$. This is very similar to the BFN Springer
theory developed in [42, 26]. In the commutative variant, we obtain a graded
module over the graded algebra, and hence a quasi-coherent sheaf over its
$\operatorname{\mathrm{Proj}}$ construction defined by (2), which proves
Theorem 1.1.

Finally, we outline a simple construction of the above action. Recall that the
homology of the affine Grassmanian version of the Springer fibers
${\mathrm{Sp}}_{t^{j}\gamma}\subset G(\mathcal{K})/G(\mathcal{O})$ can be
described in terms of the action of the finite Weyl group:

(3)
$H_{*}({\mathrm{Sp}}_{t^{j}\gamma})=H_{*}({\widetilde{\mathrm{Sp}}}_{t^{j}\gamma})^{W},\quad
H_{*}({\mathrm{Sp}}_{t^{j-1}\gamma})=H_{*}({\widetilde{\mathrm{Sp}}}_{t^{j}\gamma})^{\epsilon}[-2\dim
G/B].$

where $\epsilon$ is the sign representation of $W$ and $[-2\dim G/B]$ denotes
shift in homological degree. We explain the details of the isomorphisms in the
Lemma 2.2. Given a class in $H_{*}({\mathrm{Sp}}_{t^{j-1}\gamma})$, we can
identify it with a class in $H_{*}({\mathrm{Sp}}_{t^{j}\gamma})^{\epsilon}$,
then act by an antisymmetric polynomial (using the commutative version of the
double affine action of [88]) and get a class in
$H_{*}({\mathrm{Sp}}_{t^{j}\gamma})^{W}=H_{*}({\mathrm{Sp}}_{t^{j}\gamma})$.
This construction gives a map

(4) $A_{G}\otimes H_{*}({\mathrm{Sp}}_{t^{j-1}\gamma})\to
H_{*}({\mathrm{Sp}}_{t^{j}\gamma})$

where $A_{G}={}_{0}\mathcal{A}^{\hbar=0}_{1}$ is the space of antisymmetric
polynomials.

It is unclear if this approach can be used to define the action of the full
graded algebra $\bigoplus_{d=0}^{\infty}{}_{0}\mathcal{A}^{\hbar=0}_{d}$, the
main obstacles are:

  * •

It is unclear how to verify the relations between the products of elements of
${}_{0}\mathcal{A}^{\hbar=0}_{1}$ inside ${}_{0}\mathcal{A}^{\hbar=0}_{d}$

  * •

For $G\neq GL_{n}$, it is unclear if the algebra is generated in degree 1.

To avoid these obstacles, we have abandoned this approach altogether and
instead used the machinery of Coulomb branches throughout the paper.
Nevertheless, a posteriori we conclude that the action of the degree 1 part of
the algebra agrees with (4), and hence all necessary relations are satisfied.

####  1.4.2. Outline of the paper

In Section 2 we define the affine Springer fibers and some background
material. The sheaves we construct live on a partial resolution of
$T^{*}T^{\vee}/W$, which is introduced in Section 3. In Section 4, we study
the trigonometric Cherednik algebra and a natural ${\mathbb{Z}}$-algebra built
out of it, which is the algebraic main part of the construction. In Sections
5,6 and 7 we study the affine Springer fibers using Coulomb algebra machinery,
in particular constructing a geometric ${\mathbb{Z}}$-algebra action and
comparing it to the one in Section 4. In Section 8 we prove that the sheaves
we construct are coherent whenever $\gamma$ is homogeneous. In Section 8.2, we
study some homogeneous examples in detail.

### Acknowledgments

The authors would like to thank Roman Bezrukavnikov, Tudor Dimofte, Pavel
Etingof, Pavel Galashin, Niklas Garner, Ivan Losev, Ngô Bao Châu, Hiraku
Nakajima, Wenjun Niu, José Simental Rodriguez, Eric Vasserot, Monica Vazirani,
Ben Webster, Alex Weekes and Zhiwei Yun for useful discussions. We thank the
organizers of the MRSI program “Enumerative Geometry Beyond Numbers” in 2018
for providing an ideal working environment which motivated us to start this
project.

The work of E. G. was partially supported by the NSF grant DMS-1760329. The
work of A. O. was partially supported by the NSF grants DMS-1352398 and
DMS-1760373. The work of O. K. was partially supported by the Finnish Academy
of Science and Letters grant for postdoctoral research abroad.

##  2\. Affine Springer fibers

###  2.1. Definitions

Let $G/\mathbf{k}$ be a reductive group over a field $\mathbf{k}$. We assume
$\mathbf{k}=\overline{\mathbf{k}}$ and that the characteristic is zero or
large enough (no attempt will be made to give bounds). Fix some pinning
$T\subset B\subset G$ of the root system of $G$. Define $N=\dim(G/B)$. Let
$\mathcal{K}=\mathbf{k}(\\!(t)\\!)$ and $\mathcal{O}=\mathbf{k}\llbracket
t\rrbracket$. Write $G(\\!(t)\\!)=G(\mathcal{K})$ and $G\llbracket
t\rrbracket=G(\mathcal{O})$. Denote also $\mathfrak{g}=\operatorname{Lie}(G)$
and $\mathfrak{g}(\\!(t)\\!)=\mathfrak{g}(\mathcal{K}),\mathfrak{g}\llbracket
t\rrbracket=\mathfrak{g}(\mathcal{O})$.

Let $\mathbf{P}$ be a standard parahoric subgroup of $G(\mathcal{K})$. Let
$\operatorname{Fl}^{\mathbf{P}}_{G}=G(\mathcal{K})/\mathbf{P}$ be the
corresponding partial affine flag variety. When $\mathbf{P}=\mathbf{I}$ is the
Iwahori subgroup of $G(\mathcal{K})$ corresponding to $B$, we simply write
$\operatorname{Fl}_{G}:=\operatorname{Fl}^{\mathbf{I}}$ for the affine flag
variety and when $\mathbf{P}=G(\mathcal{O})$ we write
$\operatorname{Gr}_{G}:=\operatorname{Fl}^{G(\mathcal{O})}$ for the affine
Grassmannian. Since it will usually be clear from the context, we will also
omit the subscript $G$.

For any $\mathbf{P}$ and $\gamma\in\mathfrak{g}(\mathcal{K})$ , define the
affine Springer fiber

${\mathrm{Sp}}_{\gamma}^{\mathbf{P}}:=\\{g\mathbf{P}|\operatorname{Ad}(g^{-1})\gamma\in\operatorname{Lie}(\mathbf{P})\\}\subset\operatorname{Fl}^{\mathbf{P}}.$

When $\mathbf{P}=G(\mathcal{O})$ we omit the superscript and when
$\mathbf{P}=\mathbf{I}$ we write
${\widetilde{\mathrm{Sp}}}_{\gamma}={\mathrm{Sp}}_{\gamma}^{\mathbf{I}}$.

The space ${\mathrm{Sp}}_{\gamma}^{\mathbf{P}}$ is a sub-ind-scheme of
$\operatorname{Fl}^{\mathbf{P}}$. It is always nonreduced, but since it makes
no difference to us, we will only work with the reduced structure of
${\mathrm{Sp}}_{\gamma}^{\mathbf{P}}$. ${\mathrm{Sp}}_{\gamma}^{\mathbf{P}}$
is finite-dimensional if and only if $\gamma$ is regular semisimple [47]. We
shall mostly focus on the case $\gamma$ is regular semisimple from now on. We
also assume $\gamma$ is compact, i.e. contained in some Iwahori subgroup, or
equivalently that the affine Springer fiber is nonempty. The ind-scheme
${\mathrm{Sp}}_{\gamma}$ is locally of finite type, and by results of [47]
there exists a free abelian group $L$ acting on
${\mathrm{Sp}}_{\gamma}^{\mathbf{P}}$ freely and a projective scheme
$S\subset{\mathrm{Sp}}_{\gamma}^{\mathbf{P}}$ such that $L\cdot
S={\mathrm{Sp}}_{\gamma}^{\mathbf{P}}$. The free abelian group $L$ can be
identified with the cocharacter lattice $X_{*}(C_{G(\mathcal{K})}(\gamma))$.
In particular, $L$ is trivial if and only if $\gamma$ is elliptic. In this
case, ${\mathrm{Sp}}_{\gamma}^{\mathbf{P}}$ is a projective variety.

######  Remark 2.1.

By the Jordan decomposition, we can write any $\gamma$ as the sum of commuting
semisimple and nilpotent elements: $\gamma=\gamma_{s}+\gamma_{n}$. Therefore,
we can reduce the study of general $\gamma$ to nilpotent and semisimple
$\gamma$, as
${\mathrm{Sp}}_{\gamma}={\mathrm{Sp}}_{\gamma_{s}}\cap{\mathrm{Sp}}_{\gamma_{n}}$.
While it is also interesting to study non-regular semisimple elements, much
about this case can in principle be extracted from the regular semisimple case
by the fact that the centralizer of a semisimple element is a reductive group
over $\mathcal{K}$. In the present work, we are mostly concerned with
$H^{L_{\gamma}}_{*}({\mathrm{Sp}}_{\gamma})$ for $L_{\gamma}$ a compact
subgroup of the centralizer of $\gamma$. For example, the natural action of
the dDAHA on the coordinate ring of the universal centralizer in Theorem 3.21
can be interpreted as taking $\gamma=0$ and
$L_{\gamma}=G_{\mathcal{O}}\subseteq G_{\mathcal{K}}$.

However, the nilpotent elements seem more mysterious from our point of view.
It is clear e.g. by the convergence of the corresponding orbital integrals
that the centralizers of nilpotent elements are quite large. In the case where
$\gamma$ is nilpotent, it is not even known if there is a Levi factor of the
centralizer. It does make sense to ignore the centralizer (or at most use
compact subgroups thereof) and use finite-dimensional approximation to study
the Borel-Moore homology of ${\mathrm{Sp}}_{\gamma}$ even in the nilpotent
cases.

###  2.2. The Springer action

Assume for now that $\mathbf{k}={\mathbb{C}}$ or that we are using étale
cohomology over $\overline{{\mathbb{Q}}}_{\ell}\cong{\mathbb{C}}$ for
$\ell\neq\text{char}(p)$. One of the remarkable things about
${\widetilde{\mathrm{Sp}}}_{\gamma}$ is that
$H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})=H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma},{\mathbb{C}})$
has an action of the extended affine Weyl group $\widetilde{W}=W\ltimes
X_{*}(T)$ as shown by Lusztig [57] (in the adjoint case) and Yun [88] (in
general), analogously to the Weyl group action in the cohomology of classical
Springer fibers.

######  Lemma 2.2.

Let $\gamma\in\mathfrak{g}(\mathcal{K})$ be an element such that
$\gamma=t\gamma_{0}$ for a regular semisimple compact element
$\gamma_{0}\in\mathfrak{g}(\mathcal{K})$. Then under the Springer action of
$W\subset\widetilde{W}$ on $H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})$, we
have a natural identification

(5)
$H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})^{W}=H_{*}({\mathrm{Sp}}_{\gamma})$

and an isomorphism

(6) $H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})^{\epsilon}\cong
H_{*}({\mathrm{Sp}}_{\gamma_{0}})[-2N].$

Here $[-2N]$ means a shift in homological degree and $N=\dim G/B$.

###### Proof.

The first part is due to [89, Section 2.6], and the second part is well-known
but not found in the literature, so we give a proof here.

First recall the construction of the Springer action for the subgroup
$W\subset\widetilde{W}$. Since $\mathbf{I}\subset G\llbracket t\rrbracket$, we
have natural projections $\operatorname{Fl}\to\operatorname{Gr}$ and
${\widetilde{\mathrm{Sp}}}_{\gamma}\to{\mathrm{Sp}}_{\gamma}$. For any
$\mathbf{P}$, write $P=\mathbf{P}/t\mathbf{P}$ and $\mathfrak{p}$ for its Lie
algebra. There are natural maps of fpqc sheaves
${\mathrm{Sp}}_{\gamma}^{\mathbf{P}}\to[\mathfrak{p}/P],$ which send the
cosets $g\mathbf{P}$ to the respective images of $g^{-1}\gamma g$ under the
projections $\operatorname{Lie}(\mathbf{P})\to\mathfrak{p}$. In particular
there is a cartesian diagram

(7)
${{\widetilde{\mathrm{Sp}}}_{\gamma}}$${\left[\widetilde{\mathfrak{g}}/G\right]=[\mathfrak{b}/B]}$${{\mathrm{Sp}}_{\gamma}}$${\left[\mathfrak{g}/G\right]}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\varphi}$

where the right-hand side is naturally identified with the Grothendieck-
Springer resolution for $G$. Since $\gamma=t\gamma_{0}$, it is clear that the
image of $\varphi$ will be contained in
$[\mathcal{N}/G]\subset[\mathfrak{g}/G]$ and the image of $\varphi^{\prime}$
will be contained in $[\widetilde{\mathcal{N}}/G]$. The restriction
$S:=\pi^{\prime}_{*}{\mathbb{C}}|_{\mathcal{N}}$ is perverse, and is called
the Springer sheaf. It is in fact isomorphic to a direct sum of IC complexes
on nilpotent orbits, and it is known by classical Springer theory that

$\operatorname{End}_{\text{Perv}^{G}(\mathcal{N})}(S)\cong{\mathbb{C}}[W].$

In particular, there is a map

${\mathbb{C}}[W]\to\operatorname{End}_{\text{Perv}({\mathrm{Sp}}_{\gamma})}(\varphi^{*}S)$

and hence an action of $W$ on $H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})$.
Decomposing the regular representation of $W$, we see that there is some IC
complex $\mathcal{F}$ on $[\mathcal{N}/G]$ corresponding to the sign
representation $\epsilon$. From classical Springer theory it follows that this
complex is isomorphic as a perverse sheaf to the shifted skyscraper sheaf
${\mathbb{C}}_{\\{0\\}}[-2N]$.

By proper base change,

$H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})^{\epsilon}=\operatorname{Hom}(\varphi^{*}\mathcal{F},\pi_{*}{\mathbb{C}})\cong\operatorname{Hom}(\varphi^{*}\mathcal{F},\varphi^{*}S).$

Now note that

$\varphi^{-1}(0)=\\{gG(\mathcal{O})\in{\mathrm{Sp}}_{\gamma}|g^{-1}\gamma
g\equiv 0\;(\text{mod }t)\\}={\mathrm{Sp}}_{\gamma_{0}}.$

This is a closed subspace, so $\varphi^{*}\mathcal{F}$ is isomorphic to the
(shifted) extension by zero of the constant sheaf on
${\mathrm{Sp}}_{\gamma_{0}}$. In particular,
$R\Gamma(\varphi^{*}\mathcal{F})\cong H_{*}({\mathrm{Sp}}_{\gamma_{0}})$. On
the other hand, by the adjunction

$\operatorname{Hom}(\varphi^{*}\mathcal{F},\varphi^{*}S)=\operatorname{Hom}(\mathcal{F},\varphi_{*}\varphi^{*}S)$

it is clear that $\operatorname{Hom}(\varphi^{*}\mathcal{F},\varphi^{*}S)\cong
R\Gamma(\varphi^{*}\mathbb{C}_{\\{0\\}})[-2N]$. Thus
$H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})^{\epsilon}\cong
H_{*}({\mathrm{Sp}}_{\gamma_{0}})[-2N]$.

∎

We can rephrase Lemma 2.2 as follows. We have the Leray filtration on the
Borel-Moore homology of ${\widetilde{\mathrm{Sp}}}_{\gamma}$ such that

(8)
$\operatorname{gr}H_{k}({\widetilde{\mathrm{Sp}}}_{\gamma})=\bigoplus_{i+j=k}H_{i}({\mathrm{Sp}}_{\gamma},R^{j}\pi_{*}{\mathbb{C}}),\
0\leq j\leq 2N.$

######  Corollary 2.3.

a) The $W$-invariant part of the homology of
${\widetilde{\mathrm{Sp}}}_{\gamma}$ is canonically isomorphic to the $j=0$
part of (8):

$H_{k}({\widetilde{\mathrm{Sp}}}_{\gamma})^{W}=H_{k}({\mathrm{Sp}}_{\gamma},R^{0}\pi_{*}{\mathbb{C}})=H_{k}({\mathrm{Sp}}_{\gamma},{\mathbb{C}}).$

b) The associated graded of the $W$-antiinvariant part is isomorphic to the
$j=2N$ part of (8). In other words, the restriction of the obvious map

$H_{k}({\widetilde{\mathrm{Sp}}}_{\gamma})^{\epsilon}\to
H_{k-2N}({\mathrm{Sp}}_{\gamma},R^{2N}\pi_{*}{\mathbb{C}})\cong
H_{k-2N}({\mathrm{Sp}}_{\gamma_{0}})$

is an isomorphism.

