1.1 Real–symmetric correspondence

Let $G_{\mathbb R}$ be a real form of a connected complex reductive group $G$ . Let $X=G/K$ be the associated symmetric variety under Cartan’s bijection, where $K$ is the complexification of a maximal compact subgroup $K_{\mathbb R}\subset G_{\mathbb R}$ .

A fundamental feature of the representation theory of the real group $G_{\mathbb R}$ is that many results of an analytic nature have equivalent purely algebraic formulations in terms of the corresponding symmetric variety $X$ . We will call this broad phenomenon the real–symmetric correspondence. It allows one to use algebraic tools on the symmetric side to study questions on the real side and, conversely, to use analytic tools on the real side to study questions on the symmetric side. Famous examples include Harish-Chandra’s reformulation of admissible representations of real groups in terms of $({\mathfrak g},K)$ -modules, the Kostant–Sekiguchi correspondence between real and symmetric nilpotent orbits, and the Matsuki correspondence between $G_{\mathbb R}$ - and $K$ -orbits on the flag manifold of $G$ .

In [Reference Chen and NadlerCN18], the first and third authors established a real–symmetric correspondence relating the dg derived category of spherical constructible sheaves on the real affine Grassmannian ${\textrm{Gr}}_{G_{\mathbb R}}$ of $G_{\mathbb R}$ and the dg derived category of spherical constructible sheaves on the loop space $\mathfrak{L}X$ of $X$ . We are interested in applying this real–symmetric correspondence to study questions in the real and relative geometric Langlands programs.

In the present paper, we consider the question of a geometric Satake equivalence for real groups and symmetric varieties. We focus on the case where the real group is the quaternionic group $G_{\mathbb R}={\textrm{GL}}_n({\mathbb H})$ with associated symmetric variety $X={\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ . We prove the derived geometric Satake equivalence for ${\textrm{GL}}_n({\mathbb H})$ relating the dg constructible derived category of the quaternionic affine Grassmannian with the dg coherent derived category of a quotient stack associated to the Gaitsgory–Nadler dual group $\check G_X$ of $X$ (which is $\check G_X={\textrm{GL}}_n$ in this case). Via the real–symmetric correspondence, we obtain a derived geometric Satake equivalence for the symmetric variety ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ . As an application, we compute the stalks of the IC-complexes for spherical orbit closures in the quaternionic affine Grassmannian and the loop space of ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ . We show that the stalks are given by the Kostka–Foulkes polynomials for ${\textrm{GL}}_n$ but with all degrees doubled.

We explain how these equivalences fit into the general framework of a geometric Langlands correspondence for real groups, due to Ben-Zvi and the third author, and of the relative Langlands duality conjecture, due to Ben-Zvi, Sakellaridis, and Venkatesh.

From the point of view of real groups, the quaternionic group ${\textrm{GL}}_n({\mathbb H})$ offers in some sense the simplest possible geometry: just as complex Grassmannians are simpler than real Grassmannians (Schubert cells are $2d$ versus $d$ real-dimensional), quaternionic Grassmannians are simpler still than complex Grassmannians (Schubert cells are $4d$ versus $2d$ real-dimensional). On the other hand, the geometry of the symmetric variety ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ is more complicated than that of ${\textrm{GL}}_{2n}$ . The real–symmetric correspondence allows us to use the simpler quaternionic geometry of ${\textrm{GL}}_n({\mathbb H})$ to answer questions about the more complicated geometry of ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ .

We now describe the paper in more detail. We work throughout over the complex numbers, except where we specifically consider real forms. All topological sheaves are with respect to the classical topology with complex coefficients.

1.2 Reminder on derived Satake for ${\textrm{GL}}_{2n}$

Let $\mathfrak{L}{\textrm{GL}}_{2n}$ and $\mathfrak{L}^+{\textrm{GL}}_{2n}$ be the Laurent loop group and Taylor arc group of ${\textrm{GL}}_{2n}$ . The affine Grassmannian ${\textrm{Gr}}_{2n}=\mathfrak{L}{\textrm{GL}}_{2n}/\mathfrak{L}^+{\textrm{GL}}_{2n}$ for ${\textrm{GL}}_{2n}$ is the ind-variety classifying ${\mathbb C}[[t]]$ -lattices in ${\mathbb C}((t))^{n}$ . The arc group $\mathfrak{L}^+{\textrm{GL}}_{2n}$ acts naturally on ${\textrm{Gr}}_{2n}$ , and we denote by $D^b(\mathfrak{L}^+{\textrm{GL}}_{2n}\backslash {\textrm{Gr}}_{2n})$ the monoidal dg-category of $\mathfrak{L}^+{\textrm{GL}}_{2n}$ -equivariant constructible complexes on ${\textrm{Gr}}_{2n}$ with monoidal structure given by convolution.

