Coulomb branches of noncotangent type

Let $X$ be an algebraic variety over $\mathbb{C}$. We say that $X$ is singular symplectic (or $X$ has symplectic singularities) if

(1) $X$ is a normal Poisson variety;

(2) There exists a smooth dense open subset $U$ of $X$ on which the Poisson structure comes from a symplectic structure. We shall denote by $\omega$ the corresponding symplectic form.

(3) There exists a resolution of singularities $\pi\colon\widetilde{X}\to X$ such that $\pi^{*}\omega$ has no poles on $\widetilde{X}$.

We say that $X$ is a conical symplectic singularity if in addition to (1)-(3) above one has a $\mathbb{C}^{\times}$-action on $X$ which acts on $\omega$ with some positive weight and which contracts all of $X$ to one point.

A symplectic resolution $\pi\colon\widetilde{X}\to X$ is a proper and birational morphism $\pi$ such that $\pi^{*}\omega$ extends to a symplectic form on $\widetilde{X}$. Here is one example. Let $\mathfrak{g}$ be a semi-simple Lie algebra over $\mathbb{C}$ and let $\mathcal{N}_{\mathfrak{g}}\subset\mathfrak{g}^{*}$ be its nilpotent cone. Let $\mathcal{B}$ denote the flag variety of $\mathfrak{g}$. Then the Springer map $\pi\colon T^{*}\mathcal{B}\to\mathcal{N}_{\mathfrak{g}}$ is proper and birational, so if we let $X=\mathcal{N}_{\mathfrak{g}},\widetilde{X}=T^{*}\mathcal{B}$ we get a symplectic resolution.

The idea of symplectic duality is this: often conical symplectic singularities come in “dual” pairs $(X,X^{*})$ (the assignment $X\to X^{*}$ is by no means a functor; we just have a lot of interesting examples of dual pairs). What does it mean that $X$ and $X^{*}$ are dual? This is in general not easy to tell, but many geometric questions about $X$ should be equivalent to some other geometric questions about $X^{*}$. For example, we should have $\dim H^{\bullet}(\widetilde{X},\mathbb{C})=\dim H^{\bullet}(\widetilde{X}^{*},\mathbb{C})$ (but these spaces are not supposed to be canonically isomorphic). We refer the reader to [BPW], [BLPW] for more details. There should be a lot of other connections between $X$ and $X^{*}$ which will take much longer to describe; we refer the reader to loc.cit. for the description of these properties as well as for examples.

One source of dual pairs $(X,X^{*})$ comes from quantum field theory in the following way. Physicists have a notion of 3-dimensional ${\mathcal{N}}=4$ super-symmetric quantum field theory. Any such theory $\mathcal{T}$ is supposed to have a well-defined moduli space of vacua $\mathcal{M}(\mathcal{T})$. This space is complicated, but it should have two special pieces called the Higgs and the Coulomb branch; we shall denote these by $\mathcal{M}_{H}(\mathcal{T})$ and $\mathcal{M}_{C}(\mathcal{T})$. They are supposed to be (singular) symplectic complex algebraic varieties (in fact, they don’t even have to be algebraic but for simplicity we shall only consider examples when they are).

Let $G$ be a complex reductive algebraic group and let $\mathbf{M}$ be a symplectic vector space with a Hamiltonian action of $G$. Then to the pair $(G,\mathbf{M})$ one is supposed to associate a theory $\mathcal{T}(G,\mathbf{M})$ provided that $\mathbf{M}$ satisfies certain anomaly cancellation condition, which can be formulated as follows. The representation $\mathbf{M}$ defines a homomorphism $G\to{\mathop{\operatorname{\rm Sp}}}(\mathbf{M})$ and thus a homomorphism $\pi_{4}(G)\to\pi_{4}({\mathop{\operatorname{\rm Sp}}}(\mathbf{M}))=\mathbb{Z}/2\mathbb{Z}$. The anomaly cancellation condition is the condition that this homomorphism is trivial. Without going to further details at the moment we would like to emphasize the following:

1) Any $\mathbf{M}$ of the form $T^{*}\mathbf{N}=\mathbf{N}\oplus\mathbf{N}^{*}$ where $\mathbf{N}$ is some representation $G$ satisfies this condition.