######  Remark 2.4.

The result is also true in cohomology, cohomology with compact supports and BM
homology (where we replace the constant sheaf by
$\omega_{{\widetilde{\mathrm{Sp}}}_{\gamma}}$) by identical reasoning.

######  Remark 2.5.

The stabilizer
$K_{\gamma}:=\operatorname{\mathrm{Stab}}_{G(\mathcal{K})}(\gamma)$ acts
naturally on ${\widetilde{\mathrm{Sp}}}_{\gamma}$, inducing an action of the
component group in cohomology. In particular, we have the Springer action for
equivariant versions of any of the above theories, as well as a commuting
action of the component group of the centralizer.

###  2.3. Extended symmetry

In addition to the action of $\widetilde{W}$ on
$H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})$ there is a degenerate action of
the character lattice $X^{*}(T)$ on
$H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})$ defined as follows. There is a
natural map $\mathbf{I}\to T$ realizing $T$ as the reductive quotient of
$\mathbf{I}$. In particular, each character $\chi:T\to\mathbb{G}_{m}$ gives a
map $\mathbf{I}\to\mathbb{G}_{m}$ and in particular a $\mathbb{G}_{m}$-torsor
$\mathcal{L}(\chi)$ on $\operatorname{Fl}$. As a line bundle, we can write
this as

$G\times^{\mathbf{I},\chi}\mathbb{A}^{1}\to\operatorname{Fl}.$

Cap product with $c_{1}(\mathcal{L}(\chi))$ defines an action of
$X^{*}(T)\otimes_{\mathbb{Z}}{\mathbb{C}}$ on
$H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})$.

######  Theorem 2.6.

Let $X_{*}(T)\subset\widetilde{W}$ be the translation part of the extended
affine Weyl group. Then the Springer action of $X_{*}(T)$ and the action of
$X^{*}(T)$ defined above commute.

###### Proof.

This is proved in [88, Corollary 6.1.7]. ∎

######  Remark 2.7.

Note that this is not true in equivariant Borel-Moore homology, as one gets
essentially relations in the degenerate DAHA.

Note that we can identify the action of $X_{*}(T)$ with the action of
${\mathbb{C}}[T^{\vee}]$ and the action of
$X^{*}(T)\otimes_{\mathbb{Z}}{\mathbb{C}}$ with the action of
${\mathbb{C}}[\mathfrak{t}]$, and summarize the above results in the following

######  Proposition 2.8.

The non-equivariant Borel-Moore homology
$H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})$ is a (right) module over
${\mathbb{C}}[T^{*}T^{\vee}]\rtimes W$.

######  Proposition 2.9.

The non-equivariant Borel-Moore homology $H_{*}({\mathrm{Sp}}_{\gamma})$ is a
module over ${\mathbb{C}}[T^{*}T^{\vee}]^{W}$.

###### Proof.

By Lemma 2.2 we have
$H_{*}({\mathrm{Sp}}_{\gamma})=H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})^{W}$,
and by Proposition 2.8 $H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})$ has an
action of ${\mathbb{C}}[T^{*}T^{\vee}]$. By symmetrizing, we get the action of
${\mathbb{C}}[T^{*}T^{\vee}]^{W}$ on $H_{*}({\mathrm{Sp}}_{\gamma})$. ∎

We record the following lemma here.

######  Lemma 2.10.

The action of $X^{*}(T)\otimes_{\mathbb{Z}}{\mathbb{C}}$ on
$H_{*}({\mathrm{Sp}}_{\gamma})$ from Section 2.3 decreases the Leray
filtration (8) by two.

###### Proof.

This follows from the fact that the action comes from cap product with Chern
classes of the line bundles $L(\chi)$ constructed in the beginning of this
section. ∎

Let $\Delta$ be the top-dimensional class in $H^{2N}(G/B)$. By the isomorphism
$H^{*}(G/B)\cong{\mathbb{C}}[\mathfrak{t}]_{W}$, we can identify $\Delta$ with
an antisymmetric polynomial in ${\mathbb{C}}[\mathfrak{t}]$, namely
$\Delta=\prod_{\alpha\in\Phi^{+}}y_{\alpha}$. Moreover, note that we may write
antisymmetric polynomials as
${\mathbb{C}}[\mathfrak{t}]^{\epsilon}=\Delta\cdot{\mathbb{C}}[\mathfrak{t}]^{W}$.

######  Lemma 2.11.

We have that

$i_{*}(H_{k-2N}({\mathrm{Sp}}_{t^{-1}\gamma}))=\Delta\cdot
H_{k}({\widetilde{\mathrm{Sp}}}_{\gamma})^{\epsilon}.$

where $i:{\mathrm{Sp}}_{t^{-1}\gamma}\to{\mathrm{Sp}}_{\gamma}$ is the natural
inclusion.

###### Proof.

By Lemma 2.10 the operator $\Delta$ preserves the decomposition (8) and
decreases the $j$-grading by $2N$. Since $j\leq 2N$, the action of $\Delta$
kills all the summands in (8) with $j<2N$, so that

$\Delta\cdot H_{k}({\widetilde{\mathrm{Sp}}}_{\gamma})^{\epsilon}=\Delta\cdot
H_{k-2N}({\mathrm{Sp}}_{\gamma},R^{2N}\pi_{*}{\mathbb{C}}).$

At $p\in{\mathrm{Sp}}_{\gamma}$ where $\pi^{-1}(p)$ is a proper subset in the
flag variety, $R^{2N}\pi_{*}{\mathbb{C}}|_{p}=0$. On the other hand,
$\pi^{-1}(p)$ is the full flag variety if and only if $p\in
i({\mathrm{Sp}}_{t^{-1}\gamma})$ and in this case
$\Delta:R^{2N}\pi_{*}{\mathbb{C}}\to R^{0}\pi_{*}{\mathbb{C}}$ is the
isomorphism. Therefore

$\Delta\cdot
H_{k-2N}({\mathrm{Sp}}_{\gamma},R^{2N}\pi_{*}{\mathbb{C}})=i_{*}(H_{k-2N}({\mathrm{Sp}}_{t^{-1}\gamma})).$

∎

###  2.4. The lattice action

Let $G_{\gamma}$ be the stabilizer of $\gamma$ in $G(\mathcal{K})$. Obviously,
$G_{\gamma}$ acts on ${\mathrm{Sp}}_{\gamma}$, giving an action (of its
component group) on the homology of ${\mathrm{Sp}}_{\gamma}$. Since the action
in this case is proper [47], we also get an action on the Borel-Moore homology
$H_{*}({\mathrm{Sp}}_{\gamma})$.

The $W$-invariant translation part of the stabilizer action restricts to an
action of the ”spherical part” ${\mathbb{C}}[X_{*}(T)]^{W}$ on
$H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})$. This action commutes with the
Springer action, and the local main theorem of [89] identifies the spherical
part of the Springer action with the spherical part of the lattice action.
More precisely,

######  Theorem 2.12.

[89] One can define a canonical homomorphism
${\mathbb{C}}[X_{*}(T)]^{W}\twoheadrightarrow{\mathbb{C}}[\pi_{0}(G_{\gamma})]$
and the spherical part of the Springer action on
$H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})$ factors through this map.

By Propositions 2.8 and 2.9, the (BM) homology of ${\mathrm{Sp}}_{\gamma}$ and
the homology of ${\widetilde{\mathrm{Sp}}}_{\gamma}$ define quasicoherent
sheaves $\mathcal{F}_{\gamma}^{\prime}$ and
$\widetilde{\mathcal{F}}_{\gamma}^{\prime}$ on $(T^{*}T^{\vee})/W$.

######  Lemma 2.13.

These quasicoherent sheaves are actually coherent, and set-theoretically
supported on the Lagrangian subvariety $\\{0\\}\times T^{\vee}\subset
T^{*}T^{\vee}$. Moreover, the dimension of their support equals the rank of
the centralizer.

###### Proof.

It is clear that the action of $\mathfrak{t}$ is nilpotent, and symmetric
functions in $\mathfrak{t}$ act by 0, as they do in $H_{*}(G/B)$.

By Theorem 2.12 the homology of ${\mathrm{Sp}}_{\gamma}$ is finitely generated
over ${\mathbb{C}}[T^{\vee}]$ and the lattice part of the centralizer acts
freely on the components by Kazhdan-Lusztig [47]. Therefore the support is
exactly $\\{0\\}\times T^{\vee}_{\gamma}/W$, which has dimension the rank of
the cocharacter lattice of the centralizer of $\gamma$. ∎

###  2.5. Equivariant versions, endoscopy

We can in fact upgrade this construction with the addition of equivariance to
the picture. The centralizer $G_{\gamma}/\mathcal{K}$ has a smooth integral
model $J_{\gamma}$ over $\mathcal{O}$, see e.g. [67]. The stabilizer action
factors through the local Picard group
$P_{\gamma}=G_{\gamma}(\mathcal{K})/J_{\gamma}(\mathcal{O})$ whose underlying
reduced scheme is finite-dimensional and locally of finite type. Consider the
connected component of the identity $P_{\gamma}^{0}$ of this group scheme.
This is a linear algebraic group over ${\mathbb{C}}$ whose maximal reductive
quotient contains a split maximal torus of rank
$\text{rank}(X^{*}(G_{\gamma}))$. Call this torus $T_{\gamma}$. It also acts
on ${\mathrm{Sp}}_{\gamma}$, and we may take the equivariant BM homology as in
[51, Section 3]. The construction of the Springer action etc. from this
section go through $T_{\gamma}$-equivariantly. For simplicity, assume
$T_{\gamma}\hookrightarrow T$. Then we have:

######  Proposition 2.14.

The equivariant BM homology $H_{*}^{T_{\gamma}}({\mathrm{Sp}}_{\gamma})$ is a
quasi-coherent sheaf on $T^{*}T^{\vee}/W$. Its support is contained in the
subvariety $T^{*}T^{\vee}_{\gamma}/W\subset T^{*}T^{\vee}/W$. In particular,
if $\gamma$ is elliptic, the support is zero-dimensional.

###### Proof.

The equivariant cohomology of a point $H^{*}_{T_{\gamma}}(pt)$ acts on
$H_{*}^{T_{\gamma}}(Sp_{\gamma})$, giving that the support is contained in
$\mathfrak{t}_{\gamma}\times T^{\vee}/W$. Results of Yun [89] then imply that
the spherical part of the Springer action is given by
${\mathbb{C}}[\Lambda_{\gamma}]^{W}\cong{\mathbb{C}}[T_{\gamma}^{\vee}]^{W}$
just as in Theorem 2.12. This gives that the support lies in
$T^{*}T^{\vee}_{\gamma}/W$ (and in fact projects surjectively to
$T^{\vee}_{\gamma}/W$). ∎

######  Remark 2.15.

For split equivalued elements, or whenever we have equivariant formality of
the $H^{*}_{T_{\gamma}}(pt)$-action, the proof shows the support is all of
$T^{*}T^{\vee}_{\gamma}/W$.

If $G\neq GL_{n}$, we note that in the determination of the above support we
run into issues of endoscopy. Already for $G=SL_{n}$, the center $\mu_{n}$
will be contained in $T_{\gamma}$ for any $\gamma$. For example for elliptic
$\gamma$, we get support at $n$ points in $T^{*}T^{\vee}/W$. Let us now
illustrate how this plays out in the case of general $G$ and $\gamma$
elliptic. Recall that we may decompose

$H_{*}^{K_{\gamma}}({\mathrm{Sp}}_{\gamma})=\bigoplus_{\kappa\in
X_{*}(T)}H_{*}^{K_{\gamma}}({\mathrm{Sp}}_{\gamma})_{\kappa}$

where either $K_{\gamma}=\\{1\\}$ or $K_{\gamma}=T_{\gamma}$ (or any subgroup
of $G_{\gamma}$ for which the definitions make sense) as above. The
homological statement of the Fundamental Lemma, proved by Ngô, is essentially
that

(9) $H_{*}^{K_{\gamma}}({\mathrm{Sp}}_{\gamma})_{\kappa}\cong
H_{*}^{K_{\gamma}}({\mathrm{Sp}}^{H}_{\gamma_{H}})_{st}$

for some affine Springer fiber ${\mathrm{Sp}}^{H}_{\gamma_{H}}$ of an
endoscopic group of $G$. Here ”$st$” denotes the stable part, or in other
words the part of the BM homology where the lattice acts unipotently (see e.g.
[89]). Since $H_{*}({\mathrm{Sp}}_{\gamma})$ is finite-dimensional, by using
Lemma 2.13, the above $\kappa$-decomposition and Eq. (9), we deduce the
following version of the fundamental lemma.

######  Theorem 2.16.

The sheaf $\mathcal{F}_{\gamma}^{\prime}$ is supported at finitely many
points. Each stalk $(\mathcal{F}_{\gamma}^{\prime})_{(0,\kappa)}$ is
isomorphic to the stalk at $(0,1)$ of an ”endoscopic sheaf” on
$T^{*}T^{\vee}_{H}/W_{H}$ for some endoscopic group $H$ of $G$.

##  3\. The commuting variety

###  3.1. The commuting scheme

In this section, we introduce the partial resolution of the commuting variety
we will be considering. The construction is algebraic in nature. We show the
partial resolution coincides with
$\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}})$ in the case
$G=GL_{n}$. In general, we show it is a normal variety and conjecture that
locally its singularities are modeled by the ${\mathbb{Q}}$-factorial
terminalizations constructed by Losev in [54, 1]. Finally, we introduce a
certain open chart of the partial resolution, which turns out to be isomorphic
to the universal centralizer of $\check{G}$.

As above, let $\check{\mathfrak{g}}$ be the Lie algebra of $\check{G}$,
$\check{\mathfrak{t}}=\mathfrak{t}^{*}$ is the Lie algebra of $\check{T}$ and
$W$ is the Weyl group. We define two versions of the commuting scheme:
$\mathfrak{C}_{\check{\mathfrak{g}}}^{\prime}$ is a subscheme of
$\check{\mathfrak{g}}\times\check{\mathfrak{g}}$ cut out by the equation
$[x,y]=0$, while
$\mathfrak{C}_{\check{G}}^{\prime}=\mathfrak{C}_{\check{G},\check{\mathfrak{g}}}^{\prime}$
is a subscheme of $\check{G}\times\check{\mathfrak{g}}^{*}$ cut out by the
equation $\operatorname{Ad}_{g}(x)=x$.

Define
$\mathfrak{C}_{\check{\mathfrak{g}}}:=\mathfrak{C}_{\check{\mathfrak{g}}}^{\prime}/\\!\\!/\check{G}=\operatorname{Spec}\
{\mathbb{C}}[\mathfrak{C}_{\check{\mathfrak{g}}}^{\prime}]^{\check{G}}$ and
$\mathfrak{C}_{\check{G}}=\mathfrak{C}^{\prime}_{\check{G}}/\\!\\!/\check{G}=\operatorname{Spec}\
{\mathbb{C}}[\mathfrak{C}_{\check{G}}^{\prime}]^{\check{G}}$. It is a long-
standing open question if these schemes are reduced. We collect some facts
about $\mathfrak{C}_{\mathfrak{g}}$ and $\mathfrak{C}$ here.

