It is an established practice to take old theorems about ordinary homology, and generalise them so as to obtain theorems about generalised homology theories.J. F. Adams, [2]

Let $G$ be a connected and simply-connected semisimple group over $\cc$, let $G_c$ be a maximal compact subgroup of $G(\cc)$, and let $T$ be a maximal torus. The derived geometric Satake equivalence of Bezrukavnikov-Finkelberg localizes to an equivalence between a full subcategory of $\Loc_{G_c}(\Omega G_c; \cc)$ and $\QCoh(\ld{\g}^{\reg}[2]/\ld{G})$, which can be thought of as a version of the geometric Satake equivalence “over the regular locus”. In this article, we study the story when $\Loc_{T_c}(\Omega G_c; \cc)$ is replaced by the $\infty$-category of $T$-equivariant local systems of $A$-modules over $\Gr_G(\cc)$, where $A$ is a complex-oriented even-periodic $\Eoo$-ring equipped with an oriented group scheme $\GG$. We show that upon rationalization, $\Loc_{T_c}(\Omega G_c; A)$, which was studied variously by Arkhipov-Bezrukavnikov-Ginzburg and Yun-Zhu when $A = \cc[\beta^{\pm 1}]$, can be described in terms of the spectral geometry of various Langlands-dual stacks associated to $A$ and $\GG$. For example, this implies that if $A$ is an elliptic cohomology theory with elliptic curve $E$, then $\Loc_{T_c}(\Omega G_c; A) \otimes \QQ$ can be described via the moduli stack of $\ld{B}$-bundles of degree $0$ on $E^\vee$.

§ INTRODUCTION

Let $G$ be a simply-connected semisimple algebraic group or a torus over $\cc$. Many deep results in geometric representation theory are concerned with describing the “topological”/A-side category of D-modules on algebraic (ind-)schemes associated to $G$ (such as the flag variety, the nilpotent cone, the affine Grassmannian, the affine flag variety, etc.) in terms of representation-theoretic/algebro-geometric B-side data associated to $\ld{G}$, the Langlands dual. These equivalences can be interpreted as refinements of the Fourier/Mellin transform. By the Riemann-Hilbert equivalence, the A-side category of D-modules on $X$ may be interpreted instead as categories of constructible sheaves of $\cc$-vector spaces on $X(\cc)$. The goal of this manuscript is to study analogues of some of these equivalences when we instead consider the category of constructible sheaves of $A$-module spectra on $X(\cc)$, where $A$ is a complex-oriented even-periodic $\Eoo$-ring (such as topological K-theory $\KU$, or an elliptic cohomology theory).

§.§ Summary of content

In this article, we take a few steps towards establishing a chromatic homotopy-theoretic analogue of the derived geometric Satake equivalence.

Let $B$ be a Borel subgroup of $G$.

Let $\cK$ denote $\cc\ls{t}$, and let $\co$ denote $\cc\pw{t}$.
The affine Grassmannian $\Gr_G$ is defined as the sheafification of the functor of points $\CAlg_\cc \ni R \mapsto G(R \otimes_\cc \cK)/G(R \otimes_\cc \co)$. It has the property that $\Gr_G(\cc)$ is homotopy equivalent to $\Omega G_c \simeq \Omega^2 BG_c$, where $G_c$ is a maximal compact subgroup of $G(\cc)$; see [64]. (Note that $G_c$ is homotopy equivalent to $G(\cc)$, so for most of the topological parts of this article, the distinction between them will be irrelevant.) The classical geometric Satake equivalence says:

The abelian category $\mathrm{Perv}_{G(\co)}(\Gr_G; \QQ)$ of $G(\co)$-equivariant perverse sheaves on $\Gr_G$ is equivalent to $\Rep(\ld{G}_\QQ)$, where $\ld{G}_\QQ$ is the Langlands dual group[This denotes the base-change to $\QQ$ of the Chevalley scheme over $\Z$, i.e., the split reductive group scheme whose root datum coincides with the root datum of $\ld{G}_\cc$.] over $\QQ$.

In [8], building on work of Ginzburg [36], Bezrukavnikov-Finkelberg proved a derived analogue of the geometric Satake equivalence:

There is an equivalence $\DMod_{G(\co)}(\Gr_G) \simeq \QCoh(\ld{\g}_\cc[2]/\ld{G}_\cc)$ of $\cc$-linear $\infty$-categories, where $\ld{\g}_\cc[2]$ is the derived $\cc$-scheme $\spec \Sym_\cc(\ld{\g}^\ast_\cc[-2])$.

The Bezrukavnikov-Finkelberg equivalence leads to a simpler equivalence on the level of local systems: $\Loc_{G_c}(\Omega G_c; \cc) \simeq \QCoh(\ld{\g}_\cc^\reg[2]/\ld{G}_\cc)$. This can be proved using <cit.> and <cit.>. This statement over the regular locus in fact plays a key role in proving the derived geometric Satake equivalence.

Our goal in this article (partly inspired by Adams' quote above, the work [47] of Hopkins-Kuhn-Ravenel corresponding to the diametric case of $G$ being a finite group, and the

discussion in [90] and <ref>) is to begin exploring the analogous story when $\cc$ is replaced by a generalized cohomology theory. Specifically, we will replace $\cc$ with an even-periodic $\Eoo$-ring equipped with specific additional data. The idea of considering other coefficient cohomology theories in the context of geometric representation theory is not new; see [38] for an early discussion of such ideas, as well as [19, 56, 93] for more recent work in this direction.

Part of the reason the derived contributions are vital to generalizing the geometric Satake equivalence is that when one considers sheaves with coefficients in a $2$-periodic $\Eoo$-ring (or any $\Eoo$-ring with nonzero homotopy in positive degrees), contributions from higher cohomology are circulated to degree $0$. For instance, the result of Bezrukavnikov-Finkelberg implies that $\Shv^c_{G(\co)}(\Gr_G(\cc); \cc[\beta^{\pm 1}]) \simeq \QCoh(\ld{\g}_\cc[2]/\ld{G}_\cc) \otimes_\cc \cc[\beta^{\pm 1}]$ where $|\beta|=2$; but this is in turn equivalent to $\QCoh(\ld{\g}_\cc/\ld{G}_\cc) \otimes_\cc \cc[\beta^{\pm 1}]$, which is not the $2$-periodification of $\Rep(\ld{G}_\cc)$. However, let us note that in the setting of relative geometric Langlands (as discussed in [85]), $2$-periodification is a rather destructive procedure: the particular shifts involved on the coherent side are extremely important, since they provide a geometric analogue of the point of evaluation of the Langlands dual L-function.

We will study a variant of a result of Arkhipov-Bezrukavnikov-Ginzburg (ABG) from [1], which is closely related to the geometric Satake equivalence. Namely, let $I = G(\co) \times_G B$ denote the Iwahori subgroup of $G(\co)$. Then:

There is an equivalence $\DMod_I(\Gr_G) \simeq \IndCoh((\tilde{\ld{\cN}} \times_{\ld{\g}} \{0\})/\ld{G})$, where $\tilde{\ld{\cN}} = T^\ast(\ld{G}/\ld{B})$ is the Springer resolution. This is in turn equivalent to $\QCoh(\tilde{\ld{\g}}_\cc[2]/\ld{G}_\cc)$ by Koszul duality, where $\tilde{\ld{\g}}_\cc[2] = \ld{G} \times^{\ld{B}} \ld{\fr{b}}[2]$ is a shifted analogue of the Grothendieck-Springer resolution.

As in <ref>, the ABG equivalence leads to a simpler equivalence on the level of local systems: $\Loc_{T_c}(\Omega G_c; \cc) \simeq \QCoh(\tilde{\ld{\g}}_\cc^\reg/\ld{G}_\cc)$. Upon $2$-periodification, we therefore see that $\Loc_{T_c}(\Omega G_c; \cc[\beta^{\pm 1}]) \simeq \QCoh(\tilde{\ld{\g}}_\cc^\reg/\ld{G}_\cc) \otimes_\cc \cc[\beta^{\pm 1}]$. Again, this statement over the regular locus in fact plays a key role in proving the ABG equivalence.

Note that pullback along the inclusion of a point into $\Gr_G(\cc)$ defines a symmetric monoidal functor $\Shv^c_I(\Gr_G(\cc); \cc[\beta^{\pm 1}]) \to \Shv^c_I(\ast; \cc[\beta^{\pm 1}])$, and there is an equivalence $\Loc_{T_c}(G_c; \cc[\beta^{\pm 1}]) \simeq \End_{\Loc_{T_c}(\Omega G_c; \cc[\beta^{\pm 1}])}(\Loc_{T_c}(\ast; \cc[\beta^{\pm 1}]))$. Using the ABG theorem, one can prove an equivalence

\begin{equation}\label{abg-consequence}
\Loc_{T_c}(G_c; \cc[\beta^{\pm 1}]) \simeq \QCoh(\ld{\fr{t}} \times_{\tilde{\ld{\g}}/\ld{G}} \ld{\fr{t}}) \otimes_\cc \cc[\beta^{\pm 1}],
\end{equation}

where the map $\ld{\fr{t}} \to \tilde{\ld{\g}}/\ld{G}$ is given by the Kostant slice, and $T_c$ acts on $G_c$ by conjugation.

The goal of this article is to study a generalization of <ref> and <ref>.
Fix a complex-oriented even-periodic $\Eoo$-ring $A$, and let $\GG$ be an oriented group scheme in the sense of [62]. If $T$ is a torus and $X$ is a sufficiently nice $T$-space, one can define a $\pi_0 A$-linear $\infty$-category $\Loc_{T}^\gr(X; A)$ of “genuine $T$-equivariant $\Mod_A$-valued local systems[This is meant in the fully $\infty$-categorical sense, so it depends on the entire homotopy type (i.e., the entire fundamental $\infty$-groupoid) of $X$, and not just on the fundamental groupoid of $X$. For instance, in the nonequivariant setting, $\Loc(X; A)$ is the $\infty$-category $\Fun(X, \Mod_A)$ where $X$ is viewed as a Kan complex.] on $X$”; see <ref> and <ref>.
Let $\cM_T$ denote the Hom-stack $\Hom(\bX^\bull(T), \GG)$, and let $\cM_{T,0}$ denote its underlying stack over $\pi_0 A$. For instance, if $\GG_0$ is an elliptic curve, $\cM_{T,0}$ can be identified with the moduli stack (scheme) of $T$-bundles on $E$ of degree $0$ equipped with a trivialization at the zero section.
Let $\GG_0^\vee$ denote the group scheme $\Hom(\GG_0, B\GG_m)$ (this is a slight variant of the construction studied in [67]).

Then, one of our main results is the following; we will unwind the statement in special cases below.

[See <ref> for a precise statement]
Suppose that $G$ is a connected and simply-connected semisimple algebraic group or a torus over $\cc$, and let $T$ act on $G$ by conjugation. Let $G_c$ denote the maximal compact subgroup of $G(\cc)$, and fix a principal nilpotent element of $\ld{\fr{n}}$. Fix a complex-oriented even-periodic $\Eoo$-ring $A$, and let $\GG$ be an oriented group scheme in the sense of [62].
Assume that the underlying $\pi_0 A$-scheme $\GG_0$ is $\GG_a$, $\GG_m$, or an elliptic curve $E$.
Let $\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)^\reg$ denote the moduli stack of regular $\ld{B}$-bundles of degree zero on $\GG_{0,\QQ}^\vee$. Then, there is an $\E{2}$-monoidal equivalence of $\pi_0 A_\QQ$-linear $\infty$-categories

$$\Loc_{T_c}^\gr(\Omega G_c; A) \otimes \QQ \simeq \QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)^\reg).$$

We view the above result as a first step towards describing $\Shv^c_{T_c}(\Omega G_c; A) \otimes \QQ$ in a manner analogous to [1]. We hope to complete this description in a sequel to this article, and further use the above result to revisit (the $2$-periodification of) the ABG equivalence. The basic point in the proof of <ref> is the computation of the $T_c$-equivariant $A$-homology $\pi_0 C^{T_c}_\ast(\Omega G_c; A)$ in terms of the Langlands dual $\ld{G}$. It is likely that the rationalization in <ref> is unnecessary, but we have not attempted to verify this.

Essentially the same argument shows that there is an $\E{2}$-monoidal equivalence of $\pi_0 A_\QQ$-linear $\infty$-categories

$$\Loc_{G_c}^\gr(\Omega G_c; A) \otimes \QQ \simeq \QCoh(\Bun_{\ld{G}}^{0,\ss}(\GG_{0,\QQ}^\vee)^\reg),$$

where $\Bun_{\ld{G}}^{0,\ss}(\GG_{0,\QQ}^\vee)^\reg$ denotes the moduli stack of regular semistable $\ld{G}$-bundles of degree zero. For simplicity, we will only focus on $T_c$-equivariant local systems.

When $G$ is a torus, it is easy to establish an analogue of the geometric Satake equivalence, even before rationalization: if $T$ is a torus over $\cc$, let $\ld{T}_A := \spec A[\bX_\ast(T)]$ denote the dual torus over $A$. Then there is an $\E{2}$-monoidal $A$-linear equivalence $\Loc_T(\Gr_T(\cc); A) \simeq \QCoh(\cL_\GG B\ld{T}_A)$; see <ref>. One can also “quantize” by considering loop-rotation equivariance, which results in a $\GG$-analogue of the algebra of differential operators on $\ld{T}$; see <ref> for more. In <ref>, we discuss the story for a torus where $A$ is replaced by the sphere spectrum $S$ — already in this case, homotopy-theoretic considerations prevent one from describing $\Loc_T(\Gr_T(\cc); S)$ in terms of the algebraic geometry of some spectral stack over the sphere spectrum.

The reason that the left-hand side of <ref> is not merely $\Loc_{T_c}(\Omega G_c; \QQ) \otimes_\QQ A_\QQ$ (which could then be described by <ref>) is that the rationalization of equivariant $A$-(co)homology is essentially never isomorphic to equivariant $A \otimes \QQ$-(co)homology. This is the key reason for why <ref> is not a consequence of the results of Arkhipov-Bezrukavnikov-Ginzburg. This perspective also features in [18]. For example, if $X$ is a finite CW-complex equipped with an action of a group $H$, then $\KU^\ast(X) \otimes \QQ \cong \H^\ast(X; \QQ) \otimes_\QQ \QQ[\beta^{\pm 1}]$, but $\KU^\ast_H(X) \otimes \QQ$ is generally not isomorphic to $\H^\ast_H(X; \QQ) \otimes_\QQ \QQ[\beta^{\pm 1}]$. Indeed, they already differ if $X$ is a point: in this case, $\KU^\ast_H(X) \otimes \QQ$ is the rationalization of the representation ring of $H$, which is generally not isomorphic to $\H^\ast_H(X; \QQ) \otimes_\QQ \QQ[\beta^{\pm 1}]$ (for instance, if $H$ is finite, the latter is $\QQ[\beta^{\pm 1}]$).

<ref> is closely related to the following instantiation of Langlands duality:

In the above setup (so that the underlying $\pi_0 A$-scheme $\GG_0$ is $\GG_a$, $\GG_m$, or an elliptic curve $E$), there is a “$\GG$-Kostant slice” $\kappa: (\cM_{\ld{T},0})_\QQ \to \Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)^\reg$ over $\pi_0 A_\QQ$ such that there is an equivalence of $\pi_0 A_\QQ$-linear $\infty$-categories:

$$\Loc_{T_c}^\gr(G_c; A) \otimes \QQ \simeq \QCoh((\cM_{\ld{T},0})_\QQ \times_{\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)} (\cM_{\ld{T},0})_\QQ).$$

Here, $T_c$ acts on $G_c$ by conjugation.

Let $K_0(\Rep(G_c))$ denote the (complex) representation ring of $G_c$. In [17], Brylinski and Zhang proved that there is an isomorphism $\KU_{G_c}^\ast(G_c) \cong \Omega^\ast_{K_0(\Rep(G_c))/\Z} \otimes_\Z \Z[\beta^{\pm 1}]$.
When $A = \KU$, one can use the Hochschild-Kostant-Rosenberg theorem to view the variant of <ref> for $\Loc_{G_c}^\gr(G_c; \KU) \otimes \QQ$ as a categorification of the Brylinski-Zhang isomorphism. See <ref> for further discussion. In <ref>, we also use Hochschild homology to describe a generalization of the $\hbar=0$ case of <cit.>, which computes the equivariant cohomology of $\Omega G_c$.

Motivated by <cit.>, one can heuristically interpret <ref> as describing a version of mirror symmetry for the wrapped Fukaya category of the symplectic orbifold $T^\ast(G_c/_\ad T_c)$, albeit with coefficients in the complex-oriented even-periodic $\Eoo$-ring $A$.

Let us discuss <ref> individually for each case $\GG_0 = \GG_a, \GG_m$, and an elliptic curve.

* When $A = \QQ[\beta^{\pm 1}]$, <ref> describes an equivalence between $\Loc_{T_c}^\gr(\Omega G_c; \QQ[\beta^{\pm 1})$ and $\QCoh(\tilde{\ld{\g}}^\reg/\ld{G})$. This is a rather formal consequence of the following observation proved in <ref>:

There is a “Kostant section” $\kappa: \ld{\fr{t}} \to \tilde{\ld{\g}}/\ld{G}$ and a Cartesian square

\spec \H^T_0(\Gr_G(\cc); \QQ[\beta^{\pm 1}]) \cong (T^\ast \ld{T})^\bl \ar[r] \ar[d] & \fr{t} \cong \ld{\fr{t}} \ar[d]^-\kappa\\
\ld{\fr{t}} \ar[r]_-\kappa & \tilde{\ld{\g}}/\ld{G},

where $(T^\ast \ld{T})^\bl$ is a particular affine blowup of $T^\ast \ld{T} \cong \ld{T} \times \fr{t}$.

This can be viewed as an analogue of <cit.> and <cit.>, and it can be used to reprove the rationalization of <cit.>. There is an isomorphism $\tilde{\ld{\g}}/\ld{G} \cong \ld{\fr{b}}/\ld{B}$, and in characteristic zero, this can be identified with $\Bun_{\ld{B}}^0(B\GG_a)$, viewed as the shifted tangent bundle of $B\ld{B}$. Moreover, there is an isomorphism $\spec \H^T_0(\Gr_G(\cc); \QQ[\beta^{\pm 1}]) \cong (T^\ast \ld{T})^\bl$, and $(T^\ast \ld{T})^\bl$ admits a $W$-action (via the $W$-action on $\ld{T}$ and $T^\ast_{\{1\}} \ld{T} \cong \fr{t}$) such that $(T^\ast \ld{T})^\bl\mmod W \cong \spec \H^G_0(\Gr_G(\cc); \QQ)$ is isomorphic to the group scheme of regular centralizers in $\ld{\fr{g}}$. See [9] for further discussion.

In this case, <ref> says that if $T$ acts on $G$ by conjugation, then there is an equivalence

$$\Loc_{T_c}^\gr(G_c; \QQ[\beta^{\pm 1}]) \simeq \QCoh(\ld{\fr{t}}_\QQ \times_{\tilde{\ld{\g}}_\QQ/\ld{G}_\QQ} \ld{\fr{t}}_\QQ).$$

Similarly, if $G$ acts on itself by conjugation, one obtains an equivalence

$$\Loc_{G_c}^\gr(G_c; \QQ[\beta^{\pm 1}]) \simeq \QCoh(\ld{\fr{t}}_\QQ\mmod W \times_{\ld{\g}_\QQ/\ld{G}_\QQ} \ld{\fr{t}}_\QQ\mmod W).$$

These equivalences can be de-periodified (<ref>).
Motivated by <cit.>, these equivalences suggest viewing $\ld{\fr{t}} \times_{\tilde{\ld{\g}}/\ld{G}} \ld{\fr{t}}$ (resp. $\ld{\fr{t}}\mmod W \times_{\ld{\g}/\ld{G}} \ld{\fr{t}}\mmod W$) as a (derived) mirror to the symplectic orbifold $T^\ast(G_c/_\ad T_c)$ (resp. $T^\ast(G_c/_\ad G_c)$). Concretely, these results show that if $f$ is a regular nilpotent element of $\ld{\g}$ and $Z_f(\ld{B})$ is its centralizer in $\ld{B}$, then there is an equivalence

$$\Loc^\gr(G_c; \QQ[\beta^{\pm 1}]) \simeq \QCoh(Z_f(\ld{B}_\QQ));$$

therefore, $Z_f(\ld{B}_\QQ)$ is a mirror to $G(\cc) = T^\ast(G_c)$ viewed as a symplectic manifold. These results are not new, and can easily be deduced from the work of Bezrukavnikov-Finkelberg [8] and Yun-Zhu [92]. Notice that if $G_c = T_c$, then we are simply stating that there is an equivalence $\Loc(T_c; \QQ[\beta^{\pm 1}]) \simeq \QCoh(\ld{T})$, given by taking monodromy.

Upon adding loop rotation equivariance, there is an equivalence between $\Loc_{T_c \times S^1_\rot}^\gr(\Omega G_c; \cc)$ and a particular localization of the universal category $\ld{\co}^\univ = U_\hbar(\ld{\g})\modc^{\ld{N}, (\ld{T},w)}$ from <cit.>; this is a consequence of <ref> and <ref>.

