We will assume without loss of generality that $A = \KU$. By <ref>, there is an equivalence $\pi_0 C_\ast^T(\Gr_G(\cc); A) = \pi_0 \cf_T(\Gr_G(\cc))^\vee \simeq \co_{(T^\ast_{\GG_m} \ld{T})^\bl}$. It follows that $\LMod_{\pi_0 C_\ast^T(\Gr_G(\cc); A)} \simeq \QCoh((T^\ast_{\GG_m} \ld{T})^\bl)$.

It therefore suffices to show that over a field $k$ of characteristic zero, there is an equivalence $\End_{\QCoh(\tilde{\ld{G}}/\ld{G})}(\QCoh(\ld{T})) \simeq \QCoh((T^\ast_{\GG_m} \ld{T})^\bl)$.

As in <ref>, there is an equivalence $\End_{\QCoh(\tilde{\ld{G}}/\ld{G})}(\QCoh(\ld{T})) \simeq \QCoh(\ld{T} \times_{\tilde{\ld{G}}/\ld{G}} \ld{T})$, so it suffices to establish the existence of a Cartesian square

\begin{equation}\label{kthy-intersection-kostant}
\xymatrix{
(T^\ast_{\GG_m} \ld{T})^\bl \ar[r] \ar[d] & \ld{T} \ar[d]^-\sigma \\
\ld{T} \ar[r]_-\sigma & \tilde{\ld{G}}/\ld{G}.
\end{equation}

Again, one can reduce to the case when $\ld{G}$ has semisimple rank $1$ by the argument of <cit.>.
Every split reductive group of semisimple rank $1$ is isomorphic to the product of a split torus with $\SL_2$, $\PGL_2$, or $\GL_2$.
We will illustrate the calculation when $\ld{G} = \SL_2$, and describe an alternative simpler calculation in the case $\ld{G} = \PGL_2$ later.

View a point in $\tilde{\ld{G}}$ as a pair $(x\in \SL_2, \ell\subseteq \cc^2)$ such that $x$ preserves $\ell$. The Steinberg slice $\sigma:\ld{T} \cong \GG_m \to \tilde{\SL}_2$ is the map sending $\lambda \in \GG_m$ to the pair $(x, \ell)$ with

$$x = \begin{pmatrix}
\lambda + \lambda^{-1} & -1 \\
1 & 0
\end{pmatrix}, \ \ell = \left[\lambda: 1\right].$$

Note that this indeed a well-defined point in $\tilde{\SL}_2$, since one can check that $x$ preserves $\ell$. This calculation of $\sigma(\lambda)$ is essentially immediate from the requirement that the following diagram commutes:

\GG_m \cong \ld{T} \ar[r]^-\sigma \ar[d]_-{\lambda \mapsto \lambda + \lambda^{-1}} & \tilde{\SL}_2 \ar[d]\\
\AA^1 \cong \ld{T}\mmod W \ar[r]^-\sigma_-{\lambda\mapsto \begin{psmallmatrix}
\lambda & -1 \\
1 & 0
\end{psmallmatrix}} & \SL_2.

Moreover, the $\SL_2$-action on $\tilde{\SL}_2$ sends $g\in \SL_2$ and $(x,\ell)$ to $(\Ad_g(x), g\ell)$. If $g = \begin{psmallmatrix}
a & b \\
c & d
\end{psmallmatrix}$, one can directly compute that $g$ commutes with $\begin{psmallmatrix}
\lambda + \lambda^{-1} & -1 \\
1 & 0
\end{psmallmatrix}$ if and only if $a = c(\lambda + \lambda^{-1}) + d$ and $b=-c$. Therefore, $g = \begin{psmallmatrix}
c(\lambda + \lambda^{-1}) + d & -c \\
c & d
\end{psmallmatrix}$ for $c,d\in k$. In order for $\det(g) = 1$, we need

$$c^2 + d^2 + cd(\lambda + \lambda^{-1}) = 1.$$

As long as $\lambda\neq \pm 1$, both $x$ and $g$ can be simultaneously diagonalized by $\begin{psmallmatrix}
\lambda & \lambda^{-1} \\
1 & 1
\end{psmallmatrix}$: the diagonalization of $x$ is $\begin{psmallmatrix}
\lambda & 0 \\
0 & \lambda^{-1}
\end{psmallmatrix}$, and the diagonalization of $g$ is $\begin{psmallmatrix}
c\lambda + d & 0 \\
0 & c\lambda^{-1} + d
\end{psmallmatrix}$. If $t = c\lambda+d$, then $c\lambda^{-1}+d = t^{-1}$ by the above determinant relation. We also have that $a = t - \tfrac{\lambda(t-t^{-1})}{\lambda - \lambda^{-1}}$ and $c = \tfrac{t-t^{-1}}{\lambda - \lambda^{-1}}$. This shows that $\GG_m \times_{\tilde{\SL}_2/\SL_2} \GG_m \cong \spec k[\lambda^{\pm 1}, t^{\pm 1}, \tfrac{t-t^{-1}}{\lambda - \lambda^{-1}}]$ (even if if $k$ is of characteristic $2$).

An alternative argument for the Cartesian square <ref> can be given using the multiplicative Kostant slice, which gives a different section of the map $G \to G\mmod G$. The multiplicative Kostant slice is significantly more accessible, and the resulting <ref> is what we will generalizing below to other cohomology theories.

Let $e\in \fr{n}$ be a principal nilpotent element. Then the map $\GG_a \to G$ corresponding to $e$ factors through the map $\GG_a = B \to \SL_2$; we will denote the image of the standard generator $\begin{psmallmatrix}
1 & 0\\
1 & 1
\end{psmallmatrix}\in B^-$ under the map $\SL_2 \to G$ by $f\in G$. Let $Z_G(e)^\circ$ be the connected component of the identity in the centralizer of $e$ in $G$. Define the multiplicative Kostant slice $\cS_\mu$ by $Z_G(e)^\circ \cdot f \subseteq G$. Since $G$ is assumed to be simply-connected, the composite $\cS_\mu \to G \to G\mmod G \cong T\mmod W$ is an isomorphism. We will often denote the inclusion of the Kostant slice by $\kappa: T\mmod W \to G$.
Let $\tilde{\cS}_\mu$ denote the fiber product $\tilde{\cS}_\mu \times_G \tilde{G}$, so that the composite $\tilde{\cS}_\mu \to \tilde{G} \to T$ is an isomorphism; we will denote the inclusion of $\tilde{\cS}_\mu$ as a map $\kappa: \tilde{\cS}_\mu \cong T \to \tilde{G}$.

As with the additive Kostant slice, we will only care about the composite $T \to \tilde{G} \to \tilde{G}/G$ below, so we will also denote it by $\kappa$. If we identify $\tilde{G}/G \cong B/B$, then the map $\kappa$ admits a simple description: it is the composite $T \to B \to B/B$ which sends $x\mapsto xf$. Just as in <cit.>, there is a unique map $\mu: T\cdot f \to N$ such that $\Ad_{\mu(x)}(x) \in Z_G(e)^\circ\cdot f$, and the image of any $x\in T$ under the map $T \to T\mmod W \xar{\kappa} G$ can be identified with $\Ad_{\mu(xf)}(xf)$.

The main result of [31] states that any two sections of the map $G \to T\mmod W$ are conjugate. For instance, the multiplicative Kostant section ${T}\mmod W \cong \AA^1 \to \SL_2$ sending $\lambda \mapsto \begin{psmallmatrix}
\lambda-1 & \lambda-2 \\
1 & 1
\end{psmallmatrix}$ and the Steinberg section ${T}\mmod W \cong \AA^1 \to \SL_2$ sending $\lambda \mapsto \begin{psmallmatrix}
\lambda & -1 \\
1 & 0
\end{psmallmatrix}$ are conjugated into each other by the matrix $\begin{psmallmatrix}
1 & -1\\
0 & 1
\end{psmallmatrix}$.

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 $\kappa^\ast$, then there is an equivalence

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

Following the argument of <ref>, we only need to prove the Cartesian-ness of <ref>, where the map $\ld{T} \to \tilde{\ld{G}}/\ld{G}$ is chosen to be the multiplicative Kostant slice instead of the Steinberg slice. Again, we only review the calculation for $\ld{G} = \SL_2$; this was done in [9]. For convenience, we will drop the “check”s. As before, there are “two” ways to compute in the case $G = \SL_2$. First, we describe the argument essentially present in [9] (which works over a base field of characteristic not $2$). If $\lambda\in \GG_m$, we denote $\lambda + \lambda^{-1}\in \AA^1$ by $f(\lambda)$. The Kostant slice $\kappa:\ld{T} \cong \GG_m \to \tilde{\SL}_2$ is the map sending $\lambda \in \GG_m$ to the pair $(x, \ell)$ with

$$x = \begin{pmatrix}
f(\lambda)-1 & f(\lambda)-2 \\
1 & 1
\end{pmatrix}, \ \ell = \left[\lambda-1: 1\right].$$

Note that this indeed a well-defined point in $\tilde{\SL}_2$, since one can check that $x$ preserves $\ell$: the key point is the conic relation

$$2\lambda = f(\lambda)-\sqrt{f(\lambda)^2-4}.$$

Indeed, this calculation of $\kappa(\lambda)$ is essentially immediate from the requirement that the following diagram commutes:

\GG_m \cong \ld{T} \ar[r]^-\kappa \ar[d]_-{\lambda \mapsto f(\lambda)} & \tilde{\SL}_2 \ar[d]\\
\AA^1 \cong \ld{T}\mmod W \ar[r]^-\kappa_-{\lambda\mapsto \begin{psmallmatrix}
\lambda-1 & \lambda-2 \\
1 & 1
\end{psmallmatrix}} & \SL_2.

Moreover, the $\SL_2$-action on $\tilde{\SL}_2$ 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 directly compute that $\Ad_g(x) = x$ if and only if $b = c(f(\lambda) - 2)$ and $a-d = (f(\lambda) - 2)c$, in which case $g$ also preserves $\ell$. Therefore, $g = \begin{psmallmatrix}
(f(\lambda) - 2)c + d & (f(\lambda)-2)c \\
c & d
\end{psmallmatrix}$ for $c,d\in k$. In order for $\det(g) = 1$, we need

$$d^2 + c(f(\lambda)-2)(d-c) = 1.$$

Both $x$ and $g$ can be simultaneously diagonalized (if $f(\lambda) \neq \pm 2$); note that $\lambda+\lambda^{-1}$ is an eigenvalue of $x$. If $t$ is an eigenvalue of $g$, then we have $c = \tfrac{t-t^{-1}}{\lambda - \lambda^{-1}}$ and $d = \tfrac{t^2\lambda + 1}{t(\lambda+1)}$.
When $k$ is not of characteristic $2$, this shows that $\GG_m \times_{\tilde{\SL}_2/\SL_2} \GG_m \cong k[\lambda^{\pm 1}, t^{\pm 1}, \tfrac{t-t^{-1}}{\lambda - \lambda^{-1}}]$, as desired.

For the “second” method of calculation when $G = \SL_2$ (which works in arbitary characteristic), we use the fact that $\kappa: T \to \tilde{G}/G$ can be identified with the composite $T \to B \to B/B$ sending $x\mapsto xf$. Then, $T \times_{B/B} T$ is isomorphic to the subvariety of $T \times B$ consisting of pairs $(x,g)$ with $x\in T$ (identified with the matrix $\begin{psmallmatrix}
x & 0\\
0 & x^{-1}
\end{psmallmatrix}$) and $\Ad_g(xf) = xf$. Note that $xf$ is the matrix $\begin{psmallmatrix}
x & 0 \\
x^{-1} & x^{-1}
\end{psmallmatrix}$. If $g = \begin{psmallmatrix}
a & 0 \\
b & a^{-1}
\end{psmallmatrix}\in B$, then

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

Therefore, $\Ad_g(xf) = xf$ if and only if

$$a^{-2} x^{-1} + ba^{-1}(x-x^{-1}) = x^{-1},$$

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

We can also run this argument in the case $G = \PGL_2$ (again in arbitary characteristic). Again, $T \times_{B/B} T$ is isomorphic to the subvariety of $T \times B$ consisting of pairs $(x,g)$ with $x\in T$ (identified with the matrix $\begin{psmallmatrix}
x & 0\\
0 & 1
\end{psmallmatrix}$) and $\Ad_g(xf) = xf$. Note that $xf$ is the matrix $\begin{psmallmatrix}
x & 0 \\
1 & 1
\end{psmallmatrix}$. If $g = \begin{psmallmatrix}
a & 0 \\
b & 1
\end{psmallmatrix}\in B$, then

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

Therefore, $\Ad_g(xf) = xf$ if and only if

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

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

In the second argument for the Cartesian square <ref>, we may replace the symbol $\lambda$ by the symbol $e^\lambda$; then, $e^\lambda-1$ is the exponential of the multiplicative formal group law. In particular, the defining equation for the line $\ell$ in the cases of $\GG = \GG_a, \GG_m$ precisely describes the exponential for the formal completion $\hat{\GG}$ of $\GG$ at the identity.

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{kthy-tmmodw-bfm}
\xymatrix{
(T^\ast_{\GG_m} \ld{T})^\bl \mmod W \ar[r] \ar[d] & \ld{T}\mmod W \ar[d]^-\kappa\\
\ld{T}\mmod W \ar[r]_-\kappa & \ld{G}/\ld{G},
\end{equation}

where the top-left corner can be identified with $\spec C^G_0(\Gr_G(\cc); \KU) \otimes \QQ$. We can take the fiber product of <ref> with itself over <ref> to obtain a Cartesian square

\begin{equation}\label{kthy-springer-kostant}
\xymatrix{
(T^\ast_{\GG_m} \ld{T})^\bl \times_{(T^\ast_{\GG_m} \ld{T})^\bl \mmod W} (T^\ast_{\GG_m} \ld{T})^\bl \ar[r] \ar[d] & \ld{T} \times_{\ld{T}\mmod W} \ld{T} \ar[d]^-\kappa\\
\ld{T} \times_{\ld{T}\mmod W} \ld{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 $\End_{\QCoh((\tilde{\ld{G}} \times_{\ld{G}} \tilde{\ld{G}})/\ld{G})}(\QCoh(\ld{T} \times_{\ld{T}\mmod W} \ld{T}))$ can be identified with $\LMod_{\pi_0 C^T_\ast(\Fl_G(\cc); \KU)} \otimes \QQ$.
This can be viewed as a “once-looped” version of a K-theoretic analogue of Bezrukavnikov's equivalence from [7].

We expect that most of the steps of <ref> can be replicated to study $\LMod_{C^{\tilde{T}}_\ast(\Gr_G(\cc); \KU)} \otimes \QQ$. More precisely, let $d\in \Z$, and fix a symmetric bilinear form $(-,-): \Lambda \times \Lambda \to \tfrac{1}{d}\Z$ such that whose Gram matrix is the associated Cartan matrix (i.e., $(\alpha_i, \alpha_j)$ is the $a_{ij}$ entry of the associated Cartan matrix). We then have the quantum group $U_q(\g)$ defined over $\Z[q^{\pm 1}]$  associated to the pairing $\Lambda \times \Lambda \to \Z[q^{\pm 1}]$ sending $\lambda, \mu \mapsto q^{-(\lambda, \mu)}$.
Following <cit.>, define the quantum universal category ${\co}^\univ_q$ as the $\infty$-category of $(U_q(\g), U_q(\fr{t}))$-bimodules whose diagonal $U_q(\fr{b})$-action is integrable.