Let $\mathfrak{gl}_{2n}$ be the Lie algebra of ${\textrm{GL}}_{2n}$ . Write $\textrm{Sym}(\mathfrak{gl}_{2n}[-2])$ for the symmetric algebra of $\mathfrak{gl}_{2n}[-2]$ viewed as a dg-algebra with trivial differential. The group ${\textrm{GL}}_{2n}$ acts on $\textrm{Sym}(\mathfrak{gl}_{2n}[-2])$ via the adjoint action, and we denote by $D^{{\textrm{GL}}_{2n}}_{\textrm{perf}}(\textrm{Sym}(\mathfrak{gl}_{2n}[-2]))$ the monoidal dg-category of perfect ${\textrm{GL}}_{2n}$ -equivariant dg-modules over $\textrm{Sym}(\mathfrak{gl}_{2n}[-2]))$ with monoidal structure given by the (derived) tensor product of dg-modules.

One of the versions of the derived Satake equivalence in [Reference Bezrukavnikov and FinkelbergBF08] says that there is an equivalence of monoidal dg-categories

\begin{equation*}\Psi :D^b(\mathfrak{L}^+{\textrm{GL}}_{2n}\backslash {\textrm{Gr}}_{2n})\simeq D^{{\textrm{GL}}_{2n}}_{\textrm{perf}}(\textrm{Sym}(\mathfrak{gl}_{2n}[-2]))\end{equation*}

extending the geometric Satake equivalence $\textrm{Perv}({\textrm{Gr}}_{2n})\simeq \textrm{Rep}({\textrm{GL}}_{2n})$ , where $\textrm{Perv}({\textrm{Gr}}_{2n})\subset D^b(\mathfrak{L}^+{\textrm{GL}}_{2n}\backslash {\textrm{Gr}}_{2n})$ is the subcategory of $\mathfrak{L}^+{\textrm{GL}}_{2n}$ -equivariant perverse sheaves on ${\textrm{Gr}}_{2n}$ and $\textrm{Rep}({\textrm{GL}}_{2n})\subset D^{{\textrm{GL}}_{2n}}_{\textrm{perf}}(\textrm{Sym}(\mathfrak{gl}_{2n}[-2]))$ is the subcategory of representations of ${\textrm{GL}}_{2n}$ .Footnote ¹

1.3 Derived Satake for the quaternionic group ${\textrm{GL}}_n({\mathbb H})$

Let ${\textrm{GL}}_{n}({\mathbb H})\subset {\textrm{GL}}_{2n}$ be the real form given by the rank $n$ quaternionic group. Let $\mathfrak{L}{\textrm{GL}}_n({\mathbb H})$ and $\mathfrak{L}^+{\textrm{GL}}_n({\mathbb H})$ be the real Laurent loop group and Taylor arc group for ${\textrm{GL}}_n({\mathbb H})$ . By the real affine Grassmannian for the quaternionic group ${\textrm{GL}}_n({\mathbb H})$ , we will mean the ind semi-analytic variety ${\textrm{Gr}}_{n,{\mathbb H}}=\mathfrak{L}{\textrm{GL}}_n({\mathbb H})/\mathfrak{L}^+{\textrm{GL}}_n({\mathbb H})$ classifying ${\mathbb H}[[t]]$ -lattices in ${\mathbb H}((t))^{n}$ .Footnote ²

The real arc group $\mathfrak{L}^+{\textrm{GL}}_n({\mathbb H})$ acts naturally on ${\textrm{Gr}}_{n,{\mathbb H}}$ , and we denote by $D^b(\mathfrak{L}^+{\textrm{GL}}_{n}({\mathbb H})\backslash {\textrm{Gr}}_{n,{\mathbb H}})$ the monoidal dg-category of $\mathfrak{L}^+{\textrm{GL}}_{n}({\mathbb H})$ -equivariant constructible complexes on ${\textrm{Gr}}_{n,{\mathbb H}}$ with monoidal structure given by convolution. The $\mathfrak{L}^+{\textrm{GL}}_n({\mathbb H})$ -orbits on ${\textrm{Gr}}_{n,{\mathbb H}}$ are all even real-dimensional (in fact, $4d$ real-dimensional; see § 4.4), and hence middle perversity makes sense. We denote by $\textrm{Perv}_{}({\textrm{Gr}}_{n,{\mathbb H}})$ the category of $\mathfrak{L}^+{\textrm{GL}}_{n}({\mathbb H})$ -equivariant perverse sheaves on ${\textrm{Gr}}_{n,{\mathbb H}}$ . In [Reference NadlerNad05], the third author established a real geometric Satake equivalence, giving an equivalence of monoidal abelian categories $\textrm{Perv}({\textrm{Gr}}_{n,{\mathbb H}})\simeq \textrm{Rep}({\textrm{GL}}_n)$ in the case at hand.