2) The anomaly cancellation condition is a “$\mathbb{Z}/2\mathbb{Z}$-condition” (later on we are going to formulate it more algebraically).

The corresponding Coulomb branches are much trickier to define. Physicists had some expectations about those but no rigorous definition in general (only some examples). The idea is that at least in the conical case the pair $(\mathcal{M}_{H}(\mathcal{T}),\mathcal{M}_{C}(\mathcal{T}))$ should produce an example of a dual symplectic pair. A mathematical approach to the definition of Coulomb branches was proposed in [N]. A rigorous definition of the Coulomb branches ${\mathcal{M}}_{C}(G,{\mathbf{M}})$ is given in [BFN1] under the assumption that $\mathbf{M}=T^{*}\mathbf{N}=\mathbf{N}\oplus\mathbf{N}^{*}$ for some representation $\mathbf{N}$ of $G$.¹¹1In addition to the Coulomb branch ${\mathcal{M}}_{C}(G,{\mathbf{M}})$, in [BFN1, Remark 3.14] the authors define the $K$-theoretic Coulomb branch ${\mathcal{M}}_{C}^{K}(G,{\mathbf{M}})$ under the same assumption (physically, it should correspond to the Coulomb branch of the corresponding 4d gauge theory of ${\mathbb{R}}^{3}\times S^{1}$). We would like to emphasize that at this point we are not able to extend this construction to arbitrary symplectic ${\mathbf{M}}$ with anomaly cancellation condition. The varieties $\mathcal{M}_{C}(G,\mathbf{M})$ are normal, affine, Poisson, generically symplectic and satisfy the monopole formula. We expect that they are singular symplectic, but we can not prove this in general, cf. [We]. The main ingredient in the definition is the geometry of the affine Grassmannian ${\operatorname{Gr}}_{G}$ of $G$. In [BFN1, BFN2, BFN3] these varieties are computed in many cases (in particular, in the case of so called quiver gauge theories — it turns out that one can associate a pair $(G,\mathbf{N})$ to any framed quiver). The quantizations of these varieties are also studied, as well as their (Poisson) deformations and (partial) resolutions.

Let ${\mathcal{K}}={\mathbb{C}}(\!(t)\!)\supset{\mathcal{O}}={\mathbb{C}}[\![t]\!]$. The affine Grassmannian ind-scheme ${\operatorname{Gr}}_{G}=G_{\mathcal{K}}/G_{\mathcal{O}}$ is the moduli space of $G$-bundles on the formal disc equipped with a trivialization on the punctured formal disc. One can consider the derived Satake category $D_{G_{\mathcal{O}}}({\operatorname{Gr}}_{G})$.²²2In fact we are going to work with a certain renormalized version of it, cf. §2.1. This is a monoidal category which is monoidally equivalent to $D^{G^{\vee}}(\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{g}}^{\!\scriptscriptstyle\vee}[-2]))$: the derived category of dg-modules over $\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{g}}^{\!\scriptscriptstyle\vee}[-2])$ endowed with a compatible action of $G^{\vee}$ (the monoidal structure on this category is just given by tensor product over $\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{g}}^{\!\scriptscriptstyle\vee}[-2])$); we shall denote the corresponding functor from $D_{G_{\mathcal{O}}}({\operatorname{Gr}}_{G})$ to $D^{G^{\vee}}(\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{g}}^{\!\scriptscriptstyle\vee}[-2])$ by $\Phi_{G}$. In [BFN3] we have attached to any $\mathbf{N}$ as above a certain ring object $\mathcal{A}_{G,\mathbf{M}}$ in $D_{G_{\mathcal{O}}}({\operatorname{Gr}}_{G})$ (here as before we set $\mathbf{M}=T^{*}\mathbf{N}$) such that the algebra of functions on $\mathcal{M}_{C}(G,\mathbf{M})$ is equal to $H^{\bullet}_{G_{\mathcal{O}}}({\operatorname{Gr}}_{G},\mathcal{A}_{G,\mathbf{M}})$ (this cohomology has an algebra structure coming from the fact that $\mathcal{A}_{G,{\mathbf{M}}}$ is a ring object).