Let $(W, \Delta)$ be a crystallographic root system, let $\Lambda^\vee = \Z\Phi$ denote the associated root lattice, and let $T = \spec \Z[\Lambda]$ denote the associated torus. Each $\alpha\in W$ defines an operator $s_\alpha$ on $\co_T$. Define the multiplicative nil-Hecke algebra $\cH(T, W)$ as the subalgebra of $\mathrm{Frac}(\co_T) \rtimes \QQ[W]$ generated by $\co_T$ and the operators $T_\alpha = \tfrac{1}{e^\alpha-1}(s_\alpha-1)$. (Also see [26] for a study of a multiplicative analogue of Soergel theory.) Then, there are relations

$$T_\alpha^2 = T_\alpha, \ (T_\alpha T_\beta)^{m_{\alpha,\beta}} = (T_\beta T_\alpha)^{m_{\alpha,\beta}} , \ x\cdot T_\alpha = T_\alpha \cdot s_\alpha(x) + T_\alpha(x), \ \alpha\in \Delta.$$

Recall that $m_{\alpha_i\alpha_j}$ is $2$, $3$, $4$, $6$, $\infty$ if $a_{ij} a_{ji}$ is $0$, $1$, $2$, $3$, $\geq 4$ (respectively). This algebra was also studied in <cit.>.
Note that if $\lambda \in \Lambda$ (corresponding to the function $e^\lambda$ on $T$), we have $T_\alpha(e^\lambda) = [\langle \alpha^\vee, \lambda\rangle]_{e^\alpha} e^\lambda$, where $[\langle \alpha^\vee, \lambda\rangle]_{e^\alpha}$ denotes the $q$-integer $\tfrac{q^{\langle \alpha^\vee, \lambda\rangle} - 1}{q-1}$ with $q = e^\alpha$.

Given the discussion in <ref> relating loop-rotation equivariance in $K$-theory to $q$-deformations, as well as <ref>, we expect:

There is a Kostant functor $\kappa: \ld{\co}^\univ_q \to \QCoh(\ld{T}_\QQ \times \GG_m^q)$ (where $\GG_m^q = \spec \QQ[q^{\pm 1}]$) such that there is a $\QQ[q^{\pm 1}]$-linear equivalence

\begin{equation}\label{quantum-abg}
\LMod_{\pi_0 C^{\tilde{T}}_\ast(\Gr_G(\cc); \KU)} \otimes \QQ \simeq \End_{\ld{\co}^\univ_q}(\QCoh(\ld{T}_\QQ \times \GG_m^q)).
\end{equation}

Similarly, if $\HC_q(\ld{G})$ denotes the category of <cit.>, there is a Kostant functor $\kappa: \HC_q(\ld{G}) \to \QCoh(\ld{T}_\QQ\mmod W \times \GG_m^q)$ and a $\QQ[q^{\pm 1}]$-linear equivalence

\begin{equation}\label{quantum-ginzburg}
\LMod_{\pi_0 C^{G\times S^1_\rot}_\ast(\Gr_G(\cc); \KU)} \otimes \QQ \simeq \End_{\HC_q(\ld{G})}(\QCoh(\ld{T}_\QQ\mmod W \times \GG_m^q)).
\end{equation}

At the moment, we are only able to describe the left-hand side in terms of combinatorial data. Let $e = \tfrac{1}{\# W}\sum_{w\in W} w$ be the symmetrizer idempotent. Using <ref> and <cit.> (see also <ref>), one can show that $\pi_0 C^{\tilde{T}}_\ast(\Gr_G(\cc); \KU) \otimes \QQ$ is isomorphic to $\co_{\ld{T}} \otimes_{\co_{\ld{T}\mmod W}} e\cH(\tilde{\ld{T}}, \tilde{W}^\aff)e$, where the parameter $q\in \pi_0 \KU_{\GG_m^\rot} \cong \Z[q^{\pm 1}]$ corresponds to the coordinate on $\GG_m^q\subseteq \tilde{\ld{T}}$ viewed as an element of $\cH(\tilde{\ld{T}}, \tilde{W}^\aff)e$. Similarly, $\pi_0 C^{G \times S^1_\rot}_\ast(\Gr_G(\cc); \KU) \otimes \QQ$ is isomorphic to $e\cH(\tilde{\ld{T}}, \tilde{W}^\aff)e$. The conjectural equivalence <ref> then reduces to proving an (also conjectural) equivalence

\begin{equation}\label{simpler-quantum-ginzburg}
%\End_{\ld{\co}_q^\univ}(\QCoh(\ld{T} \times \GG_m^q)) & \simeq \LMod_{\cH(\tilde{\ld{T}}, \tilde{W}^\aff)e}, \\
\End_{\HC_q(\ld{G})}(\QCoh(\ld{T}\mmod W \times \GG_m^q)) \simeq \LMod_{e\cH(\tilde{\ld{T}}, \tilde{W}^\aff)e}.
\end{equation}

This may be understood as a quantum analogue of <cit.>. Note that the above equivalences are now statements which are squarely on one side of Langlands duality. In the case $G = \SL_2$, we described $C^{G\times S^1_\rot}_\ast(\Gr_G(\cc); \KU) \otimes \QQ$ (and hence $e\cH(\tilde{\ld{T}}, \tilde{W}^\aff)e$) below in <ref>; it might be possible to use this calculation to compare with $\End_{\ld{\co}_q^\univ}(\QCoh(\ld{T} \times \GG_m^q))$ for $\ld{G} = \PGL_2$. A positive resolution to <cit.> should be the key input into proving <ref>.

For general $G$, just as $(T\times \ld{T})^\bl$ is birational to $T \times \ld{T}$, the map from the algebra of $q$-difference operators on $\ld{T}$ to $\cH(\tilde{\ld{T}}, \tilde{W}^\aff)e$ is an isomorphism after a particular localization. One therefore expects $\ld{\co}_q^\univ$ and $\HC_q(\ld{T})$ to generically be equivalent. This is indeed true, and can be seen using <cit.> (although the functor $\ld{\co}_q^\univ \to \HC_q(\ld{T})$ in loc. cit. is not our expected functor $\kappa$).

Since $\ld{G}/\ld{G} = \Map(S^1, B\ld{G})$, the canonical orientation of $S^1$ defines a $1$-shifted symplectic structure on $\ld{G}/\ld{G}$ via <cit.>. The quasi-classical limit (i.e., $q\to 1$) of the conjectural equivalence <ref> gives the following strengthening of <ref>. (This strengthening can be proved independently of <ref>.)

Observe that the Kostant slice $\ld{T}\mmod W \to \ld{G}/\ld{G}$ is a Lagrangian morphism. It follows that the self-intersection $\ld{T}\mmod W \times_{\ld{G}/\ld{G}} \ld{T}\mmod W$ admits the structure of a symplectic stack by <cit.>. Since this fiber product is isomorphic to $(T^\ast_{\GG_m} \ld{T})^\bl\mmod W$ by <ref>, we obtain a Poisson bracket on $\co_{(T^\ast_{\GG_m} \ld{T})^\bl\mmod W} \cong \pi_0 C^G_\ast(\Gr_G(\cc); \KU)$. This structure can be seen topologically, at least after a completion: using one of the main results of [52], the Borel-equivariant analogue/completion $C_\ast(\Gr_G(\cc); \KU)^{hG_c}$ of $C^G_\ast(\Gr_G(\cc); \KU)$ can be identified with the $\E{3}$-center of $\pi_0 C_\ast(\Gr_G(\cc); \KU)$. This defines a $2$-shifted Poisson bracket on $\pi_0 C_\ast(\Gr_G(\cc); \KU)^{hG_c}$, which can be identified with the ($0$-shifted, via the $2$-periodicity of $\KU$) Poisson bracket on $\co_{(T^\ast_{\GG_m} \ld{T})^\bl\mmod W}$.

Following <ref>, one can also hope for a result analogous to <ref> when $q \rightsquigarrow \zeta_p$ is specialized to a primitive $p$th root of unity. Namely, consider the $\infty$-category $\LMod_{C^{T\times \mu_{p,\rot}}_\ast(\Gr_G(\cc); \KU)}$, where $\mu_{p,\rot} \subseteq S^1_\rot$ acts by loop rotation. Note that $C^{T \times \mu_{p,\rot}}_\ast(\Gr_G(\cc); \KU)$ is a module over $\KU^{h\Cp}$, and $\pi_\ast \KU^{h\Cp} \cong \Z\pw{q-1}[\beta^{\pm 1}]/(q^p-1)$. Inverting $q-1$, we find that $C^{T \times \mu_{p,\rot}}_\ast(\Gr_G(\cc); \KU)[\frac{1}{q-1}]$ is a module over $\KU^{h\Cp}[\frac{1}{q-1}] \simeq \KU^{t\Cp} \simeq \QQ(\zeta_p)[\beta^{\pm 1}]$. We then expect the following (likely simpler) analogues of <ref> and <ref>:

There are Kostant functors $\kappa: \ld{\co}^\univ_{\zeta_p} \to \QCoh(\ld{T}_{\QQ(\zeta_p)})$ and $\kappa: \HC_{\zeta_p}(\ld{G}) \to \QCoh(\ld{T}_{\QQ(\zeta_p)}\mmod W)$ such that there are $\QQ(\zeta_p)$-linear equivalences

\begin{align*}%\label{zetap-abg}
\LMod_{\pi_0 C^{T \times \mu_{p,\rot}}_\ast(\Gr_G(\cc); \KU)[\frac{1}{q-1}]} & \simeq \End_{\ld{\co}^\univ_{\zeta_p}}(\QCoh(\ld{T}_{\QQ(\zeta_p)})),\\
\LMod_{\pi_0 C^{G \times \mu_{p,\rot}}_\ast(\Gr_G(\cc); \KU)[\frac{1}{q-1}]} & \simeq \End_{\HC_{\zeta_p}(\ld{G})}(\QCoh(\ld{T}_{\QQ(\zeta_p)}\mmod W)).
\end{align*}

Note that there is no rationalization necessary on the left-hand sides.

As with <ref>, <ref> reduces to proving the (also conjectural) equivalence

\begin{align*}
%\End_{\ld{\co}^\univ_{\zeta_p}}(\QCoh(\ld{T}_{\QQ(\zeta_p)})) & \simeq \LMod_{\cH_{\zeta_p}(\tilde{\ld{T}}, \tilde{W}^\aff)e}, \\
\End_{\HC_{\zeta_p}(\ld{G})}(\QCoh(\ld{T}_{\QQ(\zeta_p)}\mmod W)) & \simeq \LMod_{e\cH_{\zeta_p}(\tilde{\ld{T}}, \tilde{W}^\aff)e},
\end{align*}

where $\cH_{\zeta_p}(\tilde{\ld{T}}, \tilde{W}^\aff)$ denotes the algebra obtained from $\cH(\tilde{\ld{T}}, \tilde{W}^\aff)$ by setting $q$ (arising from the loop rotation torus in $\tilde{\ld{T}}$) to $\zeta_p$.

§.§ The elliptic Kostant slice

Fix a (classical) $\QQ$-algebra $k$ for the remainder of this section. Let $E$ be a (smooth) elliptic curve over $k$, let $\Bun_B^0(E)$ denote the moduli stack of $B$-bundles on $E$ of degree $0$, and let $\Bun_T^0(E)$ denote the scheme of $T$-bundles on $E$ of degree $0$. We will also make use of the stack $\Bun_G^\ss(E)$ of semistable $G$-bundles on $E$.

Say that a $B$-bundle $\cP_B$ on $E$ is regular if $\dim \Aut(\cP_B) = \rank(G)$. Let $\Bun_B^0(E)^\reg$ denote the open substack of $\Bun_B^0(E)$ defined by the regular $B$-bundles. Similarly, if $\cP\in \Bun_G^\ss(E)$ is a semistable $G$-bundle on $E$, we say that $\cP$ is regular if $\dim \Aut(\cP) = \rank(G)$. Let $\Bun_G^\ss(E)^\reg \subseteq \Bun_G^\ss(E)$ denote the open substack of regular semistable $G$-bundles.

For $\cP_T\in \Bun_T^0(E)$, write $\Delta_\cP$ to denotes the set of those simple roots $\alpha\in \Delta$ such that the $\alpha$-component of $\cP_T$ is trivial. We will also write $N_\cP = \prod_{\alpha\in \Phi^- \cap \Delta_\cP} N_\alpha\subseteq N$.

The map $\Bun_B^0(E) \to \Bun_T^0(E)$ admits a canonical unique section $\kappa: \Bun_T^0(E) \to \Bun_B^0(E)$ landing in $\Bun_B^0(E)^\reg$.

Let $\cP$ be a semistable $G$-bundle on $E$.
By <cit.>, the regularity of $\cP$ is equivalent to the condition that for any (or some) $B$-reduction $\cP_B$ of $\cP$ of degree $0$, the associated $N$-bundle $\cP_B/T$ is induced from an $N_\cP$-bundle with nontrivial associated $N_\alpha$-bundle for each $\alpha\in \Delta_\cP$. Moreover, every geometric fiber of the map $\Bun_G^\ss(E) \to \Hom(\bX^\ast(T), E)\mmod W$ to the coarse moduli space of $\Bun_G^\ss(E)$ contains a unique regular semistable $G$-bundle. Also see <cit.>, where a similar result is stated.

Following <cit.>, set

$$\tilde{\Bun}_G^\ss(E)^\reg \cong \Bun_G^\ss(E)^\reg \times_{\Hom(\bX^\ast(T), E)\mmod W} \Hom(\bX^\ast(T), E).$$

Let $\Bun_B^0(E)^\reg$ denote the moduli stack of $B$-bundles on $E$ of degree $0$.
It then follows from the isomorphism $\tilde{\Bun}_G^\ss(E) \cong \Bun_B^0(E)$ of <cit.> and the equality $\dim \Aut(\cP) = \dim \Aut(\cP_B)$ that there is an isomorphism $\tilde{\Bun}_G^\ss(E)^\reg \cong \Bun_B^0(E)^\reg$. In particular, every geometric fiber of the map $\Bun_B^0(E) \to \Hom(\bX^\ast(T), E) = \Bun_T^0(E)$ contains a unique regular $B$-bundle of degree $0$.

The existence of $\kappa$ is a consequence of <cit.>, which is a refinement of <cit.>. Since we will not need the full strength of <cit.> outside of this proof, we will only briefly recall the necessary notation and statements. In loc. cit., the scheme $\Bun_T^0(E)$ is denoted by $Y$. Let $\tilde{\Bun}_G(E)$ denote the Kontsevich-Mori compactification of $\tilde{\Bun}_G^\ss(E) \cong \Bun_B^0(E)$; see <cit.>. Let $\Theta$ denote the theta-line bundle over $\Bun_T^0(E)$ of <cit.>, and let $\tilde{\chi}: \tilde{\Bun}_G(E) \to \Theta^{-1}/\GG_m$ denote the map constructed in <cit.>. Then, <cit.> shows that there is a map $\Theta^{-1} \to \tilde{\Bun}_G^\ss(E)$ landing in $\tilde{\Bun}_G^\ss(E)^\reg$ such that the composite

$$\Theta^{-1} \to \tilde{\Bun}_G^\ss(E) \xar{\tilde{\chi}} \Theta^{-1}/\GG_m$$

is the canonical map. Composing with the zero section of $\Theta^{-1}$, we obtain a map

$$\Bun_T^0(E) \cong 0_{\Theta^{-1}} \to \Theta^{-1} \to \tilde{\Bun}_G^\ss(E)^\reg \cong \Bun_B^0(E).$$

This is the desired map $\kappa$.

We will refer to the map $\kappa: \Bun_T^0(E) \to \Bun_B^0(E)$ from <ref> as the elliptic Kostant slice.

Let $G = \SL_2$, so that a $B$-bundle on $E$ is just a rank $2$ vector bundle $\cV$ with $\det(\cV) = 0$, equipped with a full flag. Then, the map $\kappa: \Pic^0(E) \to \Bun_B^0(E)$ sends a line bundle $\cL$ to the trivial filtration $\co_E \subseteq \co_E \oplus \cL$ if $\cL^2 \neq \co_E$; and to the Atiyah extension $\cL \subseteq \cf_2 \twoheadrightarrow \cL^{-1}$ from [4] if $\cL^2 \cong \co_E$. This extension is defined by a nontrivial element of $\Ext^1_E(\cL, \cL^{-1}) \cong \H^1(E; \cL^{-2})$.
This can either be shown by unwinding the construction of the section $\kappa$ via <cit.>, or directly by noting that the description above provides the unique regular $B$-bundle lifting $\cL$.

We will need the following lemma below.

