# Noncommutative resolutions and intersection cohomology for quotient
singularities

Tudor Pădurariu  Department of Mathematics, Columbia University, 2990
Broadway, New York, NY 10027<EMAIL_ADDRESS>
###### Abstract.

For a large class of good moduli spaces $X$ of symmetric stacks $\mathcal{X}$,
we define noncommutative motives $\mathbb{D}^{\text{nc}}(X)$ which can be
regarded as categorifications of the intersection cohomology of $X$. These
motives are summands of noncommutative resolutions of singularities
$\mathbb{D}(X)\subset D^{b}(\mathcal{X})$ of $X$. The category $\mathbb{D}(X)$
is a global analogue of the noncommutative resolutions of singularities of
$V\sslash G$ for $V$ a symmetric representation of a reductive group $G$
constructed by Špenko–Van den Bergh.

##  1\. Introduction

###  1.1. Intersection cohomology

Intersection cohomology $\mathrm{IH}^{\cdot}(X,\mathbb{Q})$ and the BBDG
Decomposition Theorem [5] are important tools in the study of the topology of
algebraic varieties. They have many applications in representation theory, see
[32], or more recently in construction of representation of
$\mathcal{W}$-algebras due to Braverman–Finkelberg–Nakajima [8] and in the
study of (Kontsevich–Soibelman) Cohomological Hall algebras [28], [15]. It is
an important problem to define a K-theoretic version of intersection
cohomology and it is expected that such a theory will have applications in
representation theory. More generally, one can try to construct
categorifications of intersection cohomology.

###  1.2. Noncommutative resolutions

A natural place to look for such categorifications is inside noncommutative
resolutions (NCRs) of $X$. There are more NCRs than usual resolution of
singularities. A conjecture of Bondal–Orlov [7, Section 5] says that there
exists a minimal NCR, i.e. a category admissible inside any other NCR of $X$.

We look for natural candidates of minimal NCRs $\mathbb{D}$ of good moduli
spaces $X$ of Artin stacks $\mathcal{X}$. Inspired by the Decomposition
Theorem, we expect the periodic cyclic homology
$\text{HP}_{\cdot}\left(\mathbb{D}\right)$ to have
$\bigoplus_{i\in\mathbb{Z}}\mathrm{IH}^{\cdot+2i}(X,\mathbb{C})$ as a direct
summand. We then try to construct noncommutative motives
$\mathbb{D}^{\text{nc}}$ inside $\mathbb{D}$ whose periodic cyclic homology is
$\bigoplus_{i\in\mathbb{Z}}\mathrm{IH}^{\cdot+2i}(X,\mathbb{C})$.

###  1.3. NCRs for good moduli spaces

We make three possible assumptions on a stack $\mathcal{X}$:

A. $\mathcal{X}$ is an algebraic stack locally of finite type over
$\mathbb{C}$ with a good moduli space $\pi:\mathcal{X}\to X$ (as defined by
Alper [1]) such that $\pi$ has affine diagonal.

B. In addition to the above, assume that $\dim\,X=\dim\,\mathcal{X}$.

C. In addition to the above, assume that $\pi$ is generically an isomorphism.

Let $\mathcal{X}$ a smooth stack which satisfies Assumption A and let $p\in
X(\mathbb{C})$. By work of Alper–Hall–Rydh [2, Theorem 1.2], there exists a
smooth affine scheme $A$ with an action of a reductive group $G$ such that the
following diagram is cartesian

(1) ${\mathcal{X}}$${A/G}$${X}$${A\sslash
G,}$$\scriptstyle{\pi}$$\scriptstyle{e}$$\scriptstyle{e}$

and $e$ is an étale map which contains $p$ in its image. The NCRs we are using
are a global version of the NCRs constructed by Špenko–Van den Bergh [38] for
quotients $V/G$ for $V$ a representation of $G$.

######  Theorem 1.1.

Let $\mathcal{X}$ be a symmetric stack satisfying Assumption B with good
moduli space $X$. Then there exists a subcategory $\mathbb{D}(\mathcal{X})$
admissible in $D^{b}(\mathcal{X})$ which is an NCR of $X$. Its complement is
generated by complexes supported on attracting stacks/ $\Theta$-stacks of
$\mathcal{X}$.

Assume that $\mathcal{X}=V/G$ where $V$ is a symmetric $G$-representation,
i.e. the weights $\beta$ and $-\beta$ of $G$ have the same multiplicity in
$V$. Let $M$ be the weight lattice of $G$, let
$M_{\mathbb{R}}=M\otimes\mathbb{R}$, and let $W$ be the Weyl group of $G$. For
$\lambda:\mathbb{C}^{*}\to G$ a cocharacter, define

$n_{\lambda}:=\langle\lambda,\det\mathbb{L}_{\mathcal{X}}^{\lambda>0}\rangle,$

where $\mathbb{L}_{\mathcal{X}}$ is the cotangent complex of $\mathcal{X}$.
Let $\delta\in M_{\mathbb{R}}^{W}$. The NCR constructed by Špenko–Van den
Bergh [38] is as follows: $\mathbb{D}_{\delta}(\mathcal{X})$ is the full
subcategory of $D^{b}(\mathcal{X})$ generated by complexes $\mathcal{F}$ such
that for any cocharacter $\lambda:\mathbb{C}^{*}\to G$:

(2)
$-\frac{n_{\lambda}}{2}+\langle\lambda,\delta\rangle\leqslant\langle\lambda,\mathcal{F}|_{0}\rangle\leqslant\frac{n_{\lambda}}{2}+\langle\lambda,\delta\rangle.$

Next, let $\mathcal{X}$ be as in Assumption B. Recall the description (1) for
a point $p\in X$. Locally analytically near a point
$x\in\mathcal{X}(\mathbb{C})$, the stack $\mathcal{X}$ is isomorphic to an
open analyic subset of $N/G$, where $G$ is the stabilizer of $x$ and $N$ is
its normal bundle in $\mathcal{X}$. We assume that all such representations
are symmetric, and call stacks $\mathcal{X}$ with this property symmetric.
Note that symmetric stacks are smooth. Let
$\ell\in\text{Pic}(\mathcal{X})_{\mathbb{R}}$. For a morphism
$\lambda:B\mathbb{C}^{*}\to\mathcal{X}$ with image in
$\mathcal{X}(\mathbb{C})$, define

$n_{\lambda}:=\text{wt}\,\lambda^{*}(\text{det}\,\mathbb{L}_{\mathcal{X}}^{\lambda>0}).$

The category $\mathbb{D}_{\ell}(\mathcal{X})$ is the full subcategory of
$D^{b}(\mathcal{X})$ such that for any map
$\lambda:B\mathbb{C}^{*}\to\mathcal{X}$ with image in
$\mathcal{X}(\mathbb{C})$, we have

$-\frac{n_{\lambda}}{2}+\text{wt}\,\lambda^{*}\ell\leqslant\text{wt}\,\lambda^{*}(\mathcal{F})\leqslant\frac{n_{\lambda}}{2}+\text{wt}\,\lambda^{*}\ell.$

To show $\mathbb{D}_{\ell}(\mathcal{X})$ is a smooth and proper category over
$X$, we show in Subsection 3.3 that $\mathbb{D}_{\ell}(\mathcal{X})$ is
admissible inside the Kirwan resolution of $\mathcal{X}$, a DM stack
constructed by Edidin–Rydh [17] which recovers the Kirwan resolution in the
local case [26]. The analogous result for quotients $Y/G$ was proved by
Špenko–Van den Bergh [40]. Our approach is different from theirs and uses the
variation of GIT techniques from [4], [20], [19]. In order for
$\pi^{*}:D^{b}(X)\to D^{b}(\mathcal{X})$ to have image in
$\mathbb{D}_{\ell}(\mathcal{X})$, we choose $\ell$ to be trivial and drop it
from the notation of $\mathbb{D}(\mathcal{X})$.

###  1.4. Intersection cohomology of symmetric good moduli spaces

Let $\mathcal{X}$ be a symmetric stack satisfying Assumption B from Subsection
1.3. Define

(3)
$\textbf{B}:=\text{image}\,\left(\bigoplus_{\mathcal{S}}\mathrm{H}^{\cdot}(\mathcal{Z})\xrightarrow{p_{*}q^{*}}\mathrm{H}^{\cdot}(\mathcal{X})\right),$

over all attracting stacks $\mathcal{S}$ (different from $\mathcal{X})$ with
fixed stack and associated maps
$\mathcal{Z}\xleftarrow{q}\mathcal{S}\xrightarrow{p}\mathcal{X}$ and the
singular cohomology is with $\mathbb{Q}$ coefficients. We denote by
$\textbf{P}^{\leqslant 0}\mathrm{H}^{\cdot}(\mathcal{X})$ the zeroth piece of
the perverse filtration on $\mathrm{H}^{\cdot}(\mathcal{X})$ induced by the
map $\pi:\mathcal{X}\to X$. Observe that if $\mathcal{X}$ satisfies Assumption
C, then $\textbf{P}^{\leqslant
0}\mathrm{H}^{\cdot}(\mathcal{X})\cong\mathrm{IH}^{\cdot}(X)$. In Section 4,
we show that:

######  Theorem 1.2.

For $\mathcal{X}$ a symmetric stack satisfying Assumption B, there is a
decomposition

$\mathrm{H}^{\cdot}(\mathcal{X})=\textbf{P}^{\leqslant
0}\mathrm{H}^{\cdot}(\mathcal{X})\oplus\textbf{B}.$

The above result follows from a version of the BBDG Decomposition Theorem for
stacks, see Proposition 4.2. The exceptional loci are covered by the
attracting stacks, and by the symmetric assumption on $\mathcal{X}$, the
images of classes from attracting stacks are in positive perverse degree.

###  1.5. Noncommutative motives for symmetric good moduli spaces

Let $\mathcal{X}$ be a symmetric stack satisfying Assumption B from Subsection
1.3. In Section 5.3 we define a noncommutative motive with rational
coefficients $\mathbb{D}^{\text{nc}}(\mathcal{X})=(\mathbb{D}(\mathcal{X}),e)$
where $e$ is an idempotent in
$\mathrm{K}_{0}\left(\text{rep}\left(\mathbb{D}(\mathcal{X}),\mathbb{D}(\mathcal{X})\right)\right)_{\mathbb{Q}}$,
see Subsection 2.5 for a brief discussion of noncommutative motives. Define

$\textbf{B}\mathrm{K}_{\cdot}(\mathcal{X}):=\text{image}\,\left(\bigoplus_{\mathcal{S}}\mathrm{K}_{\cdot}(\mathcal{Z})\xrightarrow{p_{*}q^{*}}\mathrm{K}_{\cdot}(\mathcal{X})\right),$

where the sum is an in (3) and $\mathrm{K}_{\cdot}$ is rational K-theory. In
Section 5, we show an analogue on Theorem 1.2 in K-theory:

######  Theorem 1.3.

For $\mathcal{X}$ a symmetric stack satisfying Assumption B, there is a
decompositions

$\mathrm{K}_{\cdot}(\mathcal{X})=\mathrm{K}_{\cdot}(\mathbb{D}^{\text{nc}}(\mathcal{X}))\oplus\textbf{B}\mathrm{K}_{\cdot}(\mathcal{X}).$

The category $\mathbb{D}(\mathcal{X})$ contains, in general, complexes on
attracting stacks, but it may be indecomposable as a triangulated category.
For $\lambda:B\mathbb{C}^{*}\to\mathcal{X}$ with associated fixed stack
$\mathcal{Z}$, let $\mathbb{D}(\mathcal{Z})_{b}$ be the subcategory of
$\mathbb{D}(\mathcal{Z})$ of complexes on which $\lambda$ acts with weight
$b=\frac{n_{\lambda}}{2}$. The motive $\mathbb{D}^{\text{nc}}(\mathcal{X})$ is
a complement of the images

$m_{\lambda}:=p_{*}q^{*}:\mathbb{D}(\mathcal{Z})_{b}\rightarrow\mathbb{D}(\mathcal{X})$

from fixed substacks $\mathcal{Z}$. By a result of Thomason [42, Corollary
2.17], one can compute rational K-theory using an étale cover, so it suffices
to check the statement in the local case $\mathcal{X}=A/G$, where $A$ is a
smooth affine scheme. The main tool in understanding these images for
different fixed substacks is a product-coproduct type compability that we
briefly explain. Let $\lambda$ and $\mu$ be two dominant cocharacters, let S
be a set of cocharacters $\nu$ refining $\mu$ and $\lambda$, see Subsection
5.1.5. Then the following diagram commutes:

${\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)}$${\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))}$${\bigoplus_{\textbf{S}}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\nu})_{b}\right)}$${\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\mu})_{b}\right).}$$\scriptstyle{\bigoplus_{\textbf{S}}\Delta^{\lambda}_{\nu}}$$\scriptstyle{m_{\lambda}}$$\scriptstyle{\Delta_{\mu}}$$\scriptstyle{\bigoplus_{\textbf{S}}\widetilde{m^{\mu}_{\nu}}}$

Here, $\Delta$ are restriction maps and $\widetilde{m}$ denotes a twist of the
multiplication. We check that the diagram commutes by a direct computation
using shuffle formulas for $m$ and $\Delta$.

###  1.6. Intersection K-theory

Let $\mathcal{X}$ be a symmetric stack satisfying Assumption C and let $X$ be
its good moduli space. Then $\textbf{P}^{\leqslant
0}\mathrm{H}^{\cdot}(\mathcal{X})\cong\mathrm{IH}^{\cdot}(X)$. Given Theorems
1.2 and 1.3, we propose to call
$\mathrm{K}_{\cdot}(\mathbb{D}^{\text{nc}}(\mathcal{X}))=:\mathrm{IK}_{\cdot}(X)$
the intersection K-theory of $X$. There is a Chern character map

$\text{ch}:\mathrm{IK}_{\cdot}(X)\to\mathrm{IH}^{\cdot}(X).$

Further, from the construction of $\mathrm{IK}_{\cdot}(X)$, we obtain a
natural surjection

$\mathrm{K}_{\cdot}(\mathcal{X})\twoheadrightarrow\mathrm{IK}_{\cdot}(X).$

The analogous statement in cohomology was proved by Kirwan [27]. Further, in
Subsection 5.6.2, we show that if the Kirwan resolution is a scheme, then

$\mathrm{HP}_{i}\left(\mathbb{D}^{\text{nc}}(\mathcal{X})\right)\cong\bigoplus_{j\in\mathbb{Z}}\mathrm{IH}^{i+2j}(X,\mathbb{C}).$

We use the construction of K-theory of quiver varieties to prove a version of
a Poincaré-Birkhoff-Witt Theorem for K-theoretic Hall algebras of quivers with
potential [35, Theorem 1.2]. For example, for a quiver $Q$, the K-theoretic
Hall algebra $\text{KHA}(Q,0)$ for zero potential is generated by spaces which
are twisted versions of $\mathrm{IK}_{\cdot}(X(d))$, where $X(d)$ is the
coarse space of representation of $Q$ of dimension $d$.

###  1.7. Previous work on intersection K-theory.

There are other approaches of defining intersection K-theory in particular
cases. Cautis [11], Cautis–Kamnitzer [12] have an approach for
categorification of intersection sheaves for certain subvarieties of the
affine Grassmannian.For varieties with a cellular stratification, Eberhardt
proposed a definition in [16].

A related problem is defining intersection Chow groups. Corti–Hanamura [13],
[14] proposed two approaches towards intersection Chow groups for general
varieties $X$, one which proves a version of the decomposition theorem under
some conjectures. In [35], we propose a definition of intersection (graded)
$\text{gr}^{\cdot}\mathrm{K}$-theory of any singular variety $X$ which is a
summand of $\text{gr}^{\cdot}\mathrm{K}_{\cdot}(Y)$ for any resolution of
singularities $Y\to X$; the associated graded is taken with respect to the
codimension filtration, so its zeroth level is a version of intersection Chow
groups. De Cataldo–Migliorini [9] proposed a definition of intersection Chow
motive for singularities with a semismall resolution.

###  1.8. Future directions

The definitions of $\mathbb{D}^{\text{nc}}(\mathcal{X})$ and
$\mathrm{IK}_{\cdot}(X)$ are for symmetric stacks $\mathcal{X}$. It is worth
trying to find analogues of these constructions beyond symmetric stacks. One
idea is to look for symmetric substacks
$\mathcal{X}^{\prime}\subset\mathcal{X}$ whose complement is a union of
$\Theta$-strata and thus have good moduli spaces $X^{\prime}$. It is true that
$\mathrm{IH}^{\cdot}(X)$ is a direct summand of
$\mathrm{IH}^{\cdot}(X^{\prime})$, but we do not know how to characterize the
difference between them.

The discussion in Subsection 1.2 serves as motivation for our work, but we do
not make progress towards the Bondal–Orlov conjecture. In general,
$\mathbb{D}(\mathcal{X})$ and $\mathbb{D}^{\text{nc}}(\mathcal{X})$ are
different. However, in the cases in which they are equal, for example for
quotients $V/T$ where $T$ is a torus and $V$ is a $T$-representation, it is
natural to guess that $\mathbb{D}(V/T)$ is minimal in the sense of
Bondal–Orlov. The category $\mathbb{D}(V/T)$ is indecomposable [39, Appendix
A], so it should be minimal if the Bondal–Orlov conjecture is true for this
particular class of singularities.

We plan to return to these questions in future work.

###  1.9. Acknowledgements

I thank Davesh Maulik for numerous conversations related to the present paper.
I thank the Institute of Advanced Studies for support during the preparation
of the paper. This material is based upon work supported by the National
Science Foundation under Grant No. DMS-1926686.

##  2\. Preliminaries

###  2.1. Conventions and notations

All spaces considered are over $\mathbb{C}$. All schemes considered are of
finite type over $\mathbb{C}$. All points considered are $\mathbb{C}$-points
unless otherwise stated.

For $A$ a scheme with an action of a reductive group $G$, we denote the
quotient stack by $A/G$ and its coarse space by $A\sslash G$.