One of the main goals of this paper is to construct the ring object $\mathcal{A}_{G,\mathbf{M}}$ for arbitrary symplectic representation $\mathbf{M}$ satisfying the anomaly cancellation condition.³³3Another construction of the Coulomb branch of a $3d\ {\mathcal{N}}=4$ gauge theory in the noncotangent case was proposed by C. Teleman [T]. In fact, we can construct the ring object $\mathcal{A}_{G,\mathbf{M}}$ for any symplectic $\mathbf{M}$ but instead of being an object of the derived Satake category $D_{G_{\mathcal{O}}}({\operatorname{Gr}}_{G})$ it will be an object of a certain twisted version of it. More precisely, the representation $\mathbf{M}$ defines certain determinant line bundle $\mathcal{D}_{\mathbf{M}}$ on ${\operatorname{Gr}}_{G}$ which is equipped with certain multiplicative structure; we shall denote by $\mathcal{D}_{\mathbf{M}}^{0}$ the total space of this bundle without the zero section. The line bundle $\mathcal{D}_{\mathbf{M}}$ is also $G_{\mathcal{O}}$-equivariant. In particular, for any $\tau\in{\mathbb{C}}$ one can consider the category $D_{\tau}^{G_{\mathcal{O}}}({\operatorname{Gr}}_{G})$ of $G_{\mathcal{O}}$-equivariant sheaves on $\mathcal{D}_{\mathbf{M}}^{0}$ which are ${\mathbb{C}}^{\times}$-monodromic with monodromy $q=e^{2\pi i\tau}$. This category is again monoidal (because of the above multiplicative structure on $\mathcal{D}_{\mathbf{M}}$). If $\tau$ is a rational number and $\mathcal{D}_{\mathbf{M}}^{\tau}$ exists as a multiplicative line bundle on ${\operatorname{Gr}}_{G}$, the twisted category $D_{\tau}^{G_{\mathcal{O}}}({\operatorname{Gr}}_{G})$ is naturally equivalent to $D_{G_{\mathcal{O}}}({\operatorname{Gr}}_{G})$ (as a monoidal category).

In this paper we shall construct a ring object $\mathcal{A}_{G,\mathbf{M}}\in D_{-1/2}^{G_{\mathcal{O}}}({\operatorname{Gr}}_{G})$. It turns out (see Proposition 4.1.1) that the anomaly cancellation condition is equivalent to the existence of a multiplicative square root of $\mathcal{D}_{\mathbf{M}}$. So, we can construct the ring object $\mathcal{A}_{G,\mathbf{M}}$ but it will be untwisted only if the anomaly cancellation condition is satisfied. In particular, we can take its $G_{\mathcal{O}}$-equivariant cohomology (and thus define the algebra of functions on the corresponding Coulomb branch) only under the anomaly cancellation assumption.

In fact in order to construct the ring object $\mathcal{A}_{G,\mathbf{M}}$ for any $G$ and $\mathbf{M}$ it is enough to do it when $G={\mathop{\operatorname{\rm Sp}}}(2n)$ and $\mathbf{M}=\mathbb{C}^{2n}$ is its tautological representation. The reason is as follows. Assume first that ${\mathbf{M}}=T^{*}{\mathbf{N}}$ and let $i\colon G^{\prime}\to G$ be a homomorphism of connected reductive groups. It induces a morphism $\widetilde{i}\colon{\operatorname{Gr}}_{G^{\prime}}\to{\operatorname{Gr}}_{G}$, and it follows from the construction of [BFN3] that $\mathcal{A}_{G^{\prime},\mathbf{M}}=\widetilde{i}^{!}\mathcal{A}_{G,\mathbf{M}}$. Assuming that the same is true for arbitrary $\mathbf{M}$ and since the symplectic representation $\mathbf{M}$ is the same as a homomorphism $G\to{\mathop{\operatorname{\rm Sp}}}(\mathbf{M})$ we see that the case $G={\mathop{\operatorname{\rm Sp}}}(\mathbf{M})$ is universal in the sense that the object $\mathcal{A}_{G,\mathbf{M}}$ in general should just be equal to the $!$-pullback of $\mathcal{A}_{{\mathop{\operatorname{\rm Sp}}}(\mathbf{M}),\mathbf{M}}$.⁴⁴4This was first observed by V. Drinfeld.