Let $I\subseteq \Phi^-$ be a subset, and let $\Bun_T^0(E)_I$ denote the subscheme of $\Bun_T^0(E)$ defined by those bundles $\cP_T$ whose $\alpha$-component is trivial precisely for $\alpha\in I$. Let $N_I\subseteq N$ be the smallest unipotent subgroup which is invariant under $T$-conjugation and which contains $N_\alpha$ for every $\alpha\in I$. Then the natural map

$$\Bun_{TN_I}^0(E) \times_{\Bun_T^0(E)} \Bun_T^0(E)_I \to \Bun_B^0(E) \times_{\Bun_T^0(E)} \Bun_T^0(E)_I$$

is an isomorphism.

Let $\cP_I$ denote the universal $T$-bundle over $\Bun_T^0(E)_I$, so that $\Bun_B^0(E) \times_{\Bun_T^0(E)} \Bun_T^0(E)_I$ is the stack of $B$-bundles $\cP_B$ such that $\cP_B/N \cong \cP_T$; therefore, it is isomorphic to the stack $\Bun_N^{\cP_I}$ in the notation of <cit.>. Similarly, $\Bun_{TN_I}^0(E) \times_{\Bun_T^0(E)} \Bun_T^0(E)_I \cong \Bun_{N_I}^{\cP_I}$. To show that these stacks are isomorphic, consider the filtration

$$N_\ell \subseteq N_{\ell-1} \subseteq \cdots \subseteq N_2 \subseteq N_1 = N$$

by root height (recall that the height of a root is the sum of its simple root components), so that it is invariant under $T$-conjugation, and there is an induced filtration

$$N_{I,\ell} \subseteq N_{I, \ell-1} \subseteq \cdots \subseteq N_{I,2} \subseteq N_{I,1} = N_I.$$

Then, $N_j\subseteq N$ is normal and $N_{j-1}/N_j$ is central in $N/N_j$ (and similarly for $N_{I,j}$); this implies that $\Bun_{N/N_j}^{\cP_I}$ is a $\Bun_{N_{j-1}/N_j}^{\cP_I}$-torsor over $\Bun_{N/N_{j-1}}^{\cP_I}$. Similar statements hold for $\Bun_{N_I/N_{I,j}}^{\cP_I}$. To show that $\Bun_{N_I}^{\cP_I} \to \Bun_N^{\cP_I}$ is an isomorphism, it therefore suffices to show that the induced map $\Bun_{N_{I,j-1}/N_{I,j}}^{\cP_I} \to \Bun_{N_{j-1}/N_j}^{\cP_I}$ is an isomorphism. Let $\cN = \cP_I \times^T N$, $\cN_I = \cP_I \times^T N_I$, etc., so that $\cN_{j-1}/\cN_j$ is a direct sum of line bundles of degree zero. By choice of $N_I$, the inclusion of the trivial line bundle summands into $\cN_{j-1}/\cN_j$ factors through the map $\cN_{I,j-1}/\cN_{I,j} \to \cN_{j-1}/\cN_j$. The desired isomorphism then follows from the observation that if $U$ is a vector group with $\GG_m$-action, then $\Bun_U^\cL$ is a point if $\cL$ is a nontrivial line bundle of degree zero (because then $\H^1(E; U(\cL)) = 0$).

For instance, suppose that $I = \emptyset$, so that $\Bun_T^0(E)_\emptyset$ denotes the open subscheme of $T$-bundles of degree zero whose $\alpha$-component is nontrivial for every negative root $\alpha$. The isomorphism $\tilde{\Bun}_G^\ss(E) \cong \Bun_B^0(E)$ implies that the map $\tilde{\Bun}_G^\ss(E) \to \Bun_T^0(E)$ is an isomorphism over $\Bun_T^0(E)_\emptyset$. In particular, every point of $\Bun_T^0(E)_\emptyset$ has a canonical associated (regular) semistable $G$-bundle.
The above results continue to hold if $E$ is replaced by the constant stack $S^1$ or by $B\GG_a$ (in which case $\tilde{\Bun}_G^\ss(E)$ and $\Bun_B^0(E)$ are to be interpreted as $G/G$ and $B/B$, and $\g/G$ and $\fr{b}/B$, respectively). In the case of $S^1$, for instance, the semistable $G$-bundles obtained in this way from $\Bun_T^0(E)_\emptyset$ are precisely those which lie in the regular semisimple locus $G^{\rs}/G$; similarly for the case of $B\GG_a$.

§.§ Rationalized Langlands duality over elliptic cohomology

Let $\GG_0$ be a commutative group scheme over a ring $A_0$ (even an $\Eoo$-ring, but we will not need this). Let $\GG_0^\vee$ denote the stack $\Hom(\GG_0, B\GG_m)$.

If $\GG_0 = \GG_m$, then $\GG_0^\vee = B\Z$, i.e., is $S^1$ viewed as a constant stack. If $\GG_0$ is an abelian variety, then $\GG_0^\vee$ is the dual abelian variety. If $\GG_0 = \Z$, then $\GG_0^\vee$ is $B\GG_m$. Let $W$ denote the commutative group scheme over $\Z_{(p)}$ of $p$-typical Witt vectors. Let $W[F]$ denote the kernel of Frobenius on $W$. If $\hat{\GG}_a$ denotes the formal completion of $\GG_a$ at the origin, then $\hat{\GG}_a^\vee \cong BW[F]$ (over $\Z_{(p)}$). Since $W[F] \cong \GG_a$ over a field of characteristic zero, there is an isomorphism $\hat{\GG}_{a,\QQ}^\vee \cong B\GG_a$.

In general, there is a canonical map $\GG_0 \to (\GG_0^\vee)^\vee$, and the above examples imply that it is an isomorphism if $\GG_0$ is a finite product of abelian varieties, classifying stacks of groups of multiplicative type, and finitely generated abelian groups. If this is the case, $\GG_0$ is said to be dualizable.

Note that the pairing $\GG_0 \times \GG_0^\vee \to B\GG_m$ defines a line bundle over $\GG_0 \times \GG_0^\vee$, which we will denote by $\cP$ and call the Poincaré line bundle. If $\GG_0$ is an abelian variety, this is the usual Poincaré line bundle over $\GG_0 \times \GG_0^\vee$. If $\GG_0 = \GG_m$, the Poincaré line bundle gives the equivalence $\Rep(\Z) \simeq \QCoh(\GG_m)$ obtained by viewing $\GG_m$ as the torus associated to the monoid $\Z$.

If $\GG_0$ is a finite flat, diagonal, or constant group scheme (but not an abelian variety!), then $\GG_0^\vee$ can be identified with the classifying stack of the Cartier dual of $\GG_0$.
If $X$ is an $A_0$-scheme, let $\cL_{\GG_0} X$ denote the $\GG_0$-loop space of $X$, given by the mapping stack $\Map(\GG_0^\vee, X)$.
Then, if $\GG_0$ is replaced by its formal completion at the zero section, the $\GG_0$-loop space recovers the loop space of [67].

Fix an isomorphism $\bX^\bull(T) \cong \bX_\bull(T)$ of lattices, which will be used implicitly below without further mention. (Note that we are not asking for a $W$-equivariant isomorphism, which would not exist in general.) This gives an isomorphism $\cM_T \cong \cM_{\ld{T}}$, which we will use below as an analogue of the identification between $\ld{\fr{t}} = \fr{t}^\ast$ and $\fr{t}$ (ubiquitous in geometric representation theory). Although potentially confusing, we will see below in the proof of <ref> that this identification does not run the risk of conflating different sides of Langlands duality.

We will prove the following at the end of the section, after a discussion of some consequences.

Fix a complex-oriented even-periodic $\Eoo$-ring $A$ and an oriented commutative $A$-group $\GG$, as well as a semisimple algebraic group $\ld{G}$ over $\QQ$.
Assume that the underlying $\pi_0 A$-scheme $\GG_0$ is $\GG_a$, $\GG_m$, or an elliptic curve $E$.
Given a principal nilpotent $f\in \fr{n}$, there is a “$\GG$-Kostant slice” $\kappa: (\cM_{T,0})_\QQ \to \Bun_{\ld{B}}(\GG_{0,\QQ}^\vee)$ over $\pi_0 A_\QQ$. If $\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee) = \Bun_{\ld{B}}(\GG_{0,\QQ}^\vee) \times_{\Bun_T} \cM_T$, there is a Cartesian square

(T^\ast_\GG \ld{T})^\bl \otimes \QQ \ar[r] \ar[d] & (\cM_{T,0})_\QQ \ar[d]^-\kappa \\
(\cM_{T,0})_\QQ \ar[r]_-\kappa & \Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee).

Combining with <ref>, we obtain the following:

Suppose that $G$ is a connected and simply-connected semisimple algebraic group or a torus over $\cc$.
Assume that the underlying $\pi_0 A$-scheme $\GG_0$ is $\GG_a$, $\GG_m$, or an elliptic curve $E$.
Then there is an equivalence

%\simeq \QCoh((T^\ast_{\GG} \ld{T})^\bl)
\simeq \Mod_{\pi_0 \cf_T(\Gr_G(\cc))^\vee}(\QCoh(\cM_{T,0})) \otimes \QQ,$$

where $\QCoh((\cM_{T,0})_\QQ)$ is regarded as a $\QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee))$-module via $\kappa$.

For example, if $\GG = \hat{\GG}_a$, then $\GG_0^\vee = BW[F]$. Therefore, $\GG_{0,\QQ}^\vee = B\GG_a$, and $\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee) = \ld{\fr{b}}_\QQ/\ld{B}_\QQ \cong \tilde{\ld{\g}}_\QQ/\ld{G}_\QQ$ by <cit.>. In particular, <ref> was proved above in this case as <ref>.
If $\GG = \GG_m$, then $\GG_{0,\QQ}^\vee = B\Z = S^1$, so that $\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee) = \Map(S^1_{\KU_\QQ}, B\ld{B}_{\KU_\QQ})$ is isomorphic to the $2$-periodification of $\ld{B}_\QQ/\ld{B}_\QQ$. In particular, <ref> was proved above in this case as <ref>.
If $\GG_0$ is an elliptic curve $E$, then $\GG_0^\vee = E^\vee$, so that $\Bun_{\ld{B}}^0(\GG_0^\vee) = \Bun_{\ld{B}}^0(E^\vee)$. <ref> in this case will be proved below.

We also obtain a proof of <ref> (which we restate for convenience):
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 $G(\cc)$. 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 $E$. Then there is an equivalence of $\pi_0 A_\QQ$-linear $\infty$-categories:

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

Note that $G_c$ is connected. By <ref>, there is an equivalence $\Loc_{T_c}^\gr(G_c; A) \simeq \LMod_{\pi_0 \cf_T(\Omega G_c)^\vee}(\QCoh(\cM_{T,0}))$, so the claim follows from <ref>.

If $A = \QQ[\beta^{\pm 1}]$, the equivalence resulting from <ref> is an equivalence of $2$-periodic $\QQ$-linear $\infty$-categories. However, the equivalence can be de-periodified, and one obtains an equivalence

$$\Loc_{T_c}(G_c; \QQ) \simeq \QCoh(\ld{\fr{t}}[2]_\QQ \times_{\tilde{\ld{\g}}[2]_\QQ/\ld{G}_\QQ} \ld{\fr{t}}[2]_\QQ).$$

There is also a $G_c$-equivariant analogue:

$$\Loc_{G_c}(G_c; \QQ) \simeq \QCoh(\ld{\fr{t}}[2]_\QQ\mmod W \times_{\ld{\g}[2]_\QQ/\ld{G}_\QQ} \ld{\fr{t}}[2]_\QQ\mmod W).$$

This equivalence can be de-equivariantized, to obtain an equivalence

$$\Loc(G_c; \QQ) \simeq \QCoh(Z_f(\ld{B})),$$

where $f\in \ld{\g}$ is the image of the origin in $\ld{\fr{t}}\mmod W$ under the Kostant slice, and $Z_f(\ld{B})$ is a shifted analogue of the centralizer of $f$ in $\ld{B}$.
Note that $T^\ast G_c = G(\cc)$, so that the left-hand side can be interpreted as a relative of the $\QQ$-linearization of the wrapped Fukaya category of $T^\ast G_c$ by <cit.>. In particular, this shifted analogue of $Z_f(\ld{B})$ is a (derived) mirror to $G(\cc)$ viewed as a symplectic manifold.

The proof of <ref> above uses the Koszul duality equivalence $\Loc_{T_c}(G_c; A) \simeq \LMod_{\cf_T(\Omega G_c)^\vee}(\QCoh(\cM_{T,0}))$ of <ref>. The category $\LMod_{\cf_T(\Omega G_c)^\vee}(\QCoh(\cM_{T,0}))$ (and hence the right-hand side of <ref>) admits a “quantization” parametrized by $\GG$, given by $\LMod_{\cf_{\tilde{T}}(\Omega G_c)^\vee}(\QCoh(\cM_{T,0}))$. For instance, if $A = \QQ[\beta^{\pm 1}]$, the right-hand side of <ref> quantizes to $\End_{\ld{\co}^\univ_\hbar}(\QCoh(\tilde{\fr{t}}))$; and if $A = \KU$, the right-hand side of <ref> quantizes to $\End_{\ld{\co}^\univ_q}(\QCoh(\tilde{T}))$.
It follows from this discussion that the $\infty$-category $\Loc_{T_c}(G_c; A)$ must itself admits a quantization. We have seen a quantization of this form above in <ref>.

In fact, <ref> and <ref> suggest that $\LMod_{\cf_{\tilde{T}}(\Gr_G(\cc))^\vee}(\QCoh(\cM_{\tilde{T}})) \otimes \QQ$ should be viewed as $\End_{\cC_\GG}(\QCoh(\cM_{\tilde{T}})\otimes \QQ)$ for some $A_\QQ$-linear $\infty$-category $\co_\GG$ which is a $1$-parameter deformation of $\QCoh(\Bun_{\ld{B}}(\GG_{0,\QQ}^\vee))$. The coordinate on the group scheme $\GG$ defines a “quantization parameter” (i.e., the analogue of $\hbar$ and $q$). This putative $\infty$-category $\co_\GG$ would be an analogue of the (quantum) universal category $\co$. We do not know how to define such an $\infty$-category $\co_\GG$ at the moment; however, in future work, we plan to use the results of [23] to study an “$F$-deformation” of $U(\g)$ for certain formal group laws $F(x,y)$ (at least for $G = \SL_2, \PGL_2$). When $F$ is the multiplicative formal group, this $F$-deformation of $U(\g)$ recovers the quantum enveloping algebra $U_q(\g)$. We hope that further study of such deformations will point to a good definition of the putative $\infty$-category $\co_\GG$.

It is natural to ask for an explicit description of the $1$-parameter deformation of $\Loc_{T_c}(G_c; A)$ over $\GG$ from <ref> (i.e., not in terms of the framed $\E{2}$-structure on $\Omega G_c = \Omega^2 BG_c$). To describe this, let us view $\Loc_{T_c}(G_c; A)$ as the $\infty$-category of local systems on the orbifold $G_c/_\ad T_c$. We now need the following:

The orbifold $G_c/_\ad T_c$ is isomorphic to the the moduli stack $\Conn(S^1; \g)^\lev$ of $\g$-valued smooth connections on $S^1$ equipped with a level structure given by a $T_c$-reduction at $\{1\}\in S^1$, taken modulo gauge transformations.

Write $G_c/_\ad T_c \simeq \ast/T_c \times_{\ast/G_c} G_c/_\ad G_c$. There is an equivalence

$$G_c/_\ad G_c \simeq \ast/G_c \times_{\ast/G_c \times \ast/G_c} \ast/G_c$$

which exhibits $G_c/_\ad G_c$ as the free loop space $\cL(\ast/G_c) \cong \ast/\cL G_c$ in the $\infty$-category of orbifolds. To see this, note that $G_c/G_c\simeq G_c\backslash (G_c\times G_c)/G_c$, where $G_c\times G_c$ acts on $G_c\times G_c$ via

$$(g_1, g_2): (h_1, h_2) \mapsto (g_1 h_1 g_2^{-1}, g_1 h_2 g_2^{-1}).$$

In any case, the above equivalence implies that $G_c/_\ad G_c$ is isomorphic to the moduli stack $\Conn(S^1; \g)/\cL G_c$, where $\Conn(S^1; \g)$ is the moduli space of smooth connections on $S^1$ valued in $\g$; see <cit.>. This implies the desired claim.