See <ref> for an explicit description of $\H^{G\times S^1_\rot}_\ast(\Gr_G(\cc); \cc)$ when $G = \SL_2$. From the homotopical perspective, the action of $S^1$ by loop rotation on $\Gr_G(\cc)$ arises by viewing $\Gr_G(\cc) \simeq \Omega^\lambda BG(\cc)$, where $\lambda$ is the $2$-dimensional rotation representation of $S^1$; in other words, $\Gr_G(\cc)$ admits the structure of a framed $\E{2}$-algebra, and the action of $S^1$ is via change-of-framing.

* When $A = \KU$, <ref> describes an equivalence between $\Loc_{T_c}^\gr(\Omega G_c; \KU) \otimes \QQ$ and $\QCoh(\tilde{\ld{G}}^\reg_\QQ/\ld{G}_\QQ)$, where $\tilde{\ld{G}}_\QQ^\reg/\ld{G}_\QQ$ is the regular locus in the stacky quotient of the multiplicative Grothendieck-Springer resolution $\tilde{\ld{G}}_\QQ = \ld{G}_\QQ \times_{\ld{B}_\QQ} \ld{B}_\QQ$. As above, this is a rather formal consequence of the following observation, which is a multiplicative analogue of <cit.> and <cit.>:

There is a “Kostant section” $\kappa: \ld{T} \to \tilde{\ld{G}}/\ld{G}$ and a Cartesian square

\spec \pi_0 C^T_\ast(\Gr_G(\cc); \KU) \otimes \QQ \cong (\ld{T} \times T)^\bl \ar[r] \ar[d] & \ld{T} \ar[d]^-\kappa\\
\ld{T} \ar[r]_-\kappa & \tilde{\ld{G}}/\ld{G} \simeq \Bun_{\ld{B}}^0(S^1),

where $(\ld{T} \times T)^\bl$ is a particular affine blowup of $\ld{T} \times T$. Moreover, there is an isomorphism $\spec \pi_0 C^T_\ast(\Gr_G(\cc); \KU) \otimes \QQ$ and $(\ld{T}\times T)^\bl$.
There is also a $W$-action on $(\ld{T} \times T)^\bl$ (by the $W$-action on $T$ and $\ld{T}$) such that $(\ld{T} \times T)^\bl\mmod W \cong \spec \pi_0 C^G_\ast(\Gr_G(\cc); \KU) \otimes \QQ$ is isomorphic to the group scheme of regular centralizers in $\ld{G}$. Again, see [9] for further discussion.

In this case, <ref> says that if $T$ acts on $G$ by conjugation, then there is an equivalence

$$\Loc_{T_c}^\gr(G_c; \KU) \otimes \QQ \simeq \QCoh(\ld{T}_\QQ \times_{\tilde{\ld{G}}_\QQ/\ld{G}_\QQ} \ld{T}_\QQ).$$

Similarly, if $G$ acts on itself by conjugation, one obtains an equivalence

$$\Loc_{G_c}^\gr(G_c; \KU) \otimes \QQ \simeq \QCoh(\ld{T}_\QQ\mmod W \times_{\ld{G}_\QQ/\ld{G}_\QQ} \ld{T}_\QQ\mmod W).$$

If $\{f\}$ is a regular unipotent element of $\ld{G}_\QQ$ (determined by the image of the origin in $\ld{T}_\QQ\mmod W$ under the multiplicative Kostant slice), and $Z_f^\mu(\ld{B}_\QQ)$ is the centralizer of $f\in \ld{G}_\QQ$, then the preceding equivalence in turn implies an equivalence

$$\Loc^\gr(G_c; \KU) \otimes \QQ \simeq \QCoh(Z_f^\mu(\ld{B}_\QQ)).$$

Therefore, $Z_f^\mu(\ld{B}_\QQ)$ can be viewed as a $\KU$-theoretic mirror to $G(\cc) = T^\ast(G_c)$ viewed as a symplectic manifold.
The main input into these results are not new, and can be deduced from the work of Bezrukavnikov-Finkelberg-Mirkovic [9]. Notice that if $G_c = T_c$, then we are simply stating that there is an equivalence $\Loc(T_c; \KU) \simeq \QCoh(\ld{T}_\KU)$, given by taking monodromy.

We expect (see <ref> for a more precise statement) that upon adding loop rotation equivariance, there is an equivalence between $\Loc_{T_c \times S^1_\rot}^\gr(\Omega G_c; \KU) \otimes \QQ$ and a particular localization of the quantum universal category $\ld{\co}^\univ_q$ from <cit.>. Using the calculations in this article, this expected equivalence reduces to proving an analogue of <cit.> for the quantum group and the multiplicative nil-Hecke algebra; such a conjecture also appears as <cit.>.

We also expect (see <ref>) that there is an equivalence between $\Loc_{T_c \times \mu_{p,\rot}}^\gr(\Omega G_c; \KU)[\frac{1}{q-1}]$ and a particular localization of $\ld{\co}^\univ_{\zeta_p}$, i.e., the quantum universal category $\ld{\co}$ at a primitive $p$th root of unity.

The reader is referred to <ref> for an explicit description of $\pi_0 C^{G\times S^1_\rot}_\ast(\Gr_G(\cc); \KU) \otimes \QQ$ when $G = \SL_2$.

* Suppose $A$ is a complex-oriented even-periodic $\Eoo$-ring and $\GG$ is an oriented elliptic curve over $A$ (in the sense of [62]). Let $E = \GG_0$ be the underlying classical scheme of $\GG$ over the classical ring $\pi_0(A)$, so that $E$ is an elliptic curve, and let $E^\vee$ be the dual elliptic curve. The Cartesian squares from (a) and (b) above can be generalized to this setting (see <ref>). For simplicity, let us explain this in the case $G = \SL_2$, i.e., $\ld{G} = \PGL_2$.

Then, there is a “Kostant section” $\kappa: E = \Pic^0(E^\vee) \to \Bun_{\ld{B}}^0(E^\vee)$ which sends a line bundle $\cL$ to the trivial extension $\co_{E^\vee} \subseteq \co_{E^\vee} \oplus \cL$ if $\cL \not\cong \co_{E^\vee}$, and to the Atiyah extension $\co_{E^\vee}\subseteq \cf_2 \twoheadrightarrow \co_{E^\vee}$ from [4] if $\cL$ is trivial.
Note that by construction, the $\ld{G}$-bundle underlying $\kappa(\cL)$ is semistable of degree $0$. Moreover, <ref> says that there is a Cartesian square

(\GG_m \times E)^\bl \ar[r] \ar[d] & E \ar[d]^-\kappa\\
E \ar[r]_-\kappa & \Bun_{\ld{B}}^0(E^\vee),

where $(\GG_m \times E)^\bl$ is a particular affine blowup of $\GG_m \times E$. [The desired affine blowup $(\GG_m \times E)^\bl$ is obtained by blowing up $\GG_m\times E$ at the locus cut out by the zero sections of $\GG_m$ and $E$, and deleting the proper preimage of the zero section of $E$; see also <cit.>.]

Notice that $\GG_m\times E$ admits an action of $ W = \Z/2$, via inversion on $\GG_m$ and $E$; this extends to an action of $\Z/2$ on $(\GG_m\times E)^\bl$, and the above diagram suggests viewing $(\GG_m\times E)^\bl\mmod (\Z/2)$ as an elliptic analogue of the group scheme of regular centralizers.

Furthermore, there is an isomorphism

$$\Gamma((\GG_m \times E)^\bl; \co_{(\GG_m \times E)^\bl}) \cong \pi_0 C^T_\ast(\Gr_G(\cc); A) \otimes \QQ$$

between the coherent cohomology of $(\GG_m \times E)^\bl$ and the rationalization of the $T$-equivariant $A$-homology of $\Gr_G(\cc)$. Using this, <ref> shows that there is an equivalence between a variant of $\Loc_{T_c}^\gr(\Omega G_c; A) \otimes \QQ$ and an explicit full subcategory of $\QCoh(\Bun_{\ld{B}}^0(E^\vee))$.

In this case, <ref> says that if $T$ acts on $G$ by conjugation, then there is an equivalence

$$\Loc_{T_c}^\gr(G_c; A) \otimes \QQ \simeq \QCoh(E \times_{\Bun_{\ld{B}}^0(E^\vee)} E) \otimes_{\pi_0 A} \pi_0 A_\QQ.$$

If $\{\co_{E^\vee}\subseteq \cf_2\}\in \Bun_{\ld{B}}^0(E^\vee)$ denotes the Atiyah bundle, then let $Z_f^E(\ld{B}) := (\{\co_{E^\vee}\subseteq \cf_2\} \times_{\Bun_{\ld{B}}^0(E^\vee)} E)$ be the “centralizer in $\ld{B}$ of the regular `elli-potent' element $\{\co_{E^\vee}\subseteq \cf_2\}\in \Bun_{\ld{B}}^0(E^\vee)$”. There is then an equivalence

$$\Loc^\gr(G_c; A) \otimes \QQ \simeq \QCoh(Z_f^E(\ld{B})) \otimes_{\pi_0 A} \pi_0 A_\QQ.$$

Therefore, $Z_f^E(\ld{B})$ can be viewed as an $A$-theoretic mirror to $G(\cc) = T^\ast(G_c)$ viewed as a symplectic manifold.

One might hope that these results hold without rationalization, but we do not know how to prove such a statement. In the case of $\KU$, for instance, the key obstruction is that we do not know whether the $2$-periodification of $\tilde{\ld{G}}^\reg_\QQ/\ld{G}_\QQ$ can be lifted to a flat stack $(\tilde{\ld{G}}^\reg/\ld{G})_\KU$ over $\KU$. If it does lift, then it seems reasonable to expect a $\KU$-linear equivalence of the form $\Loc_{T_c}(\Omega G_c; A) \simeq \QCoh((\tilde{\ld{G}}^\reg/\ld{G})_\KU)$.

In <ref>, we discuss some motivation for this article stemming from the Coulomb branches of $3$d $\cN=4$, $4$d $\cN=2$, and $5$d $\cN=1$ pure gauge theories (i.e., no matter). We also give explicit generators and relations for the Coulomb branches of $3$d $\cN=4$ and $4$d $\cN=2$ pure gauge theories with gauge group $\SL_2$ (i.e., $\pi_0 C_\ast^G(\Gr_G(\cc); \QQ)$ and $\pi_0 C_\ast^G(\Gr_G(\cc); \KU)$ with $G = \SL_2$). The $4$-dimensional case is a $q$-analogue of the quantization of the Atiyah-Hitchin manifold from <cit.>.

We will use the following notation throughout; furthermore, the reader should keep in mind that everything in this article will be derived, unless explicitly mentioned otherwise.

Let $G$ be a connected (often simply-connected) semisimple group over $\cc$ (or a torus). Fix a maximal torus $T\subseteq B$ contained in a Borel subgroup of $G$. Let $U = [B,B]$ denote the unipotent radical of $B$, so that $B/U \cong T$. Let $\Phi$ be the set of roots of $G$, $\Phi^+$ the set of positive roots, and $\Delta$ a set of simple roots. Let $W$ be the Weyl group; if $w\in W$, let $\dot{w}\in N_G(T)$ denote a lift of $w$ to the normalizer of $T$ in $G$. Let

$\Lambda$ denote the weight lattice, and $\Lambda^+ = \Lambda^\pos$ the set of dominant weights.
We will also follow other standard notation in homotopy theory: for instance, $\Top$ will denote the $\infty$-category of spaces, and $\Sp$ will denote the $\infty$-category of spectra.

There has been some work done previously towards analogues of the geometric Satake equivalence with other coefficients. For instance, when $A = \KU$, a conjecture was proposed in [19]; in a similar vein, a discussion of the case $A = \KU$ is the content of the talk [56].
In [93], Yang and Zhao study a higher chromatic analogue of quantum groups, and it would be interesting to study the relationship between the present article and their work. After this paper was written, the preprint [94] was posted on the arXiv; it is concerned with ideas similar to the ones studied here.
Our work is closely related to the exciting program of Ben-Zvi–Sakellaridis–Venkatesh (see [85, 13] for an overview); we hope to describe this relationship in future work.

§.§ Acknowledgements

I'd like to acknowledge Lin Chen, Charles Fu, Tom Gannon, and Kevin Lin for helpful conversations and for entertaining my numerous silly questions. I'm also grateful to Victor Ginzburg for a very enlightening discussion, and Pavel Safronov for a useful email.
Thanks to Ben Gammage for discussions which helped shape my understanding of some of the topics in <ref>, and to Hiraku Nakajima for a very informative email exchange on the same topic.
Part of this work started after I took a class taught by Roman Bezrukavnikov; I'm very grateful to him for introducing me to [9], which led me down the beautiful road to geometric representation theory.
Last, but certainly far from least, the influence, support, advice, and encouragement of my advisors Dennis Gaitsgory and Mike Hopkins is evident throughout this project; I cannot thank them enough.

§ HOMOTOPY THEORY BACKGROUND

§.§ Review of generalized equivariant cohomology

We review the construction of generalized equivariant cohomology via spectral algebraic geometry from [58], in a form suitable for our applications. This review will necessarily be brief, since a detailed exposition may be found in loc. cit.; there is also some discussion in the early sections of [38] in the setting of ordinary (as opposed to spectral) algebraic geometry.

Fix an $\Eoo$-ring $A$ and a commutative $A$-group $\GG$, so $\GG$ defines a functor $\CAlg_A \to \Mod_{\Z,\geq 0}$ which is representable by a flat $A$-algebra. We will write $\GG_0$ to denote the resulting commutative group scheme over $\pi_0 A$.

The equivalence $\Omega^\infty: \Sp_{\geq 0} \xar{\sim} \CAlg(\Top_\ast)$ extends to an equivalence between $\Mod_{\Z,\geq 0}$ and topological abelian groups. More precisely, by the Dold-Kan correspondence and the Schwede-Shipley theorem, there are equivalences of categories

$$\Mod^{\geq 0}_\Z \simeq \mathrm{Ch}_{\geq 0}(\Z) \simeq \Fun(\Deltab^{op},\Ab) = s\Ab.$$

The image of $\Mod^{\geq 0}_\Z$ under the equivalence $\Omega^\infty: \Sp_{\geq 0} \xar{\sim} \CAlg(\Top_\ast)$ can be characterized as follows.
Let us model grouplike infinite loop spaces $X$ as functors $X:\Fin_\ast\to \Top$ such that $\pi_0 \Map_\Top(Y,X)$ is an abelian group for all spaces $Y$ (i.e., $X$ is grouplike) and such that the map $X([n])\to X([1])^n$ is an equivalence. Such an object should be in the image of $\Mod^{\geq 0}_\Z$ iff it is “strictly commutative”. One way to make this precise is as follows. Let $\mathrm{Lattice}$ denote the full subcategory of the category of abelian groups spanned by the groups $\Z^n$ with $n\geq 0$, so there is a functor $\Fin_\ast\to \mathrm{Lattice}$. Then an infinite loop space is in the image of $\Mod^{\geq 0}_\Z$ if and only if the functor $\Fin_\ast\to \Top$ classifying it factors through a finite-product-preserving functor $\mathrm{Lattice} \to \Top$. In other words, $\Mod^{\geq 0}_\Z$ is equivalent to the full subcategory spanned by the grouplike objects in the category $\Fun^\pi(\mathrm{Lattice}, \Top)$. This is a very strong condition to impose on an infinite loop space: it forces the infinite loop space to decompose as a product of Eilenberg-Maclane spaces. For example, $\CP^\infty$ admits such a factorization, but $\BU$ (with either the additive or multiplicative infinite loop space structure) does not.

A preorientation of $\GG$ is a pointed map $S^2 \to \Omega^\infty \GG(A)$ of spaces, i.e., a map $\Sigma^2 \Z \to \GG(A)$ of $\Z$-modules (by adjunction). This induces a map $\CP^\infty = \Omega^\infty \Sigma^2 \Z \to \Omega^\infty \GG(A)$ of topological abelian groups, and hence a map $\spf A^{\CP^\infty} \to \GG$ of $\Eoo$-$A$-group schemes. (Note that $\spf A^{\CP^\infty}$ need not admit the structure of a commutative $A$-group scheme: for instance, $A^{\CP^\infty}$ need not be flat over $A$.)

Given a preorientation $S^2 \to \Omega^\infty \GG(A)$, we obtain a map $\co_\GG \to C^\ast(S^2; A)$ of $\Eoo$-$A$-algebras. On $\pi_0$, this induces a map $\pi_0 \co_\GG = \co_{\GG_0} \to \pi_0 C^\ast(S^2; A)$. However, the target can be identified with the trivial square-zero extension $\pi_0 A \oplus \pi_{-2} A$, so that the preorientation defines a derivation $\co_{\GG_0} \to \pi_{-2} A$. This defines a map $\beta: \omega = \Omega^1_{\GG_0/\pi_0 A} \to \pi_{-2} A$. The preorientation is called an orientation if $\GG_0$ is smooth of relative dimension $1$ over $\pi_0 A$, and the composite

$$\pi_n(A) \otimes_{\pi_0 A} \omega \to \pi_n(A) \otimes_{\pi_0 A} \pi_{-2} A \xar{\beta} \pi_{n-2} A$$

is an isomorphism for each $n\in \Z$. This forces $A$ to be $2$-periodic (but does not force its homotopy to be concentrated in even degrees).

As discussed in <cit.>, the universal $\Eoo$-$\Z$-algebra over which the additive group scheme $\GG_a$ admits an orientation is given by $\Z[\CP^\infty][\tfrac{1}{\beta}] = \QQ[\beta^{\pm 1}]$. Therefore, we are allowed to let $\GG = \GG_a$ in the story below only when $A$ is a $2$-periodic $\Eoo$-$\QQ$-algebra. (If $A$ is not an $\Eoo$-$\Z$-algebra, one cannot in general define $\GG_a = \spec A[t]$ as a commutative $A$-group: the coproduct $A[t] \to A[x,y]$ will in general not be a map of $\Eoo$-$A$-algebras.)

We can now review the definition of $T$-equivariant $A$-cohomology when $T$ is a torus.

Fix an $\Eoo$-ring $A$ as above and a commutative $A$-group $\GG$. Given a compact abelian Lie group $T$, define an $A$-scheme $\cM_T$ by the mapping stack $\Hom(\bX^\ast, \GG)$. We will be particularly interested in the case when $T$ is a torus. Let $\cT$ be the full subcategory of $\Top$ spanned by those spaces which are homotopy equivalent to $BT$ with $T$ being a compact abelian Lie group. By arguing as in <cit.>, a preorientation of $\GG$ is equivalent to the data of a functor $\cM: \cT \to \Aff_A$ along with compatible equivalences $\cM(BT) \simeq \cM_T$. The $\Eoo$-$A$-algebra $\co_{\cM_T}$ is the $T$-equivariant $A$-cochains of a point, and will occasionally be denoted by $A_T$.

We can now sketch the construction of the $T$-equivariant $A$-cochains of more general $T$-spaces; see <cit.>. Let $T$ be a torus over $\cc$ for the remainder of this discussion, and let $\GG$ be an oriented commutative $A$-group. Let $\Top(T)$ denote the $\infty$-category of finite $T$-spaces, i.e., the smallest subcategory of $\Fun(BT, \Top)$ which contains the quotients $T/T'$ for closed subgroups $T'\subseteq T$, and which is closed under finite colimits. There is a functor $\cf_T: \Top(T)^\op \to \QCoh(\cM_T)$ which is uniquely characterized by the requirement that it preserve finite limits and sends $T/T' \mapsto q_\ast \co_{\cM_{T'}}$. Here, $q: \cM_{T'} \to \cM_T$ is the canonical map induced by the inclusion $T'\subseteq T$. If $X\in \Top(T)$, then the $T$-equivariant $A$-cochains of $X$ is the global sections $\Gamma(\cM_T; \cf_T(X))$; we will denote it by $C^\ast_T(X; A)$.

We will denote the functor $\Gamma(\cM_T; \cf_T(-)): \Top(T)^\op \to \Mod(\Gamma(\cM_T; \co_{\cM_T}))$ by $C^\ast_T(-;A): \Top(T)^\op \to \Mod(A_T)$.

If $X\in \Top(T)$, then the $T$-equivariant $A$-chains of $X$ is the quasicoherent sheaf on $\cM_T$ given by the $\co_{\cM_T}$-linear dual $\cf_T(X)^\vee$. We will denote its global sections by $C_\ast^T(X; A)$. Note that $C_\ast^T(\ast;A) \simeq A_T$, which completes to the $A$-cochains (not $A$-chains) of $BT$.

Let $A$ be an $\Eoo$-$\Z$-algebra, and let $\GG = \GG_a$; then <ref> says that $A$ must be an $\Eoo$-$\QQ[\beta^{\pm 1}]$-algebra. Suppose for simplicity that $T = \GG_m$; then $\pi_\ast C_\ast(BT; A)$ may therefore be identified with the divided power algebra $\Gamma_{\pi_\ast(A)}(\hbar^\vee)$ with $|\hbar^\vee|=2$. Since $A$ is rational, this may further be identified with the polynomial ring $\pi_\ast(A)[\hbar^\vee]$. Unfortunately, this can be confused with $\pi_{\ast}(A_T)$, albeit with the reversed grading. Although this identification is technically correct, it is rather abusive: there is no canonical way to identify $A_T$ with $C_\ast(BT; A)$ when $A$ is an $\Eoo$-$\QQ[\beta^{\pm 1}]$-algebra. We will therefore refrain from making this identification, since it is not valid for more general $\Eoo$-rings $A$.