In this paper we do the following:

1) We construct the object $\mathcal{A}_{G,\mathbf{M}}$ (as was explained above it is enough to do it in the case $G={\mathop{\operatorname{\rm Sp}}}(\mathbf{M})$).

2) We check that when $\mathbf{M}=T^{*}\mathbf{N}$ for some representation $\mathbf{N}$ of $G$, this construction coincides with the one of [BFN3].

3) In the case when $G={\mathop{\operatorname{\rm Sp}}}(\mathbf{M})$ we compute the image of $\mathcal{A}_{G,\mathbf{M}}$ under the twisted version of the derived geometric Satake equivalence (see §1.8 below). To do that we express $\mathcal{A}_{G,\mathbf{M}}$ as a Radon transform of a certain theta-sheaf [Ly, LL] for the curve $\mathbb{P}^{1}$ (the necessary facts and definitions about the Radon transform are reviewed in Appendix A). The idea that $\mathcal{A}_{G,\mathbf{M}}$ should be related to the theta-sheaf also belongs to V. Drinfeld.

Let us briefly explain the idea of the construction of $\mathcal{A}_{G,\mathbf{M}}$. Let $\mathcal{C}$ be a (dg) category endowed with a strong action of an algebraic group $H$ (e.g. one can take $\mathcal{C}$ to be the (dg-model of the) derived category of $D$-modules on a scheme $X$ endowed with an action of $H$). Let $\mathcal{F}$ be an object of $\mathcal{C}$ which is equivariant under some closed subgroup $L$ of $H$. Then one can canonically attach to $\mathcal{F}$ a ring object $\mathcal{A}_{\mathcal{F}}\in\mathrm{D}{\operatorname{-mod}}^{L}(H/L)$ (the $L$-equivariant derived category of $D$-modules on $H/L$; this category is endowed with a natural monoidal structure with respect to convolution). This object has the property that its !-restriction to any $h\in H$ is equal to $\operatorname{RHom}(\mathcal{F},\mathcal{F}^{h})$.

Here is a variant of this construction. Assume that $H$ is endowed with a central extension

which splits over $L$. Then for any $\kappa\in\mathbb{C}$ it makes sense to talk about an action $H$ on $\mathcal{C}$ of level $\kappa$. Then in the same way as above we can define $\mathcal{A}_{\mathcal{F}}\in\operatorname{D-mod}_{\kappa}^{L}(H/L)$ (here $\operatorname{D-mod}_{\kappa}$ stands for the corresponding category of twisted $D$-modules on $H/L$). The same thing works when $H$ is a group ind-scheme. We are going to apply it to the case when $H=G_{\mathcal{K}},L=G_{\mathcal{O}},\mathcal{C}={\mathcal{W}}{\operatorname{-mod}}$, where ${\mathcal{W}}$ is the Weyl algebra of the symplectic vector space ${\mathbf{M}}_{\mathcal{K}}$. The line bundle $\mathcal{D}_{\mathbf{M}}$ defines a central extension ${\widetilde{G}}_{\mathcal{K}}$ of $G_{\mathcal{K}}$, and it is well-known that the action of $G_{\mathcal{K}}$ on ${\mathbf{M}}_{\mathcal{K}}$ naturally extends to a strong action of ${\widetilde{G}}_{\mathcal{K}}$ on ${\mathcal{W}}{\operatorname{-mod}}$ of level $-1/2$. We now take $\mathcal{F}$ to be ${\mathbb{C}}[{\mathbf{M}}_{\mathcal{O}}]$.⁵⁵5This action should be thought of as a categorical analog of the Weil representation, cf. [LL]. The corresponding ring object $\mathcal{A}_{G,\mathbf{M}}$ is just (the Riemann-Hilbert functor applied to) $\mathcal{A}_{\mathcal{F}}$ for $\mathcal{F}$ as above. It is not difficult to check that when $\mathbf{M}=T^{*}\mathbf{N}$ this construction coincides with the one from [BFN3].