One natural way to quantize $\Loc_{T_c}(G_c; A)$ is therefore to consider the $\infty$-category of “$S^1_\rot \ltimes \cL G_c$-equivariant $A$-valued local systems on $\Conn(S^1; \g)^\lev$”; this is a module over $\Loc_{S^1_\rot}(\ast; A) \simeq \QCoh(\GG)$, and its fiber over the zero section of $\GG$ is $\Loc_{T_c}(G_c; A)$ itself. However, defining this $\infty$-category precisely requires additional effort, since $S^1_\rot \ltimes \cL G_c$ is not a compact group.

Let us now turn to the proof of <ref>; by <ref>, we only need to consider the case when $\GG$ is a (smooth) elliptic curve $E$. Since we are working on one side of Langlands duality, we now drop the “check”.

We will work over $\QQ$, and omit it from the notation.
Write $X$ to denote the fiber product in <ref>, so that our goal is to identify $X$ with $(T^\ast_\GG \ld{T})^\bl$. (The reader should keep in mind <ref>.) The argument of <cit.> can be used to reduce to the case when $\ld{G}$ has semisimple rank $1$.

Namely, first note that both $X$ and $(T^\ast_\GG \ld{T})^\bl$ are flat over $\cM_T$: the only nontrivial case is $(T^\ast_\GG \ld{T})^\bl$, in which case this follows from <cit.>. Let $\cM_T^\circ \hookrightarrow \cM_T$ denote the open immersion given by the complement of the union of the divisors $\cM_{T_\alpha} \hookrightarrow \cM_T$ for $\alpha\in \Phi$.
Upon localizing to $\cM_T^\circ$, both $X$ and $(T^\ast_\GG \ld{T})^\bl$ are isomorphic to $\ld{T} \times \cM_T^\circ$.
Let $\cM_T^\bull$ denote the complement of the union of all pairwise intersections of the divisors $\cM_{T_\alpha} \hookrightarrow \cM_T$ for $\alpha\in \Phi$. Then $\cM_T - \cM_T^\bull \hookrightarrow \cM_T$ is of codimension $\geq 2$.
It therefore suffices to show (by flatness of both $X$ and $(T^\ast_\GG \ld{T})^\bl$ over the normal irreducible scheme $\cM_T$) that the isomorphism $X|_{\cM_T^\circ} \cong (T^\ast_\GG \ld{T})^\bl|_{\cM_T^\circ}$ extends across the codimension $1$ points of $\cM_T - \cM_T^\circ$ (i.e., points of $\cM_T^\bull - \cM_T^\circ$).

If $y$ is a codimension $1$ point of $\cM_T$ which lies on the divisor $\cM_{T_\alpha} \hookrightarrow \cM_T$ for some $\alpha\in \Phi$, let $Z_\alpha(y)\subseteq \ld{G}$ denote the reductive subgroup of $\ld{G}$ containing $\ld{T}$ and whose nonzero roots are $\pm \alpha$.

This is a connected Levi subgroup of semisimple rank $1$. It is easy to see that the localization $(T^\ast_\GG \ld{T})^\bl_y$ depends only on $Z_\alpha(y)$.
Let $\ld{B}_\alpha^- \subseteq \ld{B}$ denote the Borel subgroup of $Z_\alpha(y)$ determined by $\ld{B}$. <ref> with $I = \{\alpha\}$ implies that the induced map from $(\cM_{T,0})_\QQ \times_{\Bun_{\ld{B}_\alpha^-}^0(E)} (\cM_{T,0})_\QQ$ to $(\cM_{T,0})_\QQ \times_{\Bun_{\ld{B}}^0(E)} (\cM_{T,0})_\QQ$ defines an isomorphism upon localizing at $y$. In particular, the localization $X_y$ also depends only on $Z_\alpha(y)$.

We are now reduced to the case when $\ld{G}$ has semisimple rank $1$. Every split reductive group of semisimple rank $1$ is isomorphic to the product of a split torus with $\SL_2$, $\PGL_2$, or $\GL_2$. Let us illustrate the calculation when $\ld{G} = \PGL_2$. The cases $\ld{G} = \SL_2, \GL_2$, and products of tori with these groups can be addressed similarly.
For notational convenience, we will drop the “check”s and write $B$ instead of $\ld{B}$, etc.; also note that since $T$ is of rank $1$, we may identify $\cM_T \cong \GG$.
Let $\cV$ denote the unique indecomposable rank $2$ “Atiyah bundle” over $E^\vee \times \GG_0$; this is an extension of the structure sheaf by the Poincaré line bundle $\cP$, which is specified by a nonzero section of $\H^1(E^\vee \times \GG_0; \cP) \cong k$. The bundle $\cV$ sits in a short exact sequence

$$0 \to \cP \to \cV \to \co_{E^\vee \times \GG_0} \to 0.$$

Any fixed basepoint $p_0\in E^\vee$ defines an isomorphism $E^\vee \cong \GG_0$, and allows us to identify $\cP$ with the line bundle on $E^\vee \times E^\vee$ corresponding to the divisor $\Delta - E^\vee \times \{p_0\} - \{p_0\} \times E^\vee$, where $\Delta$ is the diagonal. In particular, $\cP|_{E^\vee \times \{x\}} \cong \co_{E^\vee}(x-p_0)$, and is therefore only trivial when $x = p_0$. The fiber of $\cV$ over $E^\vee \times \{x\}$ is specified by a nonzero element of $\Ext^1_{E^\vee}(\co, \co(x-p_0))$; but if $\cL$ is a nontrivial line bundle, then $\H^1(E^\vee; \cL) = 0$. This implies that the map $\kappa: \GG_0 \to \Bun_B(E^\vee)$ sends a degree $0$ line bundle $\cL$ on $E^\vee$ 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 $\co_{E^\vee}\subseteq \cf_2$ if $\cL \cong \co_{E^\vee}$.

We need to understand $\Aut_B(\{\cP\subseteq \cV\})$.
If $\cL$ is a nontrivial line bundle on $E^\vee$, then $\cL$ has no sections, so $\Aut_B(\{\co_{E^\vee} \subseteq \co_{E^\vee} \oplus \cL\}) \cong \GG_m$. On the other hand, the algebra $\End(\cf_2)$ of endomorphisms of $\cf_2$ as a rank $2$ vector bundle is isomorphic to $k[\epsilon]/\epsilon^2$ as an algebra; the element $\epsilon$ acts as the composite $\cf_2 \twoheadrightarrow \co_{E^\vee} \hookrightarrow \cf_2$.
In particular, the group scheme $\Aut(\cf_2)$ of automorphisms of $\cf_2$ as a rank $2$ vector bundle is $(k[\epsilon]/\epsilon^2)^\times$.
An automorphism of $\cf_2$ preserving the flag $\co_{E^\vee} \subseteq \cf_2$ is defined by a matrix
x & y\\
0 & z
\end{psmallmatrix}$, where $x,y,z\in \Hom(\co_{E^\vee}, \co_{E^\vee})$. In order for two maps $x,z: \co_{E^\vee} \to \co_{E^\vee}$ to define an automorphism of $\cf_2$, we need $x = z$. Since we are only calculating the automorphisms of $\cf_2$ as a $\PGL_2$-bundle, the factor $x = z$ can be scaled out, and we find that $\Aut_B(\{\co_{E^\vee} \subseteq \cf_2\}) \cong \GG_a$.

The fiber of the map $\GG_0 \times_{\Bun_B(E^\vee)} \GG_0 \to \GG_0$ over $\cL\in \GG_0$ is therefore $\GG_m$ if $\cL \not \cong \co_{E^\vee}$ (i.e., away from the zero section), which degenerates to $\AA^1$ over the zero section corresponding to $\cL = \co_{E^\vee}$.

(In the case $\ld{G} = \SL_2$, the same argument shows that the fiber of the map $\GG_0 \times_{\Bun_B(E^\vee)} \GG_0 \to \GG_0$ is still $\GG_m$ if $\cL^2$ is not trivial, but the fiber over any point of $\cL\in \GG_0[2]$ is instead $\GG_a\times \mu_2$. Indeed, the image of of $\cL\in \GG_0[2]$ under the Kostant slice $\GG_0 \to \Bun_B(E^\vee)$ is the nontrivial extension

$$0 \to \cL \to \cL \otimes \cf_2 \to \cL^{-1} \to 0.$$

Note that the subgroup of $\ld{B} \subseteq \SL_2$ given by $\Aut_{\ld{B}}(\{\cL \subseteq \cf_2 \otimes \cL\})$ is of the form $\begin{psmallmatrix}
x & y\\
0 & z
\end{psmallmatrix}$, where $x\in \Hom(\cL, \cL)$, $y\in \Hom(\cL^{-1}, \cL)$, and $z\in \Hom(\cL^{-1}, \cL^{-1})$. Not every such matrix defines an automorphism of $\cf_2 \otimes \cL$; for instance, in order for two maps $x: \cL \to \cL$ and $z:\cL^{-1} \to \cL^{-1}$ to define an automorphism of $\cf_2 \otimes \cL$, we need $x = z \otimes \cL^2 = z$. In order for the resulting matrix $\begin{psmallmatrix}
x & y\\
0 & z
\end{psmallmatrix}$ to preserve the trivialization of $\det(\cV \otimes \cL)$, we need $x^2 = 1$; the function $y$ can be arbitrary. This discussion implies that $\Aut_{\ld{B}}(\{\cL \subseteq \cf_2 \otimes \cL\}) \cong \mu_2 \times \GG_a$, where the $\mu_2$ encodes $x$, and $\GG_a$ encodes $y$.)

The intersection $\GG_0 \times_{\Bun_B(E^\vee)} \GG_0$ consists of $\cL, \cL'\in \GG_0$ equipped with an isomorphism $\kappa(\cL)\cong \kappa(\cL')$ of $B$-bundles over $E^\vee$ (which in particular forces $\cL \cong \cL'$). In fact, the discussion above can be used to conclude that $\GG_0 \times_{\Bun_B(E^\vee)} \GG_0$ is isomorphic to an affine blowup of $\GG_0 \times \GG_m$, defined as the complement $U$ of the proper preimage of the zero section of $\GG_0$ inside the blowup $\fr{B}$ of $\GG_0 \times \GG_m$ at the union of the zero sections of $\GG_0$ and $\GG_m$.
(In the case $\ld{G} = \SL_2$, the fiber product $\GG_0 \times_{\Bun_B(E^\vee)} \GG_0$ is isomorphic to an affine blowup of $\GG_0 \times \GG_m$, defined as the complement $U$ of the proper preimage of the $2$-torsion $\GG_0[2]\subseteq \GG_0$ inside the blowup $\fr{B}$ of $\GG_0 \times \GG_m$ at the union of the $2$-torsion sections $\GG_0[2] \subseteq \GG_0$ and $\mu_2 \subseteq \GG_m$.)

But $U\subseteq \fr{B}$ is precisely the affine blowup $(T^\ast_\GG T)^\bl$, as desired.

The most classical instantiation of the Atiyah bundle is via the Weierstrass functions. The $\GG_a$-torsor $\cA$ over $E$ associated to $\cV$ is the complement of the section at $\infty$ of the projective line $\PP(\cV)$. If we work complex-analytically, $E^\an$ can be identified as the quotient $\cc/\Lambda$ for some rank $2$ lattice $\Lambda\subseteq \cc$. Associated to $\Lambda$ are two Weierstrass functions defined on $\cc$:

\begin{align*}
\wp(z; \Lambda) & = \frac{1}{z^2} + \sum_{\lambda \in \Lambda-\{0\}} \left(\frac{1}{(z-\lambda)^2} - \frac{1}{\lambda^2}\right), \\
\zeta(z; \Lambda) & = \frac{1}{z} + \sum_{\lambda \in \Lambda-\{0\}} \left(\frac{1}{z-\lambda} + \frac{1}{\lambda} + \frac{z}{\lambda^2}\right).
\end{align*}

Note that $\wp(z; \Lambda)$ is doubly-periodic, i.e., $\wp(z + \lambda; \Lambda) = \wp(z; \Lambda)$ for any $\lambda \in \Lambda$. Alternatively, $\wp$ defines a map $\cc \to \cc$ which factors through a map $\cc/\Lambda = E^\an \to \cc$.

Although $\zeta(z; \Lambda)$ is not doubly-periodic, an easy calculation shows that $\wp(z; \Lambda) = -\partial_z \zeta(z; \Lambda)$; so if $\lambda \in \Lambda$, then $\zeta(z+\lambda; \Lambda) - \zeta(z; \Lambda) = c(\lambda)$ for some constant $c(\lambda)$. The function $\lambda \mapsto c(\lambda)$ is evidently additive, and defines a homomorphism $\Lambda \to \cc$, which defines a $\cc$-bundle over $E^\an = \cc/\Lambda$. This $\cc$-bundle is precisely the analytification $\cA^\an$ of the $\GG_a$-torsor $\cA$. It follows that although $\zeta$ is not defined on $E^\an$, the torsor $\cA^\an$ is the universal space over $E^\an$ on which $\zeta$ is defined.

This discussion also describes the total space of the rank $2$-bundle $\cV^\an$ purely analytically. For instance, if $q\in \cc^\times$ is a unit complex number of modulus $<1$, we can identify $\Tot(\cV^\an)$ over the Tate curve $\cc^\times/q^\Z$ with the quotient

$$\Tot(\cV^\an) = \left(\cc^\times \times \cc^2\right)/\left((z,x) \sim \left(qz, \begin{psmallmatrix}
1 & 1\\
0 & 1
\end{psmallmatrix} x\right)\right).$$

§.§ Putting it together

We will now explore one corollary of <ref>.

Let $\Prst$ be the $\infty$-category of compactly generated presentable $\infty$-categories and colimit-preserving functors which preserve compact objects.
Let $\cC\in \CAlg(\Prst)$, and let $\cd\in \CAlg(\LMod_\cC(\Prst))$ whose underlying object of $\LMod_\cC(\Prst)$ is dualizable. The unit map $i^\ast: \cC \to \cd$ defines a symmetric monoidal functor ${i'}^\ast: \cd \simeq \cd \otimes_\cC \cC \to \cd \otimes_\cC \cd$, and if $i_\ast: \cd \to \cC$ denotes the right adjoint to $i^\ast$, the following diagram commutes:

\cd \ar[r]^-{i_\ast} \ar[d]_-{{i'}^\ast} & \cC \ar[d]^-{i^\ast}\\
\cd \otimes_\cC \cd \ar[r]_-{{i'}_\ast} & \cd.

In <ref>, there is a fully faithful colimit-preserving functor $\Tot(\cd^{\otimes_\cC \bull+1}) \hookrightarrow \cC$; we will denote its essential image by $\cC^\wedge_\cd$.

The assumptions in <ref> imply that the augmented cosimplicial diagram $ N(\Deltab_+) \to \Cat_\infty$ given by $\cd^{\otimes_\cC \bull+1}$ satisfies the assumptions of <cit.>. Therefore, the functor $\cC \to \Tot(\cd^{\otimes_\cC \bull+1})$ has a fully faithful left adjoint, as desired.

Regard $\QCoh((\cM_{T,0})_\QQ)$ as a $\QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee))$-algebra via $\kappa: (\cM_{T,0})_\QQ \to \Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)$. Then the completion $\QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee))^\wedge_{\QCoh((\cM_{T,0})_\QQ)}$ of <ref> with respect to $\QCoh((\cM_{T,0})_\QQ)$ can be identified with $\QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)^\reg)$.