There are natural restriction maps
${\mathbb{C}}[\mathfrak{C}_{\check{\mathfrak{g}}}]\to{\mathbb{C}}[\mathfrak{t}^{*}\times\mathfrak{t}^{*}]^{W}$
and
${\mathbb{C}}[\mathfrak{C}_{\check{G}}]\to{\mathbb{C}}[\check{T}\times\mathfrak{t}]^{W}$,
which induce maps
$(\mathfrak{t}^{*}\times\mathfrak{t}^{*})/W\to\mathfrak{C}_{\check{\mathfrak{g}}}$
and $(T^{*}\check{T})/W\to\mathfrak{C}_{\check{G}}$. The former is surjective
by the result of Joseph [46] and defines an isomorphism
$(\mathfrak{t}^{*}\times\mathfrak{t}^{*})/W\simeq[\mathfrak{C}_{\check{\mathfrak{g}}}]_{\operatorname{\mathrm{red}}}$.

In [12, Proposition 5.24], the following is proved by essentially reducing to
Joseph’s results in the rational case:

######  Theorem 3.1.

The restriction of functions induces an isomorphism
$(T^{*}\check{T})/W\simeq[\mathfrak{C}_{\check{G}}]_{\operatorname{\mathrm{red}}}$.

We note that there are alternative proofs of the theorem, such as the one
given by Gan-Ginzburg in type A:

######  Theorem 3.2 ([25]).

For $G=GL_{n}$ the scheme $\mathfrak{C}_{\mathfrak{g}}$ is reduced.

From this it is easy to deduce

######  Corollary 3.3.

For $G=GL_{n}$ the scheme $\mathfrak{C}_{G}$ is reduced.

###### Proof.

For $G=GL_{n}$ we have a natural embedding $G\subset\mathfrak{g}$ which
induced embedding
$\mathfrak{C}^{\prime}\subset\mathfrak{C}_{\mathfrak{g}}^{\prime}$ and
$\mathfrak{C}_{G}\subset\mathfrak{C}_{\mathfrak{g}}$. Since
$\mathfrak{C}_{\mathfrak{g}}$ is reduced, $\mathfrak{C}_{G}$ is reduced too. ∎

Recently, Chen-Ngô [17] proved that $\mathfrak{C}_{\mathfrak{g}}$ is reduced
for $\mathfrak{g}=\mathfrak{sp}_{2n}$ and subsequently, Losev [56] showed that
$\mathfrak{C}_{\mathfrak{g}}^{\prime}$ for $\mathfrak{g}=\mathfrak{sp}_{2n}$
is reduced as well.

###  3.2. Partial resolutions

In this subsection we define several graded commutative algebras closely
related to the commuting variety. By applying $\operatorname{\mathrm{Proj}}$
construction to these graded algebras, we recover partial resolutions of
$\mathfrak{C}_{\check{G}}$. We summarize various maps between the algebras in
the following commutative diagram:

(10) ${\bigoplus J^{d}}$${\bigoplus
I^{(d)}}$${\bigoplus{}_{0}\mathcal{A}^{\hbar=0}_{d}}$${\bigoplus
A^{d}}$${\bigoplus\mathbf{e}_{d}I^{(d)}.}$$\scriptstyle{a}$$\scriptstyle{b}$

All direct sums are over $d\geq 0$. Next, we define all of the entries in this
diagram, starting with the middle column.

We denote by $\epsilon$ the one-dimensional sign representation of $W$. We
define $A_{\mathfrak{t}}$, respectively $A$, as the subspace of
$W$-antiinvariant functions (that is, $\epsilon$-isotypic component) in
${\mathbb{C}}[\mathfrak{t}^{*}\times\mathfrak{t}]$, respectively
${\mathbb{C}}[T^{\vee}\times\mathfrak{t}]$. Also, let $J_{\mathfrak{t}}$,
respectively $J$ be the ideal in
${\mathbb{C}}[\mathfrak{t}^{*}\times\mathfrak{t}]$ (resp.
${\mathbb{C}}[T^{\vee}\times\mathfrak{t}^{*}]$) generated by
$A_{\mathfrak{t}}$ (resp. A). Now $A^{d}$ and $J^{d}$ are the powers of $A$
and $J$ inside the respective polynomial rings, with the assumption that
$A^{0}={\mathbb{C}}[T^{\vee}\times\mathfrak{t}]^{W}$ and
$J^{0}={\mathbb{C}}[T^{\vee}\times\mathfrak{t}]$.

In the right column we have a family of ideals
$I^{(d)}:=\bigcap_{\alpha\in\Phi^{+}}\langle
1-\alpha^{\vee},y_{\alpha}\rangle^{d}$. In particular, for $d=1$ we get the
defining ideal of the union of the codimension 2 hyperplanes corresponding to
the roots of $\mathfrak{g}$. It is easy to see that $I^{(d_{1})}\cdot
I^{(d_{2})}\subset I^{(d_{1}+d_{2})}$, so we have a graded algebra structure
on the direct sum of all $I^{(d)}$. Furthermore, let $\mathbf{e}_{d}$ denote
the projector to the representation $\epsilon^{d}$ in ${\mathbb{C}}[W]$, that
is, symmetrizer $\mathbf{e}$ for $d$ even and antisymmetrizer $\mathbf{e}_{-}$
for $d$ odd.

######  Lemma 3.4.

a) We have $\mathbf{e}_{d}J^{d}=A^{d}$ for all $d\geq 0$.

b) There are natural inclusions $b_{d}:J^{d}\to
I^{(d)},A^{d}\to\mathbf{e}_{d}I^{(d)}.$

###### Proof.

a) For $d=0$ this is clear from the definition, so we focus on $d>0$. Since
$J$ is the ideal generated by $A$, it is spanned by elements of the form
$a\cdot f$ for $a\in A$ and $f\in{\mathbb{C}}[T^{\vee}\times\mathfrak{t}]$,
and $J^{d}$ is spanned by elements of the form $a_{1}\cdots a_{d}\cdot f$ for
$a_{1},\ldots,a_{d}\in A$ and $f\in{\mathbb{C}}[T^{\vee}\times\mathfrak{t}]$.
Since $a_{1},\ldots,a_{d}$ are antisymmetric, we get

$\mathbf{e}_{d}a_{1}\cdots a_{d}\cdot f=a_{1}\cdots a_{d}\mathbf{e}(f)\in
A^{d}.$

b) It is easy to see that an antisymmetric polynomial vanishes on all
codimension 2 hyperplanes, so we get $A\subset I^{(1)}$ and hence $J\subset
I^{(1)}$. Therefore $J^{d}\subset(I^{(1)})^{d}\subset I^{(d)}$ and the result
follows. ∎

Finally, in the left column we have commutative Coulomb branch
${\mathbb{Z}}$-algebra $\bigoplus{}_{0}\mathcal{A}^{\hbar=0}_{d}$ to be
defined in Section 6. It is a generalization of the commutative Coulomb branch
appearing in the work of Braverman, Finkelberg and Nakajima [9, 10]. It is
defined as the convolution algebra in the equivariant Borel-Moore homology of
a certain space related to the affine Grassmannnian of $G$, and we postpone
its definition to Section 6. Here we summarize some of its basic properties
with pinpoint references to the proofs later in the paper.

######  Theorem 3.5.

The algebras ${}_{0}\mathcal{A}^{\hbar=0}_{d}$ have the following properties:

a) For all $d$ the module ${}_{0}\mathcal{A}^{\hbar=0}_{d}$ is free over
${\mathbb{C}}[\mathfrak{t}]^{W}$ and embeds into
$\mathbf{e}_{d}{\mathbb{C}}[T^{\vee}\times\mathfrak{t}]$.

b) For $d=0$, we have
${}_{0}\mathcal{A}^{\hbar=0}_{0}={\mathbb{C}}[T^{\vee}\times\mathfrak{t}]^{W}$.

c) For $d=1$, we have ${}_{0}\mathcal{A}^{\hbar=0}_{1}=A$.

d) In type $A$, we have
${}_{0}\mathcal{A}^{\hbar=0}_{d}=({}_{0}\mathcal{A}^{\hbar=0}_{1})^{d}=A^{d}$
for all $d$.

###### Proof.

a) We regard ${\mathbb{C}}[\mathfrak{t}]^{W}$ as the equivariant cohomology of
the point, and ${}_{0}\mathcal{A}^{\hbar=0}_{d}$ is realized as the
equivariant Borel-Moore homology of a certain space which admits an affine
paving by Bruhat cells. Therefore it is equivariantly formal and its
equivariant cohomology is a free module over $H^{*}_{G}(pt)$. The embedding to
${\mathbb{C}}[T^{\vee}\times\mathfrak{t}]$ is realized by the inclusion to
equivariant Borel-Moore homology of the fixed point set. See Section 6 for
more details. That we land in the $\mathbf{e}_{d}$-isotypic component follows
from the fact that the localization is defined using $T$-equivariant
cohomology and to pass to $G$-equivariant cohomology we take $W$-invariants.
See for instance [9, Remark 5.23].

b) This is a specialization of Corollary 6.3 at $\hbar=0$.

c) By part (a) we have inclusion
${}_{0}\mathcal{A}^{\hbar=0}_{1}\subset\mathbf{e}_{-}{\mathbb{C}}[T^{\vee}\times\mathfrak{t}]=A$.
By Theorem 6.5 (specialized at $\hbar=0$) this is an isomorphism.

d) This is a specialization of Theorem 6.21 at $\hbar=0$. ∎

Note that part (c) of the theorem yields the map
$A\to{}_{0}\mathcal{A}^{\hbar=0}_{1}$ and hence a family of maps
$A^{d}\to{}_{0}\mathcal{A}^{\hbar=0}_{d}$. These are denoted by $a_{d}$ in the
commutative diagram (10).

######  Corollary 3.6.

If $G$ has rank 1 then
$A^{d}\simeq{}_{0}\mathcal{A}^{\hbar=0}_{d}\simeq\mathbf{e}_{d}I^{(d)}$ for
all $d$.

###### Proof.

By Theorem 3.5(d) we get $A^{d}\simeq{}_{0}\mathcal{A}^{\hbar=0}_{d}$. On the
other hand, in rank 1 we have only one codimension 2 hyperplane and it is easy
to see that $I^{(d)}=A^{d}$. ∎

######  Lemma 3.7.

There is a natural inclusion
${}_{0}\mathcal{A}^{\hbar=0}_{d}\to\mathbf{e}_{d}I^{(d)}$ corresponding to the
dotted arrow in (10).

###### Proof.

By Theorem 3.5(a) we know that ${}_{0}\mathcal{A}^{\hbar=0}_{d}$ is contained
in the $\mathbf{e}_{d}$-isotypic component, so it is sufficient to check that
it is contained in $I^{(d)}$ or, equivalently, in $\langle
1-\alpha^{\vee},y_{\alpha}\rangle^{d}$ for all $\alpha$. For this it is
sufficient to restrict to a rank 1 subgroup of $G$, similarly to [11] and [59,
Section 3].

On the other hand, if $G$ has rank 1 then the result follows from Corollary
3.6. ∎

######  Theorem 3.8.

We have ${}_{0}\mathcal{A}^{\hbar=0}_{d}\simeq\mathbf{e}_{d}I^{(d)}$ for all
$d$.

###### Proof.

By Lemma 3.7 we have an inclusion of ${}_{0}\mathcal{A}^{\hbar=0}_{d}$ into
$\mathbf{e}_{d}I^{(d)}$. Since ${}_{0}\mathcal{A}^{\hbar=0}_{d}$ is free over
${\mathbb{C}}[\mathfrak{t}]^{W}$ and $\mathbf{e}_{d}I^{(d)}$ is torsion free,
it is sufficient to prove that the inclusion is an isomorphism outside of
codimension 2 subset.

Both ${\mathbb{C}}[\mathfrak{t}]^{W}$-modules are supported on the union of
the root hyperplanes in $\mathfrak{t}^{*}/W$. If we specialize to a generic
point in one of the hyperplanes, we can replace $G$ by its rank 1 subgroup,
and the isomorphism follows from Corollary 3.6. Therefore the two modules are
isomorphic outside of the union of pairwise intersections of hyperplanes,
which has codimension 2. ∎

We can use the above graded algebras to construct projective varieties

$\widetilde{\mathfrak{C}}_{\check{G}}:=\operatorname{\mathrm{Proj}}\bigoplus_{d}{}_{0}\mathcal{A}^{\hbar=0}_{d}\simeq\operatorname{\mathrm{Proj}}\bigoplus_{d}\mathbf{e}_{d}I^{(d)},\
Y_{G}=\operatorname{\mathrm{Proj}}\bigoplus_{d}I^{(d)}.$

By the work of Haiman [40] for $G=GL_{n}$ we have
$\widetilde{\mathfrak{C}}_{\check{G}}=\operatorname{Hilb}^{n}({\mathbb{C}}^{*}\times{\mathbb{C}})$
and $Y_{G}$ is isomorphic to the isospectral Hilbert scheme of
${\mathbb{C}}^{*}\times{\mathbb{C}}$:

${Y_{G}}$${({\mathbb{C}}^{*}\times{\mathbb{C}})^{n}}$${\operatorname{Hilb}^{n}({\mathbb{C}}^{*}\times{\mathbb{C}})}$${S^{n}({\mathbb{C}}^{*}\times{\mathbb{C}}).}$

We claim that the varieties $\widetilde{\mathfrak{C}}_{\check{G}}$ and $Y_{G}$
can be considered as the partial resolutions of the commuting variety which we
identify with $T^{*}T^{\vee}/W$.

######  Remark 3.9.

In [27], Ginzburg defines and studies the isospectral commuting variety for
general $G$. On the other hand, the variety
$Y_{G}=\operatorname{\mathrm{Proj}}\bigoplus_{d\geq 0}I_{G}^{(d)}$ is another
candidate for the isospectral commuting variety. It is natural to wonder how
the two constructions are related.

######  Remark 3.10.

In [29], Ginzburg-Kaledin prove that there are no crepant resolutions of
$T^{*}\mathfrak{t}/W$ for $W$ outside types $A,B,C$. Their definition of
symplectic resolution includes the crepant condition, so their statement is
non-existence of symplectic resolutions. Indeed, these conditions are
equivalent in this case. This non-existence of a symplectic resolution is thus
likely the case for $T^{*}T/W$ as well. We note however that from the results
of [7], it follows that $T^{*}\mathfrak{t}/W$ and $T^{*}T/W$ admit birational
maps from the universal centralizer group schemes appearing Theorem 3.21,
which are smooth for simply connected groups in any type. Note that these are
not resolutions of singularities in the usual sense, since they are not
proper.

######  Proposition 3.11.

$Y_{G}$ is normal.

###### Proof.

We will prove the homogeneous coordinate ring $\bigoplus J^{(d)}$ is
integrally closed. Indeed, $1-\alpha^{\vee},y_{\alpha}$ is a regular sequence.
The powers of an ideal generated by a regular sequence are integrally closed,
and intersection preserves integral closedness. So the symbolic blow-up
considered here is integrally closed. ∎

######  Corollary 3.12.

$\widetilde{\mathfrak{C}}_{\check{G}}$ is normal.

###### Proof.

If a normal variety $Y$ is acted upon by a finite group $\Gamma$, $Y/\Gamma$
is normal [77, Chapter II.5., top of page 128] (note that Shafarevich assumes
that $Y$ is affine, but the argument works for any variety since this is a
local property). ∎

In addition to normality, we have

######  Proposition 3.13.

The variety $\widetilde{\mathfrak{C}}_{\check{G}}$ has only terminal
singularities.

###### Proof.

This follows from [87, Theorem 15]. ∎

######  Remark 3.14.

An alternative proof of normality of $\widetilde{\mathfrak{C}}_{\check{G}}$
follows from [87, Theorem 14].