For $X$ a scheme or stack, denote by $\text{QCoh}\,(X)$ the category of
(unbounded) complexes of quasicoherent sheaves on $X$, by
$D^{b}\text{Coh}\,(X)$ its subcategory of compact objects, i.e. the derived
category of bounded complexes of coherent sheaves, and by $\text{Perf}\,(X)$
the subcategory of $D^{b}\text{Coh}\,(X)$ of perfect complexes. The functors
used in the paper are derived; we sometimes drop $R$ or $L$ from notation, for
example we write $f_{*}$ instead of $Rf_{*}$. Denote by $D_{\text{shvs}}(X)$
the category of complexes of constructible sheaves on a space $X$.

For $G$ a reductive group, fix a maximal torus and a Borel subgroup $T\subset
B\subset G$. We denote by $M$ the character lattice, by $N$ the cocharacter
lattice, by $W$ the Weyl group, by
$M_{\mathbb{R}}:=M\otimes_{\mathbb{Z}}\mathbb{R}$, and by

$\langle\,,\,\rangle:N\times M\to\mathbb{Z}$

the natural pairing; it induces a pairing between $N$ and $M_{\mathbb{R}}$. We
assume that the weights in the Lie algebra of $B$ are negative roots. In
particular, $B$ induces a choice of a dominant chamber
$M_{\mathbb{R}}^{+}\subset M_{\mathbb{R}}$. Denote by $\rho$ half the sum of
positive roots of $G$. For $\chi$ a dominant weight of $G$, denote by
$\Gamma(\chi)$ the irreducible representation of $G$ of highest weight $\chi$.
By abuse of notation, for $V$ a representation of $G$, write

$\langle\lambda,V\rangle:=\langle\lambda,\det
V\rangle=\Big{\langle}\lambda,\sum_{\beta\text{ wt of }V}\beta\Big{\rangle}.$

We denote by $w\cdot\chi$ the usual $W$-action and by
$w*\chi=w(\chi+\rho)-\rho$ the shifted $W$-action. We denote by $\mathfrak{g}$
the Lie algebra of $G$.

For a stack $\mathcal{X}$, we denote by $\mathrm{H}^{\cdot}(\mathcal{X})$ the
singular cohomology with $\mathbb{Q}$ coefficients and by
$\mathrm{K}_{\cdot}(\mathcal{X})$ the rational K-theory of $\mathcal{X}$.

###  2.2. Semi-orthogonal decompositions and noncommutative resolutions

In this Subsection, we recall some basic notions related to derived
categories. References for this material are [29], [30].

#### 2.2.1.

Let $\mathcal{T}$ be a triangulated category, and let
$\mathcal{A}_{1},\cdots,\mathcal{A}_{n}\subset\mathcal{T}$ be triangulated
subcategories. We say that there is a semi-orthogonal decomposition

$\mathcal{T}=\langle\mathcal{A}_{1},\cdots,\mathcal{A}_{n}\rangle$

if the following two conditions are satisfied:

(i) $\text{Hom}(A_{i},A_{j})=0$ for all
$A_{i}\in\mathcal{A}_{i},A_{j}\in\mathcal{A}_{j}$ and $1\leqslant j<i\leqslant
n$.

(ii) the smallest triangulated subcategory of $\mathcal{T}$ containing all
$\mathcal{A}_{i}$s is $\mathcal{T}$.

#### 2.2.2.

Let $\mathcal{T}$ be a triangulated category, and let $\mathcal{A}$ be a
subcategory. $\mathcal{A}$ is called right admissible in $\mathcal{T}$ if
there exists a semi-orthogonal decomposition
$\mathcal{T}=\langle-,\mathcal{A}\rangle$. Equivalently, the inclusion functor
$\mathcal{A}\hookrightarrow\mathcal{T}$ has a right adjoint.

#### 2.2.3.

In this paper, we say that a triangulated category $\mathcal{D}$ is smooth if
it is admissible inside $D^{b}(Y)$ for a smooth DM stack $Y$. We say that
$\mathcal{D}$ is proper over a given variety $S$ if it is admissible in
$D^{b}(T)$ for $T$ a proper DM stack over $S$.

#### 2.2.4.

Let $X$ be a variety. We say that a smooth and proper over $X$ triangulated
category $\mathcal{D}$ is a noncommutative resolution of singularities (NCR)
if there exists an adjoint pair of functors

$\displaystyle F$ $\displaystyle:\mathcal{D}\to D^{b}(X),$ $\displaystyle G$
$\displaystyle:\text{Perf}\,(X)\to\mathcal{D}$

such that $FG=\text{id}_{\text{Perf}(X)}$. The definition is slightly more
general than the definition in [30, Definition 3.1 and the paragraph after
it].

Example. Let $X$ be a variety with rational singularities and let $f:Y\to X$
be a resolution of singularities. Then $D^{b}(Y)$ is an NCR of $X$ where the
corresponding functors are $f_{*}:D^{b}(Y)\to D^{b}(X)$ and
$f^{*}:\text{Perf}\,(X)\to D^{b}(Y)$.

###  2.3. Good moduli spaces

#### 2.3.1.

Let $\mathcal{X}$ be a stack. An algebraic space $X$ with a morphism
$\pi:\mathcal{X}\to X$ is called a good moduli space if

(i) $\pi_{*}:\text{QCoh}\,(\mathcal{X})\to\text{QCoh}\,(X)$ is exact,

(ii) the induced morphism $\mathcal{O}_{X}\to\pi_{*}\mathcal{O}_{\mathcal{X}}$
is an isomorphism.

Examples.

  1. (1)

For $A$ an affine variety and $G$ a reductive group acting linearly on $A$,

$\pi:A/G\to A\sslash G=\text{Spec}\left(\mathcal{O}_{A}^{G}\right)$

is a good moduli space.

  2. (2)

Let $Y$ be a smooth projective variety, let $\beta\in H^{\cdot}(Y)$, and let
$\mathcal{L}$ be an ample line bundle on $Y$. The moduli stack
$\mathfrak{M}^{\text{ss}}_{\beta}$ of $\mathcal{L}$-Gieseker semistable
sheaves with Chern character $\beta$ has a good moduli space
$M^{\text{ss}}_{\beta}$.

For properties and further examples of good moduli space, see Alper [1]. We
assume that $X$ is a scheme in this paper. The following is [2, Theorem
$4.12$, Theorem $1.2$]:

######  Theorem 2.1.

Let $\mathcal{X}$ be a stack satisfying Assumption A. Let
$x\in\mathcal{X}(\mathbb{C})$ with stabilizer group $G_{x}$ and normal bundle
$N_{x}$. Then there exists an affine scheme $A$ with a linearizable action of
$G_{x}$, a point $a\in A(\mathbb{C})$ with stabilizer group $G_{x}$, and étale
maps $e$ and $f$ such that the following squares are cartesian:

(4) ${(N_{x}/G_{x},0)}$${(A/G_{x},a)}$${(\mathcal{X},x)}$${N_{x}\sslash
G_{x}}$${A\sslash
G_{x}}$${X.}$$\scriptstyle{e}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{e}$

We will be using the following corollary:

######  Corollary 2.2.

Let $\pi:\mathcal{X}\to X$ be a stack satisfying Assumption A. Let $p\in
X(\mathbb{C})$. There exists a quotient stack $p:\mathcal{Y}:=V/G\to
Y:=V\sslash G$ with $G$ a reductive group, $V$ a $G$-representation, and
analytic open sets $p\in U\subset X$ and $0\in\mathcal{U}\subset Y$ such that
the diagram

(5)
${\pi^{-1}(U)}$${p^{-1}(\mathcal{U})}$${U}$${\mathcal{U}.}$$\scriptstyle{\pi}$$\scriptstyle{\sim}$$\scriptstyle{p}$$\scriptstyle{\sim}$

commutes. In particular, the diagram

(6)
${\widehat{\mathcal{X}_{p}}}$${\widehat{\mathcal{Y}_{0}}}$${\widehat{X_{p}}}$${\widehat{Y_{0}},}$$\scriptstyle{\pi}$$\scriptstyle{\sim}$$\scriptstyle{p}$$\scriptstyle{\sim}$

where $\widehat{\mathcal{X}_{p}}$ is the formal completion of $\mathcal{X}$
along $\pi^{-1}(p)$, $\widehat{X_{p}}$ is formal completion of $X$ at $p$ etc.

###  2.4. Theta-stratifications

References for this Section are [20, Section 1], [19, Section 2.1], [3].

#### 2.4.1.

Let $\mathcal{X}$ be an algebraic stack of finite type over $\mathbb{C}$. Let
$\Theta=\mathbb{A}^{1}/\mathbb{G}_{m}$. The stacks
$\underline{\text{Map}}(B\mathbb{G}_{m},\mathcal{X})$ and
$\underline{\text{Map}}(\Theta,\mathcal{X})$ are algebraic stacks with natural
(evaluation) maps to $\mathcal{X}$. Their connected components are called
fixed stacks and $\Theta$-stacks or attracting stacks, respectively. There is
also a natural map

$\underline{\text{Map}}(\Theta,\mathcal{X})\to\underline{\text{Map}}(B\mathbb{G}_{m},\mathcal{X}).$

A $\Theta$-stack $\mathcal{S}$ has an associated fixed stack $\mathcal{Z}$,
and fits in a diagram

${\mathcal{S}}$${\mathcal{X}}$${\mathcal{Z}}$$\scriptstyle{q}$$\scriptstyle{p}$

where $p$ is proper and $q$ is an affine bundle map. If $p$ is an immersion,
we say that $\mathcal{S}$ is a $\Theta$-stratum.

When $\mathcal{X}=V/G$ for $V$ an representation a reductive group $G$, the
fixed and $\Theta$-stacks are of the form

(7) ${V^{\lambda\geqslant 0}/G^{\lambda\geqslant
0}}$${V/G}$${V^{\lambda}/G^{\lambda}}$$\scriptstyle{q}$$\scriptstyle{p}$

where $\lambda:\mathbb{C}^{*}\to G$ is a cocharacter, $G^{\lambda}$ is the
Levi group associated to $\lambda$, $G^{\lambda\geqslant 0}$ is the parabolic
group associated to $\lambda$, $V^{\lambda}\subset V$ is the $\lambda$-fixed
locus and $V^{\lambda\geqslant 0}\subset V$ is the $\lambda$-attracting locus.
Such a $\Theta$-stack is a $\Theta$-stratum if the map $p$ is a closed
immersion, so if it is a Kempf–Ness locus in the terminology of [19, Section
2.1].

###  2.5. Noncommutative motives

We briefly explain the definition of noncommutative motives. A general
reference is [41]. Denote by dgcat the category of (small) dg categories over
$\mathbb{C}$. It has a Quillen model structure whose weak equivalences are
derived Morita equivalences. Denote by Hmo the corresponding homotopy
category. The universal category through which all additive invariants factor
(examples include cyclic homology, K-theory, and related constructions) is a
smaller (additive) category $\textbf{Hmo}_{0}$ with objects dg categories and
morphisms

$\text{Hom}_{\textbf{Hmo}_{0}}(\mathcal{A},\mathcal{B})=\mathrm{K}_{0}\left(\text{rep}(\mathcal{A},\mathcal{B})\right)$

where $\mathcal{A}$ and $\mathcal{B}$ are dg categories and
$\text{rep}(\mathcal{A},\mathcal{B})\subset
D^{b}(\mathcal{A}^{\text{op}}\otimes\mathcal{B})$ is the full subcategory of
bimodules $X$ such that $X(a,-)\in\text{Perf}\,(\mathcal{B})$ for any object
$a\in\mathcal{A}$. Consider the functor
$\mathcal{U}:\textbf{dgcat}\to\textbf{Hmo}_{0}$.

We consider the category $\textbf{Hmo}_{0;\mathbb{Q}}^{\natural}$, the
idempotent completion of the $\mathbb{Q}$-linearization of $\textbf{Hmo}_{0}$.
We call its elements (by a slight abuse) noncommutative motives; the original
definition considers the subcategory of
$\textbf{Hmo}_{0;\mathbb{Q}}^{\natural}$ generated by proper and smooth dg
categories, but in our case we need to allow proper and smooth categories over
(not necessarily proper) $X$.

###  2.6. A preliminary result

The following type of result in used by Špenko–Van den Bergh in their
construction of NCRs.

Let $A$ be a smooth affine variety with an action of a reductive group $G$ and
let $\mathcal{X}=A/G$. For a locally free sheaf $\mathcal{F}$ on
$\mathcal{X}$, its stalk at the origin is a representation $\Gamma$ of $G$. We
call $\Gamma$ the associated representation of $\mathcal{F}$.

We state the following result for future reference, the proof is same as [21,
Section 3.2] and it uses an explicit Köszul resolution for pushforward along
the map $A^{\lambda\geqslant 0}/G^{\lambda\geqslant 0}\hookrightarrow
A/G^{\lambda\geqslant 0}$ and the Borel-Bott-Weil Theorem for the map
$A/G^{\lambda\geqslant 0}\to A/G$.

######  Proposition 2.3.

Let $\lambda$ be a cocharacter of $G$ and consider the diagram of attracting
loci

$A^{\lambda}/G^{\lambda}\xleftarrow{q}A^{\lambda\geqslant
0}/G^{\lambda\geqslant 0}\xrightarrow{p}A/G.$

Let $\mathcal{F}$ be a locally free sheaf on $A^{\lambda}/G^{\lambda}$ with
associated representation $\Gamma(\chi)$ where $\chi$ is a dominant weight of
$G$. Then there is a complex

$\left(\bigoplus_{I}\mathcal{F}_{I}[|I|-\ell(w)],d\right)\to
p_{*}q^{*}\mathcal{F},$

where the terms of the complex correspond to subsets
$I\subset\\{\beta|\,\langle\lambda,\beta\rangle<0\\}$, $\mathcal{F}_{I}$ is a
locally free sheaf with associated representation

$\Gamma\left((\chi-\sigma_{I})^{+}\right),$

where $\sigma_{I}=\sum_{\beta\in I}\beta$, $(\chi-\sigma_{I})^{+}$ is the
dominant Weyl-shifted conjugate of $\chi-\sigma_{I}$ if it exists, and zero
otherwise, and $w$ is the element of the Weyl group such that
$w*(\chi-\sigma_{I})$ is dominant or zero of length $\ell(w)$.

##  3\. Noncommutative resolutions of quotient singularities

In this Section, we prove Theorem 1.1. The definition of the categories
$\mathbb{D}_{\ell}(\mathcal{X})$ are for symmetric stacks satisfying
Assumption A. In order to obtain NCRs, we need to assume that $\mathcal{X}$
satisfies Assumption C. Recall the construction of category
$\mathbb{D}(\mathcal{X})$ and the strategy of proof discussed in Subsection
1.3.

###  3.1. Local case

Let $\mathcal{X}=V/G$ where $V$ is a symmetric $G$-representation. Denote by
$X=V\sslash G$ and by $p:\mathcal{X}\to X$. We will use the notations from
Subsection 2.1. For a cocharacter $\lambda:\mathbb{C}^{*}\to G$, recall the
diagram of attracting loci (6) and define

$n_{\lambda}:=\langle\lambda,V^{\lambda>0}\rangle-\langle\lambda,\mathfrak{g}^{\lambda>0}\rangle=\langle\lambda,\det\mathbb{L}_{p}\rangle=\langle\lambda,\det\mathbb{L}_{\mathcal{X}}^{\lambda>0}\rangle.$

Let $\delta\in M_{\mathbb{R}}^{W}$, and let $\mathbb{D}_{\delta}(\mathcal{X})$
be the full subcategory of $D^{b}(\mathcal{X})$ generated by complexes
$\mathcal{F}$ such that for any cocharacter $\lambda:\mathbb{C}^{*}\to G$:

$-\frac{n_{\lambda}}{2}+\langle\lambda,\delta\rangle\leqslant\langle\lambda,\mathcal{F}|_{0}\rangle\leqslant\frac{n_{\lambda}}{2}+\langle\lambda,\delta\rangle.$

Let $\mathbb{W}\subset M_{\mathbb{R}}$ be the polytope

(8) $\mathbb{W}:=\text{sum}\,[0,\beta]\subset M_{\mathbb{R}},$

where the Minkowski sum is taken over all weights $\beta$ of $V$. The category
$\mathbb{D}_{\delta}(\mathcal{X})$ can be also described as the full
subcategory of $D^{b}(\mathcal{X})$ generated by vector bundles
$\mathcal{O}_{\mathcal{X}}\otimes\Gamma(\chi)$ where $\chi$ is a dominant
weight of $G$ such that

$\chi+\rho+\delta\in\frac{1}{2}\mathbb{W},$

where the sum is taken over all weights $\beta$ of $V$.