Let $\lambda: T \to \GG_m$ be a character, and let $T_\lambda = \ker(\lambda)$. Then the map $q: \cM_{T_\lambda} \to \cM_T$ is a closed immersion, and we will denote the ideal in $\co_{\cM_T}$ defined by this closed immersion by $\cI_\lambda$. Equivalently, let $V_\lambda$ denote the $T$-representation obtained by the projection $T \to T_\lambda$. Then $\cI_\lambda$ is given by the line bundle $\cf_T(S^{V_\lambda})$.

It is trickier to extend the definition of equivariant cochains to nonabelian groups, but a construction is sketched in <cit.>, and a detailed construction is given in [40]. We recall this for completeness; in this article, we will only be concerned with torus-equivariance. The methods of this article should work for more general compact Lie groups, but we have not studied this here.

Let $G$ be a reductive group scheme over $\cc$. Let $\Top(G)$ denote the smallest subcategory of $\Fun(BG, \Top)$ which contains the quotients $G/T'$ for closed commutative subgroups $T'\subseteq G$, and which is closed under finite colimits. Then there is a functor $C^\ast_G(-;A): \Top(G)^\op \to \Mod(A)$ which is uniquely characterized by the requirement that it preserve finite limits and sends $G/T' \mapsto A_{T'}$. According to <cit.> and <cit.>, when $G$ is connected, there is a flat $A$-scheme $\cM_G$ and a functor $\cf_G: \Top(G)^\op \to \QCoh(\cM_G)$, such that composition with the forgetful functor $\QCoh(\cM_G) \to \Mod(A)$ is the functor $C^\ast_G(-;A)$. If $X\in \Top(G)$, we will write $\cf_G(X)^\vee$ to denote the linear dual of $\cf_G(X)$ in $\QCoh(\cM_G)$, and refer to it as the $G$-equivariant $A$-chains on $X$.

Let $X$ be an ind-finite space with a $G$-action, so that $X$ can be written as the filtered colimit of a diagram $\{X_i\}$ of subspaces, each of which are in $\Top(G)$. Write $C^\ast_G(X;A)$ to denote $\varprojlim_i C^\ast_G(X_i;A)$. Similarly for $\cf_G(X)$.

Let $G$ be a connected compact Lie group, and let $T$ be a maximal torus in $G$. The flag variety $G/T$ is a $G$-space whose stabilizers are commutative, and therefore $G/T\in \Top(G)$. Therefore, $C^\ast_G(G/T; A) = A_T$. For the remainder of this text, we will make the following assumption: after inverting $|W|$, there is a (homotopy-coherent) $W$-action on $A_T$ by maps of $\Eoo$-$A$-algebras, and $A_G := C^\ast_G(\ast;A)$ is equivalent to $A_T^{hW}$ as an $\Eoo$-$A$-algebra.

§.§ Categories of equivariant local systems

Fix a complex-oriented even-periodic $\Eoo$-ring $A$ and an oriented $A$-group scheme $\GG$. Let $T$ be a compact torus. Let $X\in \Top(T)$ be a finite $T$-space. The following categorifies the $T$-equivariant $A$-cochains $C_T^\ast(X; A)$.

Let $\Loc_T(\ast; A)$ denote the $\infty$-category $\QCoh(\cM_T)$. Let $T'\subseteq T$ be a closed subgroup, so that there is an associated morphism $q: \cM_{T'} \to \cM_T$. This defines a symmetric monoidal functor $\QCoh(\cM_T) \to \QCoh(\cM_{T'})$, which equips $\QCoh(\cM_{T'})$ with the structure of a $\QCoh(\cM_T)$-module.
Let $\cLoc_T(-; A): \Top(T)^\op \to \CAlg(\shvcat(\cM_T))$ be the functor uniquely characterized by the requirement that it preserve finite limits and send $T/T' \mapsto \QCoh(\cM_{T'})$. If $X\in \Top(T)$, then the $\infty$-category $\Loc_T(X; A)$ of $T$-equivariant local systems of $A$-modules on $X$ is defined to be the global sections of the quasicoherent stack $\cLoc_T(X; A)$ on $\cM_T$. If $f: X \to Y$ is a map in $\Top(T)$, the associated symmetric monoidal functor $f^\ast: \Loc_T(Y; A) \to \Loc_T(X; A)$ (induced by taking global sections of the morphism $f^\ast: \cLoc_T(Y; A) \to \cLoc_T(X; A)$ of $\Eoo$-algebras in quasicoherent stacks over $\cM_T$) will be called the pullback. One can show that $\Loc_T(X; A)$ is a presentable stable $\infty$-category, and that $f^\ast$ preserves small colimits (so it has a right adjoint $f_\ast$, which will be called pushforward).

If $T = \{1\}$, then $\Loc_T(X; A)$ is equivalent to the $\infty$-category $\Loc(X; A) := \Fun(X, \Mod_A)$ of local systems on $X$.

Let $X$ be a finite $T$-space. The constant local system $\ul{A}_T$ is defined to be the image of $\co_{\cM_T}$ under the symmetric monoidal functor $\Loc_T(\ast; A) \simeq \QCoh(\cM_T) \to \Loc_T(X; A)$ induced by pullback along $f: X \to \ast$. Observe that if $\ul{A}_T$ denotes the constant local system, then $\End_{\Loc_T(X; A)}(\ul{A}_T) \simeq C_T^\ast(X; A)$. Indeed, $\End_{\Loc_T(X; A)}(\ul{A}_T) \simeq \Gamma(\cM_T; f_\ast f^\ast \co_{\cM_T})$, but it is easy to see that $f_\ast f^\ast \co_{\cM_T} = \cf_T(X)\in \QCoh(\cM_T)$. The desired claim then follows from <ref>.

If $T$ were a finite diagonalizable group scheme (such as $\mu_n$), the desired category $\Loc_T(X; A)$ is closely related to the $\infty$-category of $\GG$-tempered local systems on the orbispace $X\mmod T$, as described in [63]. Our understanding is that Lurie is planning to describe an extension of the work in [63] and its connections to equivariant homotopy theory in a future article. We warn the reader that <ref> is somewhat ad hoc; so the resulting category of equivariant local systems may or may not agree with the output of forthcoming work of Lurie.

Let $X$ be a $T$-space with a chosen presentation as a filtered colimit of finite $T$-spaces $X_\alpha$. Then we will write $\Loc_T(X; A)$ to denote $\lim \Loc_T(X_\alpha; A)$.

If $Y$ is a connected space, the $\infty$-category $\Loc(Y; A) = \Fun(Y, \Mod_A)$ of local systems on $Y$ is equivalent by Koszul duality to $\LMod_{C_\ast(\Omega Y; A)}$. This property of local systems is very useful, since it allows one to study of local systems using (derived) algebra. A similar property is true for $\Loc_T(X; A)$:

Let $X$ be a connected finite $T$-space. Then there is an equivalence $\Loc_T(X; A) \simeq \LMod_{\cf_T(\Omega X)^\vee}(\QCoh(\cM_T))$.

Let $s: \ast \to X$ denote the inclusion of a point. We claim that $s^\ast: \Loc_T(X; A) \to \QCoh(\cM_T)$ admits a left adjoint $s_!$. Indeed, the statement for general $X$ follows formally from the case of $X = T/T'$ for some closed subgroup $T'\subseteq T$ (so $s$ is the inclusion of the trivial coset). In this case, $s^\ast$ is the functor $\QCoh(\cM_{T'}) \to \QCoh(\cM_T)$ given by pushforward along the associated morphism $q: \cM_{T'} \to \cM_T$, so it has a left adjoint $s_!$ given by $q^\ast$. Note that $s^\ast$ also has a right adjoint; in particular, it preserves small limits and colimits. Observe now that $s_! \co_{\cM_T}$ is a compact generator of $\Loc_T(X; A)$: indeed, suppose $\cf\in \Loc_T(X; A)$ such that $\Map_{\Loc_T(X; A)}(s_! \co_{\cM_T}, \cf) \simeq 0$ as an object of $\QCoh(\cM_T)$. Because $s^\ast \cf \simeq \Map_{\Loc_T(X; A)}(s_! \co_{\cM_T}, \cf)$ in $\QCoh(\cM_T)$, we see that $s^\ast \cf \simeq 0$. Using the connectivity of $X$, we see that $\cf$ itself must be zero, which implies that $s_! \co_{\cM_T}$ is a compact generator of $\Loc_T(X; A)$. It follows from the Barr-Beck-Lurie theorem <cit.> that $\Loc_T(X; A)$ is equivalent to the $\infty$-category of left $\End_{\Loc_T(X;A)}(s_! \co_{\cM_T})$-modules in $\QCoh(\cM_T)$. But $\End_{\Loc_T(X;A)}(s_! \co_{\cM_T}) \simeq s^\ast s_! \co_{\cM_T}$, which identifies with $\cf_T(\Omega X)^\vee$.

Modifying the preceding argument shows that if $X$ is a connected finite $T$-space, there is an equivalence $\Loc_T(X; A) \simeq \coLMod_{\cf_T(X)^\vee}(\QCoh(\cM_T))$. In particular, if $X$ admits an $\E{n}$-algebra structure (compatible with the $T$-action), then $\cf_T(X)^\vee$ admits the structure of an $\E{n}$-algebra[If $\cC$ is a symmetric monoidal $\infty$-category, <cit.> can be used to show that there is an equivalence $\coCAlg(\Alg_\E{n}(\cC)) \simeq \Alg_\E{n}(\coCAlg(\cC))$.] in $\coCAlg(\QCoh(\cM_T))$, and the equivalence $\Loc_T(X; A) \simeq \coLMod_{\cf_T(X)^\vee}(\QCoh(\cM_T))$ is $\E{n}$-monoidal for the convolution tensor product on both sides. More generally, if $X$ is a $T$-space with a chosen presentation as a filtered colimit of finite $T$-spaces $X_\alpha$, there is an equivalence $\Loc_T(X; A) \simeq \coLMod_{\cf_T(X)^\vee}(\QCoh(\cM_T))$.

§.§ Filtered deformations

As usual, we will fix a complex-oriented even-periodic $\Eoo$-ring $A$ and an oriented $A$-group scheme $\GG$ throughout this section. The main idea of this section (using the double-speed Postnikov filtration) has been used to great effect in [48, 78, 80], but the focus of this section is rather different from loc. cit..

Write $\Sp^\fil$ to denote the $\infty$-category $\Fun(\Z, \Sp)$ of filtered spectra, where $\Z$ is viewed as a poset via the standard ordering. Similarly, write $\Sp^\gr$ to denote the $\infty$-category $\Fun(\Z^\ds, \Sp)$ of graded spectra, where $\Z^\ds$ denotes the discrete set of integers. There is a functor $\gr: \Sp^\fil \to \Sp^\gr$ given by taking associated graded. See [60, 80] for further discussion on filtered and graded spectra.
Recall the following equivalence from [66], which let us view a filtration as equivalent to a one-parameter deformation.

There is a symmetric monoidal equivalence $\Sp^\fil \simeq \QCoh(\AA^1/\GG_m)$, where $\AA^1/\GG_m$ is the flat spectral stack over the sphere spectrum. Under this equivalence, the functor $\gr: \Sp^\fil \to \Sp^\gr$ is given by pullback along the closed immersion $B\GG_m \hookrightarrow \AA^1/\GG_m$.
In particular, a $\Z$-filtered $\E{n}$-algebra in $\Sp$ defines an $\E{n}$-algebra in $\QCoh(\AA^1/\GG_m)$.

If $R\in \CAlg(\Sp^\fil)$, we will simply write $\Mod_R^\fil$ to denote $\Mod_R(\Sp^\fil)$. Similarly, if $R\in \CAlg(\Sp^\gr)$, we will simply write $\Mod_R^\gr$ to denote $\Mod_R(\Sp^\gr)$.
If $\cC$ is a $\Sp^\fil$-linear $\infty$-category, write $\cC^\gr$ to denote $\cC \otimes_{\Sp^\fil} \Sp^\gr$.
For $R\in \CAlg(\Sp^\fil)$, the $\infty$-category $\Mod_R^\fil$ is canonically a $\Sp^\fil$-linear $\infty$-category, and there is an equivalence

$$(\Mod_R^\fil)^\gr = \Mod_R^\fil \otimes_{\Sp^\fil} \Sp^\gr \simeq \Mod_{\gr(R)}^\gr.$$

The $\Eoo$-ring $A$ defines a canonical $\Z$-filtered $\Eoo$-algebra in $\Sp$, given by $\tau_{\geq 2\star} A$. Note that since $\tau_{\geq 2\star}: \Sp \to \Fun(\Z, \Sp)$ is a lax symmetric monoidal functor, $\tau_{\geq 2\star} A$ is an $\Eoo$-algebra in filtered spectra.
The discussion in the preceding section in turn admits a canonical one-parameter deformation.
Namely, the spectral $A$-scheme $\cM_T$ admits a filtered deformation $\cM_T^\fil$: its underlying $\pi_0 A$-scheme is just the underlying scheme of $\cM_T$, and its ring of functions is given by the sheaf $\tau_{\geq 2\star} \co_{\cM_T}$ of filtered $\tau_{\geq 2\star} A$-algebras. Motivated by the comparison to synthetic spectra in [48], we will write $\QCoh^\syn(\cM_T)$ to denote the $\Mod_{\tau_{\geq 2\star} A}^\fil$-linear $\infty$-category $\QCoh(\cM_T^\fil)$.

Similarly, if $X$ is a $T$-space, one can also consider filtered deformations of the sheaves $\cf_T(X)$ and $\cf_T(X)^\vee$. For simplicity, we will only consider the case when $\cf_T(X)$ (resp. $\cf_T(X)^\vee$) has homotopy sheaves concentrated in even degrees; in this case, the filtered deformation of $\cf_T(X)$ (resp. $\cf_T(X)^\vee$) is simply given by $\tau_{\geq 2\star} \cf_T(X)$ (resp. $\tau_{\geq 2\star} \cf_T(X)^\vee$). These are quasicoherent sheaves on $\cM_T^\fil$; since $\tau_{\geq 2\star}$ is lax symmetric monoidal, $\tau_{\geq 2\star} \cf_T(X)$ is an $\Eoo$-algebra in $\QCoh^\syn(\cM_T)$. Similarly, if $X$ is an $\E{n}$-space (compatible with the $T$-action), then $\tau_{\geq 2\star} \cf_T(X)^\vee$ is an $\E{n}$-algebra in $\QCoh^\syn(\cM_T)$.

Let $X$ be a connected finite $T$-space such that $\cf_T(\Omega X)^\vee$ is concentrated in even degrees.
Motivated by <ref>, define $\Loc_T^\syn(X; A)$ to denote $\LMod_{\tau_{\geq 2\star} \cf_T(\Omega X)^\vee}(\QCoh^\syn(\cM_T))$.
Similarly, if $Y$ is an $\E{n}$-algebra in connected $T$-spaces such that $\cf_T(Y)^\vee$ is concentrated in even degrees, define $\Loc_T^\syn(Y; A)$ to be $\coLMod_{\tau_{\geq 2\star} \cf_T(Y)^\vee}(\QCoh^\syn(\cM_T))$.

In <ref>, the definition of $\Loc_T^\syn(X; A)$ is rather ad hoc; we have not attempted to describe a general construction here, because this definition suffices for our purposes.

The key point of the preceding construction is that it allows us to interpolate between spectral and (derived) algebraic geometry. More precisely:

There is an equivalence $(\Mod_{\tau_{\geq 2\star} A}^\fil)^\gr \simeq \Mod_{\pi_0 A}$.

Base-changing the $\Sp^\fil$-linear $\infty$-category $\Mod_{\tau_{\geq 2\star} A}^\fil$ along $\gr: \Sp^\fil \to \Sp^\gr$ produces the $\Sp^\gr$-linear $\infty$-category $\Mod_{\pi_{2\star} A}^\gr$, where $\pi_{2\star} A$ is viewed as a graded $\Eoo$-ring. However, $A$ is even-periodic, so $\pi_{2\star} A \cong \pi_0(A)[\beta^{\pm 1}]$ with $\beta$ in weight $1$. This implies that $\Mod_{\pi_{2\star} A}^\gr \simeq \Mod_{\pi_0 A}$.

Let $\cM_{T,0}$ denote the underlying $\pi_0 A$-scheme of the $A$-scheme $\cM_T$.
<ref> identifies $\QCoh^\syn(\cM_T)^\gr = \QCoh^\syn(\cM_T) \otimes_{\Sp^\fil} \Sp^\gr$ with $\QCoh(\cM_{T,0})$ as $\pi_0 A$-linear $\infty$-categories.

Let $X$ be a connected finite $T$-space such that $\cf_T(\Omega X)^\vee$ is concentrated in even degrees. The preceding discussion implies that $\pi_{2\star} \cf_T(\Omega X)^\vee$ defines an $\E{1}$-algebra in $\coCAlg(\QCoh^\syn(\cM_T)^\gr)$.
Let $\Loc_T^\gr(X; A)$ denote $\LMod_{\pi_{2\star} \cf_T(\Omega X)^\vee}(\QCoh^\syn(\cM_T)^\gr)$; note that the $\Eoo$-coalgebra structure on $\cf_T(\Omega X)^\vee$ equips $\Loc_T^\gr(X; A)$ with the structure of a symmetric monoidal $\infty$-category. By $2$-periodicity, we can identify

$$\Loc_T^\gr(X; A) \simeq \LMod_{\pi_0 \cf_T(\Omega X)^\vee}(\QCoh(\cM_{T,0})).$$

Similarly, if $Y$ is an $\E{n}$-algebra in connected $T$-spaces such that $\cf_T(Y)^\vee$ is concentrated in even degrees, $\pi_{2\star} \cf_T(Y)^\vee$ defines an $\Eoo$-coalgebra in $\Alg_\E{n}(\QCoh^\syn(\cM_T)^\gr)$.
Let $\Loc_T^\gr(Y; A)$ denote $\coLMod_{\pi_{2\star} \cf_T(Y)^\vee}(\QCoh^\syn(\cM_T)^\gr)$; note that the $\E{n}$-algebra structure on $\cf_T(Y)^\vee$ equips $\Loc_T^\gr(Y; A)$ with the structure of an $\E{n}$-monoidal $\infty$-category. By $2$-periodicity, we can identify

$$\Loc_T^\gr(Y; A) \simeq \coLMod_{\pi_0 \cf_T(Y)^\vee}(\QCoh(\cM_{T,0})).$$

Both $\Loc_T^\gr(X; A)$ and $\Loc_T^\gr(Y; A)$ are $\QCoh(\cM_{T,0})$-linear $\infty$-categories, which arise as $\Loc_T^\syn(X; A)^\gr$ and $\Loc_T^\syn(Y; A)^\gr$, respectively.

§.§ GKM and complex periodic $\Eoo$-rings

We review the main result of [46], which proves a generalization of a result of Goresky-Kottwitz-MacPherson to generalized cohomology theories. This is also studied in the forthcoming work <cit.>.

Let $A$ be a complex-oriented even-periodic $\Eoo$-ring, and let $\GG$ be an oriented commutative $A$-group.

Fix a compact torus $T$. We will consider (ind-finite; see <ref>) $T$-spaces $X$ such that the following assumptions hold.

* $X$ admits a $T$-invariant stratification $\bigcup_{w\in W} X_x$ with only even-dimensional cells, with only finitely many in each dimension.

* The $T$-action on each cell $X_w = \AA^{\ell(w)}$ is via a linear action, whose weights are pairwise relatively prime.

* For each weight $\lambda$ of the $T$-action on $X_w = \AA^{\ell(w)}$, the closure of $\cc_\lambda\subseteq X_w$ is a sphere $S^\lambda$ such that $0$ and $\infty$ are fixed points of the $T$-action.

The GKM graph $\Gamma$ asssociated to an $X$ as in <ref> is defined as follows. The vertices are the (isolated) fixed points of the $T$-action, and there is an edge $x \to y$ labeled by a character $\lambda$ if $x = 0$ and $y=\infty$ in the closure $S^\lambda$ of $D(\lambda)\subseteq D(\AA^{2\ell(w)})$. Let $V$ denote the set of vertices of $\Gamma$, and $E$ the set of edges.

In <ref>, the map $\cf_T(X) \to \Map(V, \co_{\cM_T}) \simeq \cf_T(X^T)$ induces an injection on homotopy sheaves, and the following diagram is an equalizer on $\pi_0$:

$$\cf_T(X) \to \Map(V, \co_{\cM_T}) \rightrightarrows \prod_{\alpha\in E} \co_{\cM_{T_\alpha}}.$$

Here, the two maps are induced by the inclusion of the source and target of $\alpha: x \to y$.