The first main result of this paper is the following equivalence of monoidal dg derived categories, to be called derived Satake for ${\textrm{GL}}_{n}({\mathbb H})$ .

Theorem 1.1 (See Theorem5.5). There is an equivalence of monoidal dg-categories

\begin{equation*}\Psi _{\mathbb H}:D^b(\mathfrak{L}^+{\textrm{GL}}_{n}({\mathbb H})\backslash {\textrm{Gr}}_{n,{\mathbb H}})\simeq D^{{\textrm{GL}}_n}_{\textrm{perf}}(\textrm{Sym}(\mathfrak {gl}_n[-4]))\end{equation*}

extending the real geometric Satake equivalence $\textrm{Perv}({\textrm{Gr}}_{n,{\mathbb H}})\simeq \textrm{Rep}({\textrm{GL}}_n)$ .

A key ingredient in the proof of Theorem1.1 (as in the proof of the abelian quaternionic geometric Satake) is a nearby cycles functor

(1.1) \begin{equation} \mathrm R:D^b(\mathfrak{L}^+{\textrm{GL}}_{2n}\backslash {\textrm{Gr}}_{2n})\to D^b(\mathfrak{L}^+{\textrm{GL}}_{n}({\mathbb H})\backslash {\textrm{Gr}}_{n,{\mathbb H}}) \end{equation}

associated to a real form of the Beilinson–Drinfeld Grassmannian with generic fibers isomorphic to the complex affine Grassimannian ${\textrm{Gr}}_{2n}$ and special fiber isomorphic to the quaternionic affine Grassimannian ${\textrm{Gr}}_{n,{\mathbb H}}$ (see § 4.5). Note that, unlike in the complex algebraic setting, the nearby cycles functor $\mathrm R$ is not $t$ -exact: it maps perverse sheaves to direct sums of shifts of perverse sheaves (see Proposition 4.5). As a corollary of the proof, we obtain the following spectral description of the nearby cycles functor.

Consider the graded scheme

\begin{equation*}\widetilde {\mathfrak{gl}}_{2n}=\bigg\{\bigg(\begin{array}{c@{\quad}c} A[0]& B[-2]\\[2pt] C[2]& D[0] \end{array}\bigg) \bigg| A,B,C,D\in \mathfrak{gl}_n\bigg\}.\end{equation*}

We have the natural embedding of (even graded) schemes

(1.2) \begin{equation} \tau :\mathfrak{gl}_n[4]\to \widetilde {\mathfrak{gl}}_{2n}[2], \quad \tau (C[4]) =\bigg(\begin{array}{c@{\quad}c} 0& \textrm{Id}_n\\ C[4]& 0 \end{array}\bigg), \end{equation}

where $\textrm{Id}_n$ is the rank $n$ identity matrix. Note that the map $\tau$ is ${\textrm{GL}}_n$ adjoint-equivariant, where ${\textrm{GL}}_n$ acts on $\widetilde {\mathfrak{gl}}_{2n}[2]$ via the diagonal embedding ${\textrm{GL}}_n\to {\textrm{GL}}_{2n}$ . Hence pullback along $\tau$ provides a functor

\begin{equation*}\tau ^*:D^{{\textrm{GL}}_{n}}_{\textrm{perf}}\big(\textrm{Sym}(\widetilde {\mathfrak{gl}}_{2n}[-2])\big)\to D^{{\textrm{GL}}_{n}}_{\textrm{perf}}(\textrm{Sym}(\mathfrak{gl}_{n} [-4])).\end{equation*}

Here we view the rings of functions on $\mathfrak{gl}_n[4]$ and $\widetilde {\mathfrak{gl}}_{2n}[2]$ as the dg symmetric algebras $\textrm{Sym}(\mathfrak{gl}_{2n}[-4])$ and $\textrm{Sym}\big(\widetilde {\mathfrak{gl}}_{2n}[-2]\big)$ with trivial differential. Introduce the functor

\begin{equation*}\Phi :D^{{\textrm{GL}}_{2n}}_{\textrm{perf}}(\textrm{Sym}(\mathfrak{gl}_{2n}[-2]))\stackrel {}\longrightarrow D^{{\textrm{GL}}_{n}}_{\textrm{perf}}\big(\textrm{Sym}(\widetilde {\mathfrak{gl}}_{2n}[-2])\big)\stackrel {\tau ^*}\longrightarrow D^{{\textrm{GL}}_{n}}_{\textrm{perf}}(\textrm{Sym}(\mathfrak{gl}_{n}[-4]))\end{equation*}