######  Remark 3.15.

Fix some symplectic ${\mathbb{Q}}$-factorial terminalization of
$T^{*}T^{\vee}/W$, as constructed in [2, 66]. Denote it by
$\widetilde{X}_{\check{G}}$. The formal Poisson deformations of
$\widetilde{X}_{\check{G}}$ are parameterized by
$H^{2}(\widetilde{X}^{\mathrm{reg}}_{\check{G}},{\mathbb{C}})$. There is a Lie
algebra version
$\widetilde{X}_{\check{\mathfrak{g}}}\to\mathfrak{t}\oplus\mathfrak{t}^{*}/W$
where $\widetilde{X}_{\check{\mathfrak{g}}}$ is the ${\mathbb{Q}}$-factorial
terminalization constructed in [54]. This is a conical symplectic partial
resolution with ${\mathbb{Q}}$-factorial singularities such that
$\operatorname{\mathrm{codim}}(\widetilde{X}_{\check{\mathfrak{g}}}-\widetilde{X}_{\check{\mathfrak{g}}}^{\mathrm{reg}})\geq
4$. By [66], this implies that $\widetilde{X}_{\check{\mathfrak{g}}}$ is
terminal. In this case, it is possible to prove [66, 54] that the filtered
quantizations of $\widetilde{X}_{\check{\mathfrak{g}}}$ are also parametrized
by $H^{2}(\widetilde{X}^{\mathrm{reg}}_{\check{\mathfrak{g}}},{\mathbb{C}})$.
We do not know if this is the case for $\widetilde{X}_{\check{G}}$, since the
results in the Lie algebra case heavily use the fact that
$\mathfrak{t}\oplus\mathfrak{t}^{*}/W$ has conical symplectic singularities.

Finally, we note that similar to [1, Proposition 2.1.] the normal intermediate
partial resolutions

$\widetilde{X}_{\check{G}}\to X_{\check{G}}\to T^{*}T^{\vee}/W$

are classified by faces of the ample cone of $\widetilde{X}$ such that for a
given face $F$ and a rational point $f\in F$ a positive rational multiple of
$f$ is the first Chern class of an ample line bundle on the corresponding
partial resolution.

It seems reasonable thus to expect that
${\mathfrak{P}}=H^{2}(\widetilde{X}^{\mathrm{reg}}_{\check{G}},{\mathbb{C}})$
parametrizes both the filtered quantizations of $\widetilde{X}_{\check{G}}$ as
well as the partial resolutions between $\widetilde{X}_{\check{G}}\to
T^{*}T^{\vee}/W$, just as it does in the Lie algebra case. Further, it seems
by Corollary 3.12 and Proposition 3.13 reasonable to expect that
$\widetilde{\mathfrak{C}}_{\check{G}}$ equals a partial resolution constructed
this way, and that the singularities of $\widetilde{X}_{\check{G}}\to
T^{*}T^{\vee}/W$ are locally modeled on those of
$\widetilde{X}_{\check{\mathfrak{g}}}$.

To support this remark, we note the following about the local structure of our
algebras.

######  Lemma 3.16.

Upon completion at $a\in T^{\vee}$, the Coulomb branch algebra
$({}_{i}\mathcal{A}_{d}^{\hbar})^{\wedge
a}\cong\mathbb{H}_{G,c+i\hbar,\hbar}^{\wedge a}$ is isomorphic to
$\mathbb{H}^{\mathrm{rat},\wedge a}_{\mathfrak{g},c+i\hbar,\hbar}$ for some
$\mathfrak{g}$ coming from the Borel-de Siebenthal algorithm for $G$.

###### Proof.

The first part is [22, Proof of Theorem 3.2.]. Since we are dealing with the
equal parameters case, as in e.g. [1], and the parameters behave as in [23,
Section 2.8.] upon completion, the result follows. ∎

###  3.3. The universal centralizer

In the above, we have defined the partial resolution
$\widetilde{\mathfrak{C}}_{\check{G}}$ using $\operatorname{\mathrm{Proj}}$
construction, and have limited understanding of its geometry outside of type
$A$. Nevertheless, in this subsection we define an affine open chart in
$\widetilde{\mathfrak{C}}_{\check{G}}$ and prove that it coincides with the
trigonometric version of the universal centralizer of [7, 67]. It also appears
as a Coulomb branch for zero matter, and will be used later in Section 8.

We let $G$ be arbitrary for now. Let
$\Delta:=\prod_{\alpha\in\Phi}y_{\alpha}\in A_{G}$ be the Vandermonde
determinant.

######  Definition 3.17.

Let

$U_{\Delta}\subset\widetilde{\mathfrak{C}}_{\check{G}}$

be defined as the distinguished open subset given by the element $\Delta\in
A_{G}$. By definition, $U_{\Delta}$ is the affine variety whose coordinate
ring is the degree zero part of the localization of
$\bigoplus_{d=0}^{\infty}{}_{0}\mathcal{A}_{d}$ in $\Delta$.

######  Remark 3.18.

Note that this is different from the preimage of of $\\{\Delta\neq 0\\}$ under
the natural composition of maps $\widetilde{\mathfrak{C}}_{\check{G}}\to
T^{*}T^{\vee}/W\to\mathfrak{t}/W$. On this locus the first map is an
isomorphism.

We now describe this chart. In [7], two trigonometric versions of the
universal centralizer are studied. The one of interest to us is defined as
follows, see loc. cit. for more details.

######  Definition 3.19.

The universal centralizer of $\check{G}$ is the variety

$\mathfrak{B}_{\check{\mathfrak{g}}}^{\check{G}}:=\\{(g,s)\in\check{G}\times\check{\mathfrak{g}}^{*}|\mathrm{ad}_{g}(s)=s,\;g\text{
is regular }\\}\sslash\check{G}$

######  Remark 3.20.

In [7], this variety is denoted
$\mathfrak{Z}_{\check{\mathfrak{g}}}^{\check{G}}$. There is also another
version of the trigonometric universal centralizer
$\mathfrak{Z}_{\check{G}}^{\check{\mathfrak{g}}}$ with the roles of
$\mathfrak{g},G$ swapped. It has the nicer geometric property of being
symplectically isomorphic to $T^{*}(T^{\vee}/W)$ when $G$ is adjoint (so
$\check{G}$ is simply connected).

In [7] explicit description of the coordinate ring of
$\mathfrak{Z}_{\check{\mathfrak{g}}}^{\check{G}}$ is given. We also have the
following Coulomb branch description of
$\mathfrak{Z}_{\check{\mathfrak{g}}}^{\check{G}}$.

######  Theorem 3.21.

${\mathbb{C}}[\mathfrak{B}_{\check{\mathfrak{g}}}^{\check{G}}]\cong
H_{*}^{G(\mathcal{O})}(\operatorname{Gr}_{G})$, which is also the Coulomb
branch for $(G,0)$ (see Section 5.)

We now state and prove the main theorem of this section. The reader may want
to skip the proof at a first reading and see Section 5 first.

######  Theorem 3.22.

We have

${\mathbb{C}}[\mathfrak{B}_{\check{\mathfrak{g}}}^{\check{G}}]\cong\left(\left(\bigoplus_{d}{}_{0}\mathcal{A}^{\hbar=0}_{d}\right)_{\Delta}\right)_{0}.$

In particular, there is a natural isomorphism
$\mathfrak{B}_{\check{\mathfrak{g}}}^{\check{G}}\cong U_{\Delta}$.

###### Proof.

By [9, 4(vi)] there is a natural inclusion of the Coulomb branch algebra for
$(G,\mathfrak{g})$ to the Coulomb branch algebra for $(G,0)$. Using the same
geometric argument, which is essentially the the pullback map in BM homology
under $\operatorname{Gr}_{G}\hookrightarrow{}_{i}\mathcal{R}_{j}$, we also get
a map ${}_{i}\mathcal{A}_{j}^{\hbar=0}\hookrightarrow
H_{*}^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$ for any $i,j$.

Both the left and right hand sides are equipped with a Bruhat filtration
coming from the interpretation as Hom-spaces in the category of line
operators, cf. Section 5 and Theorem 3.21. Both of these spaces have bases
over $H^{*}_{\check{G}}(pt)$ indexed by $\lambda\in X_{*}(T)$, which up to
leading order are given by the formulas in Theorem 6.16 and [9, 3(x)(a) and
Prop. 6.6.]. Since the generalized root hyperplanes in the adjoint case are
just the root hyperplanes, it is clear that upon localization outside these
hyperplanes the two spaces have the same bases.

But we have even more control over the pole orders at the root hyperplanes.
Namely, the coefficient of $u^{\lambda}$ in
$[\mathcal{R}_{G,0}^{\leq\lambda}]$, meaning the class associated to $\lambda$
for zero matter in the analog of Lemma 6.7 is just
$\frac{1}{\prod_{\alpha}y_{\alpha}^{\max(0,\alpha(\lambda))-1}}$, and the
coefficient in front of $u^{\lambda}$ in the analogous expression for adjoint
matter only has $y_{\alpha}$ vanishing to order at most $d$ in the numerator.
Dividing elements of ${}_{0}\mathcal{A}_{d}^{\hbar=0}$ by $\Delta^{d}$ for
$d\geq 0$ then gives an injective map to the zero matter Coulomb branch
algebra, and multiplying by $f\in H_{G}^{*}(pt)$ if necessary, we see that we
get all the $H_{G}^{*}(pt)$-basis elements of
$H^{G_{\mathcal{O}}}(\operatorname{Gr}_{G})$, so that this is an isomorphism.

∎

######  Remark 3.23.

Alternatively, one can use the blow-up description of
$\mathfrak{B}_{\check{\mathfrak{g}}}^{\check{G}}$ [7, Section 4] to get a more
geometric proof of this result.

######  Remark 3.24.

We can interpret the proof of this Theorem as follows. As stated in the proof,
there are natural embeddings

${}_{j}\mathcal{A}_{i}^{\hbar=0}\hookrightarrow
H_{*}^{G(\mathcal{O})}(\operatorname{Gr}_{G})$

see for example Section 6.2 or [9, Lemma 5.11]. These embeddings realize
rational functions of the form $f(x,y)/\Delta^{j-i}$, where
$f\in{}_{j}\mathcal{A}^{\hbar=0}_{i}$, as functions on the open chart
$U_{\Delta}$.

When $G=GL_{n}$, this construction is closely related to the construction of
the open chart ”$U_{(1^{n})}$” on $\operatorname{Hilb}^{n}({\mathbb{C}}^{2})$
given by Haiman in [41, Corollary 2.7.].

###  3.4. Explicit antisymmetric polynomials

In Theorem 6.21 we will need an explicit construction of a
${\mathbb{C}}$-basis in the space $A$ of antisymmetric ($S_{n}$-antiinvariant)
polynomials for $G=GL_{n}$, in order to compare our Coulomb branch
construction with the one above. The exposition follows ideas of Haiman in
[41]. The reader is advised to skip this section on a first reading.

We denote by $\mathrm{Alt}$ the action of the antisymmetric projector
$\mathbf{e}_{-}$ on polynomials. Let
$S=\\{(a_{1},b_{1}),\ldots(a_{n},b_{n})\\}$ be an arbitrary $n$-element subset
of ${\mathbb{Z}}_{\geq 0}\times{\mathbb{Z}}$. We define

$\Delta_{S}(y_{1},u_{1},\ldots,y_{n},u_{n})=\mathrm{Alt}\left(y_{1}^{a_{1}}u_{1}^{b_{1}}\cdots
y_{n}^{a_{n}}u_{n}^{b_{n}}\right)=\frac{1}{n!}\det\left(y_{i}^{a_{j}}u_{i}^{b_{j}}\right).$

For a composition $\alpha$ with $\sum\alpha_{i}=n$, we can consider the set

$S_{\alpha}=\\{(0,0),\ldots,(\alpha_{1}-1,0),(0,1),\ldots(\alpha_{2}-1,1),\ldots\\}$

and denote $\Delta_{S_{\alpha}}=\Delta_{\alpha}$. In particular,

$\Delta=\prod_{i<j}(y_{i}-y_{j})=\Delta_{(n)}.$

Given a composition $\alpha$, write
$\lambda(\alpha)=(0^{\alpha_{1}},1^{\alpha_{2}},\ldots)$. More generally, for
any subset $S=\\{(a_{i},b_{i})\\}\subset{\mathbb{Z}}_{\geq
0}\times{\mathbb{Z}}$, let $\lambda(S)=\mathrm{sort}(b_{i})$ where we sort in
non-decreasing order. Clearly, $\lambda(S_{\alpha})=\lambda(\alpha)$.
Furthermore, we define a collection of subsets

$S_{k}=\\{a_{i}\ :\ (a_{i},k)\in
S\\}=\left\\{a_{i_{1}^{(k)}}<\ldots<a_{i_{r_{k}}^{(k)}}\right\\}$

and a partition

$\mu_{k}(S)=(a_{i_{1}^{(k)}},a_{i_{2}^{(k)}}-1,\ldots,a_{i_{r_{k}}^{(k)}}-r_{k}+1).$

Finally, we define $\mathbf{y}_{k}=(y_{i}\ :\ \lambda_{i}=k)$.

######  Lemma 3.25.

(a) The determinants $\Delta_{S}$ span the vector space $A$.

(b) We have the following formula for the determinant $\Delta_{\alpha}$:

$\Delta_{\alpha}=c\cdot\mathrm{Alt}\left[u^{\lambda}\prod_{r<s,\lambda_{r}=\lambda_{s}}(y_{r}-y_{s})\right].$

where $\lambda=\lambda(\alpha)$.

(c) More generally, we have the following formula for the determinant
$\Delta_{S}$:

$\Delta_{S}=c\cdot\mathrm{Alt}\left[\prod_{k}s_{\mu_{k}}(\mathbf{y}_{k})u^{\lambda}\prod_{r<s,\lambda_{r}=\lambda_{s}}(y_{r}-y_{s})\right]$

where $s_{\mu_{k}}$ are Schur polynomials, $\lambda=\lambda(S)$ and $c$ is
some nonzero scalar factor depending on the size of stabilizer of $\lambda$.

###### Proof.

(a) The space ${\mathbb{C}}[T^{*}T^{\vee}]$ is spanned by the monomials
$y_{1}^{a_{1}}u_{1}^{b_{1}}\cdots y_{n}^{a_{n}}u_{n}^{b_{n}}$, so
$A=\mathbf{e}_{-}{\mathbb{C}}[T^{*}T^{\vee}]$ is spanned by their
antisymmetrizations (recall that $u_{i}$ are invertible, so $b_{i}$ are
allowed to be negative). If some of pairs $(a_{j},b_{j})$ coincide, then the
antisymmetrization vanishes, so it is sufficient to assume that
$(a_{j},b_{j})$ are pairwise distinct and form an $n$-element subset $S$.

Clearly, (b) follows from (c) since for $S=S_{\alpha}$ we have
$S_{k}={0,1\ldots,\alpha_{k}-1}$ and $\mu_{k}=(0)$.

To prove (c), observe that the function $\Delta_{S}$ is antisymmetric and all
possible monomials in $u$ are in the $S_{n}$-orbit of $\lambda$, so it is
sufficient to compute the coefficient at $u^{\lambda}$. This coefficient is
proportional to

$\mathrm{Alt}_{\operatorname{\mathrm{Stab}}(\lambda)}(y_{1}^{a_{1}}\cdots
y_{n}^{a_{n}})=\prod_{k}\mathrm{Alt}_{S_{k}}\left[\prod_{\lambda_{i}=k}y_{i}^{a_{i}}\right]=\prod_{k}\left[s_{\mu_{k}}(\mathbf{y}_{k})\cdot\prod_{r<s,\lambda_{r}=\lambda_{s}=k}(y_{r}-y_{s})\right].$

∎

######  Example 3.26.

For $\alpha=(1,2,1)$ we have
$S=\\{(0,0),(0,1),(1,1),(0,2)\\},\lambda=(0,1,1,2)$ and

$\Delta_{S}=\mathrm{Alt}\left[(y_{2}-y_{3})u_{1}^{0}u_{2}^{1}u_{3}^{1}u_{4}^{2}\right]$

For $S=\\{(5,0),(3,1),(7,1),(2,2)\\}$ we have
$\lambda=(0,1,1,2),\mu_{0}=(5),\mu_{1}=(3,6),\mu_{2}=(2)$ and

$\Delta_{S}=\mathrm{Alt}\left[s_{5}(y_{1})s_{6,3}(y_{2},y_{3})s_{2}(y_{4})(y_{2}-y_{3})u_{1}^{0}u_{2}^{1}u_{3}^{1}u_{4}^{2}\right].$

Note that $s_{5}(y_{1})=y_{1}^{5},s_{2}(y_{4})=y_{4}^{2}$ and

$s_{6,3}(y_{2},y_{3})=\frac{\vline\begin{matrix}y_{2}^{7}&y_{3}^{7}\\\
y_{2}^{3}&y_{3}^{3}\end{matrix}\vline}{y_{2}-y_{3}}=-\frac{\mathrm{Alt}(y_{2}^{3}y_{3}^{7})}{y_{2}-y_{3}}.$

##  4\. Trigonometric Cherednik algebra

###  4.1. Definitions

We define the extended torus $\widetilde{T}=T\times{\mathbb{G}}_{m}^{*}$ and
the corresponding Lie algebra
$\widetilde{\mathfrak{t}}=\mathfrak{t}\oplus{\mathbb{C}}_{\hbar}$. The
extended affine Weyl group $\widetilde{W}:=W\ltimes X_{*}(T)$ is generated by
the affine Weyl group $W^{\mathrm{aff}}=W\ltimes Q^{\vee}$ and an additional
abelian group $\Omega=X_{*}(T)/Q^{\vee}$ where $Q^{\vee}$ is the coroot
lattice. We use the affine action of $W\ltimes X_{*}(T)$ on the cocharacter
lattice $X_{*}(T)$ depending on $\hbar$. The action of $w\in\widetilde{W}$ on
$\xi\in X_{*}(T)$ will be denoted by ${}^{w}\xi$. We will denote the longest
element in $W$ by $w_{0}$.