The argument is exactly as in <cit.> (where the spaces denoted $F_i$ are points, corresponding to the origin in $\AA^{\ell(w)}$), so we only give a sketch. We will work locally on $\GG$. In this case, we need to show that the map $\cf_T(X) \to \Map(V, \co_{\cM_T}) \simeq \cf_T(X^T)$ is injective on homotopy sheaves, and the following diagram is an equalizer on $\pi_0$:

$$\cf_T(X) \to \cf_T(X^T) \rightrightarrows \prod_{\alpha\in E} \cf_{T_\alpha}.$$

For the injectivity claim, we first claim that $\cf_T(X)^{tT} \simeq \cf_T(X^T)^{tT}$. (This is a version of Atiyah-Bott localization.) Since $X$ is generated by finite colimits from $T$-orbits $T/T'$, it suffices to prove this claim when $X$ is of that form. Then $\cf_T(T/T') \simeq \cf_{T'}(\ast) = q_\ast \co_{\cM_{T'}}$; this has zero Tate construction if $T'\neq T$. On the other hand, $X^T = \emptyset$ if $T' \neq T$, so $\cf_T(X^T)^{tT} = 0$ as desired. If $T' = T$, then $X^T = \ast$, so that both sides are simply $A^{tT}$.

Note that $\cf_T(X^T)^{tT} \simeq \cf_T(X^T) \otimes_A A^{tT}$. Since $\cf_T(X)^{tT} \simeq \cf_T(X) \otimes_{\co_{\cM_T}} A^{tT}$ is a localization, it suffices to prove that the map $\cf_T(X) \to \cf_T(X)^{tT}$ induces an injection on homotopy. For this, it suffices to prove that $\cf_T(X)$ is a free $\co_{\cM_T}$-module. This is a consequence of the assumptions on $X$.

To prove the statement about the equalizer diagram, the key case is when $X = S^W$ for a $T$-representation $W$; the general case is obtained by induction on the stratification of $X$. Let $\lambda_1,\cdots,\lambda_n$ be the weights of $W$, so that $X = \bigotimes_{i=1}^n S^{\lambda_i}$. Therefore, $X$ is the quotient of $\prod_{i=1}^n S^{\lambda_i}$ by its $(2n-2)$-skeleton. Using this observation, it is not difficult to reduce to the case when $W = \lambda$ is a character of $T$.
In this case, $X = S^\lambda$ has $T$-fixed points given by $\{0,\infty\}$. There is a cofiber sequence $S(\lambda) \to \ast \to S^\lambda$, which induces a pushout square

S(\lambda)_+ \ar[r] \ar[d] & S^0 = \{\infty\}_+ \ar[d] \\
S^0 = \{0\}_+ \ar[r] & S^\lambda_+.

Therefore, we get an equalizer diagram

$$\cf_T(S^\lambda) \to \co_{\cM_T} \rightrightarrows \cf_T(S(\lambda)).$$

However, if $T_\lambda = \ker(\lambda: T \to \GG_m)$, then $S(\lambda) \simeq T/T_\lambda$, so that $\cf_T(S(\lambda)) \simeq q_\ast \co_{\cM_{T_\lambda}}$. It follows that $\cf_T(S^\lambda)$ is the fiber of the map $\co_{\cM_T} \oplus \co_{\cM_T} \to q_\ast \co_{\cM_{T_\lambda}}$ given by the following composite:

$$\co_{\cM_T} \oplus \co_{\cM_T} \xar{(x,y)\mapsto x-y} \co_{\cM_T} \to q_\ast \co_{\cM_{T_\lambda}}.$$

However, the map $\co_{\cM_T} \to q_\ast \co_{\cM_{T_\lambda}}$ is precisely given by quotienting by the ideal $\cI_\lambda$ (by <ref>). Therefore, $\cf_T(S^\lambda)$ is described by the claimed equalizer diagram.

Informally, the image on homotopy sheaves of the map $\cf_T(X) \to \Map(V, \co_{\cM_T}) \simeq \cf_T(X^T)$ consists of those $f\in \pi_\ast \co_{\cM_T}^V$ such that $f(x) \equiv f(y) \pmod{\cI_\alpha}$ for every edge $\alpha: x \to y$ in $\Gamma$. Here, $\cI_\alpha$ is as in <ref>.

§ EQUIVARIANT TOPOLOGY OF THE AFFINE GRASSMANNIAN

For a topologically minded reader, we recommend the book [18] for a nice introduction to more classical aspects of geometric representation theory.

§.§ Kac-Moody flag varieties

Fix a complex-oriented even-periodic $\Eoo$-ring $A$ and an oriented commutative $A$-group $\GG$.

Let $\cg$ be a Kac-Moody group, and let $\cP\subseteq \cg$ be a parabolic subgroup associated to a subset $J\subseteq \Delta$ of simple roots. Let $T = T_\cg/Z(\cg)$ denote the torus of $\cg/Z(\cg)$, and let $W$ be the Weyl group associated to $\cg$. Let $W_\cP$ denote the subgroup of $W$ generated by $s_{\alpha_j}$ for $\alpha_j\in J$, and let $W^J$ denote the set of minimal-length coset representatives in $W_\cg/W_\cP$.

Then $(\cg/\cP)^T \cong W^\cP$, and the Schubert decomposition $\cg/\cP = \coprod_{w\in W^\cP} \cB\dot{w}\cP/\cP$ is a $T$-invariant stratification, where $\ol{w} = \dot{w}\cP/\cP$ is the unique $T$-fixed point in the cell $\cB\dot{w}\cP/\cP$. We claim that $\cg/\cP$ satisfies the hypotheses of <ref>. Clearly, condition (a) is satisfied. For condition (b), observe that the tangent space to $\cB\ol{w}$ at $\ol{w}$ is

$$T_{\ol{w}} \cB\dot{w} \cP/\cP = \fr{b}/(\fr{b} \cap w\cdot \fr{p}) = \bigoplus_{\alpha\in \Phi^+ - w\Phi^+(\fr{p})} \fr{g}_\alpha,$$

where each $\fr{g}_\alpha$ is $1$-dimensional. The weights are therefore all distinct, so condition (b) in <ref> is satisfied. For condition (c), let $\alpha\in \Phi^+ - w\Phi^+(\fr{p})$, and let $i_\alpha: \SL_2 \to \cg$ denote the associated subgroup. The closure of $\cB_\alpha\ol{w}$ is $\SL_2\ol{w} = \PP^1$, where the point at $0$ corresponds to $\ol{w}$, and the point at $\infty$ corresponds to $\ol{s_\alpha w}$. Then the GKM graph $\Gamma$ of $\cg/\cP$ has vertices $W^\cP$ and edges $w \to s_\alpha w$ labeled by $s_\alpha\in W_\cg$. See also <cit.>.

In the following, the reader should replace the symbol “$\cf_T(\cg/\cP)$” by $\cf_T(X_{\leq w})$ where $X_{\leq w}$ is a Schubert cell in $\cg/\cP$. In this case, $X_{\leq w}$ is a finite CW-complex, so that $\cf_T(X_{\leq w})$ is a perfect $\co_{\cM_T}$-module. This implies that the $T$-equivariant homology $\cf_T(X_{\leq w})^\vee$ is the $\co_{\cM_T}$-linear dual of $\cf_T(X_{\leq w})$; note that this is not true of $\cf_T(\cg/\cP)$ when the Kac-Moody group is not of finite type. (In general, homology is a predual of cohomology, but the linear dual of cohomology does not recover homology in the non-finite case.) We define $\cf_T(\cg/\cP)^\vee$ as the direct limit of $\cf_T(X_{\leq w})^\vee$.

Since $\cg/\cP$ satisfies the hypotheses of <ref> by <ref>, we may apply <ref> to calculate $\cf_T(\cg/\cP)$. See [57] for a related discussion.

The following diagram is an equalizer on $\pi_0$:

$$\cf_T(\cg/\cP) \to \Map(W^\cP, \co_{\cM_T}) \rightrightarrows \prod_{\alpha: w \to s_\alpha w} \co_{\cM_{T_\alpha}}.$$

Here, the two maps are given by restriction and applying $s_\alpha$ to $W^\cP$.
Therefore, $\pi_0 \cf_T(\cg/\cP)$ is the sub-$\pi_0 \co_{\cM_T}$-algebra of $\Map(W^\cP, \pi_0 \co_{\cM_T})$ consisting of those maps $f: W^\cP \to \pi_0 \co_{\cM_T}$ such that

\begin{equation}\label{gkm-condition}
f(s_\alpha w) \equiv f(w) \pmod{\cI_\alpha} \text{ for all }w \in W^\cP, \alpha\in \Phi.
\end{equation}

Motivated by <ref>, we may define an algebraic generalization of $\pi_0 \cf_T(\cg/\cP)$ as follows.

Let $(W,S)$ be a Coxeter system, and let $V = \RR^S$ denote the associated geometric representation. For $s\in S$, let $\alpha_s$ denote the associated vector, let $\Phi = \{w(\alpha_s) | s\in S, w\in W\}$ be the set of roots, and let $\Phi^+\subseteq \Phi$ denote the set of positive roots. Let $\Lambda = \Z\Phi \subseteq V$ denote the associated root lattice. Fix a smooth $1$-dimensional affine group scheme $\GG_0$ over a commutative ring $R$, and let $\cM_{T,0} = \Hom(\Lambda^\vee, \GG_0)$. Given a character $\lambda$, let $c_\lambda$ denote a function which cuts out the closed subscheme $\GG_{\ker(\lambda)} \hookrightarrow \cM_{T,0}$.
Define $\KK$ to be the sub-$\co_{\cM_{T,0}}$-algebra of $\Map(W, \co_{\cM_{T,0}})$ consisting of those maps $f: W \to \co_{\cM_{T,0}}$ satisfying <ref>, i.e., such that $f(s_\alpha w) \equiv f(w) \pmod{c_\alpha}$ for $\alpha\in \Phi$ and $w\in W$.

Note that if $\lambda$ is a character, then the function $c_\lambda$ on $\cM_T$ is given by the $T$-equivariant Thom class of the representation of $T$ given by $\lambda: T \to \GG_m^\rot$. Morever, $c_\lambda$ generates $\cI_\lambda$.

Let $s_\alpha\in W$, and let $T_\alpha = \ker(\alpha)\subseteq T$. Then we have the following commutative diagram of $R$-schemes (where the non-vertical arrows are closed immersions):

\cM_{T_\alpha,0} \ar[r]^-q \ar[dr]_-q & \cM_{T,0} \ar[d]^-{s_\alpha} \\
& \cM_{T,0};

informally, $s_\alpha \equiv 1\pmod{\cI_\alpha}$.

This follows from the fact that the character lattice of $T_\alpha$ is the quotient of $\bX^\ast(T)$ by the rank $1$ sublattice generated by $\alpha$; therefore, if $\chi\in \bX^\ast(T)$, then $s_\alpha \chi|_{T_\alpha} = \chi|_{T_\alpha}$.

<ref> implies the following:

Suppose $\GG_0$ is affine. Then there is an equivalence $\pi_0 \cf_T(\cg/\cP)^\vee \simeq \co_{\cM_{T,0}}[W^\cP, \tfrac{s_\alpha-1}{c_\alpha}, \alpha\in \Phi]$ of $\pi_0 \co_{\cM_T}$-modules.

Recall that if $w\in W$, then $\inv(w)\subseteq \Phi^+$ denotes the set of positive roots $\alpha$ such that $s_\alpha w < w$. The following is then the analogue of <cit.>.

Suppose that $\GG$ is affine.
In <ref>, $\KK$ is a free $\co_{\cM_{T,0}}$-module spanned by functions $\psi_w: W \to \co_{\cM_{T,0}}$ for $w\in W$, where $\psi_w$ is uniquely characterized by the property that it satisfies <ref> and the following two properties:

\begin{align*}
\psi_w(v) & = 0 \text{ if }v<w,\\
\psi_w(w) & = \prod_{\alpha\in \inv(w)} c_\alpha.
\end{align*}

The two stated conditions define $\psi_w$ on the interval $[1, w]\subseteq W$. We will now define an extension of $\psi_w$ to the whole of $W$. We will in fact prove the following more general claim by induction on $\ell(w)$:

$(\ast)$ Let $w\in W$, and let $[1,w]^\circ = [1,w]-\{w\}$. Then any function $\psi: [1,w]^\circ \to \co_{\cM_{T,0}}$ satisfying <ref> extends to a function $[1,w] \to \co_{\cM_{T,0}}$ satisfying <ref>.

To see this, write $w = s_{i_1} \cdots s_{i_n}$, let $\alpha = \alpha_{i_1}$, and let $w' = s_\alpha w$ (so that $w'<w$). Consider the restriction of $\psi$ to $[1, w']^\circ$, so that $\psi$ itself is an extension to $[1,w']$. Define $\psi': [1,w']^\circ \to \co_{\cM_{T,0}}$ by the formula $\psi'(v) = s_\alpha \psi(s_\alpha v)$. Then $\psi'$ also satisfies <ref>: indeed, if $\beta$ is another root, then $\psi'(s_\beta v) \equiv \psi'(v) \pmod{\cI_\beta}$ if and only if $\psi(s_\alpha s_\beta v) \equiv \psi(s_\alpha v) \pmod{s_\alpha \cI_\beta}$. However, $s_\alpha \cI_\beta = \cI_{s_\alpha(\beta)}$, while $s_\alpha s_\beta = s_{s_\alpha(\beta)} s_\alpha$. The claim therefore follows from the assumption that $\psi$ satisfies <ref>.

Since $w'<w$, the inductive hypothesis says that $\psi'$ extends to a function $\psi': [1,w'] \to \co_{\cM_{T,0}}$ which satisfies <ref>. If $v\in [1,w']^\circ$, then

$$\psi(v) - \psi'(v) = \psi(v) - s_\alpha \psi(s_\alpha v) \equiv (1 - s_\alpha) \psi(v) \pmod{\cI_\alpha}.$$

By <ref>, we see that $\psi(v) - \psi'(v) \equiv 0\pmod{\cI_\alpha}$, so we may define a function $p_v\in \co_{\cM_{T,0}}$ by the formula $\frac{\psi(v) - \psi'(v)}{c_\alpha}$. If $\beta\in \Phi^+$ is such that $s_\beta w'<w'$, then:

\begin{align*}
\psi(w') - \psi'(w') & \equiv \psi(s_\beta w') - \psi'(s_\beta w') \pmod{\cI_\beta} \\
& = c_\alpha p_{s_\beta w'} \pmod{\cI_\beta}.
\end{align*}

In particular, there is a function $p_{w'} \in \co_{\cM_{T,0}}$ such that

$$\psi(w') - \psi'(w') \equiv c_\alpha p_{w'} \pmod{\cI_\beta}$$

for all $\beta\in \Phi^+$ such that $s_\beta w'<w'$, i.e., $\beta\in \inv(w')$. In particular,

\begin{equation}\label{pw-prime}
\psi(w') - \psi'(w') \equiv c_\alpha p_{w'} \pmod{\prod_{\beta\in \inv(w')} \cI_\beta}.
\end{equation}

Note that $s_\alpha \inv(w')$ is the set of $\beta \in \Phi^+-\{\alpha\}$ such that $s_\beta w'<w'$. Define

$$\psi(w) = s_\alpha \psi'(w') + x\prod_{\beta\in s_\alpha \inv(w')} c_\beta$$

for some $x$ that we will determine in a moment. We check that $\psi$ satisfies <ref>. Let $\alpha'\in \Phi^+$ be such that $s_{\alpha'} w<w$. Then:

* If $\alpha' = \alpha$, then

\begin{align*}
\psi(w) - \psi(s_\alpha w) & = s_\alpha \psi'(w') - \psi(w') + x\prod_{\beta\in s_\alpha \inv(w')} c_\beta \\
& \equiv s_\alpha (\psi'(w') - \psi(w')) + x\prod_{\beta\in s_\alpha\inv(w')} c_\beta \pmod{\cI_\alpha}
\end{align*}

However, <ref> implies that

$$s_\alpha (\psi(w') - \psi'(w')) \equiv c_{-\alpha} s_\alpha(p_{w'}) \pmod{\prod_{\beta\in s_\alpha \inv(w')} \cI_\beta}$$

Therefore, taking $x$ to be the negative of the residue of $s_\alpha (\psi(w') - \psi'(w')) - c_{-\alpha} s_\alpha(p_{w'})$ modulo $\prod_{\beta\in s_\alpha \inv(w')} \cI_\beta$, we see that

\begin{align*}
\psi(w) - \psi(s_\alpha w) & \equiv c_{-\alpha} s_\alpha(p_{w'}) \equiv 0 \pmod{\cI_\alpha},
\end{align*}

as desired.

* If $\alpha' \neq \alpha$, then $\alpha'\in s_\alpha \inv(w')$. Then, we have

\begin{align*}
\psi'(w') & \equiv \psi'(s_{s_\alpha(\alpha')} w') \pmod{\cI_{s_\alpha(\alpha')}} \\
& = s_\alpha \psi(s_\alpha s_{s_\alpha(\alpha')} s_\alpha w) \pmod{\cI_{s_\alpha(\alpha')}}\\
& = s_\alpha \psi(s_{\alpha'} w) \pmod{\cI_{s_\alpha(\alpha')}}.
\end{align*}

In particular, $s_\alpha \psi'(w') \equiv \psi(s_{\alpha'} w) \pmod{\cI_{\alpha'}}$. But this implies that

\begin{align*}
\psi(w) - \psi(s_{\alpha'} w) & \equiv s_\alpha \psi'(w') - \psi(s_{\alpha'} w) \pmod{\cI_{\alpha'}}\\
& \equiv 0 \pmod{\cI_{\alpha'}},
\end{align*}

as desired.

This finishes the proof of $(\ast)$.

To finish the proof of the proposition, note that the two conditions on $\psi_w$ specify it on $[1,w]$, and hence on the subset of $W$ consisting of elements of length $<\ell(w)$. By $(\ast)$, we may inductively extend $\psi_w$ to the subset of $W$ consisting of elements of length $\geq \ell(w)$, and hence to all of $W$. It remains to show that any $\psi\in \Map(W, \co_{\cM_{T,0}})$ satisfying <ref> can be written as a $\co_{\cM_{T,0}}$-linear combination of the $\psi_w$; see the second half of <cit.> for the following argument.

Let $\supp(\psi)$ denote the subset of $w\in W$ such that $f(\psi)\neq 0$. Let $v\in \supp(\psi)$ be minimal. If $\alpha\in \inv(v)$ (so $s_\alpha v<v$), then $\psi(v) \equiv \psi(s_\alpha v) = 0\pmod{\cI_\alpha}$. This implies that $\psi(v) \equiv 0 \pmod{\psi_v(v)}$. Define $\psi': W \to \pi_0 \co_{\cM_{T,0}}$ by $\psi'(w) = \psi(w) - \tfrac{\psi(v)}{\psi_v(v)} \psi_v(w)$; then $\psi'$ satisfies <ref> (since $\psi$ and $\psi_v$ do). By construction, $v\not\in \supp(\psi')$, and $\supp(\psi')-\supp(\psi)$ consists of elements which are strictly larger than $v$. Therefore, we may repeat this argument for $\psi'$, and induct; this yields the desired result.

§.§ The affine Grassmannian

Fix notation as in <ref>, and assume that $G$ is semisimple. Then we have an associated affine root datum: the affine simple roots are $\Delta_\aff = \Delta \cup\{0\}$, and the affine weight lattice is given by $\Z K \oplus \bigoplus_{\alpha_i\in \Delta_\aff} \Z \alpha_i$. (In particular, we denote the affine root by $\alpha_0$.) Thus the associated Kac-Moody algebra is $\hat{\g} = \g\ls{t} \oplus \cc \alpha_0 \oplus \cc K$, where $K$ is the central class, and $\alpha_0$ is the scaling factor.
Let $\cg$ denote the associated Kac-Moody group, and let $W^\aff = \Lambda^\vee \rtimes W$ denote the associated affine Weyl group. If $\lambda^\vee \in \Lambda^\vee$, we write $t_{\lambda^\vee}$ to denote the associated element of $W^\aff$. If $\alpha + n\alpha_0$ is an affine root and $x\in \fr{t}$, then

$$s_{\alpha+n\alpha_0}(x) = x - (\langle x, \alpha\rangle + n) \alpha^\vee = s_\alpha(x) + n\alpha^\vee.$$

Let $\cB$ denote the Iwahori subgroup, and $T_\aff$ the maximal torus of $\cg$. Then $\cg/\cB$ is the affine flag variety $\Fl_G$; similarly, $\Gr_G$ is the Kac-Moody flag variety associated to the subset $\Delta\subseteq \Delta_\aff$. Up to keeping track of the central torus, we may view $\cg$ as $G\ls{t}$, and $\cB$ as the Iwahori $I$. Thus $T = T^\aff \cap G$ is the maximal torus of $G$. Let $\tilde{T}$ denote the extended torus $T\times \GG_m^\rot$ (where $\GG_m^\rot$ is the loop rotation torus); we may identify its Lie algebra $\tilde{\fr{t}}$ with $\fr{t} \oplus \cc \alpha_0$.

Let $\alpha\in \Phi$ and $n\in \Z$. Then $n\alpha_0$ is the $\GG_m^\rot$-representation of weight $n$.

Note that $\alpha + n\alpha_0$ defines an ideal sheaf $\cI_{\alpha + n\alpha_0} \subseteq \pi_0 \co_{\cM_{\tilde{T}}} = \pi_0 \co_{\cM_T} \otimes_{\pi_0 A} \pi_0 \co_\GG$.