where the first functor is the sheared forgetful functor associated to the ${\mathbb G}_m$ -action on $\mathfrak{gl}_{2n}[-2]$ via the co-character $2\rho _L:{\mathbb G}_m\to {\textrm{GL}}_{2n}$ (see (5.17)). Here $L$ is the complexification of the Levi subgroup of the minimal parabolic subgroup of ${\textrm{GL}}_{n}({\mathbb H})$ .

Theorem 1.2 (See Theorem5.7). The following square is naturally commutative

where $\Psi$ and $\Psi _{\mathbb H}$ are the complex and quaternionic derived Satake equivalences, respectively.

Later, in § 1.6, we will discuss how Theorem1.2 fits into the general framework of duality for Hamiltonian spaces.

1.4 Derived Satake for the symmetric variety ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$

Let $\mathfrak{L}\;\textrm{Sp}_{2n}$ be the Laurent loop group of the symmetric subgroup $\textrm{Sp}_{2n}\subset {\textrm{GL}}_{2n}$ . There is a natural action of $\mathfrak{L}\;\textrm{Sp}_{2n}$ on ${\textrm{Gr}}_{2n}$ , and we denote by $D^b(\mathfrak{L}\;\textrm{Sp}_{2n}\backslash {\textrm{Gr}}_{2n})$ the dg-category of $\mathfrak{L}\;\textrm{Sp}_{2n}$ -equivariant constructible complexes on ${\textrm{Gr}}_{2n}$ .

In [Reference Chen and NadlerCN18, Theorem 8.1] it is shown that there is an equivalence of dg-categories

(1.3) \begin{equation} D^b\big(\mathfrak{L}\;\textrm{Sp}_{2n}\backslash {\textrm{Gr}}_{2n}\big)\simeq D^b(\mathfrak{L}^+{\textrm{GL}}_{n}({\mathbb H})\backslash {\textrm{Gr}}_{n,{\mathbb H}}). \end{equation}

In [Reference Chen and NadlerCN18, Theorem 9.1] it is shown that the equivalence is compatible with the natural monoidal actions of $D^b(\mathfrak{L}^+{\textrm{GL}}_{2n}\backslash {\textrm{Gr}}_{2n})$ , where the action on the right-hand side is through the nearby cycles functor (1.1). One can view the above equivalence as an example of the real–symmetric correspondence for the affine Grassmannian ${\textrm{Gr}}_{2n}$ . Combining this with Theorem1.1, we obtain a derived Satake equivalence for ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ .

Theorem 1.3. There is an equivalence of dg-categories

\begin{equation*}\Psi _X:D^b\big(\mathfrak{L}\;\textrm{Sp}_{2n}\backslash {\textrm{Gr}}_{2n}\big)\simeq D^{{\textrm{GL}}_n}_{\textrm{perf}}(\textrm{Sym}(\mathfrak{gl}_{n}[-4]))\end{equation*}

compatible with the monoidal actions of $D^b(\mathfrak{L}^+{\textrm{GL}}_{2n}\backslash {\textrm{Gr}}_{2n})\simeq D^{{\textrm{GL}}_n}_{\textrm{perf}}(\textrm{Sym}(\mathfrak{gl}_{2n}[-2]))$ .

1.5 Geometric Langlands correspondence for real groups

We discuss here how one might interpret our results in terms of geometric Langlands for real groups [Reference Ben-Zvi and NadlerBZN15]. Our results specifically relate to the curve $\mathbb P^1$ with its standard real structure with real points $\mathbb R\mathbb P^1$ (whereas connections to Langlands parameters have been explored [Reference Ben-Zvi and NadlerBZN13] for $\mathbb P^1$ with its antipodal real structure with no real points).

For complex reductive groups, it is known that the derived Satake equivalence implies the geometric Langlands correspondence over the projective line $\mathbb P^1$ via a Radon transform. To state a version of this in the setting at hand, let $\textrm{Bun}_{{\textrm{GL}}_{2n}}(\mathbb P^1)$ be the moduli stack of ${\textrm{GL}}_{2n}$ -bundles over $\mathbb P^1$ , and let $\textrm{Loc}_{{\textrm{GL}}_{2n}}(\textrm{S}^2)$ be the moduli stack of Betti ${\textrm{GL}}_{2n}$ -local systems on the 2-sphere $\textrm{S}^2$ . Let $D_!(\textrm{Bun}_{{\textrm{GL}}_{2n}}(\mathbb P^1))$ be the dg-category of constructible complexes on $\textrm{Bun}_{{\textrm{GL}}_{2n}}(\mathbb P^1)$ that are extensions by zero off of a finite-type substack, and let $\textrm{Coh}(\textrm{Loc}_{{\textrm{GL}}_{2n}}(\textrm{S}^2))$ be the dg-category of coherent complexes on $\textrm{Loc}_{{\textrm{GL}}_{2n}}(\textrm{S}^2)$ with bounded cohomology.