######  Definition 4.1.

The trigonometric DAHA of $G$ is the ${\mathbb{C}}[\hbar,c]$-algebra, which as
a vector space looks like

$\mathbb{H}_{G}=\mathbb{H}_{c,\hbar}={\mathbb{C}}[\widetilde{W}]\otimes{\mathbb{C}}[X_{*}(T)\otimes_{\mathbb{Z}}{\mathbb{C}}]\otimes{\mathbb{C}}[c]\otimes{\mathbb{C}}[\hbar]$

and the algebra structure is determined as follows:

  1. (1)

Each of the tensor factors is a subalgebra, and $c,\hbar$ are central. We
denote by $\sigma_{i}$ the simple reflections in the copy of
$\widetilde{W}\subset\mathbb{H}_{G}$.

  2. (2)

$\sigma_{i}\xi-{}^{s_{i}}\xi\sigma_{i}=c\langle\xi,\alpha_{i}^{\vee}\rangle$
for all simple reflections $\sigma_{i}\in\widetilde{W}$ and $\xi\in X^{*}(T)$.

  3. (3)

For any $\omega\in\Omega\subset\widetilde{W}$,
$\omega\xi={}^{\omega}\xi\omega$

Note that here the pairing
$\langle,\rangle:\widetilde{\mathfrak{t}}\times\widetilde{\mathfrak{t}}\to{\mathbb{C}}[\hbar]$
depends on $\hbar$.

######  Example 4.2.

For $G=GL_{n}$ the group $\widetilde{W}$ is generated by simple reflections
$\sigma_{1},\ldots,\sigma_{n-1}$ (which generate $W$) and an additional
element $\pi$ (which generates $\Omega$). The lattice part of $\widetilde{W}$
is generated by

$X_{i}=\sigma_{i-1}\cdots\sigma_{1}\pi\sigma_{n-1}\cdots\sigma_{i}.$

The algebra $\mathbb{H}_{c,\hbar}$ is generated by $\sigma_{i},\pi$ and
commuting variables $y_{1},\ldots,y_{n}$, with the following relations:

$\sigma_{i}y_{i}=y_{i+1}\sigma_{i}-c,\sigma_{i}y_{i+1}=y_{i}\sigma_{i}+c,\
\sigma_{i}y_{j}=y_{j}\sigma_{i}\ (j\neq i,i+1),$ $\pi y_{i}=y_{i+1}\pi(1\leq
i\leq n-1),\ \pi y_{n}=(y_{1}+\hbar)\pi.$

######  Remark 4.3.

We will also use specializations of this algebra, to be called trigonometric
DAHAs as well when there is no risk of confusion. Let us explain how this
relates to parameter conventions in the literature. In [72], the trigonometric
DAHA is defined as above but with parameters $\delta=\hbar,c=u$. These
correspond to the generators of $H^{*}_{{\mathbb{G}}_{m}^{\mathrm{rot}}}(*)$
and $H^{*}_{{\mathbb{G}}^{\mathrm{dil}}_{m}}(*)$, respectively. In [72], the
relations $c+\nu\delta=c+\nu\hbar=0$ and $\hbar=1$ are imposed for
$\nu\in{\mathbb{C}}$. This specialization of parameters is often called ”the”
trigonometric DAHA with parameter $\nu$. We will denote it by
$\mathbb{H}_{\nu}$. If we want to emphasize the role of $G$ instead of the
parameters, we will write $\mathbb{H}_{G}$ for any of the specializations of
the Cherednik algebra.

######  Remark 4.4.

It is common in Cherednik algebra literature to specialize $\hbar$ to $1$ as
above, so that the algebra $\mathbb{H}_{c,\hbar}^{\hbar=1}$ admits a natural
filtration in powers of $\hbar$, making the full $\mathbb{H}_{c,\hbar}$ the
Rees construction for this filtration. In this language,
$\mathbb{H}_{c,\hbar}^{c=\hbar=0}$ is the associated graded of
$\mathbb{H}_{c,\hbar}^{\hbar=1}$, since $\mathbb{H}_{c,\hbar}$ is flat over
${\mathbb{C}}[c,\hbar]$. With this in mind, we will use the specialization
$c=\hbar=0$ and the associated graded interchangeably.

######  Remark 4.5.

Later on, we shall be interested in the family of Cherednik algebras
$\mathbb{H}_{c+i\hbar,\hbar}$ for $i\leq 0$ as well, and the specializations
$c+(\nu+i)\hbar=0,\hbar=1,$ i.e. $c=-\frac{m+in}{n}$.

We introduce the symmetrizer $\mathbf{e}=\frac{1}{|W|}\sum_{w\in W}w$ and the
antisymmetrizer $\mathbf{e}_{-}=\frac{1}{|W|}\sum_{w\in W}(-1)^{\ell(w)}w$ in
the group algebra of $W$. We define the spherical and antispherical
subalgebras in $\mathbb{H}_{G}$ as $\mathbf{e}\mathbb{H}_{G}\mathbf{e}$ and
$\mathbf{e}_{-}\mathbb{H}_{G}\mathbf{e}_{-}$.

Note that in the specialization $c=\hbar=0$ the structure of the algebra
simplifies dramatically:
$\mathbb{H}_{G}^{c=\hbar=0}={\mathbb{C}}[W]\ltimes{\mathbb{C}}[T^{\vee}\times\mathfrak{t}]$,
so

$\mathbf{e}\mathbb{H}_{G}^{c=\hbar=0}\mathbf{e}\cong{\mathbb{C}}[T^{\vee}\times\mathfrak{t}]^{W}.$

We will refer to this as the commutative limit, although this only gives the
limit of the spherical subalgebra the structure of a commutative algebra.

The algebra $\mathbb{H}_{G}$ has a representation

(11)
$\mathbb{H}_{c,\hbar}\hookrightarrow\text{Diff}_{\hbar}(\mathfrak{t}^{\mathrm{reg}})\rtimes{\mathbb{C}}[W]$

defined e.g. in [18, Section 2.13]. Here
$\text{Diff}_{\hbar}(\mathfrak{t}^{\mathrm{reg}})$ is the algebra of
$\hbar-$difference operators on the Lie algebra $\mathfrak{t}$, possibly with
poles along the root hyperplanes. In this representation, the generators of
$\mathbb{H}_{G}$ corresponding to simple reflections act by

$\sigma_{i}=s_{i}+\frac{c}{y_{\alpha_{i}}}(s_{i}-1)$

and $\Omega$ acts by difference operators in a standard way.

######  Example 4.6.

For $G=GL_{n}$, we get $\sigma_{i}=s_{i}+\frac{c}{y_{i}-y_{i+1}}(s_{i}-1)$ and
$\pi\cdot f(y_{1},\ldots,y_{n})=f(y_{2},\ldots,y_{n},y_{1}+\hbar)$.

###  4.2. Shift isomorphism

We will need several involutions on the algebra $\mathbb{H}_{c}$.

######  Lemma 4.7.

The map
$\Phi:w\mapsto(-1)^{\ell(w)}w_{0}w^{-1}w_{0},\xi\mapsto{}^{w_{0}}\xi,\hbar\to-\hbar$
for $w\in\widetilde{W}$ defines an involutive anti-automorphism of
$\mathbb{H}_{G}$.

###### Proof.

Suppose that $w_{0}$ sends the simple root $\alpha_{i}$ to $-\alpha_{j}$ for
some $j$, then $w_{0}s_{i}=s_{j}w_{0}$. The map $\Phi$ sends $\sigma_{i}$ to
$-\sigma_{j}$, so we get: $\Phi(\sigma_{i}\xi)=-{}^{w_{0}}\xi\sigma_{j}$,
$\Phi({}^{s_{i}}\xi\sigma_{i})=-\sigma_{j}{}^{w_{0}s_{i}}\xi$ and

$\Phi(\sigma_{i}\xi-{}^{s_{i}}\xi\sigma_{i})=\sigma_{j}{}^{s_{j}w_{0}}\xi-{}^{w_{0}}\xi\sigma_{j}$

while
$\langle\xi,\alpha_{i}^{\vee}\rangle=\langle{}^{s_{j}w_{0}}\xi,\alpha_{j}^{\vee}\rangle$,
so the equation (2) is preserved. For the equation (3), observe that
$w_{0}\omega^{-1}w_{0}=\omega$ since we assume that $\hbar\mapsto-\hbar$. ∎

######  Example 4.8.

For $G=GL_{n}$ we have

$\Phi(\sigma_{i})=-\sigma_{n-i},\ \Phi(y_{i})=y_{n+1-i},\ \Phi(\pi)=\pi,\
\Phi(\hbar)=-\hbar$

Observe that $\Phi(\mathbf{e})=\mathbf{e}_{-}$ and
$\Phi(\mathbf{e}_{-})=\mathbf{e}$. In particular, $\Phi$ exchanges spherical
subalgebra $\mathbf{e}\mathbb{H}_{c}\mathbf{e}$ with the antispherical
subalgebra $\mathbf{e}_{-}\mathbb{H}_{c}\mathbf{e}_{-}$.

######  Theorem 4.9.

a) Let $\lambda$ be a minuscule dominant coweight, then

$E_{\lambda}:=\mathbf{e}X^{\lambda}\mathbf{e}=\mathbf{e}\prod_{\langle\lambda,\alpha\rangle=1}\frac{y_{\alpha}-c}{y_{\alpha}}u^{\lambda}\mathbf{e}$

and

$F_{\lambda}:=\mathbf{e}X^{-\lambda}\mathbf{e}=\mathbf{e}\prod_{\langle\lambda,\alpha\rangle=1}\frac{y_{\alpha}+c}{y_{\alpha}}u^{-\lambda}\mathbf{e}$

Here $u^{\lambda}$ is the translation by $\hbar\lambda$.

b) For arbitrary dominant coweight $\lambda$ we get

$E_{\lambda}=\mathbf{e}\prod_{\alpha(\lambda)>0}\prod_{\ell=0}^{\alpha(\lambda)-1}\frac{(y_{\alpha}+\ell\hbar-c)}{(y_{\alpha}+\ell\hbar)}u^{\lambda}\mathbf{e}+\mathrm{lower\
order\ terms}.$

###### Proof.

We prove (b) using geometric realization of $\mathbb{H}_{G}$ as the Coulomb
branch algebra, see Proposition 6.10 below. The operator $E_{\lambda}$
corresponds (up to lower order terms) to a certain class
${}z^{*}\iota_{*}^{-1}\operatorname{gr}[\mathcal{R}^{\leq\lambda}]$ which
agrees with the formula in (b) after localization to fixed points. See
Proposition 6.10 for all details.

Alternatively, one can prove (b) using a tedious but explicit computation in
the affine Hecke algebra, see [49, Example 5.4,Theorem 5.9] and [52,
Proposition 5.13]. The translation $X^{\lambda}$ can be written as a product
of elements of $\Omega$ and simple reflections $\sigma_{i}$ in some order, and
one can control the leading term of each factor. This leads to a formula for
the leading term for $X^{\lambda}$, and its symmetrization.

In part (a) the coweight $\lambda$ is minuscule, and the formula from (b)
simplifies. First, we have either $(\alpha^{\vee},\lambda)=1$ or
$(\alpha^{\vee},\lambda)=0$ for all $\alpha^{\vee}$, so $\ell=0$ for all
nontrivial factors. Second, $\lambda$ is minimal in Bruhat order, so there
cannot be any lower order terms and the formula is exact. ∎

######  Lemma 4.10.

Suppose that $\lambda$ is an (anti)dominant coweight which is minimal in the
Bruhat order. Then

$\Phi(E_{\lambda,c})\Delta=\Delta E_{\lambda,c-\hbar}.$

######  Remark 4.11.

If $\lambda$ is minuscule, then it is indeed minimal in the Bruhat order. The
converse is not true: indeed, there are no minuscule coweights at all for root
systems $E_{8},F_{4},G_{2}$.

###### Proof.

Since $\lambda$ is minimal in Bruhat order, the formula for $E_{\lambda}$ in
Theorem 4.9(b) is exact. Now we write:

$\Phi(E_{\lambda})=\mathbf{e}_{-}u^{{}^{w_{0}}\lambda}\prod_{\alpha^{\vee}(\lambda)>0}\prod_{\ell=0}^{\alpha(\lambda)-1}\frac{(y_{\overline{\alpha}}-\ell\hbar-c)}{(y_{\overline{\alpha}}-\ell\hbar)}\mathbf{e}_{-},$

where we denote $\overline{\alpha}={}^{w_{0}}\alpha$. By replacing $\lambda$
by $\lambda^{w_{0}}$ (since we symmetrize anyway) and $\alpha$ by
${}^{w_{0}}\alpha$ (since we take product over all roots $\alpha$), we can
rewrite this product as

$\Phi(E_{\lambda})=\mathbf{e}_{-}u^{\lambda}\prod_{\alpha(\lambda)>0}\prod_{\ell=0}^{\alpha(\lambda)-1}\frac{(y_{\alpha}-\ell\hbar-c)}{(y_{\alpha}-\ell\hbar)}\mathbf{e}_{-}.$

Now $u^{\lambda}y_{\alpha}=(y_{\alpha}+\alpha(\lambda)\hbar)u^{\lambda}$,
therefore

$\Phi(E_{\lambda})=\mathbf{e}_{-}\prod_{\alpha(\lambda)>0}\prod_{\ell=0}^{\alpha(\lambda)-1}\frac{(y_{\alpha}-\ell\hbar-c+\alpha(\lambda)\hbar)}{(y_{\alpha}-\ell\hbar+\alpha(\lambda)\hbar)}u^{\lambda}\mathbf{e}_{-}=$
$\mathbf{e}_{-}u^{\lambda}\prod_{\alpha(\lambda)>0}\prod_{\ell=1}^{\alpha(\lambda)}\frac{(y_{\alpha}+\ell\hbar-c)}{(y_{\alpha}+\ell\hbar)}u^{\lambda}\mathbf{e}_{-}.$

In the last step we changed index of summation from $\ell$ to
$\alpha(\lambda)-\ell$. Now we can compute

$\Phi(E_{\lambda})\Delta=\mathbf{e}_{-}\prod_{\alpha(\lambda)>0}\prod_{\ell=1}^{\alpha(\lambda)}\frac{(y_{\alpha}+\ell\hbar-c)}{(y_{\alpha}+\ell\hbar)}u^{\lambda}\Delta\mathbf{e}=$
$\mathbf{e}_{-}\prod_{\alpha(\lambda)>0}\prod_{\ell=1}^{\alpha(\lambda)}\frac{(y_{\alpha}+\ell\hbar-c)}{(y_{\alpha}+\ell\hbar)}\frac{(y_{\alpha}+\alpha(\lambda)\hbar)}{y_{\alpha}}\Delta
u^{\lambda}\mathbf{e}=$
$\mathbf{e}_{-}\Delta\prod_{\alpha(\lambda)>0}\prod_{\ell=0}^{\alpha(\lambda)-1}\frac{(y_{\alpha}+\ell\hbar-(c-\hbar))}{(y_{\alpha}+\ell\hbar)}u^{\lambda}\mathbf{e}=\Delta
E_{\lambda,c-\hbar}.$

∎

######  Theorem 4.12.

There is a filtered algebra isomorphism
$\mathbf{e}\mathbb{H}_{c}\mathbf{e}\cong\mathbf{e}_{-}\mathbb{H}_{c-\hbar}\mathbf{e}_{-}$.