When $A$ is an $\Eoo$-$\QQ[\beta^{\pm 1}]$-algebra and $\GG = \hat{\GG}_a$, the Koszul duality equivalence of <ref> gives $\QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)) \simeq \IndCoh((\tilde{\ld{\cN}} \times_{\ld{\g}} \{0\})/\ld{G})$; we define $\IndCoh((\tilde{\ld{\cN}} \times_{\ld{\g}} \{0\})/\ld{G})_\Kost$ to be the essential image of $\QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)^\reg)$ under this equivalence. We remark that in this case, $\QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)^\reg) \simeq \QCoh(\tilde{\ld{\g}}^\reg/\ld{G})$. Similarly, if $A = \KU$ and $\GG = \GG_m$, then $\QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)^\reg) \simeq \QCoh(\tilde{\ld{G}}^\reg/\ld{G})$.

Fix a complex-oriented even-periodic $\Eoo$-ring $A$ and an oriented commutative $A$-group $\GG$.
Assume that the underlying $\pi_0 A$-scheme $\GG_0$ is $\GG_a$, $\GG_m$, or an elliptic curve $E$.
Suppose $G$ is a connected and simply-connected semisimple algebraic group over $\cc$.
Then there is an $\E{2}$-monoidal equivalence

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

The $\Eoo$-coalgebra structure on $\pi_0 \cf_T(\Gr_G(\cc))^\vee$ defines a $\QCoh(\cM_{T,0})$-coalgebra structure on $\LMod_{\pi_0 \cf_T(\Gr_G(\cc))^\vee}(\QCoh(\cM_{T,0}))$.
The right-hand side of the equivalence of <ref> also admits a $\QCoh((\cM_{T,0})_\QQ)$-coalgebra structure, being the tensor product of $\QCoh((\cM_{T,0})_\QQ)$ with itself over $\QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee))$; and it is not difficult to check that the equivalence of <ref> is one of $\QCoh((\cM_{T,0})_\QQ)$-coalgebras.
In particular, there is a commutative diagram

& \QCoh((\cM_{T,0})_\QQ) \ar[d]^-{\ast \to \Gr_G(\cc)} \ar[dl] \\
\QCoh((\cM_{T,0})_\QQ \times_{\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)} (\cM_{T,0})_\QQ) \ar[r]^-\sim & \LMod_{\pi_0 \cf_T(\Gr_G(\cc))^\vee}(\QCoh(\cM_{T,0})) \otimes \QQ\\

which defines an equivalence of cosimplicial diagrams, and hence of their totalizations. The totalization of the cosimplicial diagram built from the functor $\QCoh(\cM_{T,0}) \to \Mod_{\pi_0 \cf_T(\Gr_G(\cc))^\vee}(\QCoh(\cM_{T,0}))$ defines an equivalence

$$\Tot(\LMod_{(\pi_0 \cf_T(\Gr_G(\cc))^\vee)^{\otimes \bull}}(\QCoh(\cM_{T,0}))) \simeq \coLMod_{\pi_0 \cf_T(\Gr_G(\cc))^\vee}(\QCoh(\cM_{T,0}));$$

note that by <ref>, this is in turn equivalent to $\Loc_{T_c}^\gr(\Omega G_c; A)$.
By <ref>, we also have

$$\Tot(\QCoh((\cM_{T,0})_\QQ)^{\otimes_{\QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee))} \bull+1}) \simeq \QCoh(\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee)^\reg).$$

This gives the desired equivalence.

When $A = \QQ[\beta^{\pm 1}]$ and $\GG = {\GG}_a$, we have $\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee) = \tilde{\ld{\g}}/\ld{G}$. <ref> gives an $\E{2}$-monoidal equivalence

\begin{align*}
\IndCoh((\tilde{\ld{\cN}} \times_{\ld{\g}} \{0\})/\ld{G})_\Kost & \simeq \QCoh(\tilde{\ld{\g}}^\reg/\ld{G}) \\
& \simeq \Loc_{T_c}^\gr(\Omega G_c; \QQ[\beta^{\pm 1}]).
%\coLMod_{C^T_\ast(\Gr_G(\cc); \QQ)} \otimes_\QQ \QQ[\beta^{\pm 1}].
\end{align*}

Note that $\tilde{\ld{\g}}/\ld{G}$ is isomorphic to the quotient $\ld{G} \backslash (\ld{G} \times^{\ld{N}} \ld{\fr{b}})/\ld{T}$; and <cit.> says that $\ld{G} \times^{\ld{N}} \ld{\fr{b}}$ is the universal symplectic implosion (i.e., the symplectic implosion of $T^\ast G$). The relationship of this perspective to Langlands duality is closely related to the program of Ben-Zvi–Sakellaridis–Venkatesh [85]: namely, the Hamiltonian $\ld{G} \times \ld{T}$-space $T^\ast(\ld{G}/\ld{N})$ acts as a “kernel” for the symplectic implosion functor from Hamiltonian $\ld{G}$-spaces to Hamiltonian $\ld{T}$-spaces.

Similarly, using <ref>, one can prove an equivalence between $\Loc_{\tilde{T}_c}^\gr(\Omega G_c; \QQ[\beta^{\pm 1}])$ and a localization of $\ld{\co}^\univ_\hbar$.
There is also an $\E{2}$-monoidal equivalence

$$\QCoh(\ld{\g}^\reg/\ld{G}) \simeq
\Loc_{G_c}^\gr(\Omega G_c; \QQ[\beta^{\pm 1}]);
%\coLMod_{C^G_\ast(\Gr_G(\cc); \QQ)} \otimes_\QQ \QQ[\beta^{\pm 1}]

this follows from the analogue of <ref> for $G_c$-local systems and <cit.>, which says that the classifying stack of the group scheme $\ld{J} = \spec \H^G_\ast(\Gr_G(\cc); \QQ)$ of regular centralizers is isomorphic to $\ld{\g}^\reg/\ld{G}$.

In the case of rank $1$, one can use <ref> to show that if the circle $\SO(2)$ acts on $\SO(3)/\SO(2) = S^2$ by left multiplication, there is an equivalence

\begin{equation}\label{eq: SO2-equiv Loops S2 periodic}
\Loc_{\SO(2)}^\gr(\Omega S^2; \QQ[\beta^{\pm 1}]) \simeq \QCoh(T^\ast(\AA^2)^\reg/\SL_2).
\end{equation}

Here, $\AA^2$ is equipped with the standard action of $\SL_2$, and $T^\ast(\AA^2)^\reg$ denotes the preimage of the regular locus of $\sl_2$ under the moment map $T^\ast(\AA^2) \to \sl_2$. This is because there is an equivalence

\begin{align*}
\Loc_{\SO(2)}^\gr(\Omega S^2; \QQ[\beta^{\pm 1}]) & \simeq \Loc_{\SO(2)}^\gr(\Omega \SO(3); \QQ[\beta^{\pm 1}]) \otimes_{\Loc^\gr(\Omega \SO(2); \QQ[\beta^{\pm 1}])} \Vect_\QQ\\
& \simeq \QCoh(\tilde{\sl}_2^\reg/\SL_2 \times_{B\GG_m} \spec(\QQ)) \simeq \QCoh(\SL_2\backslash T^\ast(\SL_2/N)^\reg),
\end{align*}

where the second equivalence uses <ref> (i.e., the Arkhipov-Bezrukavnikov-Ginzburg equivalence over the regular locus). If $\ld{B}\subseteq \SL_2$ is a fixed Borel subgroup, the map $\tilde{\sl}_2^\reg/\SL_2 \to B\GG_m$ is given by the composite

$$\tilde{\sl}_2^\reg/\SL_2 \cong \ld{\fr{b}}^\reg/\ld{B} \to B\ld{B}\to B\ld{T} = B\GG_m.$$

However, $\SL_2/N \cong \AA^2 - \{0\}$, and there is an $\SL_2$-equivariant isomorphism $T^\ast(\AA^2)^\reg \cong T^\ast(\AA^2-\{0\})^\reg$. Let us remark that <ref> can be de-periodified to give an equivalence

$$\Loc_{\SO(2)}(\Omega S^2; \QQ) \simeq \QCoh(T^\ast[2](\AA^2)^\reg/\SL_2).$$

This is in fact related to the program of Ben-Zvi–Sakellaridis–Venkatesh [85] applied to the “Hecke period”; their program predicts a duality between the Hamiltonian $\PGL_2$-variety $T^\ast(\PGL_2/\GG_m)$ and the Hamiltonian $\SL_2$-variety $T^\ast(\AA^2)$.

When $A = \KU$ and $\GG = \GG_m$, we have $\Bun_{\ld{B}}^0(\GG_{0,\QQ}^\vee) = \tilde{\ld{G}}/\ld{G}$. Therefore, <ref> gives an $\E{2}$-monoidal equivalence

$$\QCoh(\tilde{\ld{G}}^\reg/\ld{G}) \simeq
\Loc_{T_c}^\gr (\Omega G_c; \KU) \otimes \QQ.
%\coLMod_{C_\ast^T(\Gr_G(\cc); \KU)} \otimes \QQ.

Note that $\tilde{\ld{G}}/\ld{G}$ is isomorphic to the quotient $\ld{G} \backslash (\ld{G} \times^{\ld{N}} \ld{B})/\ld{T}$; and <cit.> says that $\ld{G} \times^{\ld{N}} \ld{B}$ is the universal group-valued symplectic implosion (i.e., the symplectic implosion of the internal fusion double of $G$). The relationship of this perspective to Langlands duality is closely related to a quasi-Hamiltonian analogue of the program of Ben-Zvi–Sakellaridis–Venkatesh [85], which we will explore in future work.

Similarly, one can show that there is an $\E{2}$-monoidal equivalence

$$\QCoh(\ld{G}^\reg/\ld{G}) \simeq
\Loc_{G_c}^\gr (\Omega G_c; \KU) \otimes \QQ.

Were there a full $\KU$-theoretic geometric Satake equivalence, the above equivalence would be obtained by localization over the (open) regular locus of $\ld{G}$. The above equivalence is presumably related to <cit.>.

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 $E$ 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. Then $\Bun_{\ld{B}}^0(\GG_0^\vee) = \Bun_{\ld{B}}^0(E^\vee)$, and <ref> gives an $\E{2}$-monoidal $\pi_0 A_\QQ$-linear equivalence

$$\QCoh(\Bun_{\ld{B}}^0(E^\vee)^\reg) \simeq
\Loc_{T_c}^\gr (\Omega G_c; A) \otimes \QQ.
%\coLMod_{\cf_T(\Gr_G(\cc))^\vee}(\QCoh(\cM_T)) \otimes \QQ.

This may be understood as a step towards a full $A$-theoretic analogue of the ABG equivalence.

§.§ Coefficients in the sphere spectrum?

In this brief section, we study the natural question of whether there is an analogue of <ref> and <ref> with coefficients in a more general $\Eoo$-ring $R$ (e.g., the sphere spectrum). This is closely related to the discussion in <ref>, and already turns out to be rather nontrivial for a torus as soon as $R$ is not complex-orientable. As a warmup, let us make the following observation.

Fix a complex-oriented even-periodic $\Eoo$-ring $A$, and let $\GG$ be an oriented group scheme in the sense of [62] which is dualizable. Let $T$ be a torus over $\cc$, and 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 $\Shv_T(\Gr_T(\cc); A) \simeq \QCoh(\cM_T \times B\ld{T}_A)$. Fixing an isomorphism $\cM_T \cong \cM_{\ld{T}}$ as in <ref> makes this category equivalent to $\QCoh(\cL_\GG B\ld{T}_A)$.

Note that there is an $\E{2}$-monoidal equivalence $\Shv_T(\Gr_T(\cc); A) \simeq \Shv_{T_c}(\Omega T_c; A)$. Since the $T_c$-action on $\Omega T_c$ is trivial and $\Omega T_c \cong \bX_\ast(T)$ as $\Eoo$-spaces, we obtain an $\E{2}$-monoidal equivalence

$$\Shv_T(\Gr_T(\cc); A) \simeq \Fun(\bX_\ast(T), \Loc_{T_c}(\ast; A)) \simeq \Fun(\bX_\ast(T), \Mod_A) \otimes_{\Mod_A} \QCoh(\cM_T).$$

The first claim now follows from the equivalence $\QCoh(B\ld{T}_A) \simeq \Fun(\bX_\ast(T), \Mod_A)$. Fixing an isomorphism $\cM_T \cong \cM_{\ld{T}}$ and using that $\cL_\GG B\ld{T}_A \cong B\ld{T}_A \times \cM_{\ld{T}}$, we see that $\Shv_T(\Gr_T(\cc); A)$ can be identified with $\QCoh(\cL_\GG B\ld{T}_A)$, as desired.

Crucial to the argument of <ref> was the equivalence $\Loc_{T_c}(\ast; A) \simeq \QCoh(\cM_T)$. If $R$ is a general $\Eoo$-ring, then such a statement will generally only be true (for an appropriate definition of $\Loc_{T_c}(\ast; R)$) when $R$ is close to being complex-oriented. For example:

The methods of this article show that there is an analogue of <ref> for $\KO$:

$$\Loc_{T_c}^\gr(G_c; \KO) \otimes \QQ \simeq \QCoh((\cM_{\ld{T},0})_\QQ \times_{\Bun_{\ld{B}}^0(\GG_0^\vee)} (\cM_{\ld{T},0})_\QQ).$$

Here, $\GG$ is the universal spectral multiplicative group over $B\Z/2$. Similarly, using the definition of genuine $T$-equivariant $\TMF$ from [39], one can also obtain an analogue of <ref> (where $\GG$ is replaced by the universal oriented spectral elliptic curve over the moduli stack of oriented spectral elliptic curves from <cit.>).

See also <cit.> for a variant of the following:

Let $\Sp_{T_c}$ denote the $\infty$-category of genuine ${T_c}$-equivariant spectra, and let $i_{T_c}^\ast: \Sp_{T_c} \to \Sp$ be the lax symmetric monoidal right adjoint to the unique symmetric monoidal colimit-preserving functor $\Sp \to \Sp_{T_c}$. Suppose $R$ is an $\Eoo$-ring such that there is an $\Eoo$-algebra $R_{T_c}\in \CAlg(\Sp_{T_c})$ given by “genuine ${T_c}$-equivariant $R$-cohomology”. Then, $\Loc_{T_c}(\ast; R)$ might be understood to mean $\Mod_{R_{T_c}}(\Sp_{T_c})$. We are interested in the following question: when is $\Loc_{T_c}(\ast; R)$ equivalent (as a symmetric monoidal category) to the $\infty$-category of modules over some $\Eoo$-ring $B$? It is not difficult to see that if this happens, then the $\Eoo$-ring $B$ will simply be $i_{T_c}^\ast(R_{T_c})$. (One could more generally ask when $\Loc_{T_c}(\ast; R)$ is equivalent to the $\infty$-category of quasicoherent sheaves on some spectral $R$-stack; but this obscures the key homotopical point.)

Let us suppose for simplicity that ${T_c}$ is of rank $1$, i.e., that $T_c = S^1$. Recall that the $\infty$-category $\Sp_{S^1}$ is compactly generated by $S^0$ (with the trivial $S^1$-action) and $(S^1/\mu_n)_+$ for $n\geq 2$. If $\lambda$ denote the $1$-dimensional complex representation of $\mu_n$, there is a cofiber sequence $(S^1/\mu_n)_+ \to S^0 \to S^{\lambda^n}$; so $\Sp_{S^1}$ is compactly generated by $S^0$ and $S^{\lambda^n}$ for $n\geq 2$. It follows that $\Loc_{S^1}(\ast; R) \simeq \Mod_{R_{S^1}}(\Sp_{S^1})$ is compactly generated by $R_{S^1}$ and $R_{S^1} \otimes S^{\lambda^n}$ for $n\geq 2$.
If $R$ is complex-oriented, there is an equivalence $R_{S^1} \otimes S^{\lambda^n} \simeq \Sigma^2 R_{S^1}$. This lets us conclude that $\Loc_{S^1}(\ast; R)$ is compactly generated by the single unit object $R_{S^1}$, so that <cit.> implies that $\Loc_{S^1}(\ast; R) \simeq \Mod_{i_{S^1}^\ast(R_{S^1})}$.