<ref> gives an explicit description of $\pi_0 \cf_{T^\aff}(\Fl_G)$ and $\pi_0 \cf_{T^\aff}(\Gr_G)$. Using that

\begin{align*}
(\Fl_G)^T & = (\Fl_G)^{\tilde{T}} = W^\aff\\
(\Gr_G)^T & = (\Gr_G)^{\tilde{T}} = W^\aff/W \cong \Lambda^\vee,
\end{align*}

this further immediately specializes to the following explicit description of $\pi_0 \cf_{\tilde{T}}(\Fl_G)$ and $\pi_0 \cf_{\tilde{T}}(\Gr_G)$:

The following statements are true:

* We may identify $\pi_0 \cf_{\tilde{T}}(\Fl_G) \cong \pi_0 \cf_{\GG_m^\rot}(\Fl_G/I)$ with $\KK$ from <ref>, i.e., as the sub-$\pi_0 \co_{\cM_{\tilde{T}}}$-algebra of $\Map(W^\aff, \pi_0 \co_{\cM_{\tilde{T}}})$ consisting of those maps $f: W^\aff \to \pi_0 \co_{\cM_{\tilde{T}}}$ such that

\begin{equation}
f(s_{\alpha+n\alpha_0}(w)) \equiv f(w) \pmod{\cI_{\alpha + n\alpha_0}}
\end{equation}

for all $w \in W^\aff, \alpha\in \Phi, n\in \Z$.

* We may identify $\pi_0 \cf_{\tilde{T}}(\Gr_G) \cong \pi_0 \cf_{\GG_m^\rot}(\Gr_G/I)$ as the sub-$\pi_0 \co_{\cM_{\tilde{T}}}$-algebra of $\Map(\Lambda^\vee, \pi_0 \co_{\cM_{\tilde{T}}})$ consisting of those maps $f: \Lambda^\vee \to \pi_0 \co_{\cM_{\tilde{T}}}$ such that

\begin{equation}
f(s_{\alpha+n\alpha_0}(\lambda)) \equiv f(\lambda) \pmod{\cI_{\alpha + n\alpha_0}}
\end{equation}

for all $\lambda \in \Lambda^\vee, \alpha\in \Phi, n\in \Z$.

The following statements are true:

* We may identify $\pi_0 \cf_T(\Fl_G)$ as the sub-$\pi_0 \co_{\cM_T}$-algebra of $\Map(W^\aff, \pi_0 \co_{\cM_T})$ consisting of those maps $f: W^\aff \to \pi_0 \co_{\cM_T}$ such that

\begin{equation}
f(s_{\alpha+n\alpha_0}(w)) \equiv f(w) \pmod{\cI_\alpha}
\end{equation}

for all $w \in W^\aff, \alpha\in \Phi, n\in \Z$.

* We may identify $\pi_0 \cf_T(\Gr_G)$ as the sub-$\pi_0 \co_{\cM_T}$-algebra of $\Map(\Lambda^\vee, \pi_0 \co_{\cM_T})$ consisting of those maps $f: \Lambda^\vee \to \pi_0 \co_{\cM_T}$ such that

\begin{equation}\label{gkm-grg}
f(s_{\alpha+n\alpha_0}(\lambda)) \equiv f(\lambda) \pmod{\cI_\alpha}
\end{equation}

for all $\lambda \in \Lambda^\vee, \alpha\in \Phi, n\in \Z$.

The image of $s_{\alpha+n\alpha_0}$ under the identification $W^\aff/W \cong \Lambda^\vee$ is the right coset $s_{\alpha+n\alpha_0} W$. However, $s_{\alpha+n\alpha_0} s_\alpha$ is translation by $n\alpha^\vee$. If $k$ is a commutative ring, we may view $k[\Lambda^\vee]$ as the $\Eoo$-ring of functions on $\ld{T}_k$; the element $n\alpha^\vee\in \Lambda^\vee$ corresponds to the function $e^{n\alpha^\vee}$. Therefore, <ref> can be restated as

$$f((e^{n\alpha^\vee}-1)(\lambda)) \equiv 0 \pmod{\cI_\alpha}.$$

If $\GG$ is affine, then $\pi_0 \cf_T(\Gr_G)$ is the $\pi_0 \co_{\cM_T}$-linear dual of $\pi_0 \co_{\cM_T}[\Lambda^\vee][\tfrac{e^{n\alpha^\vee}-1}{c_\alpha}]_{n\geq 1}$. However, note that for any $n\geq 1$, we may write

$$\tfrac{e^{n\alpha^\vee}-1}{c_\alpha} = \tfrac{e^{\alpha^\vee}-1}{c_\alpha} + \tfrac{e^{(n-1)\alpha^\vee}-1}{c_\alpha} + c_\alpha \tfrac{e^{\alpha^\vee}-1}{c_\alpha} \tfrac{e^{(n-1)\alpha^\vee}-1}{c_\alpha}.$$

This implies that

$$\pi_0 \cf_T(\Gr_G) \cong \Map_{\QCoh(\cM_{T,0})}(\pi_0 \co_{\cM_T}[\Lambda^\vee][\tfrac{e^{\alpha^\vee}-1}{c_\alpha}], \pi_0 \co_{\cM_T}).$$

Let $\lambda\in \Lambda^{\vee,\pos}$ be a dominant coweight, and let $\Lambda^{\vee,\pos}_{\leq \lambda}$ denote the subset of $\Lambda^{\vee,\pos}$ consisting of those dominant weights which are at most $\lambda$. Then we may identify

$$(\Gr_G^{\leq \lambda})^T = W\cdot \Lambda^{\vee,\pos}_{\leq \lambda} \subseteq \Lambda^\vee = (\Gr_G)^T,$$

which allows us to calculate that if $\GG$ is affine, then

$$\pi_0 \cf_T(\Gr_G^{\leq \lambda}) \cong \Map_{\QCoh(\cM_{T,0})}(\pi_0 \co_{\cM_T}[W\cdot \Lambda^{\vee,\pos}_{\leq \lambda}][\tfrac{e^{\alpha^\vee}-1}{c_\alpha}], \pi_0 \co_{\cM_T}).$$

In the above expression, $\alpha$ ranges over $\Phi \cap W\cdot \Lambda^{\vee,\pos}_{\leq \lambda}$; in other words, $\alpha$ is of the form $w\alpha_i$ with $\alpha_i\in \Delta$ such that $\alpha_i\leq \lambda$.

Recall from <ref> that $\cf_T(\Gr_G)^\vee$ is defined to be the direct limit of $\cf_T(\Gr_G^{\leq \lambda})^\vee$.
We trust the reader to make the appropriate modifications below as needed (which we have not done to avoid an overbearance of notation), so that the calculation of the $T$-equivariant homology $\cf_T(\Gr_G)^\vee$ in <ref> by taking the linear dual of $\cf_T(\Gr_G)$ does not suffer from completion issues. This can be done, for instance, by working with the $\Lambda^{\vee,\pos}$-filtered $\co_{\cM_T}$-module $\{\cf_T(\Gr_G^{\leq \lambda})^\vee\}$.
In order for the colimit $\cf_T(\Gr_G)^\vee$ of the $\Lambda^{\vee,\pos}$-filtered module $\{\cf_T(\Gr_G^{\leq \lambda})^\vee\}$ to admit the structure of an $\E{2}$-$\co_{\cM_T}$-algebra, it suffices to show that $\{\cf_T(\Gr_G^{\leq \lambda})^\vee\}$ admits the structure of an $\E{2}$-algebra in $\Lambda^{\vee,\pos}$-filtered module; this is proved in <ref> below.

The $\Lambda^{\vee,\pos}$-indexed Schubert filtration $\{\Gr_G^{\leq \lambda}(\cc)\}$ naturally admits the structure of an $\E{2}$-algebra in $\Fun(\Lambda^{\vee,\pos}, \Top)$.

This can be proved in essentially the same way as <cit.>; let us sketch the argument. We will utilize <cit.>, which states that if $\cC$ is a symmetric monoidal $\infty$-category, then a nonunital $\E{2}$-algebra object in $\cC$ is equivalent to the datum of a locally constant $\mathrm{N}(\mathrm{Disk}(\cc))_\mathrm{nu}$-algebra object in $\cC$. Concretely, this amounts to specifying an object $A(D)\in \cC$ for every disk $D\subseteq \cc$ and coherent maps $\bigotimes_{i=1}^n A(D_i)\to A(D)$ for every inclusion $\coprod_{i=1}^n D_i\to D$ of disks, such that for every embedding $D\subseteq D'$ of disks, the induced map $A(D)\to A(D')$ is an equivalence.

In this case, $\cC = \Fun(\Lambda^{\vee,\pos}, \Top)$, and the object $A(D)\in \Fun(\Lambda^{\vee,\pos}, \Top)$ assigned to a disk $D\subseteq \cc$ may be defined via the Beilinson-Drinfeld Grassmannian $\Gr_{G,\Ran}$. Namely, the Beilinson-Drinfeld Grassmannian admits (by construction) a morphism $\Gr_{G, \Ran} \to \Ran_{\AA^1}$; upon taking complex points, we obtain a map $\Gr_{G, \Ran}(\cc) \to \Ran(\cc)$. If $S\subseteq \cc$ is a subset, then the preimage of $\Ran(S)\subseteq \Ran(\cc)$ defines a subspace $\Gr_{G, \Ran}(S\subseteq \cc)\subseteq \Gr_{G, \Ran}(\cc)$. The filtration of $\Gr_G$ via the Bruhat decomposition extends to a filtration $\Gr_{G, \Ran, \leq \mu}$ of $\Gr_{G, \Ran}$ by dominant coweights $\mu\in \Lambda^{\vee,\pos}$; see <cit.>. Finally, the object $A(D)\in \Fun(\Lambda^{\vee,\pos}, \Top)$ associated to a disk $D\subseteq \cc$ is the functor $\Lambda^{\vee,\pos}\to \Top$ sending $\mu\in \Lambda^{\vee,\pos}$ to $\Gr_{G, \Ran, \leq \mu}(D\subseteq \cc)$.

Suppose $\coprod_{i=1}^n D_i\to D$ is an inclusion of disks. The induced map $\bigotimes_{i=1}^n A(D_i)\to A(D)$ is defined as follows. Let $\mu\in \Lambda^{\vee,\pos}$; for every $n$-tuple $(\mu_1, \cdots, \mu_n)$ with $\sum_{i=1}^n \mu_i\leq \mu$, we need to exhibit maps $\bigotimes_{i=1}^n A(D_i)(\mu_i)\to A(D)(\mu)$ satisfying the obvious coherences. But

$$\bigotimes_{i=1}^n A(D_i)(\mu_i) = \prod_{i=1}^n \Gr_{G, \Ran, \leq \mu_i}(D_i\subseteq \cc),$$

so it suffices to show that if $\mu_1 + \mu_2 \leq \mu$, then there are maps $\Gr_{G, \Ran, \leq \mu_1}(D_1\subseteq \cc) \times \Gr_{G, \Ran, \leq \mu_2}(D_2\subseteq \cc)\to \Gr_{G, \Ran, \leq \mu}(D\subseteq \cc)$. The argument for this is exactly as in <cit.>.

We next need to show that the $\mathrm{N}(\mathrm{Disk}(\cc))_\mathrm{nu}$-algebra $A$ defined above is locally constant, i.e., that if $D\subseteq D'$ is an embedding of disks, then $A(D)\to A(D')$ is an equivalence of functors $\Lambda^{\vee,\pos}\to \Top$. This follows from <cit.>. To conclude, it suffices (by <cit.>) to establish the existence of a quasi-unit for the functor $A:\Lambda^{\vee,\pos}\to \Top$, i.e., a map $1_{\Fun(\Lambda^{\vee,\pos}, \Top)}\to A$ which is both a left and right unit up to homotopy. Since the unit in $\Fun(\Lambda^{\vee,\pos}, \Top)$ is the functor sending $\mu\in \Lambda^{\vee,\pos}$ to the point $\ast$, a quasi-unit is the datum of a map $\ast \to \Gr_{G, \leq \mu}(\cc)$ for each $\mu\in \Lambda^{\vee,\pos}$. As in the proof of <cit.>, this can be taken to be the inclusion of the point corresponding to the trivial $G$-bundle over $\AA^1$ with the canonical trivialization away from the origin.

With <ref> in mind, we can now use <ref> to compute the $T$-equivariant homology of $\Gr_G$.

There is an equivalence in $\Alg_\E{2}(\coCAlg(\QCoh(\cM_T)))$:

$$\cf_T(\Gr_T(\cc))^\vee \cong \co(\ld{T}_{A} \times_{\spec(A)} \cM_T).$$

Since the action of $T$ on $\Gr_T(\cc)$ is trivial, we have a canonical equivalence $\cf_T(\Gr_T(\cc))^\vee \simeq \Gr_T(\cc)_+ \otimes \cf_T(\ast)^\vee$. By definition, $\cf_T(\ast)^\vee \simeq \co_{\cM_T}$. We conclude that $\cf_T(\Gr_T(\cc))^\vee$ is equivalent as an $\E{2}$-$A$-algebra to $C_\ast(\Gr_T(\cc); A) \otimes_A \co_{\cM_T}$. Since $BT(\cc) \simeq B^2\Lambda^\vee$, there is an equivalence $\Gr_T(\cc) \simeq \Lambda^\vee$ of $\E{2}$-spaces. Therefore, $C_\ast(\Gr_T(\cc); A) \simeq A[\Lambda^\vee]$ as $\E{2}$-$A$-algebras, which is $\co(\ld{T}_A)$. This implies the desired claim.

Can <ref> be upgraded to an equivalence of $\E{3}$-$A$-algebras for a geometrically defined $\E{3}$-algebra structure on $\cf_T(\Gr_T(\cc))^\vee$? This additional structure is crucial for a statement of the geometric Satake correspondence which is $\E{3}$-monoidal.

Let $T^\ast_\GG \ld{T}_A$ denote $\ld{T}_{A} \times_{\spec(A)} \cM_T$, and let $T^\ast_\GG \ld{T}$ denote its underlying scheme (over $\cM_{T,0}$).

Note that if $\GG = \GG_a$, then $T^\ast_\GG \ld{T}$ is the cotangent bundle of $\ld{T}$, while if $\GG = \GG_m$, then $T^\ast_\GG \ld{T} = \ld{T} \times T$. If $\fr{B}_\GG$ denotes the blowup of $T^\ast_\GG \ld{T}$ at the closed subscheme given by $\cM_{T_\alpha,0}$ and the zero set of $e^{\alpha^\vee}-1$ for $\alpha\in \Phi$, then define $(T^\ast_\GG \ld{T})^\bl$ as the complement of the proper preimage of $\cM_{T_\alpha,0}$ in $\fr{B}_\GG$ for $\alpha\in \Phi$.

Let $G$ be a connected semisimple algebraic group over $\cc$. Then there is a $W$-equivariant isomorphism
$\spec \pi_0 \cf_T(\Gr_G(\cc))^\vee \cong (T^\ast_\GG \ld{T})^\bl$ of schemes over $\cM_{T,0}$, where the left-hand side denotes the relative $\spec$.

There is an $\E{2}$-map $\Gr_T(\cc) \to \Gr_G(\cc)$, which induces an $\E{2}$-map $\cf_T(\Gr_T(\cc))^\vee\to \cf_T(\Gr_G(\cc))^\vee$. This is given by dualizing the map $r: \cf_T(\Gr_G(\cc)) \to \cf_T(\Gr_T(\cc))$ of $\E{2}$-coalgebras in $\QCoh(\cM_T)$. The non-$W$-equivariant claim now follows from <ref>, since $r$ induces an injection on $\pi_0$, and the (cocommutative) Hopf algebra structure on $\pi_0 \cf_T(\Gr_T(\cc))$ is given by the dual of the equivalence of <ref>. Proving $W$-equivariance requires a bit more work, but can easily be incorporated by keeping track of the $W$-action throughout the above discussion.

The $T$-equivariant and $G$-equivariant $A$-cohomologies of $\Gr_G(\cc)$ are significantly easier to compute in terms of the stack $\cM_G$ (without any reference to root data); see <ref>. In particular, see <ref> for an alternative argument for <cit.> using Hochschild homology and the Hochschild-Kostant-Rosenberg theorem.

Suppose $A = \KU$, so that $\GG = \GG_m$ and $c_\alpha$ is $e^\alpha-1$.
It follows from <ref> that replacing $T$ with $\ld{T}$, we get an isomorphism between $\pi_0 \cf_{\ld{T}}(\Gr_\ld{G}(\cc))^\vee$ and $\pi_0 (T_{A} \times_{\spec(A)} \ld{T}_A)[\tfrac{e^{\alpha}-1}{e^{\alpha^\vee}-1}, \alpha\in \Phi]$. Therefore, $\pi_0 \cf_{{T}}(\Gr_{G}(\cc))^\vee$ and $\pi_0 \cf_{\ld{T}}(\Gr_\ld{G}(\cc))^\vee$ are both obtained from the blowup $\fr{B}_{\GG_m}$ of $T^\ast_\GG \ld{T}$ by deleting the proper preimage of two different closed subschemes which are Langlands dual to each other. In particular, the Langlands self-duality of the blowup $\fr{B}_{\GG_m}$ swaps the affine pieces $\spec \pi_0 \cf_{T}(\Gr_{G}(\cc))^\vee$ and $\spec \pi_0 \cf_{\ld{T}}(\Gr_\ld{G}(\cc))^\vee$ in $\fr{B}_{\GG_m}$.

When $G = \SL_2$ or $\PGL_2$, we can explicitly verify <ref> at least after base-changing along $C_T^\ast(\ast; A) \to C^\ast(\ast; A)$. We will identify $\PGL_2$ with $\SO_3$ (via the $\PGL_2$-action on $\fr{pgl}_2$ which preserves the quadratic form given by the determinant). If $A = \QQ[\beta^{\pm 1}]$, for instance, <ref> says:

\begin{align*}
\pi_0 C_\ast^{S^1}(\Omega S^3; \QQ[\beta^{\pm 1}]) & \cong \QQ[x,y^{\pm 1}, \tfrac{y-1}{x}], \\
\pi_0 C_\ast^{S^1}(\Omega \SO(3); \QQ[\beta^{\pm 1}]) & \cong \QQ[x,y^{\pm 1}, \tfrac{y^2-1}{2x}].
\end{align*}

After killing $x$, the fraction $\tfrac{y-1}{x}$ (resp. $\tfrac{y^2-1}{x}$) defines a polynomial generator, and so we have

\begin{align*}
\pi_0 C_\ast(\Omega S^3; \QQ[\beta^{\pm 1}]) & \cong \QQ[\tfrac{y-1}{x}], \\
\pi_0 C_\ast(\Omega \SO(3); \QQ[\beta^{\pm 1}]) & \cong \QQ[y^{\pm 1}, \tfrac{y^2-1}{2x}]/(y^2-1).
\end{align*}

The second of these isomorphisms is compatible with the identification $\Omega \SO(3) \simeq \Z/2 \times \Omega S^3$ arising from the isomorphism $S^3/(\Z/2) \cong \SO(3)$ (but note that the equivalence $\Omega \SO(3) \simeq \Z/2 \times \Omega S^3$ is not one of $\E{1}$-spaces). Similarly, if $A = \KU$, <ref> says:

\begin{align*}
\pi_0 C_\ast^{S^1}(\Omega S^3; \KU) & \cong \Z[x^{\pm 1},y^{\pm 1}, \tfrac{y-1}{x-1}], \\
\pi_0 C_\ast^{S^1}(\Omega \SO(3); \KU) & \cong \Z[x^{\pm 1},y^{\pm 1}, \tfrac{y^2-1}{x^2-1}].
\end{align*}

After killing $x-1$, the fraction $\tfrac{y-1}{x-1}$ (resp. $\tfrac{y^2-1}{x^2-1}$) defines a polynomial generator, and so we have

\begin{align*}
\pi_0 C_\ast(\Omega S^3; \KU) & \cong \Z[\tfrac{y-1}{x-1}], \\
\pi_0 C_\ast(\Omega \SO(3); \KU) & \cong \Z[y^{\pm 1}, \tfrac{y^2-1}{x^2-1}]/(y^2-1).
\end{align*}

Again, this is compatible with the identification $\Omega \SO(3) \simeq \Z/2 \times \Omega S^3$.

In the case $G = \SL_2$, we refer the reader to <ref> and <ref> for an explicit description of $\H^{G\times S^1_\rot}_\ast(\Gr_G(\cc); \cc)$ and $\KU^{G\times S^1_\rot}_0(\Gr_G(\cc)) \otimes \cc$.

§.§ Quantized equivariant homology of $\Gr_T$

We now explore the equivariant homology of $\Gr_T$ in more detail; no GKM theory is required here, but several interesting algebraic structures turn up.
Let us begin by recalling that <ref> gives a $W$-equivariant equivalence $\cf_T(\Gr_T(\cc))^\vee \cong \co(\ld{T}_{A} \times_{\spec(A)} \cM_T)$, which can be thought of as giving an equivalence between $\ld{T}_{A} \times_{\spec(A)} \cM_T$ and the “$\E{2}$-$\cM_T$-scheme $\spec \cf_T(\Gr_T(\cc))^\vee$”. This admits a natural deformation given by the loop-rotation equivariant homology $\cf_{\tilde{T}}(\Gr_T(\cc))^\vee$. Since $\tilde{T} = T \times \GG_m^\rot$, there is an equivalence $\cM_{\tilde{T}} \simeq \cM_T \times \GG$, where the second factor is identified as $\cM_{\GG_m^\rot}$.