This subsection is somewhat digressive from the point of view of the main body of this paper. We include it here for completeness and in order to indicate some future research directions.

Finally, we are able to describe the image of the universal ring object under the twisted Satake equivalence (answering a question of V. Drinfeld). First, it turns out that for $G={\mathop{\operatorname{\rm Sp}}}({\mathbf{M}}),\ {\mathfrak{g}}={\mathfrak{sp}}({\mathbf{M}})$, there is a monoidal equivalence $\Phi_{G}\colon D_{-1/2}^{G_{\mathcal{O}}}({\operatorname{Gr}}_{G})\mathbin{\vphantom{j^{X^{2}}}\smash{\overset{\sim}{\vphantom{\rule{0.0pt}{1.99997pt}}\smash{\longrightarrow}}}}D^{G}(\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{g}}[-2]))$ [DLYZ]. Second, $\Phi_{G}({\mathcal{A}}_{G,{\mathbf{M}}})\cong{\mathbb{C}}[\operatorname{Whit}_{G}(T^{*}G)]$ (Whittaker reduction of the shifted cotangent bundle of $G$ with respect to the left action. The cohomological grading arises from the one on ${\mathbb{C}}[T^{*}G]={\mathbb{C}}[G]\otimes\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{g}})$, where the generators in ${\mathfrak{g}}$ are assigned degree 2, while ${\mathbb{C}}[G]$ is assigned degree 0).

Note that under the non-twisted Satake equivalence $\Phi_{G^{\vee}}\colon D_{G^{\vee}_{\mathcal{O}}}({\operatorname{Gr}}_{G^{\vee}})\mathbin{\vphantom{j^{X^{2}}}\smash{\overset{\sim}{\vphantom{\rule{0.0pt}{1.99997pt}}\smash{\longrightarrow}}}}D^{G}(\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{g}}[-2]))$, we have $\Phi_{G^{\vee}}({\boldsymbol{\omega}}_{{\operatorname{Gr}}_{G^{\vee}}})\simeq\Phi_{G}({\mathcal{A}}_{G,{\mathbf{M}}})$. This answer to Drinfeld’s question was proposed by D. Gaiotto.

Also, if we consider $G^{\vee}\cong{\mathop{\operatorname{\rm SO}}}({\mathbf{M}}^{\prime})$ for a $2n+1$-dimensional vector space ${\mathbf{M}}^{\prime}$ equipped with a nondegenerate symmetric bilinear form, then ${\mathbf{M}}\otimes{\mathbf{M}}^{\prime}$ carries a natural symplectic form and a natural action of $G\times G^{\vee}$. We have an isomorphism $\Phi_{G}({\mathcal{A}}_{G,{\mathbf{M}}})\cong{\mathbb{C}}[\operatorname{Whit}_{G^{\vee}}({\mathbf{M}}\otimes{\mathbf{M}}^{\prime})]$ (with residual action of $G$. The cohomological grading arises from the one on $\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathbf{M}}\otimes{\mathbf{M}}^{\prime})$ where all the generators are assigned degree 1).