In contrast to the above discussion, if $R$ is not complex-oriented (or more generally does not admit a finite flat cover by a complex-oriented ring), then $\Loc_{T_c}(\ast; R)$ stands little chance of being compactly generated by the unit object. For example, if $R$ is the sphere spectrum, then $\Loc_{T_c}(\ast; R) \simeq \Sp_{T_c}$ is not compactly generated by the unit object. Note, however, that the Barr-Beck-Lurie theorem (<cit.>) implies $\Sp_{S^1}$ is equivalent to the $\infty$-category of left modules over the $\E{1}$-ring $\End_{\Sp_{S^1}}\left(S^0 \oplus \bigoplus_{n\geq 2} (S^1/\mu_n)_+\right)$; this is not an $\Eoo$-ring.

In particular, if $\ld{T}_S := \spec S[\bX_\ast(T)]$ denotes the dual torus over the sphere spectrum, then one can run part of the proof of <ref> to conclude that

$$\Shv_T(\Gr_T(\cc); S) \simeq \Fun(\bX_\ast(T), \Loc_{T_c}(\ast; S)) \simeq \Fun(\bX_\ast(T), \Sp_{T_c}) \simeq \Sp_{\ld{T}_c} \otimes \QCoh(B\ld{T}_S).$$

Here, we have identified $\Sp_{T_c} \simeq \Sp_{\ld{T}_c}$ (see <ref>). The discussion in <ref> shows that it is not clear how to view the right-hand side in terms of quasicoherent sheaves on some spectral stack. In particular, we see that already in the case of a torus, the coherent side of “derived geometric Satake with spherical coefficients” starts to deviate from the standard form of derived geometric Satake. It seems as though the appropriate analogue of the coherent side involves some combination of Hausmann's global group laws [45] and the spectral moduli stack of oriented formal groups (see [43, 78]). We hope to approach this in future work via $T$-equivariant complex cobordism $\MU_T$.

At the moment, derived geometric Satake with spherical coefficients for a general reductive group over $\cc$ seems to require more technical setup than is currently available in the literature (although a version of the geometric Casselman-Shalika equivalence of [28] was discussed in <cit.>).

§ RELATIONSHIP TO BRYLINSKI-ZHANG

<ref> is closely related to the results of Brylinski-Zhang (see [17]). To explain this, we begin by recasting the results of [17] in the language of <ref>.

Let $G$ be a simply-connected compact Lie group. Then the main result of [17] says that there is an isomorphism $\KU_G^\ast(G) \cong \Omega^\ast_{K_0(\Rep(G))/\Z} \otimes_\Z \Z[\beta^{\pm 1}]$, where $K_0(\Rep(G))$ is the (complex) representation ring of $G$. If $G$ is not necessarily simply-connected, there is also an isomorphism $\H_G^\ast(G; \QQ) \cong \Omega^\ast_{\H^\ast(BG; \QQ)/\QQ}$.

These can be simultaneously generalized by the following:

Let $A$ be a complex-oriented even-periodic $\Eoo$-ring, and let $\GG$ be an oriented commutative $A$-group. Let $G$ be a simply-connected compact Lie group, and suppose that the functor $\cf_G: \Top(G)_\conn^\op \to \QCoh(\cM_G)$ of $G$-equivariant $A$-cochains
on connected finite $G$-spaces

is symmetric monoidal[Note that this assumption will fail if $G$ is not connected!].
Then there is an equivalence

$$\Gamma(\cM_G; \cf_G(G)) \simeq \HH(\cM_G/A).$$

Indeed, since $G$ is connected, we have $G \simeq \Omega(BG)$.

Recall from <ref> that $G/G$ is the free loop space of $\ast/G$ in the category of orbifolds.
The assumption on $\cf_G$ now implies that

$$\cf_G(G) \simeq \cf_G(\ast) \otimes_{\cf_{G\times G}(\ast)} \cf_G(\ast) \simeq \co_{\cM_G} \otimes_{\co_{\cM_G} \otimes \co_{\cM_G}} \co_{\cM_G}.$$

Therefore, $\Gamma(\cM_G; \cf_G(G))$ is precisely the Hochschild homology of $\cM_G$.

One can view $\Gamma(\cM_G; \cf_G(G))$ as endomorphisms of the unit object in $\Loc_G(G; A)$, so that $\Mod_{\Gamma(\cM_G; \cf_G(G))}$ behaves as a completion of $\Loc_G(G; A)$.

In some cases, the Hochschild-Kostant-Rosenberg spectral sequence degenerates integrally. Then, $\pi_\ast \HH(\cM_G/A)$ can be identified with the $2$-periodification of the (derived) Hodge cohomology of the underlying stack of $\cM_G$ over $\pi_0(A)$. This applies, for instance, when $A = \KU$; in this case, $\cM_G$ is a lift to $\KU$ of $\spec K_0(\Rep(G)) \cong T\mmod W$, and <ref> is precisely the calculation of [17].

<ref> can be continued further to study the $G$-equivariant $A$-cohomology of $\Omega G$, if we additionally assume that the functor $\cf_G: \Ind(\Top(G))_\conn^\op \to \QCoh(\cM_G)$ on connected ind-finite $G$-spaces is symmetric monoidal. Indeed, observe that there is an equivalence

$$G\backslash \Omega G \simeq G\backslash \cL G/G \simeq \ast/G \times_{\ast/\cL G} \ast/G$$

of orbifolds. But $\ast/\cL G \simeq \cL(\ast/G) \simeq \ast/G \times_{\ast/G \times \ast/G} \ast/G$, so that $G\backslash \Omega G \simeq \Map(S^2, \ast/G)$, i.e., the cotensoring of $\ast/G$ by $S^2$ (in unpointed orbifolds).
Using the assumption on $\cf_G$, we therefore conclude that the $G$-equivariant $A$-cohomology of $\Omega G$ can be identified with the factorization homology

$$\Gamma(\cM_G; \cf_G(\Omega G)) \simeq \int_{S^2} \cM_G\in \CAlg_A$$

taken internally to $A$-modules.

The preceding discussion also computes the $T$-equivariant $A$-cohomology of $\Omega G$. To explain this, write $p: \cM_T \to \cM_G$ to denote the canonical map. The above discussion shows that
there is an equivalence

$$T\backslash \Omega G \simeq \ast/T \times_{\ast/G \times_{\ast/G \times \ast/G} \ast/G} \ast/G$$

of orbifolds, so that $p_\ast \cf_T(\Omega G)$ can be identified with the factorization homology over $S^2$ of $\cM_G$ with coefficients in the $\E{2}$-module $p_\ast \co_{\cM_T}$. In other words, there is an equivalence

$$\Gamma(\cM_T; \cf_T(\Omega G)) \simeq \int_{S^2} (\cM_G; p_\ast \co_{\cM_T})\in \CAlg_A.$$

This approach is rather robust: for instance, if $K\subseteq G$ is a closed subgroup such that $G/K$ is a finite space, there are equivalences of orbifolds

$$G\backslash \cL(G/K) \simeq \Omega(G/K)/K \simeq (\ast \times_{\ast \times_{\ast/G} \ast/K} \ast)/K \simeq \ast/K \times_{\ast/K \times_{\ast/G} \ast/K} \ast/K.$$

Under the same hypotheses as <ref>, this implies that $\Gamma(\cM_G; \cf_G(\cL(G/K)))$ is isomorphic to the relative Hochschild homology $\HH(\cM_K/\cM_G)$. One can recover <ref> by noting that if $H$ is a simply-connected compact Lie group and $K = H\subseteq H \times H = G$, the Hochschild homology $\HH(\cM_H/\cM_H \times \cM_H)$ of the diagonal embedding $\Delta: \cM_H \hookrightarrow \cM_H \times \cM_H$ is precisely the factorization homology $\int_{S^2} \cM_H$.

The relationship of the Brylinski-Zhang isomorphism to <ref> can now be explained as follows.

Continue to assume that $G$ is a simply-connected compact Lie group.
If $A = \QQ[\beta^{\pm 1}]$, then there is an equivalence

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

where all objects on the coherent side are defined over $\QQ$. Since $\ld{\fr{t}}\mmod W \times_{\ld{\g}/\ld{G}} \ld{\fr{t}}\mmod W \cong (T^\ast \ld{T})^\bl\mmod W$ is isomorphic to the group scheme of regular centralizers in $\ld{\g}$, we will write write $\ld{J}_{\GG_a}$ to denote $\ld{\fr{t}}\mmod W \times_{\ld{\g}/\ld{G}} \ld{\fr{t}}\mmod W$. The above equivalence therefore states that

\begin{equation}\label{loc-g-q}
\Loc_G^\gr(G; \QQ[\beta^{\pm 1}]) \simeq \QCoh(\ld{J}_{\GG_a}).
\end{equation}

On the other hand, by <cit.>, the Lie algebra of $\ld{J}_{\GG_a}$ over $\ld{\fr{t}}\mmod W$ is isomorphic to $T^\ast(\ld{\fr{t}}\mmod W)$. Therefore, <ref> and the Hochschild-Kostant-Rosenberg theorem gives an isomorphism

$$\H^\ast_G(G; \QQ[\beta^{\pm 1}]) \cong \pi_\ast \HH(\ld{\fr{t}}\mmod W / \QQ) \otimes_\QQ \QQ[\beta^{\pm 1}] \cong \co_{T[-1](\ld{\fr{t}}\mmod W)} \otimes_\QQ \QQ[\beta^{\pm 1}].$$

In particular, there is an equivalence

\begin{equation}\label{coherent sheaves Q-coh}
\Mod_{\H_G^0(G; \QQ[\beta^{\pm 1}])} \simeq \QCoh(T[-1](\ld{\fr{t}}\mmod W)) \otimes_\QQ \QQ[\beta^{\pm 1}].
\end{equation}

By Koszul duality, the right-hand side is equivalent to the $2$-periodification of the $\infty$-category of ind-coherent sheaves over the formal completion of $\ld{J}_{\GG_a}$ at the zero section.
One can view the resulting description of $\Mod_{\H_G^0(G; \QQ[\beta^{\pm 1}])}$ as a infinitesimal version of the equivalence <ref>. By construction, the equivalence <ref> is just a restatement of the Brylinski-Zhang isomorphism $\H_G^\ast(G; \QQ) \cong \Omega^\ast_{\H^\ast(BG; \QQ)/\QQ}$.

We can also specialize <ref> to this case: we have

\begin{equation}\label{fact homology S2}
\H^\ast_G(\Omega G; \QQ) \cong \pi_\ast \left(\int_{S^2} \cM_G\right).
\end{equation}

Here, $\cM_G = \spec C^\ast(BG; \QQ)$ is the derived $\QQ$-scheme whose underlying graded $\QQ$-scheme is $\ld{\fr{t}}[2]\mmod W = \spec \H^\ast(BG; \QQ)$.
Since $\QQ$ is a field of characteristic zero and $G$ is assumed to be connected, $\H^\ast(BG; \QQ)$ is a polynomial algebra on generators in even degrees; this implies that $C^\ast(BG; \QQ)$ is formal as an $\Eoo$-$\QQ$-algebra[This follows from the fact that the free $\Eoo$-$\QQ$-algebra on classes in even degrees can be identified with the polynomial $\QQ$-algebra, i.e., is itself formal.]. In particular, we may identify $\cM_G = \ld{\fr{t}}[2]\mmod W$.
Just as the Hochschild homology of $\ld{\fr{t}}[2]\mmod W$ can be identified with the ring of functions on $T[-1](\ld{\fr{t}}[2]\mmod W)$, a version of the Hochschild-Kostant-Rosenberg theorem implies that the factorization homology over $S^2$ can be identified with the ring of functions on the $(-2)$-shifted tangent bundle

$$T[-2](\ld{\fr{t}}[2]\mmod W) = \spec \Sym_{\ld{\fr{t}}[2]\mmod W} (\Omega^1_{\ld{\fr{t}}[2]\mmod W}[2]).$$

Now[We will not need such a general statement, but we recall it since it is very useful in many other contexts, too.], if $R$ is a (simplicial) commutative ring and $M$ is a connective $R$-module, there is a décalage isomorphism[The décalage isomorphism only applies to simplicial commutative algebras $R$, and not general $\Eoo$-$\Z$-algebras (in part because of issues in defining the derived functors of $\Sym$ and $\Gamma$). This is the reason why we conspicuously shifted from working with coefficients in $\QQ[\beta^{\pm 1}]$ to working with coefficients in $\QQ$. For instance, observe that if $R$ was instead a $\QQ[\beta^{\pm 1}]$-algebra (hence not a simplicial commutative ring) and $M$ is an $R$-module, it is not possible to distinguish between $M[2]$ and $M$. Although this might seem like a useless point, the observation that working with $2$-periodic coefficients is inherently destructive is important to clarifying why divided power structures appear in the $G$-equivariant cohomology of $\Omega G$ when one considers more general coefficients.] (see <cit.>) $\Sym^j_R(M[2]) \cong \Gamma^j_R(M)[2j]$, where $\Gamma^j$ denotes (the left derived functor of) the $j$th divided power construction.
Therefore, we see that $\Sym_{\ld{\fr{t}}[2]\mmod W} (\Omega^1_{\ld{\fr{t}}[2]\mmod W}[2])$ can be identified with a shearing (which we will simply denote by $[2\bull]$) of the divided power algebra $\Gamma_{\ld{\fr{t}}[2]\mmod W} (\Omega^1_{\ld{\fr{t}}[2]\mmod W})$. In other words, there is an isomorphism

$$\pi_\ast \left(\int_{S^2} \ld{\fr{t}}[2]\mmod W\right) \cong \Gamma_{\ld{\fr{t}}[2]\mmod W} (\Omega^1_{\ld{\fr{t}}[2]\mmod W})[2\bull];$$

the shearing on the right-hand side is undone by $2$-periodifying the left-hand side. Therefore, we obtain an isomorphism

$$\H^\ast_G(\Omega G; \QQ) \otimes_\QQ \QQ[\beta^{\pm 1}] \cong \Gamma_{\ld{\fr{t}}\mmod W} (\Omega^1_{\ld{\fr{t}}\mmod W}).$$

Up to this point, the fact that the coefficients are $\QQ$ (as opposed to a general $\Z$-algebra with some small primes inverted) has not been used outside of the formality of $C^\ast(BG; \QQ)$. Using it now, we see that the divided power algebra can be identified with a symmetric algebra, in which case the above formula implies that $\H^\ast_G(\Omega G; \QQ) \otimes_\QQ \QQ[\beta^{\pm 1}]$ can be identified with the ring of functions on the tangent bundle $T(\ld{\fr{t}}\mmod W)$. This should be compared to <cit.> with $\hbar = 0$; see <cit.> and <cit.>. A similar argument using the $S^1$-action on $S^2$ by rotation can be used to recover (the $2$-periodification of) the full quantized statement of <cit.>.

The above discussion implies a more general statement. Namely, suppose that $R$ is a (classical) commutative ring such that <ref> applies to $G$-equivariant $R$-cohomology — in particular, such that there is an isomorphism

\begin{equation}\label{general fact homology S2}
\H^\ast_G(\Omega G; R) \cong \pi_\ast \left(\int_{S^2} \cM_G\right) \in \CAlg_{\pi_\ast R}
\end{equation}

as in <ref>. (This assumption is likely to hold for rather general rings $R$.) As usual, $\cM_G = \spec C^\ast(BG; R)$ is an $\Eoo$-$R$-scheme with underlying graded $R$-scheme $\ld{\fr{t}}_R[2]\mmod W$; here, $\ld{\fr{t}}_R$ denotes the base-change of $\ld{\fr{t}}$ from $\Z$ to $R$. Suppose that $C^\ast(BG; R)$ is formal as an $\E{n}$-$R$-algebra (i.e., there is an equivalence $C^\ast(BG; R) \simeq \H^\ast(BG; R)$ as $\E{n}$-$R$-algebras); by obstruction theory, this can always be guaranteed if $n=2$ and $\H^\ast(BG; R)$ is a polynomial algebra on generators in even degrees. Then <ref> implies that $\H^\ast_G(\Omega G; R)$ is equivalent to $\pi_\ast \left(\int_{S^2} \ld{\fr{t}}_R[2]\mmod W\right)$ as $\E{n-2}$-$R$-algebras. In particular, since $C^\ast(BG; R)$ is formal as an $\E{2}$-$R$-algebra, we see that $\H^\ast_G(\Omega G; R)$ is equivalent to $\pi_\ast \left(\int_{S^2} \ld{\fr{t}}_R[2]\mmod W\right)$ as unital $R$-modules. If $C^\ast(BG; R)$ is formal as an $\E{3}$-$R$-algebra, then we can also identify $\H^\ast_G(\Omega G; R)$ as an $R$-algebra.