In this setting, the geometric Langlands correspondence for $\mathbb P^1$ constructed in [Reference LafforgueLaf09] takes the form of an equivalence

(1.4)

Moreover, it fits into a commutative diagram of equivalences

(1.5)

where the left vertical equivalence

(1.6) \begin{equation} D_!\big(\textrm{Bun}_{{\textrm{GL}}_{2n}}(\mathbb P^1)\big)\simeq D^b(\mathfrak{L}^+{\textrm{GL}}_{2n}\backslash {\textrm{Gr}}_{2n}) \end{equation}

is given by the Radon transform (see [Reference LafforgueLaf09, Proposition 2.1]) and the right vertical equivalence is given by the the Koszul duality equivalence

(1.7) \begin{equation} \textrm{Coh}(\mathfrak{gl}_{2n}[-1]/{\textrm{GL}}_{2n})\stackrel {}\simeq D^{{\textrm{GL}}_{2n}}_{\textrm{perf}}(\textrm{Sym}(\mathfrak{gl}_{2n}[-2])) \end{equation}

under the isomorphisms $\textrm{Loc}_{{\textrm{GL}}_{2n}}(\textrm{S}^2)\simeq \textrm{pt}/{\textrm{GL}}_{2n}\times _{\mathfrak{gl}_{2n}/{\textrm{GL}}_{2n}}\textrm{pt}/{\textrm{GL}}_{2n}\simeq \mathfrak{gl}_{2n}[-1]/{\textrm{GL}}_{2n}$ .

As a special case of the affine Matsuki correspondence established in [Reference Chen and NadlerCN18], we have a real group version of the equivalence (1.6) taking the form

(1.8) \begin{equation} D_!\big(\textrm{Bun}_{{\textrm{GL}}_{n}({\mathbb H})}(\mathbb {RP}^1)\big)\simeq D^b\big(\mathfrak{L}\;\textrm{Sp}_{2n}\backslash {\textrm{Gr}}_{2n}\big). \end{equation}

Here $\textrm{Bun}_{{\textrm{GL}}_{n}({\mathbb H})}(\mathbb {RP}^1)$ is the real form of $\textrm{Bun}_{{\textrm{GL}}_{2n}}(\mathbb {P}^1)$ classifying ${\textrm{GL}}_{n}({\mathbb H})$ -bundles on the real projective line $\mathbb {RP}^1$ . Combining this with the derived Satake equivalence for ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ in Theorem1.3, we obtain the following geometric Langlands correspondence for ${\textrm{GL}}_n({\mathbb H})$ .

Let $\textrm{Loc}_{{\textrm{GL}}_n}(\textrm{S}^4)$ be the moduli stack of Betti ${\textrm{GL}}_n$ -local systems on the $4$ -sphere $\textrm{S}^4$ . Note that the presentation $\textrm{S}^4=\textrm{D}^4\cup _{\textrm{S}^3}\textrm{D}^4$ (where $\textrm{D}^4$ is the four-dimensional disk in $\mathbb R^4$ ) gives an isomorphism of dg-stacks:

\begin{equation*}\textrm{Loc}_{{\textrm{GL}}_n}(\textrm{S}^4)\simeq \textrm{pt}/{\textrm{GL}}_{n}\times _{\mathfrak{gl}_{n}[-2]/{\textrm{GL}}_{n}}\textrm{pt}/{\textrm{GL}}_{n}\simeq \mathfrak{gl}_{n}[-3]/{\textrm{GL}}_{n}.\end{equation*}

From the Koszul duality $\textrm{Coh}(\mathfrak{gl}_{n}[-3]/{\textrm{GL}}_{n})\stackrel {}\simeq D^{{\textrm{GL}}_{n}}_{\textrm{perf}}(\textrm{Sym}(\mathfrak{gl}_{n}[-4]))$ , we obtain

(1.9) \begin{equation} \textrm{Coh}\big(\textrm{Loc}_{{\textrm{GL}}_n}(\textrm{S}^4)\big)\simeq \textrm{Coh}(\mathfrak{gl}_{n}[-3]/{\textrm{GL}}_{n})\simeq D^{{\textrm{GL}}_{n}}_{\textrm{perf}}(\textrm{Sym}(\mathfrak{gl}_{n}[-4])).\end{equation}

Theorem 1.5. There is an equivalence

\begin{equation*} D_!\big(\textrm{Bun}_{{\textrm{GL}}_{n}({\mathbb H})}(\mathbb {RP}^1)\big)\simeq \textrm{Coh}\big(\textrm{Loc}_{{\textrm{GL}}_n}(\textrm{S}^4)\big) \end{equation*}

that fits into a commutative diagram of equivalences as follows.