Let $\GG_0$ be a smooth $1$-dimensional group scheme over a base commutative ring, let $T$ be a compact torus, let $\Lambda$ (resp. $\Lambda^\vee$) denote the (co)character lattice of $T$, and let $\cM_{0,T} = \Hom(\Lambda, \GG_0)$.
Let $\lambda$ be a cocharacter of $T$, so that $\lambda$ defines a homomorphism $\Lambda \to \Z$, and hence a homomorphism $\lambda^\ast: \GG_0 \to \cM_{0,T}$. In turn, this defines a map

$$f^\lambda: \cM_{0,\tilde{T}} \simeq \cM_{0,T} \times \GG_0 \xar{\pr \times \lambda^\ast} \cM_{0,T}.$$

If $y$ is a local section of $\co_{\cM_{0,T}}$, we will write $\lambda^\ast(y)$ to denote the resulting local section of $\co_{\cM_{0,\tilde{T}}}$.
Let $\cd_{\ld{T}}^{\GG_0}$ denote the quotient of the associative $\co_{\GG_0}$-algebra $\co_{\cM_{0,\tilde{T}}}\pdb{x_\lambda | \lambda\in \Lambda}$ by the relations given locally by

$$x_\lambda \cdot x_\mu = x_{\lambda+\mu}, \  y \cdot x_\lambda = x_\lambda \cdot \lambda^\ast(y).$$

Here, $\lambda,\mu\in \Lambda^\vee$, and $y$ is a local section of $\co_{\cM_{0,T}}$. We will call $\cd_{\ld{T}}^{\GG_0}$ the algebra of $\GG_0$-differential operators.

The algebra $\cd_{\ld{T}}^{\GG_0}$ satisfies a Mellin transform: namely, it follows from unwinding the definition that there is an equivalence

$$\LMod_{\cd_{\ld{T}}^{\GG_0}}(\QCoh(\GG_0)) \simeq \QCoh(\cM_{0,\tilde{T}}/\Lambda),$$

where $\lambda\in \Lambda$ acts on $\cM_{0,\tilde{T}}$ via $y\mapsto \lambda^\ast y$.

If $A$ is a complex-oriented even-periodic $\Eoo$-ring and $\GG_0$ is the $\pi_0(A)$-group underlying a oriented commutative $A$-group $\GG$, we will write $\cd_{\ld{T}}^\GG$ to denote $\cd_{\ld{T}}^{\GG_0}$, and refer to it as the algebra of $\GG$-differential operators. We hope this does not cause any confusion.

There is an isomorphism $\pi_0 \cf_{\tilde{T}}(\Gr_T(\cc))^\vee \cong \cd_{\ld{T}}^\GG$ of $\pi_0 \co_\GG$-algebras.

Since $\Gr_T(\cc) \simeq \Omega T_c \simeq \Lambda^\vee$, it is easy to see that $\pi_0 \cf_{\tilde{T}}(\Gr_T(\cc))^\vee \cong \bigoplus_{\lambda\in \Lambda^\vee} \pi_0 \co_{\cM_{\tilde{T}}}$; let $x_\lambda$ be a generator of the summand indexed by $\lambda\in \Lambda^\vee$. If $\lambda\in \Lambda^\vee = \Hom(\Lambda, \Z)$, the map $\Omega T_c \to \Omega T_c$ given by multiplication-by-$\lambda$ is $T\times S^1_\rot$-equivariant for the homomorphism $T\times S^1_\rot \to T\times S^1_\rot$ given by $(t, \theta) \mapsto (t \lambda(\theta), \theta)$, where $\lambda$ is viewed as a homomorphism $S^1 \to T$. On weight lattices, this homomorphism induces the map $\Lambda \times \Z \to \Lambda \times \Z$ which sends $(\mu, n) \mapsto (\mu, n+\lambda^\vee(\mu))$. In particular, the composite $\Lambda \to \Lambda \times \Z \to \Lambda \times \Z$ sends $\mu \mapsto (\mu, \lambda^\vee(\mu))$. Applying $\Hom(-, \GG)$ to this composite precisely produces the map $f^\lambda: \cM_{\tilde{T}} \to \cM_T$ from <ref>. This implies the desired identification of $\pi_0 \cf_{\tilde{T}}(\Gr_T(\cc))^\vee$.

Let $T \cong S^1$ be a torus of rank $1$ (for simplicity).
Suppose $A = \QQ[\beta^{\pm 1}]$, so $\GG = \hat{\GG}_a$ and $\pi_0 \co_{\GG} \cong \QQ\pw{\hbar}$. Then the algebra of <ref> is the quotient of the $\QQ\pw{\hbar}$-algebra $\QQ\pw{\hbar}\pdb{y, x^{\pm 1}}$ by the relation $yx = x(y+\hbar)$. In other words, $y$ acts as the operator $\hbar x\partial_x$, so we simply have that

$$\H^{\tilde{T}}_0(\Gr_T(\cc); \QQ[\beta^{\pm 1}]) \cong \H^{\tilde{T}}_\ast(\Gr_T(\cc); \QQ) \cong \QQ\pw{\hbar}\pdb{\hbar x\partial_x, x^{\pm 1}}.$$

This has been stated previously as <cit.>.
In particular, the localization $\H^{\tilde{T}}_0(\Gr_T(\cc); \QQ[\beta^{\pm 1}])[\hbar^{-1}]$ is isomorphic to the rescaled Weyl algebra $\cd_{\ld{T}}^\hbar$; this is the motivation behind the terminology in <ref>.
Note that <ref> simply reduces to the standard Mellin transform, which gives an equivalence between $\DMod_\hbar(\ld{T})$ and $\QCoh(\fr{t}_{\QQ\pw{\hbar}}/\Lambda)$.

Again, let $T \cong S^1$ be a torus of rank $1$ (for simplicity).
Suppose $A = \KU$, so $\GG = \GG_m$ and $\pi_0 \co_\GG \cong \Z[q^{\pm 1}]$. Then the algebra of <ref> is the quotient of the $\Z[q^{\pm 1}]$-algebra $\Z[q^{\pm 1}]\pdb{y^{\pm 1}, x^{\pm 1}}$ by the relation $yx = qxy$. (This is also known as the “quantum torus”.) In other words, $y$ acts as the operator $q^{x\partial_x}$ sending $f(x) \mapsto f(qx)$, so we simply have that

$$\KU^{\tilde{T}}_0(\Gr_T(\cc)) \cong \Z[q^{\pm 1}]\pdb{q^{x\partial_x}, x^{\pm 1}}.$$

This is closely related to the $q$-Weyl algebra $\cd_q = \Z[q^{\pm 1}]\pdb{\Theta, x^{\pm 1}}/(\Theta x = x(q\Theta+1))$ for $\ld{T} = \GG_m$: indeed, since the logarithmic $q$-derivative $\Theta = x\nabla_q$ is given by the fraction $\frac{q^{x\partial_x}-1}{q-1}$, the pullback of $\cd_{\ld{T}}^\GG$ along $\GG_m-\{1\} \hookrightarrow \GG_m$ is isomorphic to the algebra $\cd_q[\frac{1}{q-1}]$.
Note that <ref> gives a “$q$-Mellin transform”, i.e., an equivalence between $\LMod_{\KU^{\tilde{T}}_0(\Gr_T(\cc))}$ and $\QCoh((\GG_m)_{\Z[q^{\pm 1}]}/\Z)$, where $\Z$ acts on $(\GG_m)_{\Z[q^{\pm 1}]}$ by sending $y\mapsto qy$.

Using <ref>, there is an equivalence $\Loc_{T_c}(T_c; A) \simeq \LMod_{\cf_T(\Gr_T(\cc))^\vee}$. Since $\pi_0 \cf_{\tilde{T}}(\Gr_T(\cc))^\vee \cong \cd_{\ld{T}}^\GG$ is a “quantization” of $\pi_0 \cf_T(\Gr_T(\cc))^\vee \cong \co_{T^\ast_\GG \ld{T}}$ (i.e., an associative deformation of $T^\ast_\GG \ld{T}$ along $\GG$), and <ref> implies an equivalence of $\E{1}$-$A_\QQ$-algebras $\cf_{\tilde{T}}(\Gr_T(\cc))^\vee \otimes \QQ \cong \cd_{\ld{T}}^\GG \otimes_{\pi_0 A} A_\QQ$, we see that $\LMod_{\cd_{\ld{T}}^\GG} \otimes_{\pi_0 A} A_\QQ$ defines a “quantization” of $\Loc_{T_c}(T_c; A) \otimes \QQ$.

Let us briefly outline the relationship between the algebra $\cd_{\ld{T}}^{\GG_0}$ of <ref> and the $F$-de Rham complex of [23].

For the purpose of this discussion, we will assume that $T \cong S^1$ is a torus of rank $1$, so that $\ld{T} \cong \GG_m$. We will also fix an invariant differential form on the formal completion $\hat{\GG}_0$ of $\GG_0$ at the zero section, so that there is an isomorphism $\hat{\GG}_0 \cong \spf R\pw{t}$ of formal $R$-schemes. Let $F(x,y)$ denote the resulting formal group law over $R$, and define the $n$-series of $F$ by

$$[n]_F := \overbrace{F(t, F(t, F(t, \cdots F(t, t) \cdots )))}^n.$$

We will often write $x+_Fy = x+_\GG y$ to denote $F(x,y)$.
Let $\hat{\cd}_{\ld{T}}^{\GG_0}$ denote the completion of $\cd_{\ld{T}}^{\GG_0}$ at the zero section of $\cM_{0,\tilde{T}} \cong \cM_{0,T} \times \GG_0$.

Let $\hat{\GG}_0$ be a $1$-dimensional formal group over a commutative ring $R$, and let $\Cart(\hat{\GG}_0)$ denote its Cartier dual (see <cit.> for more on Cartier duals of formal groups). Then there is an equivalence of categories $\QCoh(\hat{\GG}_0) \simeq \QCoh(B\Cart(\hat{\GG}_0))$ sending the convolution tensor product on the left-hand side to the usual tensor product on the right-hand side. Under this equivalence, the functor $\QCoh(\hat{\GG}_0) \to \Mod_R$ given by restriction to the zero section is identified with the functor $\QCoh(B\Cart(\hat{\GG}_0)) \to \Mod_R$ given by pullback along the map $\spec(R) \to B\Cart(\hat{\GG}_0)$.

There is a canonical action of $\hat{\cd}_{\ld{T}}^{\GG_0}$ on $(\GG_m)_{R\pw{t}} = \spf R\pw{t}[x^{\pm 1}]$ such that $R\pw{t}[x^{\pm 1}] \otimes_{\hat{\cd}_{\ld{T}}^{\GG_0}} R\pw{t}[x^{\pm 1}]$ is isomorphic to the two-term complex

$$C^\bull = (R\pw{t}[x^{\pm 1}] \to R\pw{t}[x^{\pm 1}]dx), \ x^n \mapsto [n]_F x^n dx$$

from <cit.>.

Since $T$ is of rank $1$, there is an isomorphism $\cM_{0,T} \cong \GG_0$, and hence an isomorphism $\hat{\cM}_{0,T} \cong \hat{\AA}^1$ of formal $R$-schemes, where $\hat{\cM}_{0,T}$ denotes the completion of $\cM_{0,T}$ at the zero section. Let $y$ be a local coordinate on $\cM_{0,T}$.
Then, $\hat{\cd}_{\ld{T}}^{\GG_0}$ is isomorphic to the quotient of the associative $\hat{\co}_{\GG_0}$-algebra $\hat{\co}_{\GG_0 \times \cM_{0,T}}\pdb{x^{\pm 1}}$ subject to the relation $yx = x(y +_\GG t)$.
The $t$-adic filtration on $\hat{\cd}_{\ld{T}}^{\GG_0}$ therefore has associated graded $\gr(\hat{\cd}_{\ld{T}}^{\GG_0}) \cong \hat{\co}_{\cM_{0,T}}[x^{\pm 1}]\pw{\ol{t}}$, where $\ol{t}$ lives in weight $1$. View $R$ as a $\co_{\cM_{0,T}}$-algebra via the zero section, i.e., the augmentation $\co_{\cM_{0,T}} \to R$. Then, the action of $\gr(\hat{\cd}_{\ld{T}}^{\GG_0})$ on $R[x^{\pm 1}]\pw{\ol{t}}$ is induced by the augmentation $\hat{\co}_{\cM_{0,T}} \to R$. The isomorphism $\hat{\cM}_{0,T} \cong \hat{\AA}^1$ of formal $R$-schemes then implies an isomorphism $R \otimes_{\co_{\cM_{0,T}}} R \cong R[\epsilon]/\epsilon^2$ with $\epsilon$ in homological degree $1$.
It follows that

$$R\pw{\ol{t}}[x^{\pm 1}] \otimes_{\gr(\hat{\cd}_{\ld{T}}^{\GG_0})} R\pw{\ol{t}}[x^{\pm 1}] \simeq R\pw{\ol{t}}[x^{\pm 1}][\epsilon]/\epsilon^2,$$

where $\ol{t}$ is in weight $1$ and degree $0$, and $\epsilon$ is in weight $0$ and degree $1$.

By <ref>, the $t$-adic filtration on $\hat{\cd}_{\ld{T}}^{\GG_0}$ is equivalent to the data of a $\Cart(\hat{\GG}_0)$-action on $R\pw{\ol{t}}[x^{\pm 1}] \otimes_{\gr(\hat{\cd}_{\ld{T}}^{\GG_0})} R\pw{\ol{t}}[x^{\pm 1}] \simeq R\pw{\ol{t}}[x^{\pm 1}][\epsilon]/\epsilon^2$. This in turn is equivalent to the data of a differential

$$\nabla: R\pw{\ol{t}}[x^{\pm 1}] \to R\pw{\ol{t}}[x^{\pm 1}]\cdot \epsilon$$

satisfying a $\hat{\GG}_0$-analogue of the Leibniz rule: if[Note that $\nabla$ has to be homogeneous in the degree of the monomial in $x$, as can be seen by keeping track of the $x$-weight.] $\nabla(x^n) = f(n) x^n \epsilon$ for some $f(n)\in R\pw{t}$, then $f(n+m) = f(n) +_\GG f(m)$.
It therefore suffices to determine $\nabla(x)$; but the relation $yx = x(y +_\GG t)$ forces $\nabla(x) = tx\epsilon$. This implies that

$$\nabla(x^n) = (\overbrace{t +_\GG \cdots +_\GG t}^n) x^n \epsilon = [n]_F x^n \epsilon,$$

as desired.

When $\GG_0 = \hat{\GG}_a$ over[Of course, one can work over $\Z$ too; we just chose $\QQ$ to continue with <ref>.] $\QQ$, the complex $C^\bull$ is

$$C^\bull = (\QQ\pw{\hbar}[x^{\pm 1}] \to \QQ\pw{\hbar}[x^{\pm 1}]dx), \ x^n \mapsto n\hbar x^n dx.$$

Indeed, since $yx = x(y+\hbar)$, we have $yx^n = x^n(y+n\hbar)$; since $t = \hbar$ in this case, we have $x^n\mapsto n\hbar x^n \epsilon$. This is evidently a $\hbar$-rescaling of the classical de Rham complex of $\GG_m$.

When $\GG_0 = \GG_m$ over $\Z$, the complex $C^\bull$ is

$$C^\bull = (\Z\pw{q-1}[x^{\pm 1}] \to \Z\pw{q-1}[x^{\pm 1}]dx), \ x^n \mapsto (q^n-1) x^n dx.$$

Indeed, since $yx = x(qy)$, we have $yx^n = x^n (q^n y)$, and hence

$$(y-1)x^n = x^n(q^n y - 1) = x^n((y-1) +_F (q^n-1)),$$

where $F(z,w) = z + w + zw$ is the multiplicative formal group law; since $t = q-1$ in this case, we have $x^n \mapsto (q^n-1) x^n \epsilon$. The complex $C^\bull$ is a $(q-1)$-rescaling of the $q$-de Rham complex of $\GG_m$ from [86].

The complex of <ref> is not quite the $F$-de Rham complex of <cit.>; rather, if $\eta_t$ denotes the décalage functor of [12] with respect to the ideal $(t)\subseteq R\pw{t}$, the $F$-de Rham complex is given by the décalage $\eta_t C^\bull$. In particular, the complex of <ref> is isomorphic to the $F$-de Rham complex after inverting $t$. One can modify the algebra $\cd_{\ld{T}}^{\GG_0}$ of <ref> (by performing a noncommutative analogue of an affine blowup/deformation to the normal cone[For instance, in the case of <ref>, this procedure simply adjoins the fraction $\frac{y}{\hbar}$; in the case of <ref>, this procedure simply adjoins the fraction $\frac{y-1}{q-1}$.]) such that the relative tensor product as in <ref> is the $F$-de Rham complex itself. Since it is not needed for this article, we will not describe this modification here.

<ref> says that $\hat{\cd}_{\ld{T}}^{\GG_0}$ is Koszul dual to the complex $C^\bull$. Forthcoming work of Arpon Raksit shows that the décalage $\eta_t C^\bull$ can be recovered from the “even filtration” (in the sense of [48]) on the periodic cyclic homology $\HP(\tau_{\geq 0} A[x^{\pm 1}]/\tau_{\geq 0} A)$. See also the discussion in <cit.>.
Using similar techniques, one can show that $C^\bull$ can be recovered from the even filtration on the negative cyclic homology $\HC^-(A[x^{\pm 1}]/A) = \HH(A[x^{\pm 1}]/A)^{hS^1}$.

Recalling that $T = S^1$, this $\Eoo$-$A$-algebra is simply $\HC^-(A[\Omega T]/A)$. The Hochschild homology $\HH(A[\Omega T]/A) \simeq A \otimes \THH(S[\Omega T])$ is $S^1$-equivariantly equivalent to the $A$-chains $C_\ast(\cL T; A)$ on the free loop space of $T$. (For a reference, see <cit.>.) The $A$-chains $A[\cL T]$ is $S^1$-equivariantly Koszul dual[This Koszul duality essentially stems from the (nonequivariant) decomposition $\cL T \simeq T \times \Omega T$.] to $A[\Omega T]^{hT}$; this can be identified as a completion of $\cf_T(\Omega T)^\vee$ at the zero section of $\cM_T$. In other words, $\HC^-(A[\Omega T]/A)$ is Koszul dual to the completion of $\cf_{T\times S^1_\rot}(\Omega T)^\vee$ at the zero section of $\cM_T \times \GG$. This is the topological source of the Koszul duality of <ref>.

In <ref>, we mentioned that the Koszul duality between $\GG$-differential operators and the $F$-de Rham complex manifests in topology as the Koszul duality between $\cf_{T \times S^1_\rot}(\Omega T)^\vee$ and $\HC^-(A[\Omega T]/A)$. There is clearly nothing special about $T$ in this Koszul duality: given a sufficiently robust theory of $G$-equivariant $A$-(co)homology (see the discussion surrounding <ref>), there is also a Koszul duality between $\cf_{G \times S^1_\rot}(\Omega G)^\vee$ and $\HC^-(A[\Omega G]/A) = A[\cL G]^{hS^1}$. When $A = \cc[\beta^{\pm 1}]$, <cit.> states that $\cf_{G \times S^1_\rot}(\Omega G)^\vee$ can be identified with (the $2$-periodification of) the bi-Whittaker reduction $\ld{N}^-\backslash_\chi \cd_{\ld{G}}/_\chi \ld{N}^-$. Using the results of this article, it is also possible to compute $A[\cL G]^{hS^1}$ in this manner, at least if we assume that small primes are inverted: the zeroth graded piece of the “even filtration” on $A[\cL G]^{hS^1}$ looks like the $2$-periodification of the $F$-de Rham complex of $Z_f(\ld{B})$ for a chosen principal nilpotent element $f\in \ld{\g}$. We plan to explain this in future work.

§ THE COHERENT SIDE

§.§ Langlands duality over $\QQ[\beta^{\pm 1}]$

We now turn to the coherent side of the geometric Satake equivalence. For general $\GG$, it is not obvious what the Langlands dual algebraic stack should be; we will discuss this in <ref>. As a warmup, we will focus only on $\QQ[\beta^{\pm 1}]$ in this section (this is more for pedagogical purposes than originality).

Let $G$ be a connected reductive group over $\cc$, and fix the rest of notation as in <ref>. Fix a principal nilpotent element $e\in \fr{n}$, and let $(e,f,h)$ be the associated $\sl_2$-triple in $\g$. Let $\g^e$ be the centralizer (so $\g = \g^e \oplus [e,\g]$), and let $\cS := f + \g^e\subseteq \g^\reg$ be the Kostant slice. The composite $f + \g^e \to \g \to \g\mmod G \cong \fr{t}\mmod W$ is an isomorphism, by [53].