In any case, since $R$ is not necessarily a $\QQ$-algebra, the Hochschild-Kostant-Rosenberg theorem need not give an isomorphism between $\pi_\ast \left(\int_{S^2} \ld{\fr{t}}_R[2]\mmod W\right)$ and $\Sym_{\ld{\fr{t}}[2]\mmod W} (\Omega^1_{\ld{\fr{t}}[2]\mmod W}[2])$; rather, there will always be a “HKR” filtration on $\pi_\ast \left(\int_{S^2} \ld{\fr{t}}_R[2]\mmod W\right)$ whose associated graded is given by $\Sym_{\ld{\fr{t}}[2]\mmod W} (\Omega^1_{\ld{\fr{t}}[2]\mmod W}[2])$.
If this filtration splits, we conclude that the cohomology ring $\H^\ast_G(\Omega G; R)$ will admit divided powers on the $\co_{\ld{\fr{t}}_R[2]\mmod W}$-algebra generators $\Omega^1_{\ld{\fr{t}}_R[2]\mmod W}$.
The assumption that the HKR filtration splits seems likely to hold if some primes are assumed to be units in $R$ (e.g., if $\dim(\fr{t})!\in R^\times$). Note that by virtue of the argument establishing <ref>, the divided power structure on $\H^\ast_G(\Omega G; R)$ is closely related to the $\E{3}$-algebra structure on the derived Satake category.

The preceding discussion is directly connected with a question asked by Bezrukavnikov about divided powers in the cohomology of the affine Grassmannian (see [82]). It would be interesting to determine the exact conditions under which the above assumptions on $R$ hold true (namely, $C^\ast(BG; R)$ being formal as an $\E{3}$-$R$-algebra, <ref>, and the splitting of the HKR filtration for $\int_{S^2} \ld{\fr{t}}_R[2]\mmod W$). The formality of $C^\ast(BG; R)$ seems to be the thorniest of these conditions, but we nevertheless hope that <ref> could be useful in approaching Bezrukavnikov's question.

Recall that there is an equivalence

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

where all objects on the coherent side are defined over $\QQ$. Since $\ld{T}\mmod W \times_{\ld{G}/\ld{G}} \ld{T}\mmod W \cong (T \times \ld{T})^\bl\mmod W$ is isomorphic to the group scheme of regular centralizers in $\ld{G}$, we will write write $\ld{J}_{\GG_m}$ to denote $\ld{T}\mmod W \times_{\ld{G}/\ld{G}} \ld{T}\mmod W$.
The above equivalence therefore states that

\begin{equation}\label{loc-g-ku}
\Loc_G^\gr(G; \KU) \otimes \QQ \simeq \QCoh(\ld{J}_{\GG_m}).
\end{equation}

There is a multiplicative analogue of <cit.>, which says that when $\ld{G}$ is adjoint, the Lie algebra of $\ld{J}_{\GG_m}$ over $\ld{T}\mmod W$ is isomorphic to $T^\ast(\ld{T}\mmod W)$. Therefore, <ref> and the Hochschild-Kostant-Rosenberg theorem gives an isomorphism

$$\KU^\ast_G(G) \otimes \QQ \cong \pi_\ast \HH(\ld{T}\mmod W / \Z) \otimes_\Z \QQ[\beta^{\pm 1}] \cong \co_{T[-1](\ld{T}\mmod W)} \otimes_\Z \QQ[\beta^{\pm 1}].$$

In particular, there is an equivalence

\begin{equation}\label{coherent sheaves KU-coh}
\Mod_{\KU_G^0(G)} \otimes \QQ \simeq \QCoh(T[-1](\ld{T}\mmod W)) \otimes_\Z \QQ.
\end{equation}

By Koszul duality, the right-hand side is equivalent to the $2$-periodification of the $\infty$-category of ind-coherent sheaves over the formal completion of $\ld{J}_{\GG_m}$ at the zero section.
One can view the resulting description of $\Mod_{\KU_G^0(G)} \otimes \QQ$ as a infinitesimal version of the equivalence <ref>. By construction, the equivalence <ref> is just a restatement of the Brylinski-Zhang isomorphism $\KU_G^0(G) \otimes \QQ \cong \Omega^\ast_{K_0(\Rep(G))/\Z} \otimes_\Z \QQ$.

Just as in <ref>, we can also specialize <ref> to the case of K-theory. Then, we have

\begin{equation}\label{Kthy fact homology S2}
\KU^\ast_G(\Omega G) \cong \pi_\ast \left(\int_{S^2} \cM_G\right).
\end{equation}

Here, $\cM_G = \spec \KU_G$ as a $\KU$-scheme, and the factorization homology is taken over $\KU$.

In general, $\Mod_{\pi_0 C_{G}^\ast(G; A)} \otimes \QQ$ is an “infinitesimal analogue” of $\Loc_{G}^\gr(G; A)$. The equivalence of <ref> can therefore be viewed as a infinitesimal version of the analogue of the equivalence of <ref> for $\Loc_{G}^\gr(G; A) \otimes \QQ$.

§ COULOMB BRANCHES OF PURE SUPERSYMMETRIC GAUGE THEORIES

In this brief appendix, we explain some motivation for the results of this article from the perspective of Coulomb branches of $4$d $\cN=2$ and $5$d $\cN=1$ gauge theories with a generic choice of complex structure. Our goal here is not to be precise, but instead explain some motivation for the ideas in this article. While reading this appendix, the reader should keep in mind that I know very little physics!

In [10, 70] (see also [71]), it is argued that the Coulomb branch of $3$d $\cN=4$ pure gauge theory on $\RR^3$ can be modeled by the algebraic symplectic variety $\cM_C := \spec \H^G_\ast(\Gr_G(\cc); \cc)$ over $\cc$. The calculations of [9] say that $\cM_C$ is isomorphic to $(T^\ast \ld{T})^\bl\mmod W$. This is in turn isomorphic by <cit.> to the phase space of the Toda lattice for $\ld{G}$, as well as to the moduli space of solutions of Nahm's equations on $[-1,1]$ for a compact form of $\ld{G}$ by <cit.> with an appropriate boundary condition.
The quantized Coulomb branch of $3$d $\cN=4$ pure gauge theory on $\RR^3$ is then modeled by $\cA_\epsilon := \H^{G\times S^1_\rot}_\ast(\Gr_G(\cc); \cc)$. In [9], $\cA_\epsilon$ was identified with the algebra of operators of the quantized Toda lattice for $\ld{G}$.

The physical reason for the definition of $\cA_\epsilon$ is the “$\Omega$-background” (introduced in [73]); we refer the reader to [5, 90] for helpful expositions on this topic. The essential idea is as follows: $C^G_\ast(\Gr_G(\cc); \cc)$ admits the structure of an $\Efr{3}$-algebra. In particular, the $\E{3}$-algebra structure on $C^G_\ast(\Gr_G(\cc); \cc)$ is equivariant for the action of $S^1$ on $C^G_\ast(\Gr_G(\cc); \cc)$ via loop rotation, and the action of $S^1$ on $\E{3}$ via rotation about a line $\ell\subseteq \RR^3$. Using the fact that the fixed points of the $S^1$-action on $\RR^3$ are given by the line $\ell$, it is argued in [5] that the homotopy fixed points of $C^G_\ast(\Gr_G(\cc); \cc)$ admits the structure of an $\E{1}$-$C_{S^1}^\ast(\ast; \cc)$-algebra. Furthermore, the associative multiplication on $C^{G\times S^1_\rot}_\ast(\Gr_G(\cc); \cc)$ is argued to degenerate to the $2$-shifted Poisson bracket on $\H^G_\ast(\Gr_G(\cc); \cc)$ obtained from the $\E{3}$-algebra structure. The “$\Omega$-background” is supposed to refer to the compatibility of the $S^1$-action on $C^G_\ast(\Gr_G(\cc); \cc)$ with the $S^1$-action on the $\E{3}$-operad.

From the mathematical perspective, the idea that $S^1$-actions can be viewed as a deformation quantizations has been made precise by [77, 91], and more recently in [14, 15] (at least in characteristic zero). Although often not said explicitly, the idea has been a cornerstone of Hochschild homology. (The reader can skip the following discussion, since it will not be necessary in the remainder of this section; we only include it for completeness.)

Consider a smooth $\cc$-scheme $X$, so that the HKR theorem gives an isomorphism $\HH(X/\cc) \simeq \Sym(\Omega^1_{X/\cc}[1])$. There is an isomorphism $\Sym(\Omega^1_{X/\cc}[1]) \simeq \bigoplus_{n\geq 0} (\wedge^n \Omega^1_{X/\cc})[n]$, so $\Sym(\Omega^1_{X/\cc}[1])$ can be understood as a shearing of the algebra $\Omega^\ast_{X/\cc} = \bigoplus_{n\geq 0} (\wedge^n \Omega^1_{X/\cc})[-n]$ of differential forms. The HKR theorem further states that the $S^1$-action on $\HH(X/\cc)$ is a shearing of the de Rham differential on $\Omega^\ast_{X/\cc}$. The Koszul dual of the algebra $\HH(X/\cc) \simeq \Sym(\Omega^1_{X/\cc}[1])$ is $\Sym(T_{X/\cc}[-2]) \simeq \co_{T^\ast[2] X}$; in the same way, the sheaf of differential operators on $X$ is Koszul dual to the de Rham complex of $X$. This can be drawn pictorially as follows:

\Sym(T_{X/\cc}[-2]) \simeq \co_{T^\ast[2] X} \ar@{~>}[r]^-{\text{def. quant}} \ar@{~>}[d]_-{\text{Koszul dual}} & \cd^\hbar_{X/\cc} \ar@{~>}[d]^-{\text{Koszul dual}} \\
\Sym_{\co_X}(\Omega^1_{X/\cc}[1]) \simeq \HH(X/\cc) \ar@{~>}[r]_-{S^1\text{-action}} & \text{shearing of }(\Omega^\ast_{X/k}, d_\dR).

Since the algebra $\cd_X^\hbar$ of differential operators is a quantization of $T^\ast[2] X$, this drawing illustrates that the $S^1$-action on Hochschild homology plays the role of a Koszul dual to deformation quantization.

We will keep $G = \SL_2$ as a running example in discussing Coulomb branches (see also <cit.>), so that $\ld{G} = \PGL_2$. In this case, $\cM_C \cong \spec \cc[x, t^{\pm 1}, \frac{t-1}{x}]^{\Z/2} \cong \spec \cc[x^2, t+t^{-1}, \frac{t-t^{-1}}{x}]$ by <ref> (and [9]), where $\Z/2$ acts on $\cc[x, t^{\pm 1}, \frac{t-1}{x}]$ by $x\mapsto -x$ and $t\mapsto t^{-1}$.
Let us denote $\Phi = x^2$, $U = t + t^{-1}$, and $V = \frac{t-t^{-1}}{x}$. Then

$$U^2 - \Phi V^2 = (t+t^{-1})^2 - (t-t^{-1})^2 = 4,$$

so $\cM_C$ is isomorphic to the subvariety of $\AA^3_\cc$ cut out by the equation

$$U^2 - \Phi V^2 = 4.$$

Alternatively, and perhaps more suggestively:

$$(U+2)(U-2) = \Phi V^2.$$

This is known as the Atiyah-Hitchin manifold, and was studied in great detail in [3] (see <cit.> for the definition). In <cit.>, it was shown that the Atiyah-Hitchin manifold is isomorphic to the moduli space of solutions of Nahm's equations on $[-1,1]$ for $\mathrm{PSU}(2)$ with an appropriate boundary condition.
Since a normal vector to the defining equation of $\cM_C$ is $2U\partial_U - V^2 \partial_\phi - 2V\Phi \partial_V$, the standard holomorphic $3$-form $dU \wedge d\Phi \wedge dV$ on $\AA^3_\cc$ induces a holomorphic symplectic form $\frac{d\Phi \wedge dV}{2U}$ on $\cM_C$. (This can also be written as $\frac{dU \wedge dV}{V^2}$ or as $\frac{d\Phi \wedge dU}{2\Phi V}$.) The associated Poisson bracket on $\co_{\cM_C} \cong \H^G_\ast(\Gr_G(\cc); \cc)$ agrees with the $2$-shifted Poisson bracket arising from the $\E{3}$-structure on $C^G_\ast(\Gr_G(\cc); \cc)$.

The quantized algebra $\cA_\epsilon$ was described explicitly in [8]. Let us write $\theta = \frac{1}{x}(s-1)$, where $s$ is the simple reflection generating the Weyl group of $\SL_2$. Then $\cA_\epsilon$ is generated as an algebra over $\cc\pw{\hbar}$ by $\Z/2$-invariant polynomials in $x$, $t^{\pm 1}$, and $\theta$, where $x$ is to be viewed as $t\partial_t$. Moreover, under the isomorphism $\cA_\epsilon/\hbar \cong \co_{\cM_C}$, the class $x$ is sent to $x$, and $\theta$ is sent to $\frac{t-1}{x}$. We then have the commutation relation $[x,t^{\pm 1}] = \pm \hbar t^{-1}$, induced by $[\partial_t, t] = \hbar$; see <ref>. This implies that $[x^2, t^{\pm 1}] = \hbar^2 t^{\pm 1} \pm 2\hbar t^{\pm 1} x$, which in turn implies that $\cA_\epsilon$ is the quotient of the free associative $\cc\pw{\hbar}$-algebra on $\Phi$, $U$, and $V = \frac{1}{x}(t-t^{-1})$ subject to the relations

\begin{align*}
%& = \left[x^2, \frac{1}{x}(t-t^{-1})\right] \\
%& = x(t-t^{-1}) - \frac{1}{x}(tx^2 - t^{-1}x^2)\\
%& = x(t-t^{-1}) - \frac{1}{x} ([x^2 t - \hbar^2 t - 2\hbar tx] - [x^2 t^{-1} - \hbar^2 t^{-1} + 2\hbar t^{-1} x])\\
%& = \frac{\hbar^2}{x}(t-t^{-1}) + \frac{2\hbar}{x}(t+t^{-1})x \\
%& = \frac{\hbar^2}{x}(t-t^{-1}) + \frac{2\hbar}{x}(xt - \hbar t + xt^{-1} + \hbar t^{-1})\\
%& = \frac{\hbar^2}{x}(t-t^{-1}) + 2\hbar \left(t+t^{-1} - \frac{\hbar}{x}(t-t^{-1})\right)\\
& = 2\hbar U - \hbar^2 V,\\
[\Phi, U]
%& = [x^2, t] + [x^2, t^{-1}]\\
%& = \hbar^2 t + 2\hbar tx + \hbar^2 t^{-1} - 2\hbar t^{-1} x\\
%& = \hbar^2(t+t^{-1}) - 2\hbar(t-t^{-1})x \\
%& = \hbar^2(t+t^{-1}) - 2\hbar([xt-\hbar t]-[xt^{-1}+\hbar t^{-1}]) \\
%& = 2\hbar x(t-t^{-1}) - \hbar^2(t+t^{-1})\\
& = 2\hbar \Phi V - \hbar^2 U, \\
%& = \left[t+t^{-1}, \frac{1}{x} (t-t^{-1})\right] \\
%& = \left((t+t^{-1}) \frac{1}{x} (t-t^{-1}) - \frac{1}{x} (t-t^{-1}) (t+t^{-1}) \right) - \left(\frac{1}{x} (t+t^{-1}) (t-t^{-1}) - \frac{1}{x} (t+t^{-1}) (t-t^{-1})\right) \\
%& = (t+t^{-1}) \frac{1}{x} (t-t^{-1}) - \frac{1}{x} (t+t^{-1}) (t-t^{-1})\\
%& = \left[t+t^{-1}, \frac{1}{x}\right] (t-t^{-1})\\
%& = \hbar \frac{1}{x} (t-t^{-1}) \frac{1}{x} (t-t^{-1}) \\
& = \hbar V^2,\\
%U^2 - \Phi V^2
%& = (t+t^{-1})^2 - x(t-t^{-1}) \frac{1}{x}(t-t^{-1})\\
%& = (t+t^{-1})^2 - \left((t-t^{-1})x + \hbar (t+t^{-1})\right) \frac{1}{x}(t-t^{-1})\\
%& = (t+t^{-1})^2 - (t-t^{-1})^2 + \hbar (t+t^{-1}) \frac{1}{x}(t-t^{-1})\\
%& = (t+t^{-1})^2 - (t-t^{-1})^2 - \hbar (t+t^{-1}) \frac{1}{x} (t-t^{-1})\\
%& = 4 - \hbar UV.
(U+2)(U-2) & = \Phi V^2 - \hbar UV.
\end{align*}

Note that the commutation relations for $[\Phi, U]$ and $[U,V]$ in <cit.> have typos, but it is stated correctly in <cit.>. The above is an explicit description of the nil-Hecke algebra $e\cH(\tilde{\fr{t}}, W^\aff)e$ for $\PGL_2$.