Similarly, in the universal cotangent case, when ${\mathsf{G}}={\mathop{\operatorname{\rm GL}}}({\mathbf{N}})$ for an $n$-dimensional vector space ${\mathbf{N}}$, and ${\mathsf{G}}^{\vee}\cong{\mathop{\operatorname{\rm GL}}}({\mathbf{N}}^{\prime})$ for another $n$-dimensional vector space ${\mathbf{N}}^{\prime}$, we have the untwisted Satake equivalence $\Phi_{\mathsf{G}}\colon D_{{\mathsf{G}}_{\mathcal{O}}}({\operatorname{Gr}}_{\mathsf{G}})\mathbin{\vphantom{j^{X^{2}}}\smash{\overset{\sim}{\vphantom{\rule{0.0pt}{1.99997pt}}\smash{\longrightarrow}}}}D^{\mathsf{G}}(\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{gl}}({\mathbf{N}})[-2]))$. Now ${\mathop{\operatorname{\rm Hom}}}({\mathbf{N}},{\mathbf{N}}^{\prime})\oplus{\mathop{\operatorname{\rm Hom}}}({\mathbf{N}}^{\prime},{\mathbf{N}})$ carries a natural sympectic form and a natural action of ${\mathsf{G}}\times{\mathsf{G}}^{\vee}$. We have an isomorphism $\Phi_{\mathsf{G}}({\mathcal{A}}_{{\mathsf{G}},{\mathbf{N}}})\cong{\mathbb{C}}\Big{[}\operatorname{Whit}_{{\mathsf{G}}^{\vee}}\big{(}{\mathop{\operatorname{\rm Hom}}}({\mathbf{N}},{\mathbf{N}}^{\prime})\oplus{\mathop{\operatorname{\rm Hom}}}({\mathbf{N}}^{\prime},{\mathbf{N}})\big{)}\Big{]}$ (with residual action of ${\mathsf{G}}$. The cohomological grading arises from the one on $\mathop{\operatorname{\rm Sym}}^{\bullet}\big{(}{\mathop{\operatorname{\rm Hom}}}({\mathbf{N}},{\mathbf{N}}^{\prime})\oplus{\mathop{\operatorname{\rm Hom}}}({\mathbf{N}}^{\prime},{\mathbf{N}})\big{)}$ where all the generators are assigned degree 1).

We are deeply grateful to D. Ben-Zvi, R. Bezrukavnikov, V. Drinfeld, P. Etingof, B. Feigin, D. Gaiotto, D. Gaitsgory, A. Hanany, T. Johnson-Freyd, S. Lysenko, H. Nakajima, Y. Sakellaridis, A. Venkatesh, J. Wang, E. Witten, P. Yoo and Z. Yun for many helpful and inspiring discussions. M.F. and S.R. thank the 4th Nisyros Conference on Automorphic Representations and Related Topics held in July 2019 for stimulating much of this work.

A.B. was partially supported by NSERC. G.D. was supported by an NSF Postdoctoral Fellowship under grant No. 2103387. M.F. was partially funded within the framework of the HSE University Basic Research Program and the Russian Academic Excellence Project ‘5-100’. S.R. was supported by NSF grant DMS-2101984.

Let ${\mathbf{M}}$ be a $2n$-dimensional complex vector space equipped with a symplectic form $\langle\,,\rangle$. Its automorphism group is $G={\mathop{\operatorname{\rm Sp}}}({\mathbf{M}})$.

Let ${\mathcal{K}}={\mathbb{C}}(\!(t)\!)\supset{\mathcal{O}}={\mathbb{C}}[\![t]\!]$. The affine Grassmannian ind-scheme ${\operatorname{Gr}}_{G}=G_{\mathcal{K}}/G_{\mathcal{O}}$ is the moduli space of $G$-bundles on the formal disc equipped with a trivialization on the punctured formal disc. The Kashiwara affine Grassmannian infinite type scheme ${\mathbf{Gr}}_{G}=G_{\mathcal{K}}/G_{{\mathbb{C}}[t^{-1}]}$ is the moduli space of $G$-bundles on ${\mathbb{P}}^{1}$ equipped with a trivialization in the formal neighbourhood of $0\in{\mathbb{P}}^{1}$.

The determinant line bundles over ${\operatorname{Gr}}_{G}$ and ${\mathbf{Gr}}_{G}$ are denoted by ${\mathcal{D}}$. The $\mu_{2}$-gerbe of square roots of ${\mathcal{D}}$ over ${\mathbf{Gr}}_{G}$ (resp. ${\operatorname{Gr}}_{G}$) is denoted ${\widetilde{\operatorname{Gr}}}_{G}$ (resp. ${\widetilde{\mathbf{Gr}}}_{G}$).