Let $\tilde{\g} = \fr{b} \times_B G$ be the Grothendieck-Springer resolution, so that $\tilde{\g}/G \simeq \fr{b}/B$. We will often work with $\tilde{\g}^\ast$ instead, defined as $\fr{b}^\ast \times_B G$. There is a map $\tilde{\chi}: \tilde{\g} \to \fr{t}$ which sends a pair $(x\in \Ad_g(\fr{b}))$ to the inverse image under the isomorphism $\fr{t} \to \fr{b} \to \fr{b}/\fr{n}$ of the image of $g^{-1} x\in \fr{b}$. Let $\tilde{\cS}$ denote the fiber product $\cS\times_\g \tilde{\g}$, so that $\tilde{\cS} \subseteq \tilde{\g}^\reg = \g^\reg \times_\g \tilde{\g}$. Then, Kostant's result on the Kostant slice implies formally that the composite $\tilde{\cS} \to \tilde{\g} \xar{\tilde{\chi}} \fr{t}$ is an isomorphism. We will often abusively write the inclusion of $\tilde{\cS}$ as a map $\kappa: \fr{t} \to \tilde{\g}$.

In fact, we will only care about the composite $\fr{t} \to \tilde{\g} \to \tilde{\g}/G$ below, so we will also denote it by $\kappa$. If we identify $\tilde{\g}/G \cong \fr{b}/B$, then the map $\kappa$ admits a simple description: it is the composite $\fr{t} \to \fr{b} \to \fr{b}/B$ which sends $x\mapsto f+x$. This is proved, for instance, in <cit.>, where it is shown that there is a unique map $\mu: f+\fr{t} \to N$ such that $\Ad_{\exp(\mu(x))}(x) \in f + \g^e$; this further implies that the image of any $x\in \fr{t}$ under the map $\fr{t} \to \fr{t}\mmod W \xar{\kappa} \g$ can be identified with $\Ad_{\exp(\mu(x+f))}(x+f)$.

Fix a nondegenerate invariant bilinear form on $\g$, to identify $\g$ with $\g^\ast$. The first main result of this section is the following; it is essentially equivalent to <cit.> and the rationalization of <cit.>.

Let $G$ be a connected and simply-connected semisimple algebraic group over $\cc$. Let $A$ be an $\Eoo$-$\QQ[\beta^{\pm 1}]$-algebra, and let $\GG = \GG_a$ (so $\cM_T$ is the affine space $\fr{t}[2]$ over $A$).
View $\ld{\fr{t}}^\ast$, $\ld{\fr{n}}$, $\ld{\g}$, and $\ld{B}$ as schemes over $\QQ$.
Then $\QCoh(\ld{\fr{t}}^\ast)$ admits the structure of a module over $\IndCoh((\tilde{\ld{\cN}} \times_{\ld{\g}} \{0\})/\ld{G})$, where the fiber product is (always) derived, such that there is an equivalence

$$\End_{\IndCoh((\tilde{\ld{\cN}} \times_{\ld{\g}} \{0\})/\ld{G})}(\QCoh(\ld{\fr{t}}^\ast)) \otimes_\QQ \pi_0 A \simeq \LMod_{\pi_0 C_\ast^T(\Gr_G(\cc); A)} = \Loc^\gr_{T_c}(G_c; A).$$

Recall from [1] that there is an Iwahori-Satake equivalence $\IndCoh((\tilde{\ld{\cN}} \times_{\ld{\g}} \{0\})/\ld{G}) \simeq \Shv(\Gr_G)^I$ over $\cc$, where the right-hand side is normalized appropriately. One should therefore regard <ref> as a bar construction of the restriction of this equivalence (lifted from $\cc$ to $\QQ$) to the regular locus, and more optimistically as a first step towards an alternative proof. See also <ref> for the equivalence resulting from “undoing” the bar construction.

We now turn to the proof of <ref>. For the next two results, we only work on one side of Langlands duality, so we drop the “check”s for notational simplicity. Note that $(\tilde{\ld{\cN}} \times_{\ld{\g}} \{0\})/\ld{G} \cong (\ld{\fr{n}} \times_{\ld{\g}} \{0\})/\ld{B}$; it will be more convenient to work with the latter description.

There is a Koszul duality equivalence $\QCoh(\tilde{\g}^\ast[2]/G) \simeq \IndCoh((\fr{n} \times_{\g} \{0\})/B)$.

We will give two proofs of the following fact.

Work over a field $k$ of characteristic $0$, and view $\QCoh(\fr{t}^\ast)$ as a $\QCoh(\tilde{\g}^\ast/G)$-module via the Kostant slice $\kappa: \fr{t}^\ast \to \tilde{\g}^\ast$. Then there is an equivalence
$\End_{\QCoh(\tilde{\g}^\ast/G)}(\QCoh(\fr{t}^\ast)) \simeq \QCoh((T^\ast T)^\bl)$.

We may identify $\End_{\QCoh(\tilde{\g}^\ast/G)}(\QCoh(\fr{t}^\ast))$ with $\QCoh(\fr{t}^\ast \times_{\tilde{\g}^\ast/G} \fr{t}^\ast)$. We will show, in fact, that there is a Cartesian square

\begin{equation}\label{intersection-kostant}
\xymatrix{
(T^\ast T)^\bl \ar[r] \ar[d] & \fr{t}^\ast \ar[d]^-\kappa\\
\fr{t}^\ast \ar[r]_-\kappa & \tilde{\g}^\ast/G \simeq \fr{b}^\ast/B.
\end{equation}

This is an analogue of <cit.> and <cit.>.
(Note that since $\fr{t}^\ast \to \tilde{\g}^\ast$ lands in the open locus $\tilde{\g}^{\ast, \reg}$, it does not matter whether we intersect $\fr{t}^\ast$ with itself in $\tilde{\g}^\ast/G$ or in $\tilde{\g}^{\ast, \reg}/G$; indeed, the intersection $\tilde{\g}^{\ast, \reg} \times_{\tilde{\g}^\ast} \tilde{\g}^{\ast, \reg}$ is just $\tilde{\g}^{\ast, \reg}$.) In what follows, it will be convenient (notationally) to use the chosen nondegenerate invariant bilinear form on $\g$ to identify $\fr{b}^\ast$ with the opposite Borel $\fr{b}^-$ and $N$ with its opposite unipotent, and then to flip the role of $\fr{b}$ and $\fr{b}^-$, etc.

Recall that the Kostant slice $\cS\subseteq \g$ is transverse to the regular $G$-orbits, and intersects each orbit exactly once; this implies that the image of the map $\kappa: \fr{t} \to \tilde{\g}$ is transverse to the regular $G$-orbits on $\tilde{\g}$, and intersects each orbit exactly once. In particular, if $C$ denotes the locally closed subvariety of $\tilde{\g} \times G$ consisting of pairs $(x,g)$ with $x\in \tilde{\g}^\reg$ and $\Ad_g(x) = x$, then $C\mmod G = \fr{t} \times_{\tilde{\g}/G} \fr{t}$ (so we may assume without loss of generality that $x\in \fr{t}$). To compute $C\mmod G$, one can reduce to the case when $G$ has semisimple rank $1$ by the argument of <cit.>. To work out this case, we will assume $G = \SL_2, \PGL_2$.

There are “two” ways to compute in these cases; we will describe both, because each has its own conceptual advantages when generalizing to the multiplicative case (for instance). First, we present the argument which is essentially present in [9]; for this, we will assume $G = \SL_2$. The Grothendieck-Springer resolution $\tilde{\g} = T^\ast(\AA^2-\{0\})/\GG_m$ is the total space of $\co(-1) \oplus \co(-1)$ over $\PP^1$; we will think of a point in $\tilde{\g}$ as a pair $(x\in \sl_2, \ell\subseteq \cc^2)$ such that $x$ preserves $\ell$. The Kostant slice $\kappa:\fr{t} \cong \AA^1 \to \tilde{\g}$ is the map sending $\lambda \in \AA^1$ to the pair $(x, \ell)$ with $x = \begin{psmallmatrix}
0 & \lambda^2 \\
1 & 0
\end{psmallmatrix}$ and $\ell = [\lambda: 1]$. Indeed, this is essentially immediate from the requirement that the following diagram commutes:

\AA^1 \cong \fr{t} \ar[r]^-\kappa \ar[d]_-{\lambda \mapsto \lambda^2} & \tilde{\sl}_2 \ar[d]\\
\AA^1 \cong \fr{t}\mmod W \ar[r]^-\kappa_-{\lambda\mapsto \begin{psmallmatrix}
0 & \lambda \\
1 & 0
\end{psmallmatrix}} & \sl_2.

Moreover, the $\SL_2$-action on $\tilde{\g}$ sends $g\in \SL_2$ and $(x,\ell)$ to $(\Ad_g(x), g\ell)$. If $g = \begin{psmallmatrix}
a & b \\
c & d
\end{psmallmatrix}$, we compute that

0 & \lambda^2 \\
1 & 0
\end{pmatrix} = \begin{pmatrix}
bd-ac\lambda^2 & (a\lambda)^2 - b^2 \\
d^2 - (c\lambda)^2 & ac\lambda^2 - bd
\end{pmatrix}, \ g\cdot [\lambda: 1] = [a\lambda + b: c\lambda + d].$$

From this, we see that $\Ad_g(x) = x$ if and only if $a = d$ and $b = c\lambda^2$, in which case $g$ also fixes $[\lambda: 1]$. In other words, $g = \begin{psmallmatrix}
a & c\lambda^2 \\
c & a
\end{psmallmatrix}$ with $a,c\in k$; in order for $\det(g) = 1$, we need $a^2-c^2\lambda^2=1$. When $\lambda \neq 0$, both $x$ and $g$ are diagonalized by the matrix $\tfrac{1}{2}\begin{psmallmatrix}
1 & -1 \\
-\lambda^{-1} & -\lambda^{-1}
\end{psmallmatrix}\in \SL_2$: the diagonalization of $x$ is $\begin{psmallmatrix}
\lambda & 0 \\
0 & \lambda^{-1}
\end{psmallmatrix}$, and the diagonalization of $g$ is $\begin{psmallmatrix}
t & 0 \\
0 & w
\end{psmallmatrix}$ where $2a = t+w$ and $2\lambda c = t-w$. Since we have $\det(g) = a^2 - (c\lambda)^2 = 1$, we have $w = t^{-1}$. This shows that if $k$ is not of characteristic $2$, then $\fr{t} \times_{\tilde{\sl}_2/\SL_2} \fr{t} \cong \spec k[\lambda, t^{\pm 1}, \tfrac{t-t^{-1}}{\lambda}]$.

The “second” way to reach this calculation (still with $G = \SL_2$) is to use the fact that $\kappa: \fr{t} \to \tilde{\g}/G$ can be identified with the composite $\fr{t} \to \fr{b} \to \fr{b}/B$ sending $x\mapsto f+x$. Then, $\fr{t} \times_{\fr{b}/B} \fr{t}$ is isomorphic to the subvariety of $\fr{t} \times B$ consisting of pairs $(x,g)$ with $x\in \fr{t}$ and $\Ad_g(x+f) = x+f$. If $g = \begin{psmallmatrix}
a & 0 \\
b & a^{-1}
\end{psmallmatrix}\in B$, then

$$\Ad_g \begin{pmatrix}
x & 0 \\
1 & -x
\end{pmatrix} = \begin{pmatrix}
x & 0 \\
2a^{-1}bx + a^{-2} & -x
\end{pmatrix}.$$

Therefore, $\Ad_g(x+f) = x+f$ if and only if

$$2a^{-1}bx + a^{-2} = 1,$$

which forces $b = \tfrac{a-a^{-1}}{2x}$. This implies that $\fr{t} \times_{\fr{b}/B} \fr{t}$ is isomorphic to $\spec k[x, a^{\pm 1}, \tfrac{a-a^{-1}}{x}]$, as desired.

We will now do the calculation with $G = \PGL_2$ via the second method. Again, $\fr{t} \times_{\fr{b}/B} \fr{t}$ is isomorphic to the subvariety of $\fr{t} \times B$ consisting of pairs $(x,g)$ with $x\in \fr{t}$ (identified with the matrix $\begin{psmallmatrix}
x & 0 \\
0 & 0
\end{psmallmatrix} \in \gl_2$) and $\Ad_g(x+f) = x+f$. If $g = \begin{psmallmatrix}
a & 0 \\
b & 1
\end{psmallmatrix}\in B$, then

$$\Ad_g \begin{pmatrix}
x & 0 \\
1 & 0
\end{pmatrix} = \begin{pmatrix}
x & 0 \\
(bx + 1) a^{-1} & 0
\end{pmatrix}.$$

Therefore, $\Ad_g(x+f) = x+f$ if and only if

$$(bx + 1) a^{-1} = 1,$$

which forces $b = \tfrac{a-1}{x}$. This implies that $\fr{t} \times_{\fr{b}/B} \fr{t}$ is isomorphic to $\spec k[x, a^{\pm 1}, \tfrac{1-a}{x}]$, as desired.

As in the first proof of <ref>, it will be convenient to use the chosen nondegenerate invariant bilinear form on $\g$ to identify $\fr{b}^\ast$ with the opposite Borel $\fr{b}^-$ and $N$ with its opposite unipotent, and then to flip the role of $\fr{b}$ and $\fr{b}^-$, etc. We will prove the following variant of <ref>, which in turn implies the desired result: view $\QCoh(\fr{t}^\ast\mmod W)$ as a $\QCoh({\g}^\ast/G)$-module via the Kostant slice. Then there is an equivalence
$\End_{\QCoh({\g}^\ast/G)}(\QCoh(\fr{t}^\ast\mmod W)) \simeq \QCoh((T^\ast T)^\bl\mmod W)$.

Let $\chi$ be a nondegenerate character on $\fr{n}^-$. The $N^-$-action on $G$ via conjugation induces a Hamiltonian $N^-$-action on $T^\ast G$; let $N^- {}_\chi\backslash (T^\ast G)/_\chi N^-$ denote the bi-Whittaker reduction of $T^\ast G$ with respect to this $N^-$-action at the character $\chi\in \fr{n}^{-,\ast}$. Then $(T^\ast T)^\bl\mmod W \cong N^- {}_\chi\backslash (T^\ast G)/_\chi N^-$; see <cit.>, for instance. There is a Morita equivalence between $\QCoh(\g^\ast/G)$ and $\QCoh(T^\ast G)$ (equipped with the convolution monoidal structure); under this equivalence, the $\QCoh(\g^\ast/G)$-module $\QCoh(\g^\ast/_\chi N^-)$ is sent to the $\QCoh(T^\ast G)$-module $\QCoh((T^\ast G)/_\chi N^-)$. We conclude the series of equivalences:

\begin{align*}
\QCoh((T^\ast T)^\bl\mmod W) & \simeq \QCoh(N^- {}_\chi\backslash (T^\ast G)/_\chi N^-) \\
& \simeq \End_{\QCoh(T^\ast G)} (\QCoh((T^\ast G)/_\chi N^-))\\
& \simeq \End_{\QCoh(\g^\ast/G)}(\QCoh(\g^\ast/_\chi N^-)).
\end{align*}

However, Kostant's theorem identifies $\g^\ast/_\chi N^-$ with $\fr{t}^\ast\mmod W$ (viewed as a substack of $\g^\ast/G$ via the Kostant slice), which finishes the proof.

By <ref>, we have $\H_0^T(\Gr_G(\cc); A) = \pi_0 \cf_T(\Gr_G(\cc))^\vee \cong \co_{(T^\ast T)^\bl}$. It follows that $\LMod_{\H_\ast^T(\Gr_G(\cc); A)} \simeq \QCoh((T^\ast \ld{T})^\bl_A)$. Since $\End_{\IndCoh((\tilde{\ld{\cN}} \times_{\ld{\g}} \{0\})/\ld{G})}(\QCoh(\ld{\fr{t}}^\ast)) \simeq \QCoh((T^\ast \ld{T})^\bl)$ by <ref> and <ref>, we conclude the desired result.

So far, we have not emphasized the role of Whittaker reduction in the above story (except for the second proof of <ref>). However, we take a moment to describe this briefly, since it is a key aspect of Langlands duality. Recall that a theorem of Kostant's gives an isomorphism $(f+\fr{b})/N \cong \cS = f + \g^e$. In terms of Whittaker reduction, this says that $\cS \cong \g/_\chi N^-$. Since <ref> is concerned with $\tilde{\g}$ instead of $\g$, we need a slight variant of this statement. Namely, recall the map $\pi: \tilde{\g} \to \g$, let $\mu: \g \to \fr{n}$ be the moment map for the adjoint $N$-action on $\g$, and let $\tilde{\mu}$ denote the composite $\tilde{\g} \to \g \to \fr{n}$. Then $\mu^{-1}(f)$ is the variety $f+\fr{b}$, so that $\tilde{\mu}^{-1}(f)$ is the subscheme of $\tilde{\g}$ spanned by those pairs $(\fr{b}', y\in \fr{b}'\cap (f+\fr{b}))$. Kostant's result implies that there is an isomorphism $\tilde{\mu}^{-1}(f)/N^- \xar{\sim} \fr{t}$.

Whittaker reduction is a key aspect of the Langlands-dual side of <ref>: it is needed to even define the action of $\QCoh(\tilde{\g}^\ast/G)$ on $\QCoh(\fr{t}^\ast)$.

Note that <ref> implies that $\H_0(\Omega G_c; \QQ[\beta^{\pm 1}])$ can be identified with the ring of functions on the centralizer $Z_f(\ld{G})$ of a regular nilpotent element $f\in \ld{\g}$ over $\QQ$. In type $A$ at least, one can directly check that there is such an isomorphism. (Exactly the same argument works in the K-theoretic and elliptic cases, too; in the K-theoretic case, one instead considers the centralizer of a regular unipotent element $f\in \ld{G}$.) For instance, if $\ld{G} = \SL_n$, the centralizer $Z_f(\ld{G})$ is the direct product of $\mu_n$ with a connected (commutative) unipotent group $U_n$. If $(x_1, \cdots, x_{n-1})$ is a point in $U_n$ (corresponding to the element in $Z_f(\SL_n)$ given by the $n\times n$-matrix whose $j$th row is $(0, \cdots, 0, 1, x_1, \cdots, x_{n-j})$), the group operation is given by

$$(x_1, \cdots, x_{n-1}) \cdot (y_1, \cdots, y_{n-1}) = (x_1 + y_1, \cdots, x_{n-1} + x_{n-2} y_1 + \cdots + x_1 y_{n-2} + y_{n-1}).$$

The group scheme $U_n$ is isomorphic over $\QQ$ to $\GG_a^{\times n-1}$, via Newton's identities for the transformation law for expressing the power sum symmetric polynomials in terms of the elementary symmetric polynomials. For instance, the isomorphism between $U_6 \subseteq Z_f(\SL_6)$ and $\GG_a^{\times 5}$ is given by the map

\begin{align}
(x_1, \cdots, x_5) & \mapsto (x_1, x_1^2-2x_2, x_1^3-3x_1 x_2+3x_3, x_1^4 + 2x_2^2 - 4x_4-4x_2x_1^2+4x_1x_3, \nonumber\\
& \ \ \ x_1^5-5x_1^3 x_2 + 5x_1^2 x_3 - 5x_1 (x_4 - x_2^2) - 5x_2 x_3 + 5x_5). \label{eq: power sum}
\end{align}

In general, the transformation can be determined by extracting the coefficient of $(-t)^n/n$ in the power series $\log\left(\sum_{j\geq 0} x_j (-t)^j\right)$.