Define $\mathbb{A}_{\delta}$ as the subcategory of $D^{b}(\mathcal{X})$
generated by complexes $p_{\lambda*}q_{\lambda}^{*}(\mathcal{E})$ where
$\mathcal{E}$ is a complex in $D^{b}(\mathcal{X}^{\lambda})$ with

$\langle\lambda,\mathcal{E}|_{0}\rangle<-\frac{n_{\lambda}}{2}+\langle\lambda,\delta\rangle.$

The following was proved by Špenko–Van den Bergh [38, Proposition 8.4] (the
semi-orthogonal decomposition in loc. cit. holds for quotient stacks
satisfying Assumption A, the condition that $X$ has a $T$-fixed stable point
in loc. cit. is necessary to identify the summands with NCRs):

######  Theorem 3.1.

There exists a semi-orthogonal decomposition

$D^{b}(\mathcal{X})=\langle\mathbb{A}_{\delta},\mathbb{D}_{\delta}\rangle.$

The semi-orthogonal decomposition holds relative to $X$ in the following
sense: if $\mathcal{A}\in\mathbb{A}_{\delta}$ and
$\mathcal{D}\in\mathbb{D}_{\delta}$, then

$Rp_{*}\left(R\mathcal{H}om_{\mathcal{X}}(\mathcal{D},\mathcal{A})\right)=0.$

###  3.2. Global case

Assume that $\pi:\mathcal{X}\to X$ is a symmetric stack satisfying Assumption
B. Let $\ell\in\text{Pic}(\mathcal{X})_{\mathbb{R}}$. Recall the definition of
$\mathbb{D}_{\ell}(\mathcal{X})$ from Subsection 1.3. As in Step $1$, define
$\mathbb{A}_{\ell}$ as the subcategory of $D^{b}(\mathcal{X})$ generated by
complexes of sheaves $p_{*}q^{*}\left(\mathcal{E}\right)$, where $\mathcal{S}$
is a $\Theta$-stack, $\mathcal{Z}$ is its associated fixed stack with maps
$\mathcal{Z}\xleftarrow{q}\mathcal{S}\xrightarrow{p}\mathcal{X},$ and
$\mathcal{E}$ satisfies

$\text{wt}\,\lambda^{*}(\mathcal{E})<-\frac{n_{\lambda}}{2}+\langle\lambda,\delta\rangle.$

######  Theorem 3.2.

There is a semi-orthogonal decomposition

$D^{b}(\mathcal{X})=\langle\mathbb{A}_{\ell},\mathbb{D}_{\ell}\rangle.$

The semi-orthogonal decomposition holds relative to $X$ in the following
sense: if $\mathcal{A}\in\mathbb{A}_{\ell}$ and
$\mathcal{D}\in\mathbb{D}_{\ell}$, then

(9)
$R\pi_{*}\left(R\mathcal{H}om_{\mathcal{X}}(\mathcal{D},\mathcal{A})\right)=0.$

For $p\in X$, let $D^{b}_{o}\left(\widehat{\mathcal{X}_{p}}\right)$ be the
split category generated by the restrictions of complexes in
$D^{b}(\mathcal{X})$. For $U\subset X$ as in Corollary 2.2, let
$\mathcal{X}_{U}:=\pi^{-1}(U)$ and define
$D^{b}_{o}\left(\mathcal{X}_{U}\right)$ as the split category of coherent
analytic sheaves generate by restrictions of complexes in
$D^{b}(\mathcal{X})$. For $\mathcal{Y}=V/G$, the category
$D^{b}\left(\mathcal{Y}\right)$ is generated by
$\mathcal{O}_{\mathcal{Y}}\otimes\Gamma(\chi)$ for $\chi$ a dominant weight of
$G$, and thus $D^{b}\left(\widehat{\mathcal{Y}_{0}}\right)$ is generated by
$\mathcal{O}_{\widehat{\mathcal{Y}}}\otimes\Gamma(\chi)$. By Theorem 2.1, the
categories $D^{b}_{o}\left(\mathcal{X}_{U}\right)$ and
$D^{b}_{o}\left(\widehat{\mathcal{X}_{p}}\right)$ are also generated by these
vector bundles.

Further, let $\widehat{\mathbb{D}_{\ell,p}}\subset
D^{b}_{o}\left(\widehat{\mathcal{X}_{p}}\right)$ be the category generated by
restrictions of sheaves in $\mathbb{D}_{\ell}$; we define
$\widehat{\mathbb{A}_{\ell,p}}$, $\widehat{\mathbb{D}_{\ell,U}}$ etc.
similarly. For the stack $\mathcal{Y}=V/G$ from Corollary 2.2, let $\delta\in
M(G)_{\mathbb{R}}$ be the restriction of $\ell$ and we denote by
$\mathbb{A}^{\prime}_{\delta}$ and $\mathbb{D}^{\prime}_{\delta}$ the
categories from Theorem 3.1. By Corollary 2.2 and using the notation from
there, we have that

$\displaystyle\mathbb{D}_{\ell,U}$
$\displaystyle\cong\mathbb{D}^{\prime}_{\delta,\mathcal{U}},$
$\displaystyle\mathbb{A}_{\ell,U}$
$\displaystyle\cong\mathbb{A}^{\prime}_{\delta,\mathcal{U}}$

and the analogues equivalences for formal completions.

###### Proof of Theorem 3.2.

We continue with the notations from the above. Let
$\mathcal{A}\in\mathbb{A}_{\ell}$ and $\mathcal{D}\in\mathbb{D}_{\ell}$. To
show (9), it suffices to prove the statement after restriction to
$\widehat{\mathcal{X}_{p}}$ for all $p\in X$. Then
$\mathcal{A}\in\widehat{\mathbb{A}^{\prime}_{\delta,0}}$ and
$\mathcal{D}\in\widehat{\mathbb{D}^{\prime}_{\delta,0}}$ and thus the claim
follows from Theorem 3.1. To show that $\mathbb{A}_{\ell}$ and
$\mathbb{D}_{\ell}$ generate $D^{b}(\mathcal{X})$, observe that they generate
the local categories $D^{b}_{o}\left(\mathcal{X}_{U}\right)$. The claim
follows as in [38, Proposition 3.5.8]. ∎

###  3.3. NCR

We show that $\mathbb{D}_{\ell}(\mathcal{X})$ is a smooth and proper over $X$
category. More precisely, $\mathbb{D}(\mathcal{X})$ is an admissible
subcategory of the Kirwan resolution of $\mathcal{X}$ as constructed by
Edidin–Rydh [17]. In loc. cit., the authors do not use the language of
$\Theta$-strata, but their construction is natural and applied to quotient
stacks recovers Kirwan’s resolution of singularities [26, pages 475-476]. For
a stack $\mathcal{X}$ satisfying Assumption B, the Edidin–Rydh construction
provides a sequence of stacks

$\mathcal{X}=:\mathcal{X}_{0}\xleftarrow{\pi_{0}}\mathcal{X}_{1}\hookleftarrow\mathcal{X}^{\text{ss}}_{1}\xleftarrow{\pi_{1}}\cdots\xleftarrow{\pi_{n}}\mathcal{X}_{n+1}\hookleftarrow\mathcal{X}^{\text{ss}}_{n+1}=:Y,$

with the following properties:

(i) the stacks $\mathcal{X}_{i}$ have good moduli spaces $X_{i}$,

(ii) $\mathcal{X}_{i}^{\text{ss}}\subset\mathcal{X}_{i}$ is an open subset,
complement to $\Theta$-strata,

(iii) étale locally on $X_{k}$, either
$\mathcal{X}_{k}\xleftarrow{\pi_{k}}\mathcal{X}_{k+1}\hookleftarrow\mathcal{X}^{\text{ss}}_{k+1}$
are isomorphisms, or there are neighborhoods as in Theorem 2.1:

${X_{k}}$${\mathcal{X}_{k}}$${\mathcal{X}_{k+1}}$${\mathcal{X}^{\text{ss}}_{k+1}}$${A\sslash
G}$${A/G}$${\left(\text{Bl}_{A^{0}}A\right)/G}$${\left(\text{Bl}_{A^{0}}A\right)^{\text{ss}}/G}$${N\sslash
G}$${N/G}$${\left(\text{Bl}_{N^{0}}N\right)/G}$${\left(\text{Bl}_{N^{0}}N\right)^{\text{ss}}/G.}$$\scriptstyle{\pi_{k}}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$

Here, $A^{0}\subset A$ is the $G$-fixed locus, the stability condition is
given by the tautological line bundle $\mathcal{O}(1)$ on
$\text{Bl}_{A^{0}}A$, and $N$ is the normal bundle of $a$ in $A$.

(iv) The stack $Y:=\mathcal{X}^{\text{ss}}_{n+1}$ is a smooth Deligne–Mumford
stack with a proper map $Y\to X$ of relative dimension zero.

The class $\ell\in\text{Pic}(\mathcal{X})_{\mathbb{R}}$ induces classes which
we also denote by $\ell\in\text{Pic}(\mathcal{X}_{k})_{\mathbb{R}}$ for
$1\leqslant k\leqslant n+1$.

######  Proposition 3.3.

Let $0\leqslant k\leqslant n$. Assume that $\mathcal{X}_{k}$ is a symmetric
stack. Then $\mathcal{X}^{\text{ss}}_{k+1}$ is also a symmetric stack.

###### Proof.

It suffices to check the statement in the local case

$V/G\xleftarrow{\pi}\text{Bl}_{0}V/G\hookleftarrow(\text{Bl}_{0}V)^{\text{ss}}/G,$

where $G$ is a reductive group, $V$ is a symmetric $G$-representation with $0$
the only $G$-fixed point in $V$, and the linearization is given by the
tautological line bundle $\mathcal{O}(1)$.

We claim that the unstable loci of
$\text{Bl}_{0}V\cong\text{Tot}_{\mathbb{P}(V)}\left(\mathcal{O}(-1)\right)$
are determined by pairs

$(\lambda,\mathbb{P}(V_{a}))$

where $\lambda:\mathbb{C}^{*}\to G$, $V_{a}\subset V$ is the subspace on which
$\lambda$ acts with weight $a$, and $a<0$. The GIT algorithm, see [19, Section
2.1], eliminates pairs $(\lambda,Z)$ for $\lambda:\mathbb{C}^{*}\to G$, $Z$ is
a $\lambda$-fixed component on
$\text{Tot}_{\mathbb{P}(V)}\left(\mathcal{O}(-1)\right)$, and

$\text{wt}_{\lambda}\,\mathcal{O}(1)|_{Z}>0.$

The fixed loci $Z$ are $\mathbb{P}(V_{a})$ for $a\neq 0$ and
$\text{Tot}_{\mathbb{P}(V_{0})}\left(\mathcal{O}(-1)\right)$. Further, we
compute

$\text{wt}_{\lambda}\mathcal{O}(1)|_{\mathbb{P}(V_{a})}=\sum_{n\in\mathbb{Z}}(n-a)\dim
V_{n}>0.$

The representation $V$ is symmetric, so $\text{dim}\,V_{i}=\text{dim}\,V_{-i}$
for $i\in\mathbb{Z}$. We thus have that

$0<\sum_{n\in\mathbb{Z}}n\cdot\dim V_{n}-a\cdot\dim V=-a\cdot\dim V,$

so indeed $a<0$.

Finally, let $\lambda:\mathbb{C}^{*}\to G$ be a cocharacter which fixes a
point $v$ in
$\text{Tot}_{\mathbb{P}(V)^{\text{ss}}}\left(\mathcal{O}(-1)\right)$. If it
lies on a $\lambda$-fixed component $\mathbb{P}(V_{a})$ for $a\neq 0$, it is
part of an unstable locus for either $\lambda$ or $\lambda^{-1}$. Thus it lies
on $\text{Tot}_{\mathbb{P}(V_{0})}\left(\mathcal{O}(-1)\right)$; the normal
bundle is $\bigoplus_{i\in\mathbb{Z}-\\{0\\}}V_{i}$, which is symmetric. ∎

######  Proposition 3.4.

Let $0\leqslant k\leqslant n$. The category
$\mathbb{D}_{\ell}(\mathcal{X}_{k})$ is admissible in
$\mathbb{D}_{\ell}\left(\mathcal{X}^{\text{ss}}_{k+1}\right)$.

###### Proof.

To simplify the notation, let $\mathcal{X}:=\mathcal{X}_{k}$,
$\mathcal{Y}:=\mathcal{X}_{k+1}$. We will see that the unstable loci are
indexed by $(\lambda,a)$, where $\lambda:B\mathbb{G}_{m}\to\mathcal{X}$ and
$a\in\mathbb{Z}$. For each $\Theta$-stratum $\mathcal{S}_{\lambda,a}$ with
associated fixed stack $\mathcal{Z}_{\lambda,a}$, choose a real number
$w_{\lambda,a}\notin\mathbb{Z}$. By [20, Theorem 3.9], there is an admissible
subcategory $\mathbb{G}\hookrightarrow D^{b}(\mathcal{Y})$ with objects
complexes $\mathcal{F}$ such that for any $\Theta$-stratum
$\mathcal{S}_{\lambda,a}$:

$-\frac{n_{\lambda,a}}{2}+\lambda^{*}(\ell)+w_{\lambda,a}\leqslant\lambda^{*}\mathcal{F}\leqslant\frac{n_{\lambda,a}}{2}+\lambda^{*}(\ell)+w_{\lambda,a}$

with the property that

$\text{res}:\mathbb{G}\cong D^{b}(\mathcal{Y}^{\text{ss}}).$

We next characterize $\Theta$-strata and compute $n_{\lambda,a}$. We can
assume that we are in the local case

$V/G\xleftarrow{\pi}\text{Bl}_{0}V/G\cong\text{Tot}_{\mathbb{P}(V)}\left(\mathcal{O}(-1)\right)\hookleftarrow(\text{Bl}_{0}V)^{\text{ss}}/G,$

where $G$ is a reductive group, $V$ is a symmetric $G$-representation with $0$
the only $G$-fixed point in $V$, and the linearization is given by the
tautological line bundle $\mathcal{O}(1)$. By the argument of Proposition 3.3,
the unstable loci are determined by pairs

$(\lambda,\mathbb{P}(V_{a}))$

where $V_{a}$ is the $\lambda$-weight $a$ subspace of $V$ and $a<0$. The
$\lambda$-positive part of the normal bundle is

$N_{\mathbb{P}(V_{a})/\mathbb{P}(V)}^{\lambda>0}\cong\bigoplus_{i>a}V_{i}$

and $\lambda$ acts with weight $i+a$ on $V_{i}$. The length of the window
$n_{\lambda,a}$ is thus:

(10)
$n_{\lambda,a}=\langle\lambda,N_{\mathbb{P}(V_{a})/\mathbb{P}(V)}^{\lambda>0}\rangle-\langle\lambda,\mathfrak{g}^{\lambda>0}\rangle=\\\
\sum_{i>a}(i+a)\text{dim}\,V_{i}-\langle\lambda,\mathfrak{g}^{\lambda>0}\rangle>\langle\lambda,V^{\lambda>0}\rangle-\langle\lambda,\mathfrak{g}^{\lambda>0}\rangle.$

The category $\mathbb{D}_{\ell}(\mathcal{X})\subset D^{b}(\mathcal{X})$ is
defined by the conditions

(11)
$-\frac{m_{\lambda}}{2}+\lambda^{*}(\ell)\leqslant\text{wt}\,\lambda^{*}\mathcal{F}\leqslant\frac{m_{\lambda}}{2}+\lambda^{*}(\ell),$

where, in the local model $\mathcal{X}=V/G$ from above,
$m_{\lambda}=\langle\lambda,V^{\lambda>0}\rangle-\langle\lambda,\mathfrak{g}^{\lambda>0}\rangle$.

Similarly, the category $\mathbb{D}_{\ell}(\mathcal{Y}^{\text{ss}})$ of
$D^{b}(\mathcal{Y}^{\text{ss}})$ is defined by the conditions

(12)
$-\frac{m_{\lambda}}{2}+\lambda^{*}(\ell)\leqslant\text{wt}\,\lambda^{*}\mathcal{F}\leqslant\frac{m_{\lambda}}{2}+\lambda^{*}(\ell).$

Indeed, a fixed substack $\mathcal{Z}$ associated to a map
$\lambda:B\mathbb{G}_{m}\to\mathcal{X}$ is isomorphic, in the local model, to
$\text{Tot}_{\mathbb{P}(V_{0})}\left(\mathcal{O}(-1)\right)$, where
$V_{0}\subset V$ is the $\lambda$-fixed locus. This follows from the analysis
at the end of the proof of Proposition 3.3.

The $\lambda$-fixed loci of $\mathcal{Y}^{\text{ss}}$ are all above
$\lambda$-fixed loci of $\mathcal{X}$. Using (10), (11), (12), we can thus
choose weights $w_{\lambda,a}\notin\mathbb{Z}$ such that

$\displaystyle\pi^{*}:D^{b}(\mathcal{X})$ $\displaystyle\to
D^{b}(\mathcal{Y})$ $\displaystyle\mathbb{D}_{\ell}(\mathcal{X})$
$\displaystyle\subset\mathbb{D}_{\ell}\left(\mathcal{Y}^{\text{ss}}\right).$

Let $\Phi:D^{b}(\mathcal{X})\to\mathbb{D}_{\ell}(\mathcal{X})$ be the right
adjoint of $\mathbb{D}_{\ell}(\mathcal{X})\hookrightarrow D^{b}(\mathcal{X})$
which exists by Theorem 3.2, see Subsection 2.2.2. Consider the functor

$\Phi\pi_{*}:\mathbb{D}_{\ell}\left(\mathcal{Y}^{\text{ss}}\right)\to\mathbb{D}_{\ell}(\mathcal{X}),$

where recall that $\pi_{*}$ is the derived functor. We claim that $\pi^{*}$
and $\Phi\pi_{*}$ are adjoint. Let $A\in\mathbb{D}_{\ell}(\mathcal{X})$ and
$B\in\mathbb{D}_{\ell}\left(\mathcal{Y}^{\text{ss}}\right)$. Then

$\text{RHom}_{\mathcal{Y}}(\pi^{*}A,B)\cong\text{RHom}_{\mathcal{X}}(A,\pi_{*}B)\cong\text{RHom}_{\mathcal{X}}(A,\Phi\pi_{*}B).$

Finally, the functor $\pi^{*}$ is fully faithful. By the projection formula,
it suffices to show that
$\pi_{*}\mathcal{O}_{\mathcal{Y}}=\mathcal{O}_{\mathcal{X}}$, which follows
from a direct computation in the local case. Thus
$\mathbb{D}_{\ell}(\mathcal{X})$ is admissible in
$\mathbb{D}_{\ell}\left(\mathcal{Y}^{\text{ss}}\right)$. ∎

We thus obtain that:

######  Corollary 3.5.

The category $\mathbb{D}_{\ell}(\mathcal{X})$ is admissible in
$D^{b}\left(Y\right)$.

We finally show that $\mathbb{D}(\mathcal{X}):=\mathbb{D}_{0}(\mathcal{X})$ is
an NCR of $X$.

######  Proposition 3.6.

Let $\ell\in\text{Pic}(\mathcal{X})_{\mathbb{R}}$ be such that for any
cocharacter $\lambda$,

$-\frac{n_{\lambda}}{2}+\lambda^{*}(\ell)\leqslant
0\leqslant\frac{n_{\lambda}}{2}+\lambda^{*}(\ell).$

Then $\mathbb{D}_{\ell}(\mathcal{X})$ is an NCR of $X$.