###### Proof.

The spherical subalgebra $\mathbf{e}\mathbb{H}_{c-\hbar}\mathbf{e}$ is
generated by the elements

$E_{\lambda}[f]=\mathbf{e}f(w)X^{\lambda}\mathbf{e},\;\;F_{\lambda}[f]=\mathbf{e}f(w)X^{-\lambda}\mathbf{e}$

where $\lambda$ is a dominant coweight which is minimal in the Bruhat order,
and $f(w)$ is a polynomial in $w_{i}$ (see [9, Proposition 6.8.]). Similarly
to Lemma 4.10, one can check that

(12) $\Phi(E_{\lambda,c}[f])\Delta=\Delta E_{\lambda,c-\hbar}[f^{\prime}],\
\Phi(F_{\lambda,c}[f])\Delta=\Delta E_{\lambda,c-\hbar}[f^{\prime}]$

where $f^{\prime}$ is related to $f$ by a certain shift by $\hbar$.

Let $G$ be an operator in $\mathbf{e}\mathbb{H}_{c-\hbar}\mathbf{e}$, then we
can consider the operator $\Delta G\Delta^{-1}$ acting on antisymmetric
polynomials. By (12) the operator $\Delta G\Delta^{-1}$ belongs to
$\mathbf{e}_{-}\mathbb{H}_{c}\mathbf{e}_{-}$. Since $\Phi$ exchanges the
spherical and antispherical subalgebras, the operators $\Delta G\Delta^{-1}$
generate $\mathbf{e}_{-}\mathbb{H}_{c}\mathbf{e}_{-}$.

∎

######  Remark 4.13.

In [43, 74] a similar isomorphism between the spherical and antispherical
subalgebras was obtained using Dunkl representation by differential-difference
operators. It is natural to ask if the two isomorphisms are the same. They are
not, for the isomorphism in loc. cit. is given by conjugation by the
”Vandermonde in $X$”, in other words
$\prod_{\alpha\in\Phi^{+}}(1-\alpha^{\vee})\in{\mathbb{C}}[T^{\vee}]$, which
acts by identity on the operators $E_{\lambda,c}[1]$ since $X\mapsto X$ in the
differential Dunkl representation. The two isomorphisms are related by the
Harish-Chandra transform of [18]. This is similar to the fact that in the
rational case, there are two Dunkl embeddings, to
$\operatorname{Diff}(\mathfrak{h}^{reg})\rtimes W$ and
$\operatorname{Diff}((\mathfrak{h}^{*})^{reg})\rtimes W$ in which one gets
similar shift isomorphisms by either conjugation by
$\prod_{\alpha\in\Phi^{+}}y_{\alpha}$ or respectively by $\prod
x_{\alpha^{\vee}}$ [4], and the two isomorphisms are related by Cherednik’s
Fourier transform.

###  4.3. ${\mathbb{Z}}$-algebras

We now recall the definition of ${\mathbb{Z}}$-algebras, as explained e.g. in
[35, Section 5]. Note that our conventions are exactly opposite to those of
loc. cit. because it makes the Springer action in Section 7 a bit more
natural.

######  Definition 4.14.

An associative (non-unital) algebra $B=\bigoplus_{i\leq j}B_{ij}$ is a
${\mathbb{Z}}$-algebra if $B_{ij}B_{jk}\subseteq B_{ik}$ for all $i\leq j\leq
k$, $B_{ij}B_{lk}=0$ if $j\neq l$, and each $B_{ii}$ is unital such that
$1_{i}b_{ij}=b_{ij}=b_{ij}1_{ij}$ for all $b_{ij}\in B_{ij}$.

The above definition ensures that $B_{ii}$ is a unital associative algebra for
all $i$, and $B_{ij}$ is a $(B_{ii},B_{jj})$-bimodule. The
${\mathbb{Z}}$-algebra multiplication factors though the convolution of
bimodules:

${B_{ij}\bigotimes_{{\mathbb{C}}}B_{jk}}$${B_{ik}}$${B_{ij}\bigotimes_{B_{jj}}B_{jk}}$

The simplest example of ${\mathbb{Z}}$-algebras comes from
${\mathbb{Z}}$-graded algebras.

######  Example 4.15.

Suppose that $S=\oplus_{d}S_{d}$ is an associative ${\mathbb{Z}}$-graded
algebra with multiplication $S_{d}S_{d^{\prime}}\to S_{d+d^{\prime}}$. Define
$B_{ij}=S_{j-i}$ for all $i$ and $j$, then $B(S)=\bigoplus_{i\leq j}B_{ij}$ is
a ${\mathbb{Z}}$-algebra. Note that in this example the algebras $B_{ii}$ are
all isomorphic to $S_{0}$.

Our main source of ${\mathbb{Z}}$-algebras will be a filtered deformation of
Example 4.15. We say that a ${\mathbb{Z}}$-algebra $B$ is of graded type if it
has an algebra filtration (which we omit from the notations) such that
$\operatorname{gr}B=B(S)$ for some commutative graded algebra $S$. Unpacking
this definition, we get the following properties of $\operatorname{gr}B$:

  * •

$S_{0}:=\operatorname{gr}B_{ii}$ is a commutative algebra which does not
depend on $i$ up to isomorphism

  * •

$S_{j-i}:=\operatorname{gr}B_{ij}$ depends only on the difference $j-i$ up to
isomorphism

  * •

For all $i,j,k$ we have a commutative square

${\operatorname{gr}B_{ij}\bigotimes\operatorname{gr}B_{jk}}$${\operatorname{gr}B_{ik}}$${S_{j-i}\bigotimes
S_{k-j}}$${S_{k-i}}$

  * •

The left and right actions of
$S_{0}\simeq\operatorname{gr}B_{ii}\simeq\operatorname{gr}B_{jj}$ on the
bimodule $\operatorname{gr}B_{ij}$ agree.

The last bullet point is related to the Harish-Chandra property for the
bimodules $B_{ij}$, see [78, 55]. Namely, a Harish-Chandra bimodule for a
filtered algebra $A$ is a bimodule $B$ with an exhaustive filtration s.t.
$[A_{\leq i},B_{\leq j}]\subseteq B_{i+j-d}$ s.t. $\operatorname{gr}B$ is
finitely generated. The commutator condition implies that the left and right
actions of $\operatorname{gr}A$ on $\operatorname{gr}B$ agree.

Also note that we can associate a pair of schemes to a ${\mathbb{Z}}$-algebra
of graded type: the affine scheme $\operatorname{Spec}S_{0}$ and the scheme
$\operatorname{\mathrm{Proj}}S$. We have a natural morphism
$\operatorname{\mathrm{Proj}}S\to\operatorname{Spec}S_{0}$.

Next, we define modules over a ${\mathbb{Z}}$-algebra $B$. A graded vector
space $M=\oplus M_{i}$ is a $B$-module if for all $i$ and $j$ we have
multiplication maps $B_{ij}\otimes M_{j}\to M_{i}$ such that we have a
commutative diagram

${B_{ij}\otimes B_{jk}\otimes M_{k}}$${B_{ij}\otimes M_{j}}$${B_{ik}\otimes
M_{k}}$${M_{i}.}$

In particular, $M_{i}$ is a module over the algebra $B_{ii}$ for all $i$. If
$B$ is of graded type and $M$ admits a filtration compatible with a filtration
on $B$ then $\operatorname{gr}M$ is graded $S$-module for the graded algebra
$S$. In particular, $\operatorname{gr}M$ defines a quasicoherent sheaf on
$\operatorname{\mathrm{Proj}}S$.

###  4.4. ${\mathbb{Z}}$-algebras from Cherednik algebras

We now turn to defining a ${\mathbb{Z}}$-algebra
$\mathcal{B}={}_{\bullet}\mathcal{B}_{\bullet}^{\hbar}$ as follows. The
component ${}_{i}\mathcal{B}_{i}^{\hbar}$ is the spherical Cherednik algebra
$\mathbf{e}\mathbb{H}_{c+i\hbar}\mathbf{e}$ with parameter $c+i\hbar$. The
component ${}_{i}\mathcal{B}_{i+1}^{\hbar}$ is the shift bimodule

${}_{i}\mathcal{B}_{i+1}^{\hbar}=\mathbf{e}\mathbb{H}_{c+(i+1)\hbar,\hbar}\mathbf{e}_{-}$

over the algebras
${}_{i+1}\mathcal{B}_{i+1}^{\hbar}=\mathbf{e}\mathbb{H}_{c+(i+1)\hbar,\hbar}\mathbf{e}$
and

${}_{i}\mathcal{B}_{i}^{\hbar}=\mathbf{e}\mathbb{H}_{c+i\hbar,\hbar}\mathbf{e}\simeq\mathbf{e}_{-}\mathbb{H}_{c+(i+1)\hbar,\hbar}\mathbf{e}_{-}.$

The last isomorphism is given by Theorem 4.12. Finally, for more general $i<j$
we define the shift bimodules

${}_{i}\mathcal{B}_{j}^{\hbar}={}_{i}\mathcal{B}_{i+1}^{\hbar}\cdots{}_{j-1}\mathcal{B}_{j}^{\hbar}$

where $\cdot$ denotes the appropriate tensor product.

######  Lemma 4.16.

At $\hbar=c=0$ one has ${}_{i}\mathcal{B}_{j}^{\hbar=0}=A^{j-i}$, where $A$ is
the subspace of diagonally antisymmetric polynomials in
${\mathbb{C}}[T^{*}T^{\vee}]$, and this is compatible with the multiplication.
When $i=j$ this is the subspace of diagonally symmetric polynomials.

###### Proof.

Let us prove that ${}_{i}\mathcal{B}_{i+1}^{\hbar=0}=A$. Indeed,
${}_{i}\mathcal{B}_{i+1}^{\hbar=0}=\mathbf{e}_{-}\mathbb{H}\mathbf{e}\cong\mathbf{e}_{-}{\mathbb{C}}[T^{*}T^{\vee}]$
is the space of antisymmetric polynomials in ${\mathbb{C}}[T^{*}T^{\vee}]$.
Similarly,
${}_{i}\mathcal{B}_{i}^{\hbar=0}\cong\mathbf{e}{\mathbb{C}}[T^{*}T^{\vee}]={\mathbb{C}}[T^{*}T^{\vee}]^{W}$.
Now

${}_{i}\mathcal{B}_{j}^{\hbar=0}=A\otimes_{{\mathbb{C}}[T^{*}T^{\vee}]^{W}}\otimes\cdots\otimes_{{\mathbb{C}}[T^{*}T^{\vee}]^{W}}A=A^{j-i}.$

∎

######  Example 4.17.

Consider the trigonometric Cherednik algebra for $G=GL_{2}$. For $\nu=1/2$ it
has a $1$-dimensional representation $L_{1/2}(\mathrm{triv})$ with invariant
part
$\mathbf{e}L_{1/2}(\mathrm{triv})\cong\mathbf{e}_{-}L_{3/2}(\mathrm{triv})$.
Using this isomorphism, the bimodule
$\mathbf{e}\mathbb{H}_{3/2}\mathbf{e}_{-}$ sends
$\mathbf{e}\mathbb{H}_{3/2}\mathbf{e}_{-}\otimes_{\mathbf{e}_{-}\mathbb{H}_{3/2}\mathbf{e}_{-}}\mathbf{e}_{-}L_{3/2}(\mathrm{triv})\cong\mathbf{e}L_{3/2}(\mathrm{triv})$.
More generally,

$\mathbf{e}\mathbb{H}_{(2k+1)/2}\mathbf{e}_{-}\otimes_{\mathbf{e}_{-}\mathbb{H}_{(2k+1)/2}\mathbf{e}_{-}}\mathbf{e}_{-}L_{(2k+1)/2}(\mathrm{triv})\cong\mathbf{e}L_{(2k+1)/2}(\mathrm{triv})$

and the direct sum

$\bigoplus_{k\geq 0}\mathbf{e}L_{(2k+1)/2}(\mathrm{triv})$

is a module for the ${\mathbb{Z}}$-algebra $\mathcal{B}$.

###  4.5. ${\mathbb{Z}}$-algebra for $GL_{n}$

Consider now the ${\mathbb{Z}}$-algebra as introduced above. We have

######  Theorem 4.18.

For all $i\leq j$ the
${\mathbb{C}}[\hbar]\otimes{\mathbb{C}}[y_{1},\ldots,y_{n}]^{S_{n}}$-module
${}_{i}\mathcal{B}_{j}^{\hbar}$ is free.

###### Proof.

The idea of the proof is to replace the trigonometric Cherednik algebra
$\mathbb{H}_{c,\hbar}=H^{\mathrm{trig}}_{c,\hbar}$ with the rational Cherednik
algebra $H^{\mathrm{rat}}_{\hbar}$ [39]. The algebra
$H^{\mathrm{rat}}_{\hbar}$ is the quotient of
$\mathbb{C}[X_{1},\dots,X_{n}]\otimes\mathbb{C}[z_{1},\dots,z_{n}]\rtimes
S_{n}$ modulo the relations:

$[X_{i},z_{i}]=\hbar+\sum_{j\neq i}\sigma_{ij}\ i=1,\dots,n,$
$[X_{i},z_{j}]=-\sigma_{ij},i\neq j,$

where $\sigma_{ij}\in S_{n}$ is the transposition (the generators $z_{i}$ are
usually called $y_{i}$ in the rational Cherednik algebra literature).

For the rational Cherednik algebra the corresponding ${\mathbb{Z}}$–algebra
${}_{\bullet}\mathcal{B}_{\bullet}^{\mathrm{rat},\hbar=\hbar_{0}}$ was
constructed by Gordon and Stafford [35, 36] who defined a filtration on
${}_{i}\mathcal{B}_{j}^{\mathrm{rat},\hbar=\hbar_{0}}$ for any specialization
of $\hbar$ and proved that
$\operatorname{gr}{}_{i}\mathcal{B}_{j}^{\mathrm{rat},\hbar=\hbar_{0}}\simeq
A^{j-i}_{\mathrm{rat}}$ using Haiman’s results. Note that this was achieved
without relying on Haiman’s results in [28]. This implies that
${}_{i}\mathcal{B}_{j}^{\mathrm{rat},\hbar}$ is free over $\hbar$. The
freeness over ${\mathbb{C}}[y_{1},\ldots,y_{n}]^{S_{n}}$ is e.g. [35, Lemma
6.11(2)]. We remark that this freeness uses results of [4] about Morita
equivalence of Cherednik algebras.

Now the trigonometric case is obtained by Ore localization in the central
element $X_{1}\cdots X_{n}$, which commutes with the action of $\hbar$ and
$y_{i}$. This follows from

######  Lemma 4.19.

There is a natural map

${}_{i}\mathcal{B}_{j}^{\mathrm{rat},\hbar=\hbar_{0}}\to{}_{i}\mathcal{B}_{j}$

which becomes an isomorphism upon localization in $\prod X_{i}$:
$({}_{i}\mathcal{B}_{j}^{\mathrm{rat},\hbar=\hbar_{0}})_{\prod
X_{i}}\cong{}_{i}\mathcal{B}_{j}$.