(1.10)

Here the left and right vertical equivalence are the affine Matsuki correspondence ( 1.8 ) and Koszul duality ( 1.9 ), respectively, and the bottom equivalence is the derived Satake equivalence for ${\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ .

1.6 Relative Langlands duality conjectures

A far-reaching program of Ben-Zvi, Sakellaridis, and Venkatesh proposes relative Langlands duality conjectures between periods and L-functions (see e.g. [Reference Ben-Zvi, Sakellaridis and VenkateshBZSV24]). A fundamental conjecture in the program predicts that given a complex reductive group $G$ , with dual group $\check G$ , and a homogeneous spherical $G$ -variety $X$ , there exists a (graded) Hamiltonian $\check G$ -variety $\check M$ together with a moment map $\mu :\check M\to \check {\mathfrak g}^*$ equipped with a commuting ${\mathbb G}_m$ -action of weight $2$ , and an equivalence

(1.11) \begin{equation} D^b(\mathfrak{L}X/\mathfrak{L}^+G)\simeq \textrm{Coh}(\check M/\check G) \end{equation}

where $\textrm{Coh}(\check M/\check G)$ is the dg-category of $\check G$ -equivariant perfect dg-modules over the ring of functions on $\check M$ viewed as a dg-algebra with trivial differential and grading given by the above ${\mathbb G}_m$ -action. Moreover, this equivalence should be compatible with the derived Satake equivalence $D^b(\mathfrak{L}^+G\backslash {\textrm{Gr}}_G)\simeq D^{\check G}_{\textrm{perf}}(\textrm{Sym}(\check {\mathfrak g}[-2]))\simeq \textrm{Coh}(\check {\mathfrak g}^*[2]/\check G)$ , in the sense that the right convolution action of $D^b(\mathfrak{L}^+G\backslash {\textrm{Gr}}_G)$ on $D^b(\mathfrak{L}X/\mathfrak{L}^+G)$ should correspond to the tensor product action of $\textrm{Coh}(\check {\mathfrak g}^*[2]/\check G)$ on $\textrm{Coh}(\check M/\check G)$ through the moment map $\mu$ .

We now explain how the derived Satake equivalence for the symmetric variety $X={\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ fits into the general setting of relative Langlands duality. On the one hand, by [Reference Chen and NadlerCN24, Proposition 8.1], there are an isomorphism of stacks

\begin{equation*}\mathfrak{L}X/\mathfrak{L}^+{\textrm{GL}}_{2n}\cong \mathfrak{L}\;\textrm{Sp}_{2n}\backslash \mathfrak{L}{\textrm{GL}}_{2n}/\mathfrak{L}^+{\textrm{GL}}_{2n}\cong \mathfrak{L}\;\textrm{Sp}_{2n}\backslash {\textrm{Gr}}_{2n}\end{equation*}

and hence equivalences of categories

(1.12) \begin{equation} D^b(\mathfrak{L}X/\mathfrak{L}^+{\textrm{GL}}_{2n})\simeq D^b(\mathfrak{L}\;\textrm{Sp}_{2n}\backslash {\textrm{Gr}}_{2n}) \end{equation}

where $D^b(\mathfrak{L}X/\mathfrak{L}^+{\textrm{GL}}_{2n})$ is the dg-category of $\mathfrak{L}^+{\textrm{GL}}_{2n}$ -equivariant constuctible complexes on the loop space $\mathfrak{L}X$ of $X$ .