The action of $G_{\mathcal{K}}$ on ${\operatorname{Gr}}_{G}$ and ${\mathbf{Gr}}_{G}$ lifts to the action of the metaplectic group-stack ${\widetilde{G}}_{\mathcal{K}}$ on ${\widetilde{\operatorname{Gr}}}_{G}$ and ${\widetilde{\mathbf{Gr}}}_{G}$. We have a splitting $G_{\mathcal{O}}\hookrightarrow{\widetilde{G}}_{\mathcal{K}}$.

In what follows we only consider the genuine constructible sheaves on ${\widetilde{\operatorname{Gr}}}_{G}$ and ${\widetilde{\mathbf{Gr}}}_{G}$: such that $-1\in\mu_{2}$ acts on them as $-1$. We consider a dg-enhancement $D^{b}_{G_{\mathcal{O}}}({\widetilde{\operatorname{Gr}}}_{G})$ of the (genuine) bounded equivariant constructible derived category. We denote by $D_{G_{\mathcal{O}}}({\widetilde{\operatorname{Gr}}}_{G})$ the renormalized equivariant derived category defined as in [AGa, §12.2.3]. We also consider the category $D_{G_{\mathcal{O}}}({\widetilde{\mathbf{Gr}}}_{G})_{!}$ defined as in [ArG, §3.4.1] (the inverse limit over the $G_{\mathcal{O}}$-stable open subgerbes of ${\widetilde{\mathbf{Gr}}}_{G}$, cf. §A.4). It contains the IC-sheaves of the $G_{\mathcal{O}}$-orbits closures.

An open sub-gerbe ${\mathcal{T}}\hookrightarrow{\widetilde{\operatorname{Gr}}}_{G}\times{\widetilde{\mathbf{Gr}}}_{G}$ is formed by all the pairs of transversal compact and discrete Lagrangian subspaces in ${\mathbf{M}}_{\mathcal{K}}$. We denote by

the natural projections. The Radon Transform is (cf. §A.5, where its $D$-module version is denoted ${\mathop{\rm RT}}_{!}^{-1}$)

The Theta-sheaf $\Theta\in D_{G_{\mathcal{O}}}({\widetilde{\mathbf{Gr}}}_{G})_{!}$ introduced in [Ly] is the direct sum of IC-sheaves of two $G_{\mathcal{O}}$-orbits in ${\widetilde{\mathbf{Gr}}}_{G}\colon\Theta_{g}$ of the open orbit, and $\Theta_{s}$ of the codimension 1 orbit.

The dg-category of $G_{\mathcal{O}}$-equivariant $D$-modules on ${\operatorname{Gr}}_{G}$ (resp. on ${\mathbf{Gr}}_{G}$) twisted by the inverse square root ${\mathcal{D}}^{-1/2}$ is denoted ${\operatorname{D-mod}_{-1/2}^{G_{{\mathcal{O}}}}}({\operatorname{Gr}}_{G})$ (resp. ${\operatorname{D-mod}_{-1/2}^{G_{{\mathcal{O}}}}}({\mathbf{Gr}}_{G})_{!}$). More precisely, by ${\operatorname{D-mod}_{-1/2}^{G_{{\mathcal{O}}}}}({\operatorname{Gr}}_{G})$ we mean the renormalized equivariant category defined as in [AGa, §12.2.3], and ${\operatorname{D-mod}_{-1/2}^{G_{{\mathcal{O}}}}}({\mathbf{Gr}}_{G})_{!}$ is defined in §A.4. We have the Riemann–Hilbert equivalences

We denote ${\mathop{\rm RH}}^{-1}(\Theta)$ by $\varTheta\in{\operatorname{D-mod}_{-1/2}^{G_{{\mathcal{O}}}}}({\mathbf{Gr}}_{G})_{!}$, a direct sum of two irreducible $D$-modules, $\varTheta_{g}$ with the full support, and $\varTheta_{s}$ supported at the Schubert divisor.