###### Proof.

In [79] it is shown that

$\imath(w)=w,\quad\imath(X_{i})=X_{i},\quad\imath(z_{i})=X_{i}^{-1}(y_{i}-\sum_{1\leq
j\leq i}\sigma_{ji}),$

extends to the algebra homomorphism
$\imath:\mathbb{H}^{\mathrm{rat}}_{\hbar}\to\mathbb{H}^{\mathrm{trig}}_{1,\hbar}$
that becomes an isomorphism after localization by $X_{1}\cdots X_{n}$. By
hitting with $\mathbf{e}$ on both sides, this implies the statement on the
level of the spherical subalgebras. (To match parameters, we observe that
$H^{\mathrm{trig}}_{c,\hbar}\simeq H^{\mathrm{trig}}_{\lambda c,\lambda\hbar}$
for any $\lambda\in\mathbb{C}^{*}$.)

For the one-step bimodules,
${}_{i}\mathcal{B}_{i+1}^{\mathrm{rat}}=\mathbf{e}\mathbb{H}^{\mathrm{rat}}_{c+i\hbar,\hbar}\mathbf{e}_{-}$
by definition, so the result is true for $j=i+1$ as well. Finally,

${}_{i}\mathcal{B}_{j}^{\mathrm{rat}}={}_{i}\mathcal{B}^{\mathrm{rat}}_{i+1}\cdots{}_{j-1}\mathcal{B}_{j}^{\mathrm{rat}}$

and by standard properties of localization and tensor product we get the
result.

∎

Now since tensoring with ${\mathbb{C}}[X_{1},\ldots,X_{n}]_{\prod X_{i}}$ is
faithfully flat, we deduce that since ${}_{i}\mathcal{B}_{j}^{\mathrm{rat}}$
is free over

${\mathbb{C}}[y_{1},\ldots,y_{n}]^{S_{n}},$

so is ${}_{i}\mathcal{B}_{j}$. This finishes the proof of Theorem 4.18. ∎

More geometrically, the bimodule ${}_{i}\mathcal{B}_{j}^{\mathrm{rat},\hbar}$
quantizes the line bundle $\mathcal{O}(j-i)$ on the Hilbert scheme of points
on ${\mathbb{C}}^{2}$ while ${}_{i}\mathcal{B}_{j}^{\hbar}$ quantizes its
restriction to the Hilbert scheme of ${\mathbb{C}}^{*}\times{\mathbb{C}}$.

######  Corollary 4.20.

The ${\mathbb{Z}}$-algebra $\mathcal{B}$ is of graded type and for $G=GL_{n}$,
$\operatorname{gr}\mathcal{B}$ corresponds to the graded algebra
$S=\oplus_{d=0}^{\infty}A^{d}$. The corresponding algebraic varieties are
$\operatorname{\mathrm{Proj}}S=\operatorname{Hilb}^{n}({\mathbb{C}}^{*}\times{\mathbb{C}})$
and $\operatorname{Spec}S_{0}=({\mathbb{C}}^{*}\times{\mathbb{C}})^{n}/S_{n}$.

##  5\. Coulomb branches and ${\mathbb{Z}}$-algebras

In this section, we explain half of the main construction of the paper, namely
the construction of a ${\mathbb{Z}}$-algebra associated to the Coulomb branch
of the $3d$ $\mathcal{N}=4$ theory with adjoint matter, or in other words the
spherical trigonometric DAHA. Most of the results work in greater generality,
and are stated as such wherever possible. In Section 6 we specialize these
general constructions to the case of adjoint representation.

The other half of the main construction, consisting of a generalized affine
Springer theory for this ${\mathbb{Z}}$-algebra, is treated in Section 7.

###  5.1. Coulomb branches

Let $1\to G\to{\widetilde{G}}\to G_{F}\to 1$ be an extension of algebraic
groups, where $G$ is reductive and $G_{F}$ is diagonalizable. Let $N$ be an
algebraic representation of ${\widetilde{G}}$, $\mathbf{P}\subset
G(\mathcal{O})\subset G_{\mathcal{K}}$ be a (standard) parahoric subgroup and
$N_{\mathbf{P}}$ a lattice in $N_{\mathbf{P}}$ stable under $\mathbf{P}$. We
will only be interested in the case where $N=\operatorname{Ad}$,
$N_{\mathbf{P}}=\operatorname{Lie}(\mathbf{P})$.

Let $\mathcal{R}_{\mathbf{P}}:=\mathcal{R}_{G,N,\mathbf{P},N_{\mathbf{P}}}$ be
the parahoric BFN space of triples as in [26]. More precisely, we have

######  Definition 5.1.

$\mathcal{R}_{G,N,\mathbf{P},N_{\mathbf{P}}}$ is the fpqc sheaf on
$Sch/\mathbf{k}$ associating to $S$ the groupoid of tuples
$(\mathcal{P},\varphi,s,\mathcal{P}_{\mathbf{P}})$ where $\mathcal{P}$ is a
$G$-bundle on $S\times\operatorname{Spec}\mathcal{K}$,
$\mathcal{P}_{\mathbf{P}}$ is a $\mathbf{P}$-reduction of $\mathcal{P}$ over
$S\times\operatorname{Spec}\mathcal{O}$, and $\varphi$ is a trivialization
over $S\times\operatorname{Spec}\mathcal{O}$ compatible with the
$\mathbf{P}$\- structure. Moreover, $s$ is a section of the associated
$N$-bundle of $\mathcal{P}_{\mathbf{P}}$ such that $\varphi\circ s(t)\in
N_{\mathbf{P}}$.

######  Remark 5.2.

Dropping the condition that $\varphi\circ s(t)\in N_{\mathbf{P}}$, we get the
space
$\mathcal{T}_{G,N,\mathbf{P},N_{\mathbf{P}}}=G_{\mathcal{K}}\times_{\mathbf{P}}N_{\mathbf{P}}$,
which is an (ind-)vector bundle over the partial affine flag variety
$\operatorname{Fl}_{\mathbf{P}}$. In particular, if $N=\operatorname{Ad},$
this can be thought of as the cotangent bundle of
$\operatorname{Fl}_{\mathbf{P}}$.

We recall the following definitions and theorems as motivation for the
following sections. From [9, 26, 42], we have

######  Theorem 5.3.

$H_{*}^{{\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}}(\mathcal{R}_{\mathbf{P}})=:\mathcal{A}_{\mathbf{P}}^{\hslash}$
is an associative algebra with unit. It is a flat deformation of
$\mathcal{A}_{\mathbf{P}}:=H_{*}^{{\widetilde{\mathbf{P}}}}(\mathcal{R}_{\mathbf{P}})$.
When $\mathbf{P}=G_{\mathcal{O}}$, the algebra
$\mathcal{A}^{\hslash}_{G_{\mathcal{O}}}$ is a filtered quantization of
$\mathcal{A}_{G_{\mathcal{O}}}$, which is commutative. The spectrum of
$\mathcal{A}_{G_{\mathcal{O}}}$ is called the Coulomb branch of $(G,N)$.

######  Remark 5.4.

The group ${\widetilde{\mathbf{P}}}$ above equals $\mathbf{P}$ if the flavor
group $G_{F}$ is trivial. Otherwise, we define

${\widetilde{\mathbf{P}}}:=\mathrm{ev}_{0}^{-1}(G_{F}\mathrm{ev}_{0}(\mathbf{P}))$

where $\mathrm{ev}_{0}:{\widetilde{G}}_{\mathcal{O}}\to{\widetilde{G}}$ is the
map sending $t\mapsto 0$ or more generally modding out by the nilradical. In
general, we refer by $\;\widetilde{}\;$ to flavor-deformed objects.

######  Theorem 5.5 ([42, 26]).

Let $L_{v}\subset{\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}$ be the
stabilizer of $v$ in
${\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}},$ where
${\widetilde{\mathbf{P}}}$ is the preimage of
$G_{F}\mathrm{ev}_{0}(\mathbf{P})$ under
$\mathrm{ev}_{0}:{\widetilde{G}}_{\mathcal{O}}\to{\widetilde{G}}$. The algebra
$\mathcal{A}_{\mathbf{P}}^{\hslash}$ acts on
$H^{L_{v}}_{*}(M_{v}^{\mathbf{P}})$ via natural cohomological correspondences,
provided the group $L_{v}$ is compact in the $t$-adic topology.

###  5.2. A category of line defects

Heuristically, the equivariant BM homologies of the spaces of triples above
are endomorphisms of objects in a ”category of line operators” [21, 86, 85]
which is something like $G_{\mathcal{K}}$-equivariant $D$-modules on
$N_{\mathcal{K}}$.

We won’t stipulate on the definition of the actual category (see however [3]
in the adjoint case), but this category should contain objects coming from
$\eta=(U,\mathbf{P})$, where $U\subset N_{\mathcal{K}}$ is a
$\mathbf{P}$-stable lattice and $\mathbf{P}$ is a parahoric subgroup of
$G_{\mathcal{K}}$. We will simply define
$\operatorname{Hom}(\eta,\eta^{\prime})=H^{\mathbf{P}^{\prime}\rtimes{\mathbb{C}}^{\times}}_{*}({}_{\eta}\mathcal{R}_{\eta^{\prime}})$,
where

${}_{\eta}\mathcal{R}_{\eta^{\prime}}=\\{[g,s]\in
G_{\mathcal{K}}\times^{\mathbf{P}^{\prime}}U^{\prime}|gs\in U\\}.$

We will use the notation $U=N_{\mathbf{P}}$ to emphasize $N_{\mathbf{P}}$ is a
$\mathbf{P}$-stable lattice. By abuse of notation, we will also write
$\operatorname{Hom}(\eta,\eta^{\prime})$ for the flavor- or loop-rotation
deformed versions of these spaces.

######  Theorem 5.6.

There is an associative multiplication
$\operatorname{Hom}(\eta,\eta^{\prime})\otimes_{\mathbb{C}}\operatorname{Hom}(\eta^{\prime},\eta^{\prime\prime})\to\operatorname{Hom}(\eta,\eta^{\prime\prime})$
via the following modification of the BFN convolution product.

${{}_{\eta}\mathcal{R}_{\eta^{\prime}}\times{}_{\eta^{\prime}}\mathcal{R}_{\eta^{\prime\prime}}}$${p^{-1}({}_{\eta}\mathcal{R}_{\eta^{\prime}}\times{}_{\eta^{\prime}}\mathcal{R}_{\eta^{\prime\prime}})}$${q(p^{-1}({}_{\eta}\mathcal{R}_{\eta^{\prime}}\times{}_{\eta^{\prime}}\mathcal{R}_{\eta^{\prime\prime}}))}$${\mathcal{T}_{\eta^{\prime}}\times{}_{\eta^{\prime}}\mathcal{R}_{\eta^{\prime\prime}}}$${G_{\mathcal{K}}\times{}_{\eta^{\prime}}\mathcal{R}_{\eta^{\prime\prime}}}$${{}_{\eta}\mathcal{R}_{\eta^{\prime\prime}}}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{j}$$\scriptstyle{m}$$\scriptstyle{p}$

Here the maps $p,q,m$ send

$p:(g_{1},[g_{2},s])\mapsto([g_{1},g_{2}s],[g_{2},s]),\;\;q:(g_{1},[g_{2},s])\mapsto[g_{1},[g_{2},s]],$
$m:[g_{1},[g_{2},s]]\mapsto[g_{1}g_{2},s]$

and $i,j$ are inclusions of closed subvarieties.

###### Proof.

This can be proved using a straightforward modification of the proof of
associativity in [9, Section 3]. Similar results for
$\eta=(N_{\mathbf{P}},\mathbf{P})$ where $\mathbf{P}$ is an Iwahori subgroup
are mentioned in [85]. ∎

######  Corollary 5.7.

For any $\eta$ the space $\operatorname{Hom}(\eta,\eta)$ is an associative
algebra, and $\operatorname{Hom}(\eta,\eta^{\prime})$ is a bimodule over
$\operatorname{Hom}(\eta,\eta)$ and
$\operatorname{Hom}(\eta^{\prime},\eta^{\prime})$. Given $\eta,\eta^{\prime}$
and $\eta^{\prime\prime}$ we have a natural morphism of bimodules over
$\operatorname{Hom}(\eta,\eta)$ and
$\operatorname{Hom}(\eta^{\prime\prime},\eta^{\prime\prime})$:

$\operatorname{Hom}(\eta,\eta^{\prime})\bigotimes_{\operatorname{Hom}(\eta^{\prime},\eta^{\prime})}\operatorname{Hom}(\eta^{\prime},\eta^{\prime\prime})\to\operatorname{Hom}(\eta,\eta^{\prime\prime}).$

###### Proof.

We need to prove the morphism from Theorem 5.6 is bilinear over
$\operatorname{Hom}(\eta^{\prime},\eta^{\prime})$. The only axiom of a tensor
product we need to show is $m\cdot r\otimes n=m\otimes r\cdot n$ for
$m\in\operatorname{Hom}(\eta,\eta^{\prime}),n\in\operatorname{Hom}(\eta^{\prime},\eta^{\prime\prime})$
and $r\in\operatorname{Hom}(\eta^{\prime},\eta^{\prime})$, which is clear from
the associativity of the construction. ∎

The following is a generalization of [9, Lemma 5.3].

######  Theorem 5.8.

${}_{\eta}\mathcal{A}_{\eta^{\prime}}$ is flat as a left
${\mathbb{C}}[\mathfrak{t}^{*}][\hbar,c]=H_{*}^{\widetilde{T}}(pt)$-module.

###### Proof.

The associated graded for the Bruhat filtration is free by equivariant
formality. On the other hand, filtered colimits of free modules are flat. ∎

###  5.3. ${\mathbb{Z}}$-algebras and the flavor deformation

Taking a sequence $\eta_{0},\eta_{-1},\ldots$ it is clear from Theorem 5.6
that we get a ${\mathbb{Z}}$-algebra by taking

$\mathcal{A}=\bigoplus_{i\leq j}{}_{i}\mathcal{A}_{j}$

where ${}_{i}\mathcal{A}_{j}$ denotes ${}_{\eta_{i}}\mathcal{A}_{\eta_{j}}$.
That is,

######  Theorem 5.9.

The algebra $\bigoplus_{i\leq j}{}_{i}\mathcal{A}^{\hbar}_{j}$ is a
${\mathbb{Z}}$-algebra.

In the setup of BFN Coulomb branches, nontrivial ${\mathbb{Z}}$-algebras are
most easily obtained via a flavor deformation of $G$, i.e. by letting $G_{F}$
be nontrivial. We now explain this procedure for $G_{F}={\mathbb{G}}_{m}$ and
associate to $({\widetilde{G}},N)$ a ${\mathbb{Z}}$-algebra.

Specifically, we can consider a sequence $\eta_{i}=(t^{-i}U,G_{\mathcal{O}})$
for some fixed lattice $U$ (for example, $U=N_{\mathcal{O}}$). If we set
$\hbar=0$, it is easy to see that $S_{j-i}:={}_{i}\mathcal{A}^{\hbar=0}_{j}$
depends only on $j-i$ and the algebra $\mathcal{A}^{\hbar=0}$ is of graded
type as in Section 4.3. In particular, at $\hbar=0$ all commutative algebras
${}_{i}\mathcal{A}^{\hbar=0}_{i}$ are isomorphic to the commutative Coulomb
branch algebra $S_{0}=\mathcal{A}_{G_{\mathcal{O}}}$. In particular, this
construction yields a map
$\operatorname{\mathrm{Proj}}S\to\operatorname{Spec}S_{0}$, which is a variant
of the construction of a partial resolution of the Coulomb branch in [11].

###  5.4. ${\mathbb{Z}}$-algebras in the abelian case

Since it might be of independent interest and is used for computations below,
we now work out the ${\mathbb{Z}}$-algebras for the cases when $G=T$ is a
diagonalizable algebraic group and $\\{\eta_{i}\\}_{i=0}^{\infty}$ is given by
$\eta_{i}=(T_{\mathcal{O}},t^{i\phi}N_{\mathcal{O}})$ for some (flavor)
cocharacter $\phi:{\mathbb{G}}_{m}\to T$. Note that when $N=0$, the
${\mathbb{Z}}$-algebra collapses to $\mathcal{A}_{T,0}^{\hbar}[c]$ where $c$
is the flavor parameter (the generalization to more flavors is
straightforward).