###### Proof.

Consider the inclusion functor and its natural adjoint obtained by Theorem 3.2
and the discussion in Subsection 2.2.2:

$\displaystyle\iota$
$\displaystyle:\mathbb{D}_{\ell}(\mathcal{X})\hookrightarrow
D^{b}(\mathcal{X})$ $\displaystyle\Phi$
$\displaystyle:D^{b}(\mathcal{X})\to\mathbb{D}_{\ell}(\mathcal{X}).$

Consider the functors

$\displaystyle\pi_{*}\iota$ $\displaystyle:\mathbb{D}_{\ell}(\mathcal{X})\to
D^{b}(X)$ $\displaystyle\Phi\pi^{*}$
$\displaystyle:\text{Perf}\,(X)\to\mathbb{D}_{\ell}(\mathcal{X}).$

We need to show that for $\mathcal{F}\in\text{Perf}\,(X)$, we have that

$\pi_{*}\iota\Phi\pi^{*}(\mathcal{F})=\mathcal{F}.$

The complexes $\pi^{*}\mathcal{F}$ have weight zero for any cocharacter
$\lambda$, and thus they are in $\mathbb{D}_{\ell}(\mathcal{X})$. This means
that $\iota\Phi\pi^{*}\mathcal{F}=\pi^{*}\mathcal{F}$. We have that

$\pi_{*}\iota\Phi\pi^{*}(\mathcal{F})=\pi_{*}\pi^{*}(\mathcal{F})=\mathcal{F}\otimes\pi_{*}\mathcal{O}_{\mathcal{X}}=\mathcal{F}.$

The last equality follows from
$\pi_{*}\mathcal{O}_{\mathcal{X}}=\mathcal{O}_{X}$, see Subsection 2.3.
Finally, the category $\mathbb{D}_{\ell}(\mathcal{X})$ is smooth and proper
over $X$ by Corollary 3.5, and thus it is an NCR of $X$. ∎

##  4\. Intersection cohomology for quotient singularities

In this Section, $\mathcal{X}$ is a symmetric stack satisfying Assumption B.
Let

$\pi:\mathcal{X}\to X$

be the good moduli space morphism. Denote by $I$ the set of connected
components of $\underline{\text{Map}}\,(B\mathbb{G}_{m},\mathcal{X})$, by $o$
the connected component $\mathcal{X}$, and let $J:=I-o$. Further, let
$I^{\prime}$ be the set of connected components of
$\underline{\text{Map}}\,(\Theta,\mathcal{X})$, $o$ the connected component
corresponding to $\mathcal{X}$, and let $J^{\prime}:=I^{\prime}-o$. For an
attracting stack $\mathcal{S}$ in $J^{\prime}$ with associated fixed stack
$\mathcal{Z}$, consider the map

$p_{\mathcal{S}*}q_{\mathcal{S}}^{*}:\mathrm{H}^{\cdot}(\mathcal{Z})\to\mathrm{H}^{\cdot}(\mathcal{X}).$

Define

$\textbf{B}:=\text{image}\,\left(\bigoplus_{J^{\prime}}p_{\mathcal{S}*}q_{\mathcal{S}}^{*}:\mathrm{H}^{\cdot}(\mathcal{Z})\to\mathrm{H}^{\cdot}(\mathcal{X})\right).$

Recall that $D_{\text{shvs}}(X)$ is the category of complexes of constructible
sheaves on a space $X$. Let $\left(\tau^{\leqslant i},\tau^{>i}\right)$ be the
functors corresponding to the usual t-structure on $D_{\text{shvs}}(X)$ and
let $D_{\text{shvs}}^{\leqslant i}(X)$ the subcategory of sheaves with
$\tau^{>i}=0$. Let $\left({}^{p}\tau^{\leqslant i},{}^{p}\tau^{>i}\right)$ be
the functors associated to the perverse t-structure on $D_{\text{shvs}}(X)$
and denote by ${}^{p}D_{\text{shvs}}^{\leqslant i}(X)$ the subcategory of
sheaves with ${}^{p}\tau^{>i}=0$. The map $\pi$ induces a perverse filtration:

$\textbf{P}^{\leqslant
i}\mathrm{H}^{\cdot}(\mathcal{X}):=\text{image}\big{(}\mathrm{H}^{\cdot}\left(X,{}^{p}\tau^{\leqslant
i}R\pi_{*}\mathrm{IC}_{\mathcal{X}}\right)\to\mathrm{H}^{\cdot}\left(\mathcal{X},R\pi_{*}\mathrm{IC}_{\mathcal{X}}\right)\big{)}\subset\mathrm{H}^{\cdot}(\mathcal{X}).$

If $\mathcal{X}$ satisfies Assumption C, then $\textbf{P}^{\leqslant
0}\mathrm{H}^{\cdot}(\mathcal{X})\cong\mathrm{IH}^{\cdot}(X)$ by Proposition
4.6. The main result we prove in this Section is:

######  Theorem 4.1.

Let $\mathcal{X}$ be a symmetric stack satisfying Assumption B. Then

$\mathrm{H}^{\cdot}(\mathcal{X})=\textbf{P}^{\leqslant
0}\mathrm{H}^{\cdot}(\mathcal{X})\oplus\textbf{B}.$

We first explain that the BBDG Decomposition Theorem [5] implies a
Decomposition Theorem for the map $\pi$.

######  Proposition 4.2.

There is a decomposition

$R\pi_{*}\mathrm{IC}_{\mathcal{X}}\cong\bigoplus_{i\geqslant
0}{}^{p}\mathcal{H}^{i}(R\pi_{*}\mathrm{IC}_{\mathcal{X}})[-i]$

and each sheaf ${}^{p}\mathcal{H}^{i}(R\pi_{*}\mathrm{IC}_{\mathcal{X}})$ is a
direct sum of sheaves $\mathrm{IC}_{Z}(\mathcal{L})$ for $Z\subset X$ and
$\mathcal{L}$ a local system on an open smooth subset of $Z$.

An explicit computation of $R\pi_{*}\mathrm{IC}_{\mathcal{X}}$ for
$\mathcal{X}$ a stack of representation of a quiver appears in [33, Theorem
4.6], [15, Proof of Theorem 4.10].

###### Proof.

The idea, inspired by Totaro’s approximations of quotient stacks, is to
approximate the stack $\mathcal{X}$ with smooth varieties proper over $X$ and
apply the BBDG Decomposition Theorem [5]. By Corollary 2.2, it suffices to
treat the local case $\mathcal{X}=V/G$.

For $n\geqslant 1$, consider the stacks

$\mathcal{X}_{n}:=\left(V\oplus V^{\oplus n}\right)/G\times\mathbb{C}^{*},$

where $\mathbb{C}^{*}$ acts with weight zero on the first copy of $V$ and with
weight $1$ on the summand $V^{\oplus n}$, and $G$ acts naturally on all copies
of $V$. Consider the linearization
$G\times\mathbb{C}^{*}\xrightarrow{\text{pr}_{2}}\mathbb{C}^{*}$. Define the
schemes

$\displaystyle S_{n}$ $\displaystyle:=\left(V\oplus V^{\oplus
n}\right)^{\text{st}}\sslash G\times\mathbb{C}^{*}$ $\displaystyle S_{n}^{o}$
$\displaystyle:=\left(V\oplus V^{\oplus n}\right)^{\text{st},\text{nf}}\sslash
G\times\mathbb{C}^{*}.$

The superscript nf means that we take the open subset of the stable locus
$\left(V\oplus V^{\oplus n}\right)^{\text{st}}$ not fixed by any elements of
$G$. Consider the natural maps:

${\mathcal{X}_{n}}$${S_{n}^{o}}$${\mathcal{X}}$${S_{n}}$${X.}$$\scriptstyle{t_{n}}$$\scriptstyle{i_{n}}$$\scriptstyle{\ell_{n}}$$\scriptstyle{\pi}$$\scriptstyle{q_{n}}$

Denote by $\pi_{n}:=\pi t_{n}:\mathcal{X}_{n}\to X$. We discuss two
preliminary results.

######  Proposition 4.3.

We have that $\left(V\oplus V^{\oplus n}\right)^{\text{st}}=\left(V\oplus
V^{\oplus n}\right)^{\text{ss}}$.

###### Proof.

Let $\mu$ be a cocharacter of $G\times\mathbb{C}^{*}$. It is enough to show
that there are no $\mu$-fixed points on $\left(V\oplus V^{\oplus
n}\right)^{\text{ss}}$. Write $\mu=\lambda\cdot z^{b}$ for $b\in\mathbb{Z}$,
$\lambda$ a cocharacter of $G$, and $z$ the identity cocharacter of
$\mathbb{C}^{*}$.

If $b>0$, then $\left(V\oplus V^{\oplus n}\right)^{\mu}$ is unstable. If
$b<0$, then $\left(V\oplus V^{\oplus n}\right)^{\mu^{-1}}$ is unstable. If
$b=0$, then

$\left(V\oplus V^{\oplus n}\right)^{\lambda}\subset\left(V\oplus V^{\oplus
n}\right)^{\lambda\cdot z\geqslant 0},$

so $\left(V\oplus V^{\oplus n}\right)^{\lambda}$ is contained in an unstable
locus. ∎

######  Proposition 4.4.

Fix $a\in\mathbb{Z}$. For $n$ large enough, we have that

$\displaystyle{}^{p}\tau^{\leqslant a}R\pi_{n*}\mathbb{Q}_{\mathcal{X}_{n}}$
$\displaystyle\cong{}^{p}\tau^{\leqslant
a}R\pi_{n*}i_{n*}\mathbb{Q}_{S_{n}^{o}}$ $\displaystyle{}^{p}\tau^{\leqslant
a}Rq_{n*}\mathbb{Q}_{S_{n}}$ $\displaystyle\cong{}^{p}\tau^{\leqslant
a}R\pi_{n*}i_{n*}\mathbb{Q}_{S_{n}^{o}}.$

###### Proof.

We only explain the first equality; the second one is similar. All complexes
we consider have cohomology of finite dimension and are bounded on the left.
For stacks, functoriality of such complexes is discussed by Laszlo–Olsson
[24], [25]. Consider the substacks

$j_{n}:\mathcal{Z}_{n}\hookrightarrow\mathcal{X}_{n}\hookleftarrow
S_{n}^{o}:i_{n}.$

There is a distinguished triangle in $D_{\text{shvs}}(Y)$:

$j_{n!}j_{n}^{!}\mathbb{Q}_{\mathcal{X}_{n}}=j_{n*}\omega_{\mathcal{Z}_{n}}[-2\dim\mathcal{X}_{n}]\to\mathbb{Q}_{\mathcal{X}_{n}}\to
i_{n*}\mathbb{Q}_{S^{o}_{n}}\xrightarrow{[1]}.$

The complex $\omega_{\mathcal{Z}_{n}}$ is in
$D^{\geqslant-2\dim\mathcal{Z}_{n}}_{\text{shvs}}(\mathcal{Z}_{n})$, see [23,
Section V.2]. Let $c_{n}$ be the codimension of $\mathcal{Z}_{n}$ in
$\mathcal{X}_{n}$. Then $\omega_{\mathcal{Z}_{n}}[-2\dim\mathcal{X}_{n}]\in
D^{\geqslant 2c_{n}}_{\text{shvs}}(\mathcal{Z}_{n}).$ Pushforward preserves
the categories $D^{\geqslant\cdot}$, so

$R\pi_{n*}i_{n*}\omega_{\mathcal{Z}_{n}}[-2\dim\mathcal{X}_{n}]\in
D^{\geqslant 2c_{n}}_{\text{shvs}}(X).$

There is a constant $b$ only depending on $X$ such that

$R\pi_{n*}i_{n*}\omega_{\mathcal{Z}_{n}}[-2\dim\mathcal{X}_{n}]\in{}^{p}D^{\geqslant
b+2c_{n}}_{\text{shvs}}(X),$

and this implies the desired conclusion for large enough $n$. ∎

Now we continue the proof of Proposition 4.2. By Proposition 4.3, the variety
$S_{n}$ has finite quotient singularities, thus
$\mathrm{IC}_{S_{n}}\cong\mathbb{Q}_{S_{n}}[\dim S_{n}]$. By Proposition 4.4,
we have that for $m>n$ large enough,

${}^{p}\tau^{\leqslant
a}R\pi_{n*}\mathbb{Q}_{\mathcal{X}_{n}}\cong{}^{p}\tau^{\leqslant
a}Rq_{n*}\mathbb{Q}_{S_{n}}\cong{}^{p}\tau^{\leqslant
a}Rq_{m*}\mathbb{Q}_{S_{m}}.$

Fix $a\in\mathbb{Z}$. The complex $R\pi_{*}\mathbb{Q}_{\mathcal{X}}$ is a
direct summand of $R\pi_{n*}\mathbb{Q}_{\mathcal{X}_{n}}$, so
${}^{p}\tau^{\leqslant a}R\pi_{*}\mathbb{Q}_{\mathcal{X}}$ is a direct summand
of ${}^{p}\tau^{\leqslant a}R\pi_{n*}\mathbb{Q}_{\mathcal{X}_{n}}$. The
Decomposition Theorem for the maps $q_{n}$ implies the desired conclusion.

∎

For a cocharacter $\lambda$, define
$c_{\lambda}:=\dim\mathcal{X}-\dim\mathcal{X}^{\lambda\geqslant 0}$.

######  Proposition 4.5.

Assume that $\mathcal{X}=V/G$. Let $\lambda$ be a cocharacter of $G$ and let B
be a semisimple summand of
${}^{p}\mathcal{H}^{i}(R\pi_{*}\mathrm{IC}_{\mathcal{X}})$ with support
contained in the image of $X^{\lambda}\to X$. Then

$\textbf{B}\subset\text{image}\,\left({}^{p}\mathcal{H}^{i}\left(R\pi_{*}p_{*}\mathrm{IC}_{\mathcal{X}^{\lambda\geqslant
0}}[-c_{\lambda}]\right)\to{}^{p}\mathcal{H}^{i}\left(R\pi_{*}\mathrm{IC}_{\mathcal{X}}\right)\right).$

###### Proof.

We use the same notations as in the proof of Proposition 4.2. First, for any
$n$, there is an isomorphism

(13)
$Rt_{n*}\mathbb{Q}_{\mathcal{X}_{n}}\cong\mathbb{Q}_{\mathcal{X}}\otimes\mathbb{Q}[h],$

where $\mathbb{Q}[h]$ is the polynomial ring in a generator $h$ of
cohomological degree $2$. It suffices to show the statement for summands B of
$R\pi_{n*}\mathbb{Q}_{\mathcal{X}_{n}}$. Choose $n$ such that B is a summand
of ${}^{p}\mathcal{H}^{i}\left(Rq_{n*}\mathbb{Q}_{S_{n}}[\dim X]\right)$.
Define

$\displaystyle\mathcal{Y}_{n}$ $\displaystyle:=\left(V^{\lambda\geqslant
0}\oplus V^{\oplus n}\right)/G\times\mathbb{C}^{*},$ $\displaystyle W_{n}$
$\displaystyle:=\left(V^{\lambda\geqslant 0}\oplus V^{\oplus
n}\right)^{\text{st}}\sslash G\times\mathbb{C}^{*}$ $\displaystyle W_{n}^{o}$
$\displaystyle:=\left(V^{\lambda\geqslant 0}\oplus V^{\oplus
n}\right)^{\text{st},\text{nf}}\sslash G\times\mathbb{C}^{*}.$

It suffices to show that:

(14)
$\textbf{B}\subset\text{image}\,\left({}^{p}\mathcal{H}^{\cdot}\left(R\pi_{n*}p_{*}\mathrm{IC}_{\mathcal{Y}_{n}}[-c_{\lambda}]\right)\to{}^{p}\mathcal{H}^{\cdot}\left(R\pi_{n*}\mathrm{IC}_{\mathcal{X}_{n}}\right)\right).$

Consider the diagram

${S_{n}}$${W_{n}}$${X}$${X^{\lambda}.}$$\scriptstyle{q_{n}}$$\scriptstyle{p}$$\scriptstyle{r_{n}}$

By [14, Proposition 1.5], B appears in the image of

${}^{p}\mathcal{H}^{\cdot}\left(Rr_{n*}\mathrm{IC}_{W_{n}}[-c_{\lambda}]\right)\to{}^{p}\mathcal{H}^{\cdot}\left(Rq_{n*}\mathrm{IC}_{S_{n}}\right).$

Using an argument similar to Proposition 4.4 for $\mathcal{Y}_{n}$, $W_{n}$,
and $W_{n}^{o}$, the claim in (14) follows. ∎

######  Proposition 4.6.

Let $\mathcal{X}=V/G$ be a symmetric stack satisfying Assumption B. Let
$\lambda$ be a non-zero cocharacter of $G$. Then

$\displaystyle\text{image}\,\left(R\pi_{*}p_{*}\mathrm{IC}_{\mathcal{X}^{\lambda\geqslant
0}}[-c_{\lambda}]\to
R\pi_{*}\mathrm{IC}_{\mathcal{X}}\right)\subset{}^{p}D^{\geqslant
1}_{\text{shvs}}(\mathcal{X}).$

In particular, ${}^{p}\tau^{\leqslant 0}R\pi_{*}\mathrm{IC}_{\mathcal{X}}$ is
a direct sum of IC sheaves with full support.