On the other hand, it is expected that the Hamiltonian $\check G$ -space $\check M$ associated to the symmetric variety $X={\textrm{GL}}_{2n}/\textrm{Sp}_{2n}$ (note that symmetric varieties are spherical) is given by $\check M=T^*({\textrm{GL}}_{2n}/{\textrm{GL}}_n^{\textrm{}}\ltimes U,\psi )$ , the partial Whittaker reduction of $T^*{\textrm{GL}}_{2n}$ with respect to the generic homomorphism $\psi$ of the Shalika subgroup ${\textrm{GL}}^{\textrm{}}_n\ltimes U$ of ${\textrm{GL}}_{2n}$ :

(1.13) \begin{equation} {\textrm{GL}}^{\textrm{}}_n\ltimes U=\bigg\{\!\begin{pmatrix} A&\quad\! 0\\ 0&\quad\! A \end{pmatrix}\begin{pmatrix} \textrm{Id}_n&\quad\! 0\\ C&\quad\! \textrm{Id}_n \end{pmatrix}\!\bigg|\! \ A\in {\textrm{GL}}_n,\, C\in \mathfrak{gl}_n\bigg\},\quad\! \psi \left(\!\begin{pmatrix} A&\quad\! 0\\ 0&\quad\! A \end{pmatrix}\begin{pmatrix} \textrm{Id}_n&\quad\! 0\\ C&\quad\! \textrm{Id}_n \end{pmatrix}\!\right)=-\textrm{tr}(C) \end{equation}

(see the list of examples of relative duality in [Reference WangWan]).

By Lemma 3.2, there is an isomorphism $\check M\simeq {\textrm{GL}}_{2n}\times ^{{\textrm{GL}}_n^{\textrm{}}}\mathfrak{gl}_n$ such that the induced isomorphism $\check M/{\textrm{GL}}_{2n}\simeq \mathfrak{gl}_n/{\textrm{GL}}_n$ fits into a commutative diagram as follows.

Here $\mu$ is the moment map, $\tau$ is the embedding in (1.2) (disregarding the cohomological grading), and we identify $\mathfrak{gl}^*_{2n}/{\textrm{GL}}_{2n}\simeq \mathfrak{gl}_{2n}/{\textrm{GL}}_{2n}$ via the trace form. Thus, in view of (1.12), the equivalence of Theorem1.3 gives an instance of (1.11) of the formFootnote ³

\begin{equation*}D^b(\mathfrak{L}X/\mathfrak{L}^+{\textrm{GL}}_{2n})\simeq \textrm{Coh}(\check M/{\textrm{GL}}_{2n}).\end{equation*}

1.7 IC-stalks and Kostka–Foulkes polynomials

As an application of the proof of Theorem1.1, we determine the stalk cohomology of the IC-complexes for the $\mathfrak{L}^+{\textrm{GL}}_n({\mathbb H})$ -orbit closures in the quaternionic affine Grassmannian ${\textrm{Gr}}_{n,{\mathbb H}}$ and the $\mathfrak{L}\;\textrm{Sp}_{2n}$ -orbit closures in the complex affine Grassmannian ${\textrm{Gr}}_{2n}$ .

The $\mathfrak{L}^+{\textrm{GL}}_n$ -orbits (respectively, $\mathfrak{L}^+{\textrm{GL}}_n({\mathbb H})$ and $\mathfrak{L}\;\textrm{Sp}_{2n}$ -orbits) on ${\textrm{Gr}}_n$ (respectively, ${\textrm{Gr}}_{n,{\mathbb H}}$ and ${\textrm{Gr}}_{2n}$ ) are in bijection with the set of dominant coweights $\Lambda ^+_n$ of ${\textrm{GL}}_n$ ; see § 4.4. For any $\lambda \in \Lambda ^+_n$ we denote by ${\textrm{Gr}}_n^\lambda$ (respectively, ${\textrm{Gr}}_{n,{\mathbb H}}^\lambda$ and ${\textrm{Gr}}^\lambda _{2n,X}$ ) the corresponding orbit and by $\textrm{IC}(\overline {{\textrm{Gr}}_n^\lambda })$ (respectively, $\textrm{IC}(\overline {{\textrm{Gr}}_{n,{\mathbb H}}^\lambda })$ and $\textrm{IC}(\overline {{\textrm{Gr}}^\lambda _{2n,X}})$ ) the intersection cohomology complex on the orbit closure. We will write $\mathscr{H}^{\;\;i}(\mathcal {F})$ for the $i$ th cohomology sheaf of a complex $\mathcal {F}$ and $\mathscr H_x^{\;\,i}(\mathcal {F})$ for its stalk at a point $x$ .

For any pair of dominant coweights $\lambda ,\mu \in \Lambda ^+_n$ , we denote by $K_{\lambda ,\mu }(q)$ the associated Kostka–Foulkes polynomial with variable $q$ . Denote by $\rho _n$ the half-sum of positive roots of ${\textrm{GL}}_n$ . A well-known result of Lusztig [Reference LusztigLus81] says that we have $\mathscr H^{\,\,i-2\langle \lambda ,\rho _n\rangle }(\textrm{IC}(\overline {{\textrm{Gr}}_n^\lambda }))=0$ for $2\nmid i$ and

\begin{equation*}\sum _i\textrm{dim}\;\mathscr H^{\,\,2i-2\langle \lambda ,\rho _n\rangle }_x\big(\textrm{IC}\big(\overline {{\textrm{Gr}}_n^\lambda }\big)\big)q^i =q^{\langle \lambda -\mu ,\rho _n\rangle }K_{\lambda ,\mu }(q^{-1}) \quad \text {for } x\in {\textrm{Gr}}_{n}^\mu .\end{equation*}

We have the following real and symmetric analogue.