An unpublished conjecture of Gaiotto (which I learned about from Nakajima) says that the Coulomb branch of $4$d $\cN=2$ pure gauge theory over $\RR^3 \times S^1$ with a generic choice of complex structure can be modeled by $\cM_C^\fourd := \spec \KU^G_0(\Gr_G(\cc)) \otimes \cc$. Although I do not know Gaiotto's motivation for this conjecture (it is probably inspired by [89]), my attempt at heuristically justifying it goes as follows. Recall that $\Gr_G(\cc)/G(\cc\pw{t})$ can be viewed as $\Bun_G(S^2)$. It is reasonable to view $\KU_0(\Bun_G(S^2)) \otimes \cc$ as closely related to $\H_\ast(\cL \Bun_G(S^2); \cc)$, where $\cL \Bun_G(S^2)$ denotes the free loop space. Since $\cL BG \simeq B\cL G$, we have $\cL \Bun_G(S^2) \simeq \Bun_{\cL G}(S^2)$, so one might view $\H_\ast(\cL \Bun_G(S^2); \cc)$ as the ring of functions on the “Coulomb branch of $3$d $\cN=4$ pure gauge theory on $\RR^3$ with gauge group $\cL G$”.

Making precise sense of this phrase seems difficult, but one possible workaround could be the following. It is often useful to view gauge theory with gauge group $\cL G$ as “finite temperature” gauge theory with gauge group $G$. Recall that Wick rotation relates $(3+1)$-dimensional quantum field theory at a finite temperature $T$ to statistical mechanics over $\RR^3 \times S^1$ where the circle has radius $\frac{1}{2\pi T}$. This suggests that $\H_\ast(\cL \Bun_G(S^2); \cc)$ (which is more precisely to be understood as $\KU^G_0(\Gr_G(\cc)) \otimes \cc$) can be viewed as the ring of functions on the “Coulomb branch of $4$d $\cN=2$ pure gauge theory on $\RR^3 \times S^1$ with gauge group $G$”. See <cit.>. In [9], $\spec \KU^G_0(\Gr_G(\cc)) \otimes \cc$ was identified with the phase space of the relativistic Toda lattice for $\ld{G}$.

One can also define a quantization of $\cM_C^\fourd$ via $\cA_\epsilon^\fourd := \KU^{G\times S^1_\rot}_0(\Gr_G(\cc)) \otimes \cc$; this can be viewed as a model for the quantized Coulomb branch of $4$d $\cN=2$ pure gauge theory on $\RR^3\times S^1$. In [9], $\cA_\epsilon^\fourd$ was be identified with the algebra of operators of the quantized relativistic Toda lattice for $\ld{G}$.

When $G = \SL_2$, the calculations of <ref> and [9] tell us that $\cM_C^\fourd \cong \spec \cc[x^{\pm 1}, t^{\pm 1}, \frac{t-1}{x-1}]^{\Z/2} \cong \spec \cc[x+x^{-1}, t+t^{-1}, \frac{t-t^{-1}}{x-x^{-1}}]$, where $\Z/2$ acts on $\cc[x^{\pm 1}, t^{\pm 1}, \frac{t-1}{x-1}]$ by $x\mapsto x^{-1}$ and $t\mapsto t^{-1}$. Let us write $\Psi = x + x^{-1}$, $W = t+t^{-1}$, and $Z = \frac{t-t^{-1}}{x-x^{-1}}$. Then, one easily verifies that $\cM_C^\fourd$ is the subvariety of $\AA^3_\cc$ cut out by the equation

$$W^2 - (\Psi^2 - 4)Z^2 = 4.$$

Alternatively, and perhaps more suggestively:

$$(W+2)(W-2) = (\Psi+2)(\Psi-2)Z^2.$$

This may be regarded as an a multiplicative analogue of the Atiyah-Hitchin manifold. It would be very interesting to understand a relationship between this manifold and the moduli space of solutions to some analogue of Nahm's equations for $\mathrm{PSU}(2)$ with an appropriate boundary condition. The complex manifold $\cM_C^\fourd$ has a holomorphic symplectic form given by $\frac{d\Psi \wedge dZ}{W}$, which can also be written as $\frac{d\Psi \wedge dW}{(\Psi^2-4)Z}$ or as $\frac{dZ \wedge dW}{\Psi Z^2}$.

We can also describe the quantized algebra $\cA_\epsilon^\fourd$ explicitly. In this case, instead of the relation $[\partial_t, t] = \hbar$ which appeared in <ref>, we have the relation $xt = qtx$ (i.e., $xtx^{-1} t^{-1} = q$); see <ref>. In particular, $xt^{-1} = q^{-1} t^{-1} x$, $x^{-1} t = q^{-1} tx^{-1}$, and $x^{-1} t^{-1} = qt^{-1}x^{-1}$. It follows after some tedious calculation that $\cA_\epsilon^\fourd$ is the quotient of the free associative $\cc\pw{q-1}$-algebra (in fact, $\cc[q^{\pm 1}]$-algebra) on $\Psi$, $W$, and $Z = \frac{1}{x-x^{-1}} (t-t^{-1})$ subject to the relations

\begin{align*}
[\Psi, W]
%& = [x+x^{-1}, t+t^{-1}]\\
%& = (xt + xt^{-1} + x^{-1} t + x^{-1} t^{-1}) - (q^{-1} xt + q xt^{-1} + q x^{-1} t + q^{-1} x^{-1} t^{-1})\\
%& = (1-q^{-1}) xt + (1-q) xt^{-1} + (1-q) x^{-1} t + (1-q^{-1}) x^{-1} t^{-1}\\
%& = (q-1)(q^{-1} xt - xt^{-1} - x^{-1} t + q^{-1} x^{-1} t^{-1})\\
%& = (q-1)(q^{-1} xt - xt + xt - xt^{-1} - x^{-1} t + x^{-1} t^{-1} - x^{-1} t^{-1} + q^{-1} x^{-1} t^{-1}) \\
%& = (q-1)((x-x^{-1}) (t-t^{-1}) + (q^{-1}-1) (xt+x^{-1} t^{-1})) \\
%& = (q-1)(\Psi^2-4)Z + \frac{(q-1)(q^{-1}-1)}{2} ((\Psi^2-4)Z + \Psi W) \\
& = (q-1)(\Psi^2-4)Z - \frac{(q-1)^2}{2q} ((\Psi^2-4)Z + \Psi W), \\
%& = (q-1)(\Psi^2-4)Z - \frac{1}{2} \sum_{n\geq 2} (-1)^n (q-1)^n ((\Psi^2-4)Z + \Psi W), \\
[\Psi, Z]
%& = [x+x^{-1}, \frac{1}{x-x^{-1}} (t-t^{-1})]\\
%& = \frac{1}{x-x^{-1}} [x+x^{-1}, t-t^{-1}]\\
%& = \frac{1}{x-x^{-1}} ((1-q^{-1}) xt - (1-q) xt + (1-q) x^{-1} t - (1-q^{-1}) x^{-1} t^{-1})\\
%& = \frac{q-1}{x-x^{-1}} (q^{-1} xt + xt^{-1} - x^{-1} t - q^{-1} x^{-1} t^{-1})\\
%& = \frac{q-1}{x-x^{-1}} ((q^{-1}-1) xt + x(t+t^{-1}) - x^{-1} (t+t^{-1}) + (1-q^{-1}) x^{-1} t^{-1})\\
%& = \frac{q-1}{x-x^{-1}} ((x-x^{-1})(t+t^{-1}) + (q^{-1}-1)(xt - x^{-1} t^{-1}))\\
%& = (q-1)W + \frac{(q-1)(q^{-1}-1)}{x-x^{-1}} (xt - x^{-1} t^{-1})\\
%& = (q-1)W + \frac{(q-1)(q^{-1}-1)}{2}(\Psi Z + W),\\
& = (q-1)W - \frac{(q-1)^2}{2q}(\Psi Z + W),\\
%& = (q-1)W - \frac{1}{2} \sum_{n\geq 2} (-1)^n (q-1)^n (\Psi Z + W) ,\\
%& = \frac{1}{x-x^{-1}} (t-t^{-1}) (t+t^{-1}) - (t+t^{-1}) \frac{1}{x-x^{-1}} (t-t^{-1})\\
%& = [\frac{1}{x-x^{-1}}, t+t^{-1}] (t-t^{-1})\\
%& = \frac{1}{x-x^{-1}} [x-x^{-1}, t+t^{-1}] \frac{1}{x-x^{-1}} (t-t^{-1})\\
%& = (q-1)((q^{-1}-1) \frac{\Psi Z+W}{2} + \Psi Z) Z\\
%& = (q-1) \Psi Z^2 + \frac{(q-1)(q^{-1}-1)}{2} (\Psi Z + W)Z, \\
& = (q-1) \Psi Z^2 - \frac{(q-1)^2}{2q} (\Psi Z + W)Z, \\
%& = (q-1) \Psi Z^2 - \frac{1}{2} \sum_{n\geq 2} (-1)^n (q-1)^n (\Psi Z + W)Z , \\
%(\Psi^2-4) Z^2 -W^2
%& = (x-x^{-1})(t-t^{-1}) \frac{1}{x-x^{-1}} (t-t^{-1}) -(t-t^{-1})^2 - 4 \\
%& = (x-x^{-1})(t-t^{-1}) \frac{1}{x-x^{-1}} (t-t^{-1}) -(t-t^{-1})^2 - 4\\
%& = (t-t^{-1})^2 + [t-t^{-1}, x-x^{-1}] \frac{1}{x-x^{-1}} (t-t^{-1}) -(t-t^{-1})^2 - 4\\
%& = [t-t^{-1}, x-x^{-1}] Z - 4\\
%& = \frac{(q-1)}{2} ((q^{-1}-1) \Psi W - (q^{-1}+1) (\Psi^2-4) Z) Z - 4\\
%& = \frac{(q-1)^2}{2q} \Psi W Z - \frac{q-q^{-1}}{2} (\Psi^2-4)Z^2 - 4\\
%%%%%%% or instead of the final two lines, note:
%[t-t^{-1}, x-x^{-1}] & = (tx-xt - tx^{-1} + x^{-1} t - t^{-1} x + xt^{-1} + t^{-1} x^{-1} - x^{-1} t^{-1})\\
%& = ((q^{-1}-1) xt + (1-q) xt^{-1} + (1-q) x^{-1} t - (1-q^{-1}) x^{-1} t^{-1})\\
%& = (q-1)(-q^{-1} xt - xt^{-1} - x^{-1} t - q^{-1} x^{-1} t^{-1})\\
%& = (q-1)((x-x^{-1})(t-t^{-1}) - (q^{-1}+1)(xt + x^{-1} t^{-1}))\\
%& = (q-1)(\Psi^2-4) Z - \frac{(q-1)(q^{-1}+1)}{2} ((\Psi^2-4)Z + \Psi W)\\
%& = (q-1)(\Psi^2-4) Z - \frac{q^2-1}{2q} ((\Psi^2-4)Z + \Psi W)\\
%& = \frac{(q-1)^2}{2q} (\Psi^2-4) Z - \frac{q^2-1}{2q} \Psi W\\
%% negate the final answer from the previous line
%W^2 - (\Psi^2-4)Z^2 & = 4 - \frac{(q-1)^2}{2q} (\Psi^2-4)Z^2 + \frac{q^2-1}{2q} \Psi WZ.
(W+2)(W-2) & = (\Psi+2)(\Psi-2)Z^2 - \frac{(q-1)^2}{2q} (\Psi^2-4)Z^2 + \frac{q^2-1}{2q} \Psi WZ.
\end{align*}

This algebra is an explicit description of the multiplicative nil-Hecke algebra $e\cH(\tilde{T}, W^\aff)e$ from <ref> for $\PGL_2$.

Now consider an elliptic curve $E(\cc)$ over $\cc$. Motivated by <ref> and [75], one might expect that the Coulomb branch of $5$d $\cN=1$ pure gauge theory over $\RR^3 \times E(\cc)$ (with some specific complex structure) can be modeled by the complexification of the $G$-equivariant $A$-homology of $\Gr_G(\cc)$, where $A$ is an elliptic cohomology theory associated to a putative integral lift of $E$. A classical result of Tate says that there are no smooth elliptic curves over $\Z$, so $E(\cc)$ cannot literally lift to $\Z$ (i.e., $\pi_0(A)$ cannot be $\Z$). As a fix, one can more generally simultaneously consider all possible “Coulomb branches” $\cM_C^\fived := \spec A^G_0(\Gr_G(\cc)) \otimes \cc$ associated to every complex-oriented even-periodic $\Eoo$-ring $A$ equipped with an oriented elliptic curve (this is almost equivalent to considering the universal example $\spec \tmf^G_0(\Gr_G(\cc)) \otimes \cc$). We have described $\spec A^T_0(\Gr_G(\cc)) \otimes \cc$ in <ref>, from which one can calculate $\cM_C^\fived$. Similarly, one can even use <ref> to calculate $A^{T\times S^1_\rot}_0(\Gr_G(\cc)) \otimes \cc$ and $\cA_\epsilon^\fived := A^{G\times S^1_\rot}_0(\Gr_G(\cc)) \otimes \cc$, but this is already incredibly complicated for $G = \SL_2$.

Let $A$ be a complex-oriented even-periodic $\Eoo$-ring equipped with an oriented elliptic curve $\tilde{E}$, and let $E$ denote the associated elliptic curve over $\pi_0(A)\otimes \cc$. Let $(\GG_m\times E)^\bl$ denote the complement of the proper preimage of the zero section of $E$ inside the blowup of $\GG_m\times E$ at the locus cut out by the zero sections of $\GG_m$ and $E$. There is an action of $\Z/2$ on $(\GG_m\times E)^\bl$, induced by the inversion on the group structures on $\GG_m$ and $E$. If $G = \SL_2$, then <ref> can be used to show that $\cM_C^\fived = \spec A^G_0(\Gr_G(\cc)) \otimes \cc$ is isomorphic to $(\GG_m\times E)^\bl\mmod(\Z/2)$; this can be viewed as an elliptic analogue of the Atiyah-Hitchin manifold. We do not have a simple description for $\cA_\epsilon^\fived$ analogous to <ref> and <ref>.

It would be very interesting to give a physical interpretation to $A^G_0(\Gr_G(\cc)) \otimes \cc$ and $A^{G\times S^1_\rot}_0(\Gr_G(\cc)) \otimes \cc$ for other even-periodic $\Eoo$-rings $A$, although we expect this to be very difficult (since most other chromatically interesting generalized cohomology theories only exist after profinite or $p$-adic completion, and do not admit transcendental analogues). It would also be very interesting to describe the analogue of our calculations for the ind-schemes $\cR_{G,\mathbf{N}}$ introduced in [10]. By adapting the methods of <cit.>, this is approachable when $G$ is a torus. We expect it to lead to interesting geometry for nonabelian $G$.

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