###### Proof.

We show the first statement. We use induction on $\dim G$. Consider the
diagram

${\mathcal{X}^{\lambda\geqslant
0}}$${\mathcal{X}^{\lambda}}$${\mathcal{X}}$${X^{\lambda}}$${X.}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{\pi^{\lambda}}$$\scriptstyle{\pi}$

There are natural maps

$p_{*}q^{*}\mathrm{IC}_{\mathcal{X}^{\lambda}}\to
p_{*}\mathrm{IC}_{\mathcal{X}^{\lambda\geqslant
0}}[-c_{\lambda}]\to\mathrm{IC}_{\mathcal{X}}.$

The map $q$ is an affine bundle map, so
$q_{*}\mathbb{Q}_{\mathcal{X}^{\lambda\geqslant
0}}=\mathbb{Q}_{\mathcal{X}^{\lambda}}$. The stack $\mathcal{X}$ is symmetric,
so $\text{reldim}\,q=c_{\lambda}$. We thus need to show that

$\text{image}\,\left(R\pi^{\lambda}_{*}\mathrm{IC}_{\mathcal{X}^{\lambda}}\to
R\pi_{*}\mathrm{IC}_{\mathcal{X}}\right)\subset{}^{p}D^{\geqslant
1}_{\text{shvs}}(X).$

Let $\widetilde{G^{\lambda}}$ be the quotient of $G^{\lambda}$ by the torus
which acts trivially on $V^{\lambda}$. Then

$\widetilde{\pi^{\lambda}}:\widetilde{\mathcal{X}^{\lambda}}:=V^{\lambda}/\widetilde{G^{\lambda}}\to
X^{\lambda}$

is a good moduli space and $\widetilde{\mathcal{X}^{\lambda}}$ is a symmetric
stack satisfying Assumption B. We thus have that

$\displaystyle
R\widetilde{\pi^{\lambda}}_{*}\mathrm{IC}_{\widetilde{\mathcal{X}^{\lambda}}}\in{}^{p}D^{\geqslant
0}_{\text{shvs}}(\mathcal{X}^{\lambda})$ $\displaystyle
R\pi_{\lambda*}\mathrm{IC}_{\mathcal{X}^{\lambda}}\in{}^{p}D^{\geqslant
1}_{\text{shvs}}(\mathcal{X}^{\lambda}).$

The second statement follows from Proposition 4.5 and the fact that $\pi$ is
generically finite. ∎

We next discuss some preliminary computations regarding the maps
$m_{\mathcal{S}}$. For $\mathcal{X}=A/G$ for $A$ an affine scheme (but one can
assume for simplicity that $A$ is an affine space by Corollary 2.2) and
$\lambda:\mathbb{C}^{*}\to G$, denote by

$m_{\lambda}:=p_{\lambda^{-1}*}q_{\lambda^{-1}}^{*}:\mathrm{H}^{\cdot}(\mathcal{X}^{\lambda})\to\mathrm{H}^{\cdot}(\mathcal{X}).$

Let $N:=N_{a/A}$ be the normal bundle, and denote by A for the set of weights
in $N$, $\textbf{A}_{\lambda}$, $\textbf{g}_{\lambda}$ for the set of weights
(counted with multiplicities) of $N^{\lambda>0}$, $\mathfrak{g}^{\lambda>0}$
etc. For $\beta$ a weight of $G$, denote by $h_{\beta}\in\mathrm{H}^{2}(T)$.
The following computation are standard, for similar computations see [28,
Theorem 2.2], [43, Proposition 1.2]:

######  Proposition 4.7.

Let $\lambda$ be a cocharacter of $G$. Let
$x\in\mathrm{H}^{\cdot}(\mathcal{X}^{\lambda})$. Then

$m_{\lambda}(x)=\sum_{w\in
W/W^{\lambda}}w\left((-1)^{|\textbf{A}_{\lambda}|-|\textbf{g}_{\lambda}|}x\frac{\prod_{\textbf{A}_{\lambda}}h_{\beta}}{\prod_{\textbf{g}_{\lambda}}h_{\beta}}\right).$

######  Proposition 4.8.

Let $\lambda$ and $\mu$ be cocharacters of $G$ with the same associated Levi
group $L$. Then

$\text{image}\,\left(\mathrm{H}^{\cdot}(\mathcal{X}^{L})\xrightarrow{m_{\lambda}}\mathrm{H}^{\cdot}(\mathcal{X})\right)=\text{image}\,\left(\mathrm{H}^{\cdot}(\mathcal{X}^{L})\xrightarrow{m_{\mu}}\mathrm{H}^{\cdot}(\mathcal{X})\right).$

###### Proof.

Let $y\in\mathrm{H}^{\cdot}(\mathcal{X}^{L})$. By Proposition 4.7, we have
that:

$\displaystyle m_{\lambda}(y)$ $\displaystyle=\pm\sum_{w\in
W/W^{L}}w\left(x\frac{\prod_{\textbf{A}_{\lambda}}h_{\beta}}{\prod_{\textbf{g}_{\lambda}}h_{\beta}}\right)$
$\displaystyle m_{\mu}(y)$ $\displaystyle=\pm\sum_{w\in
W/W^{L}}w\left(x\frac{\prod_{\textbf{A}_{\mu}}h_{\beta}}{\prod_{\textbf{g}_{\mu}}h_{\beta}}\right).$

The representation $N$ is symmetric, so

$\\{\pm h_{\beta}|\,\beta\in N^{\lambda>0}\\}=\\{\pm h_{\beta}|\,\beta\in
N^{\mu>0}\\}=\\{h_{\beta}|\,\beta\in N/N^{L}\\},$

and thus $m_{\lambda}(y)=\pm m_{\mu}(y)$. ∎

Let $\lambda$ be a cocharacter of $G$. Let $T^{\lambda}\subset G^{\lambda}$ be
the torus which acts trivially on $A^{\lambda}$, and let
$\widetilde{G^{\lambda}}:=G^{\lambda}/T^{\lambda}$. Consider the map

$\widetilde{\pi^{\lambda}}:\widetilde{\mathcal{X}^{\lambda}}:=A^{\lambda}/\widetilde{G^{\lambda}}\to
X^{\lambda}.$

Define

$\mathrm{H}^{\cdot}(\mathcal{X}^{\lambda})^{\prime}:=\mathrm{H}^{\cdot}\left({}^{p}\tau^{\leqslant
0}R\widetilde{\pi^{\lambda}}_{*}\mathrm{IC}_{\widetilde{\mathcal{X}^{\lambda}}}\right)\otimes\mathbb{Q}[\mathfrak{t}^{\lambda}]\hookrightarrow\mathrm{H}^{\cdot}(\mathcal{X}^{\lambda}),$

where the generators of $\mathfrak{t}^{\lambda}$ have cohomological degree
$2$. For each Levi group $L$, choose a cocharacter $\lambda_{L}$ such that
$L\cong G^{\lambda_{L}}$.

######  Proposition 4.9.

We have that

B
$\displaystyle\cong\text{image}\,\left(\bigoplus_{L}\mathrm{H}^{\cdot}\left(\mathcal{X}^{\lambda_{L}}\right)\to\mathrm{H}^{\cdot}(\mathcal{X})\right)$
B
$\displaystyle\cong\text{image}\,\left(\bigoplus_{L}\mathrm{H}^{\cdot}\left(\mathcal{X}^{\lambda_{L}}\right)^{\prime}\to\mathrm{H}^{\cdot}(\mathcal{X})\right).$

###### Proof.

The first equality follows from Proposition 4.8 and the second follows by
induction and the Decomposition Theorem. ∎

###### Proof of Theorem 4.1.

Let $p\in X$ and recall the setting from Corollary 2.2. The restriction of
fixed and $\Theta$-stacks to $p^{-1}(\mathcal{U})$ and $\pi^{-1}(U)$ are in a
natural bijection. It suffices to prove the statement in the local case
$\mathcal{X}=V/G$.

Choose a splitting in the decomposition theorem for $\pi:\mathcal{X}\to X$:

$\mathrm{H}^{\cdot}(\mathcal{X})\cong\bigoplus_{i\geqslant
0}{}^{p}\mathrm{H}^{i}(\mathcal{X}),$

where
${}^{p}\mathrm{H}^{i}(\mathcal{X}):=\mathrm{H}^{\cdot}\left({}^{p}\mathcal{H}^{i}\left(R\pi_{*}\mathrm{IC}_{\mathcal{X}}\right)\right)$.
For $L$ a Levi subgroup, denote by $\mathrm{H}^{\cdot}(\mathcal{X})_{X^{L}}$
the cohomology of the summands in
$\bigoplus_{i}{}^{p}\mathcal{H}^{i}(R\pi_{*}\mathrm{IC}_{\mathcal{X}})$ with
support $X^{L}\to X$. These are all the supports that appear in the
Decomposition Theorem for the map $\pi:\mathcal{X}\to X$. Further, by
Propositions 4.5, 4.6, and 4.8,

$\text{image}\,\left(\mathrm{H}^{\cdot}\left(\mathcal{X}^{\lambda_{L}}\right)^{\prime}\to\mathrm{H}^{\cdot}(\mathcal{X})\right)=\mathrm{H}^{\cdot}(\mathcal{X})_{X^{L}}\cong\bigoplus_{i\geqslant
1}{}^{p}\mathrm{H}^{i}(\mathcal{X})_{X^{L}}.$

The conclusion follows from Proposition 4.6. ∎

##  5\. Categorification of IH for quotient singularities

In this Section, assume that $\mathcal{X}$ satisfies Assumption B and is
symmetric. We will use the category $\mathbb{D}(\mathcal{X})$ for $\ell$ zero.
For $\mathcal{X}=A/G$ as in Theorem 2.1, denote by $N=N_{a/A}$ the normal
bundle. We write $\textbf{A}_{\lambda}$, $\textbf{g}_{\lambda}$, n,
$\textbf{n}^{\mu}$ etc. for the sets of weights (counted with multiplicities)
of $N^{\lambda>0}$, $\mathfrak{g}^{\lambda>0}$, $\mathfrak{n}$,
$\mathfrak{n}^{\mu}$ etc. We (abuse notation and) denote by $N^{\lambda>0}$,
$\mathfrak{g}^{\lambda>0}$ etc. the sum of weights in $\textbf{A}_{\lambda}$,
$\textbf{g}_{\lambda}$ etc. Recall that $\mathrm{K}_{\cdot}$ denotes rational
K-theory.

###  5.1. Notations and definitions

We begin with some preliminary constructions and definitions.

#### 5.1.1.

There is a natural isomorphism

$\mathrm{K}_{\cdot}(A/G)\cong\mathrm{K}_{\cdot}(A/T)^{W}.$

#### 5.1.2.

Let $\chi$ be a weight of $T$. Denote by $\mathrm{K}_{\cdot}(A/T)_{\chi}$ the
subspace of $\mathrm{K}_{\cdot}(A/T)$ which is the image in K-theory of the
inclusion $D^{b}(A/T)_{\chi}\subset D^{b}(A/T)$ of the subcategory generated
by locally free $T$-equivariant sheaves with $T$-weight $\chi$.

#### 5.1.3.

Let $\lambda$ be a cocharacter. For $I$ a subset of $\textbf{A}_{\lambda}$,
denote by

$\sigma_{I}:=\sum_{\beta\in I}\beta.$

#### 5.1.4.

For two cocharacters $\lambda$ and $\mu$, let $I_{\mu}^{\lambda}$ be the set
of weights $\beta$ of $\textbf{A}_{\lambda}$ such that
$\langle\mu,\beta\rangle<0$; define similarly $J_{\mu}^{\lambda}$ for the
adjoint representation. We will use the notations:

$\displaystyle d^{\lambda}_{\mu}=|I^{\lambda}_{\mu}|,$ $\displaystyle
e^{\lambda}_{\mu}=|J^{\lambda}_{\mu}|,$ $\displaystyle
c^{\lambda}_{\mu}=|I^{\lambda}_{\mu}|-|J^{\lambda}_{\mu}|$ $\displaystyle
N^{\lambda}_{\mu}=\sum_{I_{\mu}^{\lambda}}\beta,$
$\displaystyle\mathfrak{g}^{\lambda}_{\mu}=\sum_{J_{\mu}^{\lambda}}\beta,$
$\displaystyle\mathcal{N}^{\lambda}_{\mu}=N^{\lambda}_{\mu}-\mathfrak{g}^{\lambda}_{\mu}.$

#### 5.1.5.

Let $\lambda$ and $\mu$ be cocharacters of $G$ with associated Levi and Weyl
groups $G^{\lambda}$, $W^{\lambda}$, $G^{\mu}$, $W^{\mu}$. Consider the set
$\textbf{S}:=W^{\lambda}\backslash W/W^{\mu}$. Let $V$ be the set of simple
weights. For $s\in\textbf{S}$, consider partitions of $V$ induced by $\lambda$
and $w\mu$:

$V=\bigsqcup_{i\in I^{\lambda}}V_{i},\,V=\bigsqcup_{j\in I^{w\mu}}V_{j}.$

The sets $I^{\lambda}$ and $I^{w\mu}$ are the sets of eigenvalues of $\lambda$
and $w\mu$ on $\mathfrak{h}=\mathbb{C}^{V}$, respectively. They are ordered by
the natural ordering of $\mathbb{Z}$. Define $V^{w}_{ij}:=V_{i}\cap V_{j}$ and
use the lexicographic order on the set $(I^{\lambda},I^{w\mu})$. Consider the
decomposition

(15) $V=\bigsqcup_{(i,j)\in(I^{\lambda},I^{w\mu})}V^{w}_{ij}.$

Similarly, consider the partitions of $V$ induced by $\mu$ and
$w^{-1}\lambda$:

$V=\bigsqcup_{i\in I^{\mu}}V_{i},\,V=\bigsqcup_{j\in I^{w^{-1}\lambda}}V_{j}.$

Define $V^{\prime w}_{ij}:=V_{i}\cap V_{j}$. Use the lexicographic order on
the set $\left(I^{\mu},I^{w^{-1}\lambda}\right)$ and consider the
decomposition

(16) $V=\bigsqcup_{(i,j)\in(I^{\mu},I^{w^{-1}\lambda})}V^{\prime w}_{ij}.$

Assume next that $\lambda$ and $\mu$ are dominant cocharacters. Let $w\in
W^{\lambda}sW^{\mu}$ be an element such that the lexicographic order on
$(I^{\lambda},I^{w_{s}\mu})$ induces a dominant cocharacter $\nu$; $w$ can be
chosen such that it does not permute elements of $V_{i}$ for $i\in
I^{\lambda}$ among themselves, and similarly for $V_{j}$ for $j\in I^{\mu}$.
We can choose $w$ to be the element of minimal length $w_{s}\in
W^{\lambda}sW^{\mu}$. Let $\nu$ be the dominant cocharacter induced by
$\left(I^{\lambda},I^{w_{s}\mu}\right)$. Further,
$\left(I^{\mu},I^{w_{s}^{-1}\lambda}\right)$ induces a dominant cocharacter
$\nu^{\prime}$.

Further, for any $w\in W^{\lambda}w_{s}W^{\mu}$, there exists $w^{\prime}\in
W^{\lambda}$ such that

$\\{w^{\prime}V^{w}_{ij}|i\in I^{\lambda},j\in
I^{w\mu}\\}=\\{V^{w_{s}}_{ij}|i\in I^{\lambda},j\in I^{w_{s}\mu}\\}.$

####  5.1.6. Example.

We discuss an example of the construction from the previous Subsection. Let
$G=GL(n)$ and assume that $\lambda$ and $\mu$ are dominant cocharacters with
parabolic groups

$\displaystyle G^{\lambda\geqslant 0}$ $\displaystyle=GL(a,b)$ $\displaystyle
G^{\mu\geqslant 0}$ $\displaystyle=GL(c,d).$

We identify the set $V$ with $\\{1,\cdots,n\\}$. The partition of $V$
corresponding to $\lambda$ is $\\{1,\cdots,a\\}\sqcup\\{a+1,\cdots,n\\}$, and
the partition of $V$ corresponding to $\mu$ is
$\\{1,\cdots,c\\}\sqcup\\{c+1,\cdots,n\\}$. The set
$\textbf{S}=\mathfrak{S}_{a}\times\mathfrak{S}_{b}\backslash\mathfrak{S}_{n}/\mathfrak{S}_{c}\times\mathfrak{S}_{d}$
parametrizes quadruplets $\left(e_{1},e_{2},e_{3},e_{4}\right)$ such that

$\displaystyle e_{1}+e_{2}$ $\displaystyle=a,$ $\displaystyle e_{3}+e_{4}$
$\displaystyle=b,$ $\displaystyle e_{1}+e_{3}$ $\displaystyle=c,$
$\displaystyle e_{2}+e_{4}$ $\displaystyle=d.$

The decomposition (15) corresponds to a partition of $V$ in four sets $V_{i}$
of cardinal $e_{i}$ for $1\leqslant i\leqslant 4$ such that $V_{1}\sqcup
V_{2}=\\{1,\cdots,a\\}$; the decomposition (16) corresponds to a partition of
$V$ in four sets $V^{\prime}_{i}$ of cardinals $e_{1}$, $e_{3}$, $e_{2}$, and
$e_{4}$ respectively, such that $V^{\prime}_{1}\sqcup
V^{\prime}_{2}=\\{1,\cdots,c\\}$. The decomposition corresponding to $\nu$ is

$V_{1}=\\{1,\cdots,e_{1}\\},\cdots,V_{4}=\\{e_{1}+e_{2}+e_{3}+1,\cdots,n\\}.$

The dominant cocharacters $\nu$ and $\nu^{\prime}$ correspond to the parabolic
groups

$\displaystyle G^{\nu\geqslant 0}$ $\displaystyle=GL(e_{1},e_{2},e_{3},e_{4})$
$\displaystyle G^{\nu^{\prime}\geqslant 0}$
$\displaystyle=GL(e_{1},e_{3},e_{2},e_{4}).$

The permutation $w_{s}\in\mathfrak{S}_{n}$ sends

$\displaystyle i$ $\displaystyle\mapsto i+e_{2}\text{ for }e_{1}+1\leqslant
i\leqslant e_{1}+e_{2},$ $\displaystyle i$ $\displaystyle\mapsto i-e_{2}\text{
for }e_{1}+e_{2}+1\leqslant i\leqslant e_{1}+e_{2}+e_{3}.$

#### 5.1.7.

Recall the definition of $\mathbb{W}$ from (8). Let $\lambda$ a cocharacter of
$G$. Assume that $n_{\lambda}$ is even and let $b_{\lambda}=n_{\lambda}/2$.
Denote by $\mathbb{F}(\lambda)$ the set of weights $\chi$ such that

$\langle\lambda,\chi\rangle=b_{\lambda}.$

Let $\chi$ be a weight satisfying the inequalities in (2) and such that
$\chi\in\mathbb{F}(\lambda)$. Then there exists $\psi$ and $w\in W$ such that
$w\psi$ is dominant,
$w\psi+\rho_{L}\in\frac{1}{2}\mathbb{W}(\mathcal{X}^{L})$, and

(17) $\chi=\frac{1}{2}N^{\lambda>0}-\frac{1}{2}\mathfrak{g}^{\lambda>0}+\psi.$

Here $\rho_{L}$ is half the sum of positive roots of
$\mathfrak{l}=\mathfrak{g}^{\lambda}$.