On the other hand, $G_c$ is a maximal compact subgroup of $\PGL_n(\cc)$, and there is a homotopy equivalence $\Omega \PGL_n(\cc) \simeq \Z/n \times \Omega \SU(n)$, so that

$$\H_0(\Omega \PGL_n(\cc); \QQ[\beta^{\pm 1}]) \simeq \QQ[x^{\pm 1}]/(x^n-1) \otimes_\Z \H_0(\Omega \SU(n); \Z[\beta^{\pm 1}]).$$

Under Langlands duality, the $\mu_n$ factor in $Z_f(\SL_n)$ comes from the first tensor factor. Similarly, $\spec \H_0(\Omega \SU(n); \Z[\beta^{\pm 1}])$ is a connected unipotent group scheme: for instance, there is a Bott periodicity equivalence $\Omega \SU \simeq \BU$ (where $\SU = \colim_{n\to\infty} \SU(n)$), so $\spec \H_0(\Omega \SU; \Z[\beta^{\pm 1}])$ can be identified with the ring of functions over the big Witt ring scheme $\WW$ over $\Z$. This group scheme is unipotent over $\Z$, and the ghost components define an isomorphism to $\prod_{\Z_{\geq 0}} \GG_a$ upon rationalization (see <cit.> for a textbook reference). The group scheme $\spec \H_0(\Omega \SU(n); \Z[\beta^{\pm 1}])$ is a quotient of $\WW$ (hence is unipotent): in fact, it is isomorphic to the group scheme $\WW_{n-1}$ of big Witt vectors of length $n-1$. Since this is rationally isomorphic to $\GG_a^{\times n-1}$, we see that

$$\spec \H_0(\Omega \PGL_n(\cc); \QQ[\beta^{\pm 1}]) \cong \mu_n \times \WW_{n-1} \cong \mu_n \times \GG_a^{\times n-1} \cong Z_f(\SL_n),$$

as desired. Note, however, that the isomorphism $\WW_{n-1} \cong U_n\subseteq Z_f(\SL_n)$ is somewhat tricky to write down in coordinates. As an example, using the formula for the ghost components in the big Witt vectors, it is easy to see that the formula <ref> implies that the isomorphism $Z_f(\SL_6) \supseteq U_6 \xar{\sim} \WW_5$ sends $(x_1, \cdots, x_5)$ to the Witt vector

$$(x_1, \cdots, x_5) \mapsto (x_1, -x_2, x_3-x_1x_2, x_1x_3-x_2x_1^2-x_4, x_5-x_1^3 x_2 + x_1^2 x_3 - x_1 (x_4 - x_2^2) - x_2 x_3).$$

One special feature of rational homology which sets it apart from K-theory or elliptic cohomology is that it can be de-periodified. On the Langlands-dual side, this equips the relevant geometric objects with a $\GG_m$-action, i.e., with a grading. Continuing <ref>, there is still an isomorphism

$$\H_\ast(\Omega \PGL_n(\cc); \QQ) \simeq \QQ[x^{\pm 1}]/(x^n-1) \otimes_\Z \H_\ast(\Omega \SU(n); \Z),$$

and there is still an isomorphism $\spec \H_\ast(\Omega \SU(n); \Z) \cong \WW_{n-1}$. Here, the grading on $\H_\ast(\Omega \SU(n); \Z)$ by half the homological degree corresponds to the $\GG_m$-action on $\WW_{n-1}$ defined as follows: if we view $\WW_{n-1}(R) = 1+R[t]/t^n \subseteq (R[t]/t^n)^\times$, the coordinate $t$ is given weight $-1$.
This defines a grading on $Z_f(\SL_n)$, which can also be described directly in general as follows (see [53]). The element $2\rho=\sum_{\alpha\in \Phi^+} \alpha\in \bX^\ast(T) \cong \bX_\ast(\ld{T})$ defines a homomorphism $2\rho: \GG_m \to \ld{T}$, which defines a $\GG_m$-action on $\ld{\g}$. This $\GG_m$-action stabilizes the Kostant section $f + \ld{\g}^e$, and hence defines a $\GG_m$-action on $Z_e(\ld{G})$; this is the grading on $\co_{Z_e(\ld{G})}$ corresponding to half the homological grading on $\H_\ast(\Omega G_c; \QQ)$.

In [9], the following analogue of <ref> is established (over $\cc$, but this does not affect the statement): there is a Cartesian square

\begin{equation}\label{tmmodw-bfm}
\xymatrix{
(T^\ast \ld{T})^\bl \mmod W \ar[r] \ar[d] & \fr{t}\mmod W \ar[d]^-\kappa\\
\fr{t}\mmod W \ar[r]_-\kappa & \ld{\g}/\ld{G},
\end{equation}

where the top-left corner can be identified with $\spec \pi_0 C^G_\ast(\Gr_G(\cc); \QQ)$. We can take the fiber product of <ref> with itself over <ref> to obtain a Cartesian square

\begin{equation}\label{springer-kostant}
\xymatrix{
(T^\ast \ld{T})^\bl \times_{(T^\ast \ld{T})^\bl \mmod W} (T^\ast \ld{T})^\bl \ar[r] \ar[d] & \fr{t} \times_{\fr{t}\mmod W} \fr{t} \ar[d]^-\kappa\\
\fr{t} \times_{\fr{t}\mmod W} \fr{t} \ar[r]_-\kappa & (\tilde{\ld{\g}} \times_{\ld{\g}} \tilde{\ld{\g}})/\ld{G}.
\end{equation}

Using <ref> and the above discussion, one can use <ref> to show that $\End_{\QCoh((\tilde{\ld{\g}} \times_{\ld{\g}} \tilde{\ld{\g}})/\ld{G})}(\QCoh(\fr{t} \times_{\fr{t}\mmod W} \fr{t}))$ can be identified with $\LMod_{\pi_0 C^T_\ast(\Fl_G(\cc); \QQ[\beta^{\pm 1}])}$. This can be viewed as a “once-looped” version of Bezrukavnikov's equivalence from [7].

One can quantize <ref> as follows.

Following [54], define the (Langlands dual) universal category $\ld{\co}_\hbar^\univ$ to be $\DMod_\hbar(\ld{G}/\ld{N})^{(\ld{G}\times \ld{T}, w)} \simeq U_\hbar(\ld{\g})\modc^{\ld{N}, (\ld{T}, w)}$. The $\infty$-category $\ld{\co}_\hbar^\univ$ is a quantization of $\QCoh(\ld{\fr{b}}^-/\ld{B}^-)$, since there are isomorphisms

$$\ld{\fr{b}}^-/\ld{B}^- \cong \tilde{\ld{\g}}/\ld{G} \cong \ld{T}\backslash T^\ast(\ld{G}/\ld{N})/\ld{G}.$$

Let $A$ be an $\Eoo$-$\cc[\beta^{\pm 1}]$-algebra, and let $G$ be a connected and simply-connected semisimple algebraic group or a torus over $\cc$. Then there is a Kostant functor $\ld{\co}^\univ_\hbar \to \QCoh(\ld{\fr{t}}^\ast \times \AA^1_\hbar)$ and a left $A\pw{\hbar}$-linear equivalence

$$\LMod_{\pi_0 C^{\tilde{T}}_\ast(\Gr_G(\cc); A)} \simeq \End_{\ld{\co}^\univ_\hbar}(\QCoh(\ld{\fr{t}}^\ast \times \AA^1_\hbar)).$$

We will assume $A = \cc[\beta^{\pm 1}]$, so that $\pi_0 C^{\tilde{T}}_\ast(\Gr_G(\cc); A)$ is a $2$-periodification of $\pi_\ast C^{\tilde{T}}_\ast(\Gr_G(\cc); \cc)$. Let $\cH(\tilde{\ld{\fr{t}}}^\ast, \tilde{W}^\aff)$ be the nil-Hecke algebra associated to $\tilde{\ld{\fr{t}}}^\ast \cong \ld{\fr{t}}^\ast \oplus \cc\alpha_0$, and let $e = \tfrac{1}{\# W}\sum_{w\in W} w\in \QQ[W]$ be the symmetrizer idempotent. Using <ref>, one can then show that $\H^{\tilde{T}}_\ast(\Gr_G(\cc); \cc)$ is isomorphic to $\co_{\ld{\fr{t}}^\ast} \otimes_{\co_{\ld{\fr{t}}^\ast \mmod W}} e\cH(\tilde{\ld{\fr{t}}}^\ast, \tilde{W}^\aff)e$, where the loop rotation parameter $\hbar$ corresponds to the affine root $\alpha_0$; see [51]. This implies that $\LMod_{\pi_0 C^{\tilde{T}}_\ast(\Gr_G(\cc); A)}$ can be identified with $\QCoh(\ld{\fr{t}}^\ast) \otimes_{\QCoh(\ld{\fr{t}}^\ast \mmod W)} \LMod_{e\cH(\tilde{\ld{\fr{t}}}^\ast, \tilde{W}^\aff)e}$.

We now construct the Kostant functor $\kappa_\hbar: \ld{\co}^\univ_\hbar \to \QCoh(\ld{\fr{t}}^\ast \times \AA^1_\hbar)$. Recall that the Kostant functor $\HC_\hbar(\ld{G}) \to \QCoh(\ld{\fr{t}}^\ast\mmod W \times \AA^1_\hbar)$ is given by the composite

$$\HC_\hbar(\ld{G}) = \DMod_\hbar(\ld{G})^{(\ld{G}\times \ld{G}, w)} \to \DMod_\hbar(\ld{G})^{(\ld{G}, w)} \to \DMod_\hbar(\ld{N}^-\backslash_\chi \ld{G})^{(\ld{G}, w)}.$$

However, the final term is equivalent to $U_\hbar(\ld{\g})\modc^{(\ld{N}^-, \chi)}$, which in turn can be identified with $\QCoh(\ld{\fr{t}}^\ast\mmod W \times \AA^1_\hbar)$ by the Skryabin equivalence (see the appendix of [76]).
Similarly, the desired Kostant functor on $\ld{\co}^\univ_\hbar$ is also given by Whittaker averaging: there is a composite

$$\ld{\co}^\univ_\hbar = \DMod_\hbar(\ld{G}/\ld{N})^{(\ld{G}\times \ld{T}, w)} \to \DMod_\hbar(\ld{G}/\ld{N})^{(\ld{T}, w)} \xar{\mathrm{Av}_!^\chi} \DMod_\hbar(\ld{N}^-\backslash_\chi \ld{G}/\ld{N})^{(\ld{T}, w)}.$$

However, the final term is equivalent by a standard argument to $\DMod_\hbar(\ld{T})^{(\ld{T}, w)} \simeq \QCoh(\ld{\fr{t}}^\ast \times \AA^1_\hbar)$. Note that by construction, the following diagram commutes:

\HC_\hbar(\ld{G}) \ar[d] \ar[r] & \QCoh(\ld{\fr{t}}^\ast\mmod W \times \AA^1_\hbar) \ar[d] \\
\ld{\co}^\univ_\hbar \ar[r] & \QCoh(\ld{\fr{t}}^\ast \times \AA^1_\hbar).

Here, the horizontal maps are given by the Kostant functors.

To finish, we need to show that $\QCoh(\ld{\fr{t}}^\ast) \otimes_{\QCoh(\ld{\fr{t}}^\ast \mmod W)} \LMod_{e\cH(\tilde{\ld{\fr{t}}}^\ast, \tilde{W}^\aff)e}$ is equivalent to $\End_{\ld{\co}^\univ_\hbar}(\QCoh(\ld{\fr{t}}^\ast \times \AA^1_\hbar))$.

There is an equivalence

$$\QCoh(\ld{\fr{t}}^\ast \times \AA^1_\hbar) \simeq \ld{\co}^\univ_\hbar \otimes_{\HC_\hbar(\ld{G})} \QCoh(\ld{\fr{t}}^\ast \mmod W \times \AA^1_\hbar),$$

so that

$$\End_{\ld{\co}^\univ_\hbar}(\QCoh(\ld{\fr{t}}^\ast \times \AA^1_\hbar)) \simeq \QCoh(\ld{\fr{t}}^\ast) \otimes_{\QCoh(\ld{\fr{t}}^\ast \mmod W)} \End_{\HC_\hbar(\ld{G})}(\QCoh(\ld{\fr{t}}^\ast\mmod W \times \AA^1_\hbar)).$$

The desired claim now follows from the observation that there is an isomorphism $\ld{N}^- {}_\chi\backslash {\cd_\ld{G}} /_\chi \ld{N}^- \cong e\cH(\tilde{\ld{\fr{t}}}^\ast, \tilde{W}^\aff)e$ given by <cit.>, which gives an equivalence between $\End_{\HC_\hbar(\ld{G})}(\QCoh(\ld{\fr{t}}^\ast\mmod W \times \AA^1_\hbar))$ and $\LMod_{e\cH(\tilde{\ld{\fr{t}}}^\ast, \tilde{W}^\aff)e}$.

In fact, one can quantize the result of [1]: namely, there is an equivalence

\begin{equation}\label{gm-rot-abg}
\DMod_{I \rtimes \GG_m^\rot}(\Gr_G)\simeq \ld{\co}_\hbar^\univ.
\end{equation}

We do not have a reference for this fact when $G$ lives over $\cc$, but it can be deduced using the equivalence of <cit.> and the arguments of [1]. I am grateful to Tom Gannon for discussions about this equivalence. (If $G$ lives over $\ol{\FF_p}$ and $\DMod$ is replaced with $\ol{\QQ_\ell}$-adic sheaves, then <ref> can be deduced from <cit.> and the parabolic-Whittaker duality for the affine Grassmannian from [16].) Just as with <ref>, <ref> may be regarded as a “once-looped” version of <ref>.
One can similarly show that there is an equivalence

\begin{equation}\label{gm-rot-bez}
\DMod_{I \rtimes \GG_m^\rot}(\Fl_G)\simeq \DMod_\hbar(\ld{N} \backslash \ld{G}/\ld{N})^{(\ld{T} \times \ld{T}, \weak)},
\end{equation}

which quantizes Bezrukavnikov's equivalence from [7]. Note that $\ld{T} \backslash T^\ast(\ld{N} \backslash \ld{G}/\ld{N}) / \ld{T}$ is isomorphic to $(\tilde{\ld{\g}} \times_{\ld{\g}} \tilde{\ld{\g}})/\ld{G}$, so that this equivalence does indeed quantize Bezrukavnikov's equivalence

$$\DMod_I(\Fl_G) \simeq \QCoh((\tilde{\ld{\g}}[2] \times_{\ld{\g}[2]} \tilde{\ld{\g}}[2])/\ld{G}).$$

If $G$ is a connected and simply-connected semisimple algebraic group or a torus over $\cc$, let $\HC_\hbar(\ld{G})$ denote the $\infty$-category $U_\hbar(\ld\g)\modc^{\ld{G},w}$. Then $\Gamma(\ld{G}; \cd_{\ld{G}})^{\ld{G}\times \ld{G}} \cong U(\ld{\g})^{\ld{G}} \cong \Sym(\ld{\fr{t}})^W$. An argument very similar to <ref> proves that there is a Kostant functor $\HC_\hbar(\ld{G}) \to \QCoh(\ld{\fr{t}}^\ast\mmod W \times \AA^1_\hbar)$ and a left $A\pw{\hbar}$-linear equivalence

\begin{equation}\label{looped-HC}
\LMod_{\pi_0 C^{G \times S^1_\rot}_\ast(\Gr_G(\cc); A)} \simeq \End_{\HC_\hbar(\ld{G})}(\QCoh(\ld{\fr{t}}^\ast\mmod W \times \AA^1_\hbar)).
\end{equation}

This is closely related to [37], [55], and <cit.>. Let $\ld{\fr{t}}\mmod \tilde{W}^\aff$ be the coarse quotient as defined in [35].
Then, the aforementioned articles provide a monoidal “Fourier transform” equivalence $\DMod(\ld{N}^- {}_\chi\backslash \ld{G} /_\chi \ld{N}^-) \simeq \IndCoh(\ld{\fr{t}}\mmod \tilde{W}^\aff)$. Note that combined with the preceding discussion, we obtain an equivalence

\begin{equation}\label{622-a}
\IndCoh(\ld{\fr{t}}\mmod \tilde{W}^\aff) \simeq \End_{\HC(\ld{G})}(\QCoh(\ld{\fr{t}}^\ast\mmod W)).
\end{equation}

There is also an equivalence (see [55])

$$\End_{\Shv_{G\times S^1_\rot}(\Gr_G; \cc)}(\QCoh(\ld{\fr{t}}^\ast\mmod W)) \simeq \LMod_{\H^{G \times S^1_\rot}_\ast(\Gr_G(\cc); \cc)} \simeq \IndCoh(\ld{\fr{t}}\mmod \tilde{W}^\aff),$$

and its relationship to <ref> is explained by the derived loop-rotation equivariant geometric Satake equivalence of [8].

In the same way, we have the following result. We expect that the techniques of [11] can be used to show that this implies the equivalences conjectured in <cit.>.

We have:

\begin{align}
\IndCoh(\ld{\fr{t}}\mmod \tilde{W}^\aff) & \simeq \End_{\DMod(\ld{N}\backslash \ld{G}/\ld{N})^{(\ld{T}\times \ld{T}, w)}}(\QCoh(\ld{\fr{t}}^\ast)), \label{622-b}
%\IndCoh(\ld{\fr{t}}\mmod \tilde{W}^\aff) & \simeq \Fun^L_{\ld{\g}\modc^\ld{N}}(\QCoh(\ld{\fr{t}}^\ast/\Lambda^\vee), \QCoh(\ld{\fr{t}}^\ast\mmod W)). \label{622-c}
\end{align}

The equivalence <ref> is proved via:

\begin{align*}
\IndCoh(\ld{\fr{t}}\mmod \tilde{W}^\aff) & \simeq \DMod(\ld{N}^- {}_\chi\backslash \ld{G} /_\chi \ld{N}^-)\\
& \simeq \End_{\DMod(\ld{G})}(\DMod(\ld{G} /_\chi \ld{N}^-))\\
& \simeq \End_{\DMod(\ld{N} \backslash \ld{G}/\ld{N})^{(\ld{T}\times \ld{T}, w)}}(\DMod(\ld{N} \backslash \ld{G} /_\chi \ld{N}^-)^{\ld{T},w}) \\
& \simeq \End_{\DMod(\ld{N} \backslash \ld{G}/\ld{N})^{(\ld{T}\times \ld{T}, w)}}(\DMod(\ld{T})^{\ld{T},w}) \\
& \simeq \End_{\DMod(\ld{N} \backslash \ld{G}/\ld{N})^{(\ld{T}\times \ld{T}, w)}}(\QCoh(\ld{\fr{t}}^\ast)).
\end{align*}

The third equivalence above uses <cit.>, and the fourth equivalence above is the well-known fact that restriction to the big cell in $\ld{G}$ defines an equivalence $\DMod(\ld{N} \backslash \ld{G} /_\chi \ld{N}^-) \xar{\sim} \DMod(\ld{N} \backslash \ld{B} /_\chi \ld{N}^-) \simeq \DMod(\ld{T})$; see <cit.>, for instance.

Since $\ld{\g}/\ld{G} = \Map(B\GG_a, B\ld{G})$, the canonical orientation of $B\GG_a$ defines a $1$-shifted symplectic structure on $\ld{\g}/\ld{G}$ via <cit.>. The quasi-classical limit (i.e., $\hbar\to 0$) of the quantized equivalence <ref> gives the following strengthening of <ref>. The Kostant slice $\ld{\fr{t}}\mmod W \to \ld{\g}/\ld{G}$ is a Lagrangian morphism by <cit.>, so that the self-intersection $\ld{\fr{t}}\mmod W \times_{\ld{\g}/\ld{G}} \ld{\fr{t}}\mmod W$ admits the structure of a symplectic stack (using <cit.>). Since this fiber product is isomorphic to $(T^\ast \ld{T})^\bl\mmod W$ by <ref>, we obtain a Poisson bracket on $\co_{(T^\ast \ld{T})^\bl\mmod W} \cong \H^G_\ast(\Gr_G(\cc); \cc)$. This structure can be seen topologically, at least after a completion: using one of the main results of [52], the Borel-equivariant analogue/completion $C_\ast(\Gr_G(\cc); \cc)^{hG_c}$ of $C^G_\ast(\Gr_G(\cc); \cc)$ can be identified with the $\E{3}$-center of $C_\ast(\Gr_G(\cc); \cc)$. This defines a $2$-shifted Poisson bracket on $\H_\ast(\Gr_G(\cc)^{hG_c}; \cc)$, which can be identified after $2$-periodification with the ($0$-shifted) Poisson bracket on $\co_{(T^\ast \ld{T})^\bl\mmod W}$.

§.§ Rationalized Langlands duality over $\KU$

Let us now discuss the $K$-theoretic analogue of <ref>. First, we discuss the story where the Kostant slice from <ref> is replaced by the “Steinberg slice”; below, we will discuss the story where the Kostant slice from <ref> is replaced by a multiplicative version of the Kostant slice.

Let $G$ be a simply-connected semisimple algebraic group or a torus. Given $w\in W$, let $N_w = N \cap w^{-1} N^- w$, so that $N_w = \prod_{\alpha\in \Phi_w} U_\alpha$, where $\Phi_w$ is the set of roots made negative by $w$. Let $w = \prod_{\alpha\in \Delta} s_\alpha\in W$ be a Coxeter element, and let $\dot{w}$ be a lift of $w$ to $N_G(T)$. Define the Steinberg slice $\Sigma = \dot{w} N_w\subseteq G$. Then [88] proved/stated that the composite $\Sigma \to G \to G\mmod G \cong T\mmod W$ is an isomorphism.
Let $\tilde{G} = B \times_{B} G$ be the multiplicative Grothendieck-Springer resolution, so that $\tilde{G}/G = B/B$. There is a map $\tilde{G} \to T$ sending a pair $x\in gBg^{-1}$ to $x\pmod{g[B,B]g^{-1}}$. Let $\tilde{\Sigma}$ denote the fiber product $\Sigma\times_G \tilde{G}$, so that the composite $\tilde{\Sigma} \to \tilde{G} \to T$ is an isomorphism. We will denote the inclusion of $\tilde{\Sigma}$ by $\sigma: T \to \tilde{G}$.

Let $G$ be a connected and simply-connected semisimple algebraic group or a torus over $\cc$. Let $A$ be an $\Eoo$-$\KU$-algebra, and let $\GG = \GG_m$ (so $\cM_T$ is the torus $T$ over $A$). View $\tilde{\ld{G}}$ as a scheme over $\QQ$. If $\QCoh(\ld{T})$ is viewed as a module over $\QCoh(\tilde{\ld{G}}/\ld{G})$ via $\sigma^\ast$, then there is an equivalence

$$\End_{\QCoh(\tilde{\ld{G}}/\ld{G})}(\QCoh(\ld{T})) \otimes_\QQ \pi_0 A_\QQ \simeq \LMod_{\pi_0 C_\ast^T(\Gr_G(\cc); A)} \otimes \QQ.$$