Under $\mathcal{T}_{j}\hookrightarrow\operatorname{Gr}_{T}\times
N_{\mathcal{K}}$ the image is naturally identified with

$\bigsqcup_{\lambda\in\operatorname{Gr}_{T}}\\{t^{\lambda}\\}\times
t^{j}t^{\lambda}N_{\mathcal{O}}$

and similarly

${}_{i}\mathcal{R}_{j}\cong\bigsqcup_{\lambda\in\operatorname{Gr}_{T}}\\{t^{\lambda}\\}\times(t^{j}t^{\lambda}N_{\mathcal{O}}\cap
t^{i}N_{\mathcal{O}}).$

Now let ${}_{i}r_{j}^{\lambda}$ be the preimage of
$\lambda\in\operatorname{Gr}_{G}$ under the projection
${}_{i}\mathcal{R}_{j}\to\operatorname{Gr}_{T}$. Suppose also $N$ is the
direct sum of the characters $\xi_{1},\ldots,\xi_{n}$ as a $T$-representation.

######  Theorem 5.10.

Under the convolution product in Theorem 5.6, we have for all
$i,j,k\in{\mathbb{Z}}$ that

${}_{i}r_{j}^{\lambda}{}_{j}r_{k}^{\mu}=\prod_{\ell=1}^{n}A_{\ell}(i,j,k,\lambda,\mu){}_{i}r_{k}^{\lambda+\mu}$

where

$A_{\ell}(i,j,k,\lambda,\mu)=\prod_{a=\max(\lambda+i,k-\mu)+1}^{\max(\lambda+i,k-\mu,j)}(\xi_{\ell}+c+(a+\xi_{\ell}(\lambda))\hbar)\prod_{b=\min(\lambda+i,k-\mu,j)+1}^{\min(\lambda+i,k-\mu)}(\xi_{\ell}+c+(b+\xi_{\ell}(\lambda))\hbar)$

###### Proof.

We may restrict to the case where the rank of $T$ is $1$. In this case, the
computation is essentially [85, Proposition 3.10], generalizing [9, Theorem
4.1]. In the notation of loc. cit. we have

${}_{i}r_{j}^{\lambda}=y_{\lambda}r(\lambda+i,j),\;{}_{j}r_{k}^{\mu}=y_{\mu}r(\mu+j,k)$

so we get

$\displaystyle{}_{i}r_{j}^{\lambda}{}_{j}r_{k}^{\mu}$
$\displaystyle=y_{\lambda}r(\lambda+i,j)r(j,k-\mu)y_{\mu}$
$\displaystyle=y_{\lambda}e\left(\frac{t^{\lambda+i}N_{\mathcal{O}}\cap
t^{k-\mu}N_{\mathcal{O}}}{t^{\lambda+i}N_{\mathcal{O}}\cap
t^{k-\mu}N_{\mathcal{O}}\cap
t^{j}N_{\mathcal{O}}}\right)e\left(\frac{t^{\lambda+i}N_{\mathcal{O}}+t^{k-\mu}N_{\mathcal{O}}+t^{j}N_{\mathcal{O}}}{t^{\lambda+i}N_{\mathcal{O}}+t^{k-\mu}N_{\mathcal{O}}}\right)y_{\mu}$

And we compute the Euler classes

$e\left(\frac{t^{\lambda+i}N_{\mathcal{O}}\cap
t^{k-\mu}N_{\mathcal{O}}}{t^{\lambda+i}N_{\mathcal{O}}\cap
t^{k-\mu}N_{\mathcal{O}}\cap
t^{j}N_{\mathcal{O}}}\right)=\prod_{a=\max(\lambda+i,k-\mu)+1}^{\max(\lambda+i,k-\mu,j)}(\xi_{\ell}+c+a\hbar)$
$e\left(\frac{t^{\lambda+i}N_{\mathcal{O}}+t^{k-\mu}N_{\mathcal{O}}+t^{j}N_{\mathcal{O}}}{t^{\lambda+i}N_{\mathcal{O}}+t^{k-\mu}N_{\mathcal{O}}}\right)=\prod_{b=\min(\lambda+i,k-\mu,j)+1}^{\min(\lambda+i,k-\mu)}(\xi_{\ell}+c+b\hbar)$

From the relation
$y_{\lambda}\chi=(\chi+\hbar\langle\chi,\lambda\rangle)y_{\lambda}$ for
$\chi\in\mathfrak{t}^{*}$ we get that

${}_{i}r_{j}^{\lambda}{}_{j}r_{k}^{\mu}=A_{\ell}(i,j,k,\lambda,\mu){}_{i}r_{k}^{\lambda+\mu}$

∎

######  Remark 5.11.

When $\hbar=c=0$, the above becomes

${}_{i}r_{j}^{\lambda}{}_{j}r_{k}^{\mu}=\prod_{\ell=1}^{n}\frac{\xi_{\ell}^{\max(\lambda+i,k-\mu,j)}}{\xi_{\ell}^{\max(\lambda+i,k-\mu)}}\cdot\frac{\xi_{\ell}^{\min(\lambda+i,k-\mu)}}{\xi_{\ell}^{\min(\lambda+i,k-\mu,j)}}{}_{i}r_{k}^{\lambda+\mu}$

######  Lemma 5.12.

All algebras in question are naturally graded with

$\deg\xi_{\ell}=2,\ \deg({}_{i}r_{j}^{\lambda})=|\lambda+i-j|.$

###### Proof.

Observe that

(13)
$|a-b|+|b-c|-|a-c|=2\left(\max(a,b,c)-\max(a,c)+\min(a,c)-\min(a,b,c)\right).$

Indeed, both sides of (13) are symmetric in $a$ and $c$ and vanish if $b$ is
between $a$ and $c$. If $b<a<c$ then we get

$(a-b)+(c-b)-(c-a)=2(a-b)=2(c-c+a-b),$

while for $a<c<b$ we get

$(b-a)+(b-c)-(c-a)=2(b-c)=2(b-c+a-a).$

By substituting $a=\lambda+i,b=j,c=k-\mu$ we can verify that the defining
equations are homogeneous since

$\deg({}_{i}r_{j}^{\lambda})=|\lambda+i-j|,\
\deg({}_{j}r_{k}^{\mu})=|k-\mu-j|=|j+\mu-k|,\
\deg({}_{i}r_{k}^{\lambda+\mu})=|(\lambda+i)-(k-\mu)|=|\lambda+\mu+i-k|.$

∎

As in [10, Appendix] and [9], the inclusion
$\operatorname{Gr}_{T}\hookrightarrow\mathcal{T}_{j}$ as a subbundle gives
rise to an injective map $z^{*}$ in equivariant Borel-Moore homology:

${}_{i}z_{j}^{*}:{}_{i}\mathcal{A}^{\hbar}_{j}\hookrightarrow
H^{\widetilde{T}_{\mathcal{O}}\rtimes{\mathbb{G}}_{m}}_{*}(\operatorname{Gr}_{T}).$

If $u_{\lambda}\in
H^{\widetilde{T}_{\mathcal{O}}\rtimes{\mathbb{G}}_{m}}_{*}(\operatorname{Gr}_{T})$
is the class of the cocharacter $\lambda$, or more algebraically the
$\hbar$-difference operator on $\mathfrak{t}$ acting on $f\in k[\mathfrak{t}]$
by $f(x)\mapsto f(x+\hbar\lambda)$, we have

######  Theorem 5.13.

Under ${}_{i}z_{j}^{*}$, we have

${}_{i}z_{j}^{*}({}_{i}r_{j}^{\lambda})=e(t^{\lambda}t^{i}N_{\mathcal{O}}/t^{\lambda}t^{i}N_{\mathcal{O}}\cap
t^{j}N_{\mathcal{O}})u_{\lambda}$

Note that the injectivity of ${}_{i}z_{j}^{*}$ is clear from the above.

##  6\. Coulomb branches and ${\mathbb{Z}}$-algebras in the adjoint case

###  6.1. From Coulomb branch to Cherednik algebra

We now discuss the adjoint case. For arbitrary $G$ and $N=\operatorname{Ad}$,
the construction of [9] yields a noncommutative resolution of
$T^{*}T^{\vee}/W$ in the sense of [84]. Instead of the spherical case, we
focus on the Iwahori case as well as the resulting ${\mathbb{Z}}$-algebras.

First of all, we claim Theorem 5.6 for
$\eta=(\mathbf{I},\operatorname{Lie}(\mathbf{I}))$ gives a realization of the
dDAHA (as conjectured in many places, including [9]) and that the resulting
action on the affine Springer fibers coincides with Yun’s action. The goal of
this section is to prove these claims, and to show that for
$\eta=(G_{\mathcal{O}},\operatorname{Lie}(G_{\mathcal{O}}))$ we similarly get
the spherical dDAHA, as expected in [9, 7] and other places.

We now state the main theorem of this section.

######  Theorem 6.1.

The Iwahori-Coulomb branch algebra

$\widetilde{\mathcal{A}}_{G,\mathbf{I}}^{\hbar}=H^{{\widetilde{\mathbf{I}}}\rtimes{\mathbb{C}}^{\times}}_{*}(\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}})$

is naturally isomorphic to $\mathbb{H}_{c,\hbar}$.

###### Proof.

There are essentially two subalgebras to find in
$\widetilde{\mathcal{A}}_{G,\mathbf{I}}^{\hbar}$,
${\mathbb{C}}[\widetilde{W}]$ and ${\mathbb{C}}[\mathfrak{t}^{*}]$. The latter
is identified with $H^{*}_{T}(pt)$, and the elements of this subalgebra act as
capping by Chern classes of the associated line bundles on
$\operatorname{Fl}_{G}$. Ditto for the equivariant parameters $c,\hbar$.

We denote by $\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq w}$ the preimage
of the Schubert cell $\overline{\operatorname{Fl}_{G}^{\leq w}}$ under the
projection $\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}\to\operatorname{Fl}_{G}$.
The algebra $\widetilde{\mathcal{A}}_{G,\mathbf{I}}^{\hbar}$ is spanned by
$[\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq w}]$ for $w\in\widetilde{W}$.
We use the maps $m,p,q$ as in Theorem 5.6. Let us study
$[\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq
w}][\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq w^{\prime}}].$ Since

$mqp^{-1}(\overline{\operatorname{Fl}_{G}^{\leq
w}}\times\overline{\operatorname{Fl}_{G}^{\leq
w^{\prime}}})\subset\overline{\operatorname{Fl}_{G}^{\leq ww^{\prime}}}$

is codimension zero (see e.g. [63, Lemma 4.4]) and $N$ is the adjoint
representation,

$mpq^{-1}(\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq
w}\times\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq
w^{\prime}})\subset\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{ww^{\prime}}$

is codimension zero, the algebra $\widetilde{\mathcal{A}}_{G,\mathbf{I}}$ is a
filtered flat deformation of

$\operatorname{gr}_{\mathrm{Bruhat}}\widetilde{\mathcal{A}}_{G,\mathbf{I}}^{\hbar}=\bigoplus_{w\in\widetilde{W}}H^{{\widetilde{\mathbf{I}}}\rtimes{\mathbb{C}}^{\times}}_{*}(\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{w})$

such that

$[\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq
w}][\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq
w^{\prime}}]=[\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq
ww^{\prime}}]+\text{ lower order terms in Bruhat order}$

Note that each $w$ has a reduced expression, say $w=s_{i_{1}}\cdots
s_{i_{j}}$. By taking the associated graded we see we must have

$[\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq
s_{i_{1}}}]\cdots[\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq
s_{i_{j}}}]=[\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq s_{i_{1}}\cdots
s_{i_{j}}}]+\text{ lower order terms}.$

This implies that the classes above the one-dimensional $\mathbf{I}$-orbits in
$\operatorname{Fl}_{G}$ generate
$\widetilde{\mathcal{A}}_{G,\mathbf{I}}^{\hbar}$ together with the equivariant
parameters. So it remains to check these satisfy the right relations. Similar
to (7) we have

(14)
${\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}}$${\left[\widetilde{\mathfrak{g}}/G\right]=[\mathfrak{b}/B]}$${\mathcal{R}_{N_{\mathcal{O}},G_{\mathcal{O}}}}$${\left[\mathfrak{g}/G\right]}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\varphi}$

On the right column, the Springer action is classically identified with the
convolution action by correspondences on the Steinberg variety. Similar to
[88, Proposition 5.2.1.], the pullback of these correspondences is identified
with the correspondence on (the equivariant homology of)
$\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}$ given by
$*[\mathcal{R}_{N_{\mathbf{I}},\mathbf{I}}^{\leq s_{i}}]$. Letting
$G_{\mathcal{O}},N_{\mathcal{O}}$ be replaced by parahorics of rank two
subgroups of $\widetilde{W}$, it is clear the relations of
${\mathbb{C}}[\widetilde{W}]$ are satisfied.

We still need to check relation (2) from Definition 4.1. This is identical to
[88, Lemma 6.4.1].

Since we have a morphism preserving the Bruhat filtration, inducing an
isomorphism on associated graded objects,
$\widetilde{\mathcal{A}}_{G,\mathbf{I}}^{\hbar}\cong\mathbb{H}_{G}$. ∎

######  Remark 6.2.

In $K$-theory, a similar proposition is proven for the full DAHA in [82,
Section 2.5].

######  Corollary 6.3.

The Coulomb branch algebra $\mathcal{A}^{\hbar}_{G,G_{\mathcal{O}}}$ of
Braverman-Finkelberg-Nakajima is isomorphic to the spherical subalgebra of
$\mathbb{H}_{G}$.

###### Proof.

Let $\mathbf{e}=|W|^{-1}\sum_{w\in W}w$. Then from (14) and the definitions it
follows that

$\mathcal{A}^{\hbar}_{G,G_{\mathcal{O}}}=\mathbf{e}\mathcal{A}^{\hbar}_{G,\mathbf{I}}\mathbf{e}$

and by Theorem 6.1 we get the result. ∎

######  Remark 6.4.

This proves the speculation in [9, Remark 6.20]. For $G=GL_{n}$ this was
proved by Kodera and Nakajima in [52].

Finally, we give a geometric realization of the ”shift” bimodules of the
trigonometric Cherednik algebra using line operators.

Let $\eta=(t^{i}\operatorname{Lie}(\mathbf{I}),\mathbf{I})$ and
$\eta^{\prime}=(t^{i}\operatorname{Lie}(\mathbf{I}\cap
tG(\mathcal{O})),\mathbf{I})$, where $\operatorname{Lie}(\mathbf{I}\cap
tG(\mathcal{O}))$ is the pronilpotent radical of
$\operatorname{Lie}(\mathbf{I})$. In the notations of 5.2 denote

${}_{i+1}\widetilde{\mathcal{R}}_{i}:={}_{\eta}\mathcal{R}_{\eta^{\prime}}$

and

${}_{i+1}\widetilde{\mathcal{A}}_{i}=H_{*}^{{\widetilde{\mathbf{I}}}\rtimes{\mathbb{C}}^{\times}}({}_{i+1}\widetilde{\mathcal{R}}_{i})$

######  Theorem 6.5.

There are natural isomorphisms of graded bimodules

${}_{i+1}\mathcal{A}_{i}\cong\mathbf{e}{}_{i+1}\widetilde{\mathcal{A}}_{i}\mathbf{e}\cong\mathbf{e}_{-}{}_{i}\widetilde{\mathcal{A}}_{i}\mathbf{e}$

###### Proof.

Let
${}_{i+1}\widetilde{\mathcal{R}}_{i}\hookrightarrow{}_{i}\widetilde{\mathcal{R}}_{i}$
be the natural inclusion. $(g,s)\in{}_{i}\widetilde{\mathcal{R}}_{i}$ belongs
to ${}_{i+1}\widetilde{\mathcal{R}}_{i}$ exactly when
$gs\in\operatorname{Lie}(\mathbf{I})\cap t\mathfrak{g}(\mathcal{O})$, or in
other words when it is in the kernel of the map