Indeed, the condition that $\chi$ satisfies the inequalities in (2) means that
there exists $w\in W$ such that $w\chi$ is dominant and

$w\chi+\rho\in\frac{1}{2}\mathbb{W}.$

This also implies that $w\lambda$ is dominant. By [21, Lemma 3.12], [35,
Corollary 2.4], there exists a weight
$\tau\in\frac{1}{2}\mathbb{W}(\mathcal{X}^{L})$ such that

$w\chi+\rho=\frac{1}{2}N^{w\lambda>0}+\tau.$

Write $\rho=\frac{1}{2}\mathfrak{g}^{w\lambda>0}+\rho_{L}$. Then there exists
$\omega$ dominant such that
$\omega+\rho_{L}\in\frac{1}{2}\mathbb{W}(\mathcal{X}^{L})$ and

$w\chi=\frac{1}{2}N^{w\lambda>0}-\frac{1}{2}\mathfrak{g}^{w\lambda>0}+\omega.$

For $\psi=w^{-1}\omega$ we obtain the desired conclusion.

#### 5.1.8.

Recall from the discussion in Subsection 1.5 that the categories
$\mathbb{D}(\mathcal{X})$ may contain complexes supported on attracting
stacks. We discuss how to characterize these complexes.

Assume that $\mathcal{X}=A/G$. Let $\lambda$ be a cocharacter. Recall that
$b_{\lambda}=n_{\lambda}/2$. Denote by $\mathbb{D}(\mathcal{X}^{\lambda})_{b}$
the subcategory of $\mathbb{D}(\mathcal{X}^{\lambda})$ generated by sheaves of
weights $\chi$ such that $\langle\lambda,\chi\rangle=b_{\lambda}$, see (17)
for their description. A cocharacter $\lambda$ determines a map

$m_{\lambda}:=\frac{1}{|W^{\lambda}|}p_{{\lambda^{-1}}*}q_{\lambda^{-1}}^{*}:\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)\to\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X})\right).$

To see that the image lies in $\mathbb{D}(\mathcal{X})$, the sheaves in
Proposition 2.3 all have $r$-invariant $\leqslant 1/2$ by the argument in [21,
Proposition 3.12]. A cocharacter $\nu:\mathbb{C}^{*}\to G$ with image in
$G^{\lambda}$ determines a cocharacter of $G^{\lambda}$ and thus a map

$m^{\lambda}_{\nu}=\frac{1}{|W^{\nu}|}p_{{\nu^{-1}}*}q_{\nu^{-1}}^{*}:\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\nu})_{b}\right)\to\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right).$

There are similarly defined maps in the global case.

A dominant cocharacter $\lambda$ of $G$ determines a restriction map:

$\Delta_{\lambda}:=\beta_{\geqslant
b_{\lambda}}p_{\lambda}^{*}:\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X})\right)\to\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right).$

Here, $\beta_{\geqslant b_{\lambda}}$ is the functor which considers the top
$\lambda$-weight component:

$\beta_{\geqslant b_{\lambda}}:D^{b}(\mathcal{X}^{\lambda\geqslant 0})\to
D^{b}(\mathcal{X}^{\lambda\geqslant 0})_{b_{\lambda}}\cong
D^{b}(\mathcal{X}^{\lambda})_{b_{\lambda}},$

see [19, Lemma 3.4, Corollary 3.17] for a defintion of the functor
$\beta_{\geqslant\cdot}$; the equivalence is induces by the functor
$q_{\lambda}^{*}$ [19, Amplification 3.18]. It has the formula from
Proposition 5.7. For dominant $\nu$ as above, there is a restriction map:

$\Delta^{\lambda}_{\nu}:\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)\to\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\nu})_{b}\right).$

#### 5.1.9.

Assume that $\mathcal{X}=A/G$. We use the notations and settings of
Subsections 5.1.4 and 5.1.5. Consider dominant $\lambda$ and $\mu$ inducing
dominant $\nu$ and $\nu^{\prime}$. Then

$\\{\beta\in N|\,\langle\lambda,\beta\rangle>0,\langle
w_{s}\mu,\beta\rangle<0\\}=\\{\beta\in
N|\,\langle\nu,\beta\rangle>0,\langle\nu^{\prime},\beta\rangle<0\\},$

and so
$c^{\lambda}_{w_{s}\mu}=c^{\nu}_{\nu^{\prime}},\mathcal{N}^{\lambda}_{w_{s}\mu}=\mathcal{N}^{\nu}_{\nu^{\prime}}.$
A weight $\beta$ of $T$ determines $q_{\beta}\in K_{0}(BT)$. Define

$\displaystyle\widetilde{\text{sw}_{s}}:\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\nu})_{b}\right)$
$\displaystyle\to\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\nu^{\prime}})_{b}\right)$
$\displaystyle y$
$\displaystyle\mapsto(-1)^{c^{\nu}_{\nu^{\prime}}}w_{s}^{-1}\left(yq^{-\mathcal{N}^{\nu}_{\nu^{\prime}}}\right)=(-1)^{c^{\lambda}_{w_{s}\mu}}w_{s}^{-1}\left(yq^{-\mathcal{N}^{\lambda}_{w_{s}\mu}}\right).$

###  5.2. Computations in K-theory

Recall the notations $I,J,I^{\prime},J^{\prime}$ from the beginning of Section
4. For $\mathcal{S}$ an attracting stack in $J$ with fixed locus
$\mathcal{Z}$, consider the map

$m_{\mathcal{S}}:=p_{*}q^{*}:\mathrm{K}_{\cdot}(\mathcal{Z})\to\mathrm{K}_{\cdot}(\mathcal{X})$

and the subspace of $K_{\cdot}(\mathcal{X})$:

$\textbf{B}\mathrm{K}_{\cdot}(\mathcal{X}):=\text{image}\left(\bigoplus_{J^{\prime}}\mathrm{K}_{\cdot}(\mathcal{Z})\to\mathrm{K}_{\cdot}(\mathcal{X})\right).$

We define $\textbf{P}\mathrm{K}_{\cdot}(\mathcal{X})$ in Subsection 5.3. The
main result we prove in this Section is:

######  Theorem 5.1.

There is a decomposition
$\mathrm{K}_{\cdot}(\mathcal{X})=\textbf{P}\mathrm{K}_{\cdot}(\mathcal{X})\oplus\textbf{B}\mathrm{K}_{\cdot}(\mathcal{X})$.

The definition of $\textbf{P}\mathrm{K}_{\cdot}(\mathcal{X})$ is based on the
following result. We restrict to the local case $\mathcal{X}=A/G$. The
following is the main ingredient in the proof of Theorem 5.1.

######  Theorem 5.2.

Let $\lambda$ and $\mu$ be two dominant cocharacters, let S be the set defined
in Subsection 5.1.5, and let $\nu$ be dominant cocharacters as constructed in
Subsection 5.1.5. The following diagram commutes:

${\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)}$${\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))}$${\bigoplus_{\textbf{S}}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\nu})_{b}\right)}$${\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\mu})_{b}\right).}$$\scriptstyle{\bigoplus_{\textbf{S}}\Delta^{\lambda}_{\nu}}$$\scriptstyle{m_{\lambda}}$$\scriptstyle{\Delta_{\mu}}$$\scriptstyle{\bigoplus_{\textbf{S}}\widetilde{m^{\mu}_{\nu}}}$

Before the start of the above result, we list some preliminary computations.
The first two are standard, for example, for the first one see [43,
Proposition 1.2], [36, Propositions 3.1 and 3.2]:

######  Proposition 5.3.

Consider the maps $A/T\xleftarrow{t}A/B\xrightarrow{s}A/G.$ Then the map
$s_{*}t^{*}:\mathrm{K}_{\cdot}(A/T)\to\mathrm{K}_{\cdot}(A/G)$ has the formula

$s_{*}t^{*}(y)=\sum_{w\in
W}w\left(\frac{y}{\prod_{\beta\in\textbf{n}}(1-q^{-\beta})}\right).$

######  Proposition 5.4.

The map
$m_{\lambda}:\mathrm{K}_{\cdot}(\mathcal{X}^{\lambda})\to\mathrm{K}_{\cdot}(\mathcal{X})$
has the formula

$m_{\lambda}(x)=\frac{1}{|W^{\lambda}|}\sum_{w\in
W/W^{\lambda}}w\left(x\frac{\prod_{\beta\in\textbf{A}_{\lambda}}(1-q^{-\beta})}{\prod_{\beta\in\textbf{g}_{\lambda}}(1-q^{-\beta})}\right).$

###### Proof.

Consider the natural maps

$\displaystyle r$ $\displaystyle:A\hookrightarrow A^{\lambda\geqslant
0}\times_{A^{\lambda}}A^{\lambda\leqslant 0}\to A^{\lambda},$ $\displaystyle
v$ $\displaystyle:A/G^{\lambda\geqslant 0}\to A/G.$

In the statement of Proposition 2.3, the sheaves $\mathcal{F}_{I}$ are
$v_{*}\left(r^{*}(\mathcal{F})(-\sigma_{I})\right)$. The claim thus follows
from Proposition 2.3, see also [36, Propositions 3.1] for a similar
computation. ∎

######  Proposition 5.5.

Let $\lambda$ and $\mu$ be cocharacters with the same associated Levi group
$L\subset G$. For $y\in\mathrm{K}_{\cdot}(\mathcal{X}^{L})$, let
$y^{\prime}:=(-1)^{c^{\lambda}_{\mu}}yq^{-\mathcal{N}^{\lambda}_{\mu}}\in\mathrm{K}_{\cdot}(\mathcal{X}^{L})$.
Then

$m_{\lambda}(y)=m_{\mu}(y^{\prime}).$

###### Proof.

Using Proposition 5.4, we have that:

$\displaystyle m_{\lambda}(y)$
$\displaystyle=\frac{1}{|W^{\lambda}|}\sum_{W/W^{L}}w\left(y\frac{\prod_{\textbf{A}_{\lambda}}(1-q^{-\beta})}{\prod_{\textbf{g}_{\lambda}}(1-q^{-\beta})}\right)$
$\displaystyle m_{\mu}(y)$
$\displaystyle=\frac{1}{|W^{\lambda}|}\sum_{W/W^{L}}w\left(y\frac{\prod_{\textbf{A}_{\mu}}(1-q^{-\beta})}{\prod_{\textbf{g}_{\mu}}(1-q^{-\beta})}\right).$

Further, we have that

$\frac{\prod_{\textbf{A}_{\mu}}(1-q^{-\beta})}{\prod_{\textbf{g}_{\mu}}(1-q^{-\beta})}=\frac{\prod_{\textbf{A}_{\lambda}}(1-q^{-\beta})}{\prod_{\textbf{g}_{\lambda}}(1-q^{-\beta})}(-1)^{c^{\lambda}_{\mu}}q^{\mathcal{N}^{\lambda}_{\mu}},$

which implies the desired conclusion. ∎

######  Corollary 5.6.

Under the hypothesis of Proposition 5.5,

$\text{image}\,\left(\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)\xrightarrow{m_{\lambda}}\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))\right)=\text{image}\,\left(\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\mu})_{b}\right)\xrightarrow{m_{\mu}}\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))\right).$

###### Proof.

Assume that
$y\in\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)$.
Let $\chi$ and $\chi^{\prime}$ be the weights of $y$ and $y^{\prime}$,
respectively. Then

$\chi^{\prime}=\chi+\mathcal{N}^{\mu}_{\lambda}.$

By the discussion in Subsection 5.1.7 and the decomposition in (17), there is
a weight $\psi$ with the properties mentioned there such that

$\displaystyle\chi$
$\displaystyle=\frac{1}{2}N^{\lambda>0}-\frac{1}{2}\mathfrak{g}^{\lambda>0}+\psi$
$\displaystyle\chi^{\prime}$
$\displaystyle=\frac{1}{2}N^{\mu>0}-\frac{1}{2}\mathfrak{g}^{\mu>0}+\psi,$

and thus
$y^{\prime}=(-1)^{c^{\lambda}_{\mu}}yq^{\mathcal{N}^{\mu}_{\lambda}}\in\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\mu})_{b}\right)$.
∎

######  Proposition 5.7.

Let $\mu$ be a dominant character and let $y\in\mathrm{K}_{\cdot}(A/T)_{\chi}$
with $\chi$ on the face $\mathbb{F}(\mu)$. Then the restriction map
$\Delta_{\mu}:\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))\to\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\mu})_{b}\right)$
sends

$\Delta_{\mu}\left(\sum_{w\in
W}w\left(\frac{y}{\prod_{\beta\in\textbf{n}}(1-q^{-\beta})}\right)\right)=\sum_{w\in
W^{\mu}}w\left(\frac{y}{\prod_{\beta\in\textbf{n}^{\mu}}(1-q^{-\beta})}\right).$

Let $y\in\mathrm{K}_{\cdot}(A/T)_{\chi}$ with $w*\chi$ not on
$\mathbb{F}(\mu)$ for any $w\in W$, then

$\Delta_{\mu}\left(\sum_{w\in
W}w\left(\frac{y}{\prod_{\beta\in\textbf{n}}(1-q^{-\beta})}\right)\right)=0.$

###### Proof.

The weight $\chi$ is dominant up to multiplication by an element of $W^{\mu}$,
so we can assume it is dominant. For $\mathcal{F}\in\mathbb{D}(\mathcal{X})$,
we have that

$\Delta_{\mu}(\mathcal{F})=\beta_{\geqslant b_{\mu}}p_{\mu}^{*}(\mathcal{F}).$

For $\mathcal{F}$ with associated representation $\Gamma_{G}(\chi)$,
$\Delta_{\mu}(\mathcal{F})$ has associated representation
$\Gamma_{G^{\mu}}(\chi)$, and the first part follows. For the second part, by
Proposition 5.5 we can replace $\chi$ with any $w*\chi$ for $w\in W$, so we
can assume that $\chi$ is dominant. Then $\chi$ is not on $\mathbb{F}(\mu)$,
so $\langle\mu,p_{\mu}^{*}(\mathcal{F})\rangle<b_{\mu},$ and the second part
thus follows. ∎

######  Proposition 5.8.

Consider $\lambda,\tau$ two dominant cocharacters and $\chi$ a dominant weight
with $\chi\in\mathbb{F}(\lambda)$. Assume there is a partial sum $\sigma$ of
weights in $\textbf{A}_{\lambda}$ and an element $w\in W$ such that
$w*\left(\chi-\sigma\right)\in\mathbb{F}(\tau)$. Let $\mu=w^{-1}\tau$. Then

$\sigma=N^{\lambda}_{\mu}+\sigma^{\prime},$

where $\sigma^{\prime}$ is a partial sum of weights in
$(N^{\mu})^{\lambda>0}$. Conversely, any such partial sum $\sigma$ has the
property that $w*\left(\chi-\sigma\right)\in\mathbb{F}(\tau)$.

###### Proof.

The weight

(18) $\left(\chi-\sigma\right)^{+}+\rho\in\frac{1}{2}\mathbb{W}$

by the same argument as in [34, Proposition 3.6], see also [21, Proof of
Theorem 3.2]. Using the description in Subsection 5.1.7, write

$\displaystyle\chi+\rho$ $\displaystyle=\frac{1}{2}N^{\lambda>0}+\psi$
$\displaystyle w*\left(\chi-\sigma\right)+\rho$
$\displaystyle=\frac{1}{2}N^{\tau>0}+\phi^{\prime},\text{ and so}$
$\displaystyle\chi-\sigma+\rho$ $\displaystyle=\frac{1}{2}N^{\mu>0}+\phi,$

where $\psi$ is a sum of weights of $N^{\lambda}$ and $\phi$ is a sum of
weights of $N^{\mu}$. For the first two relations above we use that $\lambda$
and $\tau$ are dominant. Then

(19) $\sigma=N^{\lambda}_{\mu}+\psi-\phi.$

Write

$\displaystyle\psi$ $\displaystyle=\psi^{\lambda 0}_{\mu+}+\psi^{\lambda
0}_{\mu 0}+\psi^{\lambda 0}_{\mu-}$ $\displaystyle\phi$
$\displaystyle=\phi^{\lambda+}_{\mu 0}+\phi^{\lambda 0}_{\mu
0}+\phi^{\lambda-}_{\mu 0},$

where $\psi^{\lambda 0}_{\mu+}$ is a sum of weights in $(N^{\lambda})^{\mu>0}$
etc. Then the decomposition (19) implies that

$\displaystyle\psi$ $\displaystyle=\psi^{\lambda 0}_{\mu 0},$
$\displaystyle\phi$ $\displaystyle=\phi^{\lambda 0}_{\mu
0}+\phi^{\lambda-}_{\mu 0},$ $\displaystyle\phi^{\lambda 0}_{\mu 0}$
$\displaystyle=\psi^{\lambda 0}_{\mu 0}.$

This implies that $\sigma=N^{\lambda}_{\mu}-\phi^{\lambda-}_{\mu 0}$, where
$-\phi^{\lambda-}_{\mu 0}$ is a partial sum of weights in
$(N^{\mu})^{\lambda>0}$.