The (derived) global sections ${\boldsymbol{\Gamma}}({\mathbf{Gr}}_{G},\varTheta_{g})$ and ${\boldsymbol{\Gamma}}({\mathbf{Gr}}_{G},\varTheta_{s})$ are irreducible $G_{\mathcal{O}}$-integrable ${\mathfrak{g}}_{\operatorname{aff}}$-modules of central charge $-1/2$, namely $L^{0}_{-1/2}$ and $L^{\omega_{1}}_{-1/2}$ [KT, Theorem 4.8.1]. Here ${\mathfrak{g}}={\mathfrak{sp}}({\mathbf{M}})$, and the highest component of $L^{0}_{-1/2}$ (resp. $L^{\omega_{1}}_{-1/2}$) with respect to ${\mathfrak{g}}_{\mathcal{O}}$ is the trivial (resp. defining) representation of ${\mathfrak{g}}$.⁸⁸8For a finite dimensional counterpart of this statement (about global sections of irreducible equivariant $D$-modules on the Lagrangian Grassmannian of ${\mathfrak{g}}$), see §5.3.

The (derived) global sections functors

($G_{\mathcal{O}}$-integrable ${\mathfrak{g}}_{\operatorname{aff}}$-modules with central charge $-1/2$) admit the left adjoints (see §§A.7,A.4)

According to [KT, Theorem 4.8.1(iv)], we have $\tau_{\geq 0}{\bf{Loc}}(L^{0}_{-1/2}\oplus L^{\omega_{1}}_{-1/2})=\varTheta$ (the top cohomology in the natural $t$-structure).

The symplectic form on ${\mathbf{M}}$ extends to the same named ${\mathbb{C}}$-valued symplectic form on ${\mathbf{M}}_{\mathcal{K}}\colon\langle f,g\rangle=\operatorname{Res}\langle f,g\rangle_{\mathcal{K}}dt$. We denote by ${\mathcal{W}}$ the completion of the Weyl algebra of $({\mathbf{M}}_{\mathcal{K}},\langle\,,\rangle)$ with respect to the left ideals generated by the compact subspaces of ${\mathbf{M}}_{\mathcal{K}}$. It has an irreducible representation ${\mathbb{C}}[{\mathbf{M}}_{\mathcal{O}}]$. Also, there is a homomorphism of Lie algebras ${\mathfrak{g}}_{\operatorname{aff}}\to\operatorname{Lie}{\mathcal{W}}$, see e.g. [FF]. According to [FF, rows 3,4 of Table XII at page 168], the restriction of ${\mathbb{C}}[{\mathbf{M}}_{\mathcal{O}}]$ to ${\mathfrak{g}}_{\operatorname{aff}}$ is $L^{0}_{-1/2}\oplus L^{\omega_{1}}_{-1/2}$ (even and odd functions, respectively).⁹⁹9For a finite dimensional counterpart of this statement (about restriction to ${\mathfrak{g}}$ of an irreducible module over the Weyl algebra of ${\mathbf{M}}$), see §5.3.

We consider the dg-category ${\mathcal{W}}{\operatorname{-mod}}$ of discrete ${\mathcal{W}}$-modules. More concretely, we identify ${\mathcal{W}}$ with the ring of differential operators on a Lagrangian discrete lattice ${\mathbf{L}}\subset{\mathbf{M}}_{\mathcal{K}}$, e.g. ${\mathbf{L}}=t^{-1}{\mathbf{M}}_{{\mathbb{C}}[t^{-1}]}$. Then ${\mathcal{W}}{\operatorname{-mod}}$ is the inverse limit of ${\rm D}{\operatorname{-mod}}(V)$ over finite dimensional subspaces $V\subset{\mathbf{L}}$ with respect to the functors $i_{V\hookrightarrow V^{\prime}}^{!}$. Equivalently, ${\mathcal{W}}{\operatorname{-mod}}$ is the colimit of ${\rm D}{\operatorname{-mod}}(V)$ with respect to the functors $i_{V\hookrightarrow