In other words, the theorem says that the IC-complex for the $\mathfrak{L}^+{\textrm{GL}}_{n}({\mathbb H})$ and $\mathfrak{L}K$ -orbit closures $\overline {{\textrm{Gr}}^\lambda _{n,{\mathbb H}}}$ and $\overline {{\textrm{Gr}}^\lambda _{2n,X}}$ have the same stalk cohomology as the $\overline {{\textrm{Gr}}_n^\lambda }$ ones for ${\textrm{GL}}_n$ , but with all degrees doubled.

1.8 Outline of the proof

We briefly explain the proof of Theorem1.1. Similar to the proof of the derived Satake for complex reductive groups [Reference Bezrukavnikov and FinkelbergBF08], the desired equivalence follows from the following two statements: (1) the de-equivariantized extension algebra $\textrm{Ext}^*_{D^b_{}(\mathfrak{L}^+G_{n,{\mathbb H}}\backslash {\textrm{Gr}}_{n,{\mathbb H}})}(\textrm{IC}_0,\textrm{IC}_0\star \mathcal {O}(G_n))$ is isomorphic to the dg symmetric algebra $\textrm{Sym}(\mathfrak{gl}_n[-4])$ (see Proposition 4.21) and (2) the dg algebra $\textrm{RHom}_{D^b_{}(\mathfrak{L}^+G_{n,{\mathbb H}}\backslash {\textrm{Gr}}_{n,{\mathbb H}})}(\textrm{IC}_0,\textrm{IC}_0\star \mathcal {O}(G_n))$ is formal (see Proposition 5.3).

We deduce (1) from a fully-faithfulness property of the equivariant cohomology functor. In [Reference Bezrukavnikov and FinkelbergBF08], this was proved using a general result of Ginzburg [Reference GinzburgGin91], whose proof uses Hodge theory and hence does not apply directly to the real analytic setting. We observe that in the situation of the quaternionic affine Grassmannian, the stalks of the IC-complexes satisfy a parity vanishing property and, as observed in [Reference Achar and RiderAR15], one can use parity considerations in place of Hodge theory. To prove the parity vanishing result of the IC-stalks, we show that fibers of certain convolution Grassmannians (which are basically quaternionic Springer fibers) admit pavings by quaternionic affine spaces.

The proof of (2) in [Reference Bezrukavnikov and FinkelbergBF08] also relies on Hodge theory (or theory of weights) and hence must be modified in the real setting. We observe that the nearby cycles functor (1.1) induces a surjective homomorphism from the dg-algebra $\textrm{RHom}_{D^b_{}(\mathfrak{L}^+G_{2n}\backslash {\textrm{Gr}}_{2n})}(\textrm{IC}_0,\textrm{IC}_0\star \mathcal {O}(G_{2n}))$ associated to the complex affine Grassimannian ${\textrm{Gr}}_{2n}$ to the dg-algebra $\textrm{RHom}_{D^b_{}(\mathfrak{L}^+G_{n,{\mathbb H}}\backslash {\textrm{Gr}}_{n,{\mathbb H}})}(\textrm{IC}_0,\textrm{IC}_0\star \mathcal {O}(G_n))$ . Since the former dg-algebra is formal, thanks to [Reference Bezrukavnikov and FinkelbergBF08], the desired claim follows from a general criterion of formality; see Lemma 5.4.

1.9 Organization

We briefly summarize the main goals of each section. In § 2, we recall some notation and results on constructible sheaves on a semi-analytic stack. In § 3, we study the spectral side of the quaternionic Satake equivalence, including results on quaternionic groups, symplectic groups, regular centralizers group schemes, and Whittaker reduction. In § 4, we study the constructible side of the equivalence, including the study of nearby cycles functors, parity vanishing results, fully-faithfulness of the equivariant cohomology functor, and the computation of the IC-stalks and the de-equivariantized extension algebra. Finally, in § 5, we prove the formality result and deduce the derived Satake equivalence for quaternionic groups, including a version involving nilpotent singular supports (Theorem5.5), and also the spectral description of the nearby cycles functor (Theorem5.7).