Conversely, if $\sigma=N^{\lambda}_{\mu}+\sigma^{\prime}$, the argument above
shows that $\chi-\sigma+\rho=\frac{1}{2}N^{\mu>0}+\phi$ and by (18) we have
that $w*\left(\chi-\sigma\right)\in\mathbb{F}(\tau)$. ∎

###### Proof of Theorem 5.2.

Let
$x\in\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)$. By
Proposition 5.3, let $y\in\mathrm{K}_{\cdot}(A/T)$ such that

$x=\sum_{v\in
W^{\lambda}}v\left(\frac{y}{\prod_{\beta\in\textbf{n}^{\lambda}}(1-q^{-\beta})}\right).$

We may assume that $y$ is of dominant weight $\chi$. By Proposition 5.4, we
have that

$p_{{\lambda}^{-1}*}q_{\lambda^{-1}}^{*}\sum_{v\in
W^{\lambda}}v\left(\frac{y}{\prod_{\beta\in\textbf{n}^{\lambda}}(1-q^{-\beta})}\right)=\sum_{u\in
W/W^{\lambda}}u\sum_{I\subset\textbf{A}_{\lambda}}\sum_{v\in
W^{\lambda}}v\left(\frac{(-1)^{|I|}yq^{-\sigma_{I}}}{\prod_{\beta\in\textbf{n}}(1-q^{-\beta})}\right).$

The weight of $yq^{-\sigma_{I}}$ is $\chi-\sigma_{I}$. By Propositions 5.7 and
5.8, such an element has non-zero $\mu$-restriction if and only if there
exists $w\in W$ such that

(20) $\sigma_{I}=N^{\lambda}_{w\mu}+\sigma_{I^{\prime}},$

where $I^{\prime}$ is a subset of $\textbf{A}^{w\mu}_{\lambda}$, the set of
weights $\beta$ in $\textbf{A}_{\lambda}$ such that $\langle
w\mu,\beta\rangle=0.$

Fix $w$. Let $\nu$ and $\nu^{\prime}$ be the cocharacters constructed as in
Subsection 5.1.5. The weight $N^{\lambda}_{w\mu}$ and the set
$\textbf{A}^{w\mu}_{\lambda}$ depend on the coset $W/W^{w\mu}$; if they have
associated $\nu$ as above, then they depend on $W^{\lambda}/W^{\nu}\subset
W/W^{w\mu}$. Recall the element $w^{\prime}$ as the end of Subsection 5.1.5.
The element $yq^{-\sigma_{I}}$ has weight $\chi-\sigma_{I}$. By (17), the
weight $\left(\chi-\sigma_{I}\right)^{+}$ is on $\mathbb{F}(\nu)$. Then

$\displaystyle\sum_{v\in
W^{\lambda}}v\left(\frac{y}{\prod_{\beta\in\textbf{n}^{\lambda}}(1-q^{-\beta})}\right)q^{-N^{\lambda}_{w\mu}-\sigma_{I^{\prime}}}$
$\displaystyle=\sum_{v\in
W^{\lambda}}v\left(\frac{y}{\prod_{\beta\in\textbf{n}^{\lambda}}(1-q^{-\beta})}\right)w^{\prime}\left(q^{-N^{\lambda}_{w_{s}\mu}-\sigma_{J}}\right)$
$\displaystyle=\sum_{v\in
W^{\lambda}}\left(\frac{yq^{-N^{\nu}_{\nu^{\prime}}-\sigma_{J}}}{\prod_{\beta\in\textbf{n}^{\lambda}}(1-q^{-\beta})}\right),$

where $J$ is a subset of $\textbf{A}^{w_{s}\mu}_{\lambda}$. Define

$m_{\lambda\nu}(x):=\frac{1}{|W^{\nu}|}\sum_{u\in
W/W^{\lambda}}u\sum_{J\subset\textbf{A}^{w_{s}\mu}_{\lambda}}\sum_{v\in
W^{\lambda}}v\left(\frac{(-1)^{d^{\nu}_{\nu^{\prime}}+|J|}yq^{-N^{\nu}_{\nu^{\prime}}-\sigma_{J}}}{\prod_{\beta\in\textbf{n}}(1-q^{-\beta})}\right).$

Then $m_{\lambda}=\sum_{\nu}m_{\lambda\nu}$. It suffices to show that the
following diagram commutes

${K_{\cdot}(\mathbb{D}(\mathcal{X}^{\lambda})_{b})}$${K_{\cdot}(\mathbb{D}(\mathcal{X}))}$${K_{\cdot}(\mathbb{D}(\mathcal{X}^{\nu})_{b})}$${K_{\cdot}(\mathbb{D}(\mathcal{X}^{\mu})_{b}).}$$\scriptstyle{\Delta^{\lambda}_{\nu}}$$\scriptstyle{m_{\lambda\nu}}$$\scriptstyle{\Delta_{\mu}}$$\scriptstyle{\widetilde{m^{\mu}_{\nu}}}$

Let $\tau$ be the sum of weights $\beta$ in $\mathfrak{n}$ such that
$w_{s}^{-1}\beta$ is not in $\mathfrak{n}$. Then
$-w_{s}\tau=\mathfrak{g}^{\lambda}_{w_{s}\mu}$ because the two sides are sums
over the weights in the following two sets

$\\{\beta\in\textbf{n}\text{ such that }-\beta\in
w_{s}\textbf{n}\\}=\\{\beta\in\textbf{g}_{\lambda}\text{ such that
}-\beta\in\textbf{g}_{w_{s}\mu}\\}.$

Recall the setting of Subsection 5.1.9. We have that

$\displaystyle
w_{s}^{-1}\left(\frac{(-1)^{d^{\nu}_{\nu^{\prime}}}yq^{-N^{\nu}_{\nu^{\prime}}}}{\prod_{\beta\in\textbf{n}}(1-q^{-\beta})}\right)$
$\displaystyle=\frac{(-1)^{d^{\nu}_{\nu^{\prime}}}w_{s}^{-1}\left(yq^{-N^{\nu}_{\nu^{\prime}}}\right)}{(-1)^{e^{\nu}_{\nu^{\prime}}}q^{\tau}\prod_{\beta\in\textbf{n}}(1-q^{-\beta})}$
$\displaystyle=\frac{(-1)^{c^{\nu}_{\nu^{\prime}}}w_{s}^{-1}\left(yq^{-N^{\nu}_{\nu^{\prime}}-w_{s}\tau}\right)}{\prod_{\beta\in\textbf{n}}(1-q^{-\beta})}$
$\displaystyle=\frac{(-1)^{c^{\nu}_{\nu^{\prime}}}w_{s}^{-1}\left(yq^{-\mathcal{N}^{\nu}_{\nu^{\prime}}}\right)}{\prod_{\beta\in\textbf{n}}(1-q^{-\beta})}.$

We can thus rewrite

$m_{\lambda\nu}(x):=\frac{1}{|W^{\nu}|}\sum_{J^{\prime}\subset\textbf{A}^{\mu}_{w_{s}^{-1}\lambda}}\sum_{v\in
W}v\left(\frac{(-1)^{c^{\nu}_{\nu^{\prime}}+|J^{\prime}|}w_{s}^{-1}\left(yq^{-\mathcal{N}^{\nu^{\prime}}_{\nu}}\right)q^{-\sigma_{J^{\prime}}}}{\prod_{\beta\in\textbf{n}}(1-q^{-\beta})}\right),$

and thus

$\Delta_{\mu}m_{\lambda\nu}(x)=\frac{1}{|W^{\nu}|}\sum_{J^{\prime}\subset\textbf{A}^{\mu}_{w_{s}^{-1}\lambda}}\sum_{v\in
W^{\mu}}v\left(\frac{(-1)^{c^{\nu}_{\nu^{\prime}}+|J^{\prime}|}w_{s}^{-1}\left(yq^{-\mathcal{N}^{\nu^{\prime}}_{\nu}}\right)q^{-\sigma_{J^{\prime}}}}{\prod_{\beta\in\textbf{n}^{\mu}}(1-q^{-\beta})}\right).$

This is the same as the composition of left-bottom maps:

$\displaystyle\widetilde{m^{\mu}_{\nu}}\Delta^{\lambda}_{\nu}(x)$
$\displaystyle=\widetilde{m^{\mu}_{\nu}}\Delta^{\lambda}_{\nu}\left(\sum_{v\in
W^{\lambda}}v\left(\frac{y}{\prod_{\beta\in\textbf{n}^{\lambda}}(1-q^{-\beta})}\right)\right)$
$\displaystyle=\widetilde{m^{\mu}_{\nu}}\sum_{v\in
W^{\nu}}v\left(\frac{y}{\prod_{\beta\in\textbf{n}^{\nu}}(1-q^{-\beta})}\right)$
$\displaystyle=m^{\mu}_{\nu^{\prime}}\sum_{v\in
W^{\nu}}v\left(\frac{(-1)^{c^{\nu}_{\nu^{\prime}}}w_{s}^{-1}\left(yq^{-\mathcal{N}^{\nu}_{\nu^{\prime}}}\right)}{\prod_{\beta\in\textbf{n}^{\nu}}(1-q^{-\beta})}\right)$
$\displaystyle=\frac{1}{|W^{\nu}|}\sum_{J^{\prime}\subset\textbf{A}^{\mu}_{w_{s}^{-1}\lambda}}\sum_{v\in
W^{\mu}}v\left(\frac{(-1)^{c^{\nu}_{\nu^{\prime}}+|J^{\prime}|}w_{s}^{-1}\left(yq^{-\mathcal{N}^{\nu^{\prime}}_{\nu}}\right)q^{-\sigma_{J^{\prime}}}}{\prod_{\beta\in\textbf{n}^{\mu}}(1-q^{-\beta})}\right)$

where the second equality follows from Proposition 5.7 and the last one by
Proposition 5.4. ∎

###  5.3. Primitive K-theory: the local case

Recall the setting and notations from the beginning of Subsection 5.2. In this
Subsection, we define
$\textbf{P}\mathrm{K}_{\cdot}(\mathcal{X})\subset\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))$
which appears in Theorem 5.1. Assume that $\mathcal{X}=A/G$ is a local stack.

######  Proposition 5.9.

Let $\lambda$ be a dominant cocharacter and let $\mathfrak{S}_{\lambda}\subset
W$ be the set of elements $w_{s}$ for $s\in W^{\lambda}\backslash
W/W^{\lambda}$ such that the weight $\nu$ constructed in Subsection 5.1.5 is
equal to $\lambda$. Then $\mathfrak{S}_{\lambda}$ is a group and it acts on
$\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}^{\lambda})_{b})$ via
$\widetilde{\text{sw}}$.

###### Proof.

The elements $w_{s}$ above are the elements of $W$ that induce permutations of
$I^{\lambda}$ and which do not permute elements in $V_{i}$ for $i\in
I^{\lambda}$ among themselves. These elements are clearly closed under
multiplication and taking inverses.

For the second part, consider elements $w_{1},w_{2}\in\mathfrak{S}_{\lambda}$
and let
$y\in\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)$.
Let $\widetilde{\text{sw}_{1}}$, $\widetilde{\text{sw}_{2}}$,
$\widetilde{\text{sw}_{3}}$ be the swap maps for the elements $w_{1},w_{2}$,
and $w_{2}w_{1}$, respectively. We need to show that
$\widetilde{\text{sw}_{1}}\widetilde{\text{sw}_{2}}=\widetilde{\text{sw}_{3}}$:

$\displaystyle(-1)^{c^{\lambda}_{w_{1}\lambda}}w_{1}^{-1}\left((-1)^{c^{\lambda}_{w_{2}\lambda}}w_{2}^{-1}\left(yq^{-\mathcal{N}^{\lambda}_{w_{2}\lambda}}\right)q^{-\mathcal{N}^{\lambda}_{w_{1}\lambda}}\right)=$
$\displaystyle(-1)^{c^{\lambda}_{w_{1}\lambda}+c^{\lambda}_{w_{2}\lambda}}w_{1}^{-1}w_{2}^{-1}\left(yq^{-\mathcal{N}^{\lambda}_{w_{2}\lambda}-\mathcal{N}^{w_{2}\lambda}_{w_{2}w_{1}\lambda}}\right)=$
$\displaystyle(-1)^{c^{\lambda}_{w_{2}w_{1}\lambda}}(w_{2}w_{1})^{-1}\left(yq^{-\mathcal{N}^{\lambda}_{w_{2}w_{1}\lambda}}\right).$

∎

In the example from Subsection (5.1.6), $\mathfrak{S}_{\lambda}$ is trivial
unless $a=b$, case in which it is the symmetric group $\mathfrak{S}_{2}$. More
generally, for $\lambda$ a cocharacter of $GL(n)$ with corresponding
decomposition in distinct parts $d_{1},\cdots,d_{k}$ with multiplicities
$m_{1},\cdots,m_{k}$, the group $\mathfrak{S}_{\lambda}$ is the product of
symmetric groups $\times_{i=1}^{k}\mathfrak{S}_{m_{i}}$. In the framework of
the above Proposition, denote by

$\displaystyle\text{Sym}_{\lambda}:\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)$
$\displaystyle\to\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)$
$\displaystyle x$
$\displaystyle\mapsto\sum_{\sigma\in\mathfrak{S}_{\lambda}}\sigma(x).$

Define

$\textbf{B}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)=\text{image}\left(\bigoplus_{\nu}m_{\nu}:\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\nu})_{b}\right)\to\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)\right),$

where the sum is over all non-trivial cocharacters $\nu$ of $G^{\lambda}$. We
define
$\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)$
inductively on $\dim G^{\lambda}$ such that

(21)
$\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)=\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)\oplus\textbf{B}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right).$

There are then natural surjections
$\pi_{\lambda}:\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)\twoheadrightarrow\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)$.

When $\dim G^{\lambda}=0$, then
$\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)=\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)$.
Assume that $\dim G>0$. For any Levi $L<G$, choose a dominant cocharacter
$\lambda_{L}$ such that $G^{\lambda_{L}}=L$. Denote by $m_{H}=m_{\lambda_{H}}$
etc. Define

$\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)=\bigcap_{H<L}\left(\text{Sym}_{H}\pi_{H}\Delta_{H}:\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)\to\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{H}})_{b}\right)\right).$

Denote by
$\iota:\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)\hookrightarrow\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)$
the natural map, and denote by $\Phi_{H}:=\text{Sym}_{H}\pi_{H}\Delta_{H}$.

######  Proposition 5.10.

The composition

$\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)\xrightarrow{m_{L}}\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))\xrightarrow{\pi_{L}\Delta_{L}}\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)$

is
$\pi_{L}\Delta_{L}m_{L}(x)=\frac{1}{|W^{\lambda}|}\sum_{\sigma\in\mathfrak{S}_{\lambda}}\sigma(x)$.

###### Proof.

By Theorem 5.2, we have that

$\Delta_{\lambda}m_{\lambda}(x)=\sum_{\textbf{S}}\widetilde{m_{\nu^{\prime}}}\Delta_{\nu}(x).$

For $\nu^{\prime}$ different from $\lambda$, the element
$\widetilde{m_{\nu^{\prime}}}\Delta_{\nu}(x)$ is in
$\textbf{B}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)$.
Then

$\pi_{L}\Delta_{L}m_{L}(x)=\sum_{\mathfrak{S}_{\lambda}}\widetilde{m_{\nu^{\prime}}}\Delta_{\nu}(x)=\frac{1}{|W^{\lambda}|}\sum_{\mathfrak{S}_{\lambda}}\sigma(x).$

∎

######  Proposition 5.11.

Let $L$ and $E$ be proper Levi groups of $G$ such that $E\nsubseteq L$. Let
$\lambda$ and $\mu$ be the associated cocharacters to these Levi groups. The
composition

$\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda})_{b}\right)\xrightarrow{m_{L}}\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))\xrightarrow{\Delta_{E}}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\mu})_{b}\right)\xrightarrow{\pi_{E}}\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\mu})_{b}\right)$

is zero.

###### Proof.

By Theorem 5.2, we have that

$\Delta_{E}m_{L}(x)=\sum_{\textbf{S}}\widetilde{m_{\nu^{\prime}}}\Delta_{\nu}(x).$

If $E\nsubseteq L$, there is no $s\in\textbf{S}$ such that $\nu^{\prime}=\mu$,
and so the right hand side is in
$\textbf{B}\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}^{\mu})_{b})$. ∎

We now assume the statements in (21) for $L<G$.

######  Proposition 5.12.

There is a surjection

$\bigoplus_{L<G}\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)^{\mathfrak{S}_{L}}\twoheadrightarrow\textbf{B}\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X})).$

###### Proof.

The image of
$m_{L}:\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)\to\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))$
factors through the symmetrization map

$\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)\xrightarrow{\text{Sym}_{L}}\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)^{\mathfrak{S}_{L}}\xrightarrow{m_{L}}\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))$

by Proposition 5.5. The statement follows using (21) for $L<G$ and Proposition
5.6. ∎

######  Theorem 5.13.

There is a decomposition

$\left(\iota,\bigoplus_{L<G}m_{\lambda_{L}}\right):\textbf{P}\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))\oplus\bigoplus_{L<G}\textbf{P}\mathrm{K}_{\cdot}\left(\mathbb{D}(\mathcal{X}^{\lambda_{L}})_{b}\right)^{\mathfrak{S}_{L}}\xrightarrow{\sim}\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X})).$

###### Proof.

Using Propositions 5.10 and 5.11, we see that the map is an injection. To see
it is a surjection, let $x\in\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))$ and
assume that $\Phi_{H}(x)=0$ for all $L<H<G$ and $\pi_{L}(x)\neq 0$. By
Propositions 5.10 and 5.11, there is a constant $c$ such that

$y:=x-c\cdot m_{L}\Phi_{L}(x)\in\mathrm{K}_{\cdot}(\mathbb{D}(\mathcal{X}))$

satisfies $\Phi_{H}(y)=0$ for $L\leqslant H<G$. Repeating this process, we see
that the map is indeed surjective. ∎

We now prove Theorem 5.1 in the local case.

######  Corollary 5.14.

There is a decomposition
$\mathrm{K}_{\cdot}(\mathcal{X})=\textbf{P}\mathrm{K}_{\cdot}(\mathcal{X})\oplus\textbf{B}\mathrm{K}_{\cdot}(\mathcal{X})$.

###### Proof.

This follows from Theorem 3.2, Proposition 5.5, and Theorem 5.13. ∎

###  5.4. Compatibility of decompositions along étale maps

Let $e:\mathcal{X}^{\prime}\to\mathcal{X}$ be an étale map. Let $\mathcal{S}$
be a $\Theta$-stack of $\mathcal{X}$ with associated fixed stack
$\mathcal{Z}$. Let $\mathcal{S}^{\prime}$ be a $\Theta$-stack in
$\mathcal{X}^{\prime}$ contained in $e^{-1}(\mathcal{S})$, and let
$\mathcal{Z}^{\prime}$ be its associated fixed stack. Finally, let
$w\in\mathbb{Z}$. Then

$\displaystyle e^{*}:D^{b}(\mathcal{Z})_{w}\to
D^{b}(\mathcal{Z}^{\prime})_{w},\,e^{*}:D^{b}(\mathcal{S})\to
D^{b}(\mathcal{S}^{\prime}),$ $\displaystyle
e_{*}:D^{b}(\mathcal{Z}^{\prime})_{w}\to
D^{b}(\mathcal{Z})_{w},\,e^{*}:D^{b}(\mathcal{S}^{\prime})\to
D^{b}(\mathcal{S}).$

By the construction of the categories $\mathbb{D}$ from Section 3, we obtain
functors

$\displaystyle e^{*}:\mathbb{D}(\mathcal{X})$
$\displaystyle\to\mathbb{D}(\mathcal{Y}),$ $\displaystyle
e_{*}:\mathbb{D}(\mathcal{Y})$ $\displaystyle\to\mathbb{D}(\mathcal{X}).$

By the construction of the spaces $\textbf{P}\mathrm{K}$ and
$\textbf{B}\mathrm{K}$, we see that $e_{*}$ and $e^{*}$ respect these spaces
for $\mathcal{X}$ and $\mathcal{Y}$ quotient stacks of smooth affine varieties
by reductive groups.

###  5.5. Primitive K-theory: the global case

Let $\mathcal{X}$ be a symmetric stack satisfying Assumption B and let $X$ be
its good moduli space. Consider a direct system of étale covers $\mathcal{A}$
containing étale maps $\mathcal{Y}\to\mathcal{X}$ as in Theorem 2.1. Then, by
[42, Corollary 2.17]:

(22)
$\check{\mathrm{H}}^{p}\left(\mathcal{A},\mathrm{K}_{q}(-)\right)\Rightarrow\mathrm{K}_{q-p}(\mathcal{X}),$

where $\check{\mathrm{H}}$ denotes Čech cohomology and the spectral sequence
converges strongly. It is essential that we use rational K-theory to obtain
this statement. Any $\lambda:B\mathbb{G}_{m}\to\mathcal{X}$ induces a
cocharacter $\lambda$ in local charts $\mathcal{Y}$. Further, any local
attracting locus corresponds to a map $\lambda:B\mathbb{G}_{m}\to\mathcal{X}$
and thus determines an attracting stack. Denote by $\mathcal{X}^{\lambda}$ the
corresponding fixed stack. For $\lambda:B\mathbb{G}_{m}\to\mathcal{X}$, define
$\mathrm{K}_{\cdot}^{\lambda}(\mathcal{Y}):=\mathrm{K}_{\cdot}\left(\mathcal{Y}^{\lambda}\right)$.
Then

$\check{\mathrm{H}}^{p}\left(\mathcal{A},\mathrm{K}_{q}^{\lambda}(-)\right)\Rightarrow\mathrm{K}_{q-p}\left(\mathcal{X}^{\lambda}\right).$

Thus the following spectral sequence converges strongly:

(23)
$\check{\mathrm{H}}^{p}\left(\mathcal{A},\textbf{B}\mathrm{K}_{q}(-)\right)\Rightarrow\textbf{B}\mathrm{K}_{q-p}(\mathcal{X}).$

By Theorem 5.14, (22), and (23), we thus obtain that the following spectral
sequence converges strongly and define
$\textbf{P}\mathrm{K}_{\cdot}(\mathcal{X})$ such that

$\check{\mathrm{H}}^{p}\left(\mathcal{A},\textbf{P}\mathrm{K}_{q}(-)\right)\Rightarrow\textbf{P}\mathrm{K}_{q-p}(\mathcal{X}).$

###### Proof of Theorem 5.1.

The decomposition claimed in Theorem 5.1 follows from the construction of
$\textbf{P}\mathrm{K}_{\cdot}(\mathcal{X})$ and
$\textbf{B}\mathrm{K}_{\cdot}(\mathcal{X})$ and by Theorem 5.14. ∎

###  5.6. Categorification of intersection cohomology

#### 5.6.1.

The categories $D^{b}(\mathcal{X})$ have natural dg enhancements and the
admissible subcategories $\mathbb{D}(\mathcal{X})$ also have natural dg
enhancements . Recall the definitions from Subsection 2.5. The splitting

$\mathrm{K}_{0}(\mathbb{D}(\mathcal{X}))\to\textbf{P}\mathrm{K}_{0}(\mathbb{D}(\mathcal{X}))\to\mathrm{K}_{0}(\mathbb{D}(\mathcal{X}))$

induces an idempotent of
$e_{\mathcal{\mathcal{X}}}\in\text{Hom}_{\textbf{Hmo}_{0;\mathbb{Q}}}(\mathbb{D}(\mathcal{X}),\mathbb{D}(\mathcal{X}))$.
Indeed, all the functors used to construct the above splitting are constructed
from the functors for attracting and fixed stacks

$\displaystyle p_{*}$ $\displaystyle:D^{b}(\mathcal{S})\to D^{b}(\mathcal{X})$
$\displaystyle p^{*}$ $\displaystyle:D^{b}(\mathcal{X})\to D^{b}(\mathcal{S})$
$\displaystyle q^{*}_{w}$
$\displaystyle:D^{b}(\mathcal{Z})_{w}\xrightarrow{\sim}D^{b}(\mathcal{S})_{w},$

see [19, Amplification 3.18] for the last functor. Thus the functors used are
induced by Fourier–Mukai transforms in
$\text{rep}\left(D^{b}(\mathcal{S}),D^{b}(\mathcal{X})\right)$ and
$\text{rep}\left(D^{b}(\mathcal{Z}),D^{b}(\mathcal{X})\right)$, respectively.
We thus obtain a noncommutative motive:

$\mathbb{D}^{\text{nc}}(\mathcal{X}):=\left(\mathbb{D}(\mathcal{X}),e_{\mathcal{X}}\right)\in\textbf{Hmo}_{0;\mathbb{Q}}^{\natural}.$

Consider the Chern character

$\text{ch}:\mathrm{K}_{\cdot}(\mathcal{X})\to\widehat{\mathrm{H}^{\cdot}}(\mathcal{X}):=\prod_{j\in\mathbb{Z}}\mathrm{H}^{i+2j}(\mathcal{X}).$

We write $\text{gr}^{\cdot}K_{\cdot}(\mathcal{X})$ for the associated graded
with respect to the codimension filtration [18, Definition 3.7, Section 5.4].
By Theorems 4.1 and 5.1, we obtain:

######  Corollary 5.15.

There is an inclusion
$\text{gr}^{\cdot}\mathrm{K}_{\cdot}(\mathbb{D}^{\text{nc}}(\mathcal{X}))\subset\textbf{P}^{\leqslant
0}\mathrm{H}^{\cdot}(\mathcal{X}).$

If $\mathcal{X}$ is symmetric and satisfies Assumption C, then
$\textbf{P}^{\leqslant
0}\mathrm{H}^{\cdot}(\mathcal{X})\cong\mathrm{IH}^{\cdot}(X)$. In this
situation, we define the intersection K-theory of $X$:

$\mathrm{IK}_{\cdot}(X):=\mathrm{K}_{\cdot}(\mathbb{D}^{\text{nc}}(\mathcal{X})).$

There is a natural map

$\text{ch}:\mathrm{IK}_{\cdot}(X)\to\mathrm{IH}^{\cdot}(X)$

obtained using the splittings from Theorem 4.1 and Theorem 5.1:

$\mathrm{IK}_{\cdot}(X)\hookrightarrow\mathrm{K}_{\cdot}(\mathcal{X})\xrightarrow{\text{ch}}\widehat{\mathrm{H}^{\cdot}}(\mathcal{X})\twoheadrightarrow\mathrm{IH}^{\cdot}(X).$

Directly from the definition of $\mathbb{D}^{\text{nc}}(\mathcal{X})$,
intersection K-theory satisfies a version of Kirwan surjectivity [26, Theorem
2.5]:

$\mathrm{K}_{\cdot}(\mathcal{X})_{\mathbb{Q}}\twoheadrightarrow\mathrm{IK}_{\cdot}(X).$

#### 5.6.2.

Denote by $\mathrm{K}^{\text{top}}$ the Blanc topological K-theory of a dg
category [6]. Recall that for $\mathcal{X}$ a smooth stack,
$\mathrm{K}^{\text{top}}\left(D^{b}(\mathcal{X})\right)$ recovers the Atiyah-
Segal equivariant topological K-theory [22, Theorem 3.9]. For $j\in\\{0,1\\}$,
denote by $\text{gr}^{\cdot}\mathrm{K}_{j}^{\text{top}}(\mathcal{X})$ the
associated graded with respect to filtration $F^{\geqslant i}$ induced by
$\mathrm{H}^{\geqslant j+2i}(\mathcal{X})$ via

$\text{ch}:\mathrm{K}_{j}^{\text{top}}(\mathcal{X})\to\prod_{i\in\mathbb{Z}}\mathrm{H}^{j+2i}(\mathcal{X}).$

The idempotent $e$ induces a well-defined direct summand
$\mathrm{K}^{\text{top}}\left(\mathbb{D}^{\text{nc}}(\mathcal{X})\right)$ of
$\mathrm{K}^{\text{top}}\left(\mathbb{D}(\mathcal{X})\right)$. For
$j\in\\{0,1\\}$, we have that

$\text{gr}^{i}\mathrm{K}^{\text{top}}_{j}(\mathcal{X})\cong\mathrm{H}^{j+2i}(\mathcal{X},\mathbb{Q}).$

Indeed, it suffices to show the statement for quotient stacks, case in which
both sides can be computed using Totaro’s approximations $S_{n}^{o}$ of
$\mathcal{X}$ from the proof of Proposition 4.1. The isomorphism for a scheme
$S_{n}^{o}$ holds by the Atiyah-Hirzebruch theorem. By Theorems 4.1 and 5.1,
we have that
$\text{gr}^{i}\mathrm{K}^{\text{top}}_{j}\left(\mathbb{D}^{\text{nc}}(\mathcal{X})\right)\cong\mathrm{IH}^{j+2i}(X,\mathbb{Q})$
for $j\in\\{0,1\\}$. If the natural map

(24)
$\mathrm{K}^{\text{top}}_{j}\left(\mathbb{D}^{\text{nc}}(\mathcal{X})\right)\xrightarrow{\sim}\text{gr}^{i}\mathrm{K}^{\text{top}}_{j}\left(\mathbb{D}^{\text{nc}}(\mathcal{X})\right),$

is an isomorphism, then

(25)
$\mathrm{K}^{\text{top}}_{j}\left(\mathbb{D}^{\text{nc}}(\mathcal{X})\right)\cong\mathrm{IH}^{j+2i}(X,\mathbb{Q})$

for $j\in\\{0,1\\}$. We claim that if the Kirwan resolution $Y\to X$ is a
scheme, then (24) holds. Indeed, it suffices to check (24) for
$\mathbb{D}(\mathcal{X})$ instead of $\mathbb{D}^{\text{nc}}(\mathcal{X})$. It
suffices to check that the Chern character map for $\mathbb{D}(\mathcal{X})$
is injective. By Corollary 3.5, it suffices to check that the Chern character
map for $Y$ is injective. For a scheme $Y$, the Chern character is an
isomorphism by the Atiyah-Hirzebruch theorem.

Denote by $\mathrm{HP}$ the periodic cyclic homology of a dg category. The
idempotent $e$ induces a well-defined direct summand
$\mathrm{HP}_{\cdot}\left(\mathbb{D}^{\text{nc}}(\mathcal{X})\right)$ of
$\mathrm{HP}_{\cdot}\left(\mathbb{D}(\mathcal{X})\right)$. By [22, Theorem A]
and (25), we also obtain that if the Kirwan resolution $Y\to X$ is a scheme,
then

$\mathrm{HP}_{i}\left(\mathbb{D}^{\text{nc}}(\mathcal{X})\right)\cong\bigoplus_{j\in\mathbb{Z}}\mathrm{IH}^{i+2j}(X,\mathbb{C}).$

## References

  * [1] J. Alper.  Good moduli spaces for Artin stacks.  Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2349–-2402.
  * [2] J. Alper, J. Hall, D. Rydh.  A Luna étale slice theorem for algebraic stacks.  Ann. of Math. (2) 191 (2020), no. 3, 675–-738.
  * [3] J. Alper, D. Halpern-Leistner, J. Heinloth.  Existence of moduli spaces for algebraic stacks  http://arxiv.org/abs/1812.01128, 2018.
  * [4] M. Ballard, D. Favero, L. Katzarkov.  Variation of geometric invariant theory quotients and derived categories.  J. Reine Angew. Math. 746 (2019), 235–-303.
  * [5] A.A. Beĭlinson, J. Bernstein, P. Deligne.  Faisceaux pervers.  Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Astérisque, 100, Soc. Math. France, Paris, 1982.
  * [6] A. Blanc.  Topological K-theory of complex noncommutative spaces.  Compos. Math. 152 (2016), no. 3, 489–555.
  * [7] A. Bondal, D. Orlov.  Derived categories of coherent sheaves.  Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 47–-56, Higher Ed. Press, Beijing, 2002.
  * [8] A. Braverman, M. Finkelberg, H. Nakajima.  Instanton moduli spaces and $\mathcal{W}$-algebras.  Astérisque No. 385 (2016), vii+128 pp.
  * [9] M. A. de Cataldo, L. Migliorini.  The Chow motive of semismall resolutions.  Math. Res. Lett. 11 (2004), no. 2-3, 151–170.
  * [10] M. A. de Cataldo, L. Migliorini.  The decomposition theorem, perverse sheaves and the topology of algebraic maps.  Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 4, 535–633.
  * [11] S. Cautis.  Clasp technology to knot homology via the affine Grassmannian.  Math. Ann. 363 (2015), no. 3-4, 1053–1115.
  * [12] S. Cautis, J. Kamnitzer.  Quantum K-theoretic geometric Satake: the $SL_{n}$ case.  Compos. Math. 154 (2018), no. 2, 275–327.
  * [13] A. Corti, M. Hanamura.  Motivic decomposition and intersection Chow groups. I.  Duke Math. J. 103 (2000), no. 3, 459–522.
  * [14] A. Corti, M. Hanamura.  Motivic decomposition and intersection Chow groups. II.  Pure Appl. Math. Q. 3 (2007), no. 1, Special Issue: In honor of Robert D. MacPherson. Part 3, 181–203.
  * [15] B. Davison, S. Meinhardt.  Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras.  Invent. Math. 221 (2020), no. 3, 777–871.
  * [16] J. Eberhardt.  K-motives and Koszul Duality.  https://arxiv.org/pdf/1909.11151.pdf, 2019.
  * [17] D. Edidin, D. Rydh.  Canonical reduction of stabilizers for Artin stacks with good moduli spaces.  Duke Mathematical Journal, Volume 170, pp 827–880. https://arxiv.org/abs/1710.03220.
  * [18] H. Gillet.  K-theory and intersection theory.  Handbook of K-theory. Vol. 1, 2, 235–293, Springer, Berlin, 2005.
  * [19] D. Halpern-Leistner.  The derived category of a GIT quotient.  Journal of the American Mathematical Society 28 (3), 871–912.
  * [20] D. Halpern-Leistner.  Derived $\Theta$-stratifications and the D-equivalence conjecture.  https://arxiv.org/pdf/2010.01127v1.pdf.
  * [21] D. Halpern-Leistner, S. Sam.  Combinatorial constructions of derived equivalences.  Journal of the American Mathematical Society 33 (3), 735–773.
  * [22] D. Halpern-Leistner and D. Pomerleano.  Equivariant Hodge theory and noncommutative geometry.  Geom. Topol. 24 (2020), no. 5, 2361–2433.
  * [23] B. Iversen.  Cohomology of sheaves.  Universitext. Springer–Verlag, Berlin, 1986.
  * [24] Y. Laszlo, M. Olsson.  The six operations for sheaves on Artin stacks. I. Finite coefficients.  Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 109–168.
  * [25] Y. Laszlo, M. Olsson.  The six operations for sheaves on Artin stacks. II. Adic coefficients.  Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 169–210.
  * [26] F. Kirwan.  Rational intersection cohomology of quotient varieties.  Invent. Math. 86 (1986), no. 3, 471–505.
  * [27] F. Kirwan.  Rational intersection cohomology of quotient varieties. II.  Invent. Math. 90 (1987), no. 1, 153–167.
  * [28] M. Kontsevich, Y. Soibelman.  Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants.  Comm. Num. Th. and Phys., 5(2):231–252, 2011.
  * [29] A. Kuznetsov.  Lefschetz decompositions and categorical resolutions of singularities.  Selecta Math. (N.S.) 13 (2008), no. 4, 661–-696.
  * [30] A. Kuznetsov.  Semiorthogonal decompositions in algebraic geometry.  Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, 635–660.
  * [31] A. Kuznetsov.  Hochschild homology and semiorthogonal decompositions  https://arxiv.org/pdf/0904.4330.pdf.
  * [32] G. Lusztig.  Intersection cohomology methods in representation theory.  A plenary address presented at the International Congress of Mathematicians held in Kyoto, August 1990.
  * [33] S. Meinhardt, M. Reineke.  Donaldson-Thomas invariants versus intersection cohomology of quiver moduli.  J. Reine Angew. Math. 754 (2019), 143–178.
  * [34] T. Pădurariu.  Intersection K-theory.  https://arxiv.org/abs/2103.06223.
  * [35] T. Pădurariu.  Categorical and K-theoretic Hall algebras for quivers with potential.  https://arxiv.org/abs/2107.13642, to appear in J. Inst. Math. Jussieu.
  * [36] T. Pădurariu.  K-theoretic Hall algebras of quivers with potential as Hopf algebras.  https://arxiv.org/abs/2106.05169.
  * [37] Š. Špenko, M. Van den Bergh.  Non-commutative resolutions of quotient singularities for reductive groups.  Invent. Math. 210 (2017), no. 1, 3–67.
  * [38] Š. Špenko, M. Van den Bergh.  Semiorthogonal decompositions for GIT quotients.  Selecta Math. (N.S.) 27 (2021), no. 2, Paper No. 16, 43 pp.
  * [39] Š. Špenko, M. Van den Bergh.  Non-commutative crepant resolutions for some toric singularities. II.  J. Noncommut. Geom. 14 (2020), no. 1, 73–103.
  * [40] Š. Špenko, M. Van den Bergh.  Comparing the Kirwan and noncommutative resolutions of quotient singularities.  https://arxiv.org/pdf/1912.01689.pdf, 2019.
  * [41] G. Tabuada.  Noncommutative motives.  University Lecture Series, 63. American Mathematical Society, Providence, RI, 2015. x+114 pp.
  * [42] R.W. Thomason.  Algebraic K-theory and étale cohomology.  Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437–552.
  * [43] Y. Yang, G. Zhao.  The cohomological Hall algebra of a preprojective algebra.  Proc. Lond. Math. Soc. (3) 116 (2018), no. 5, 1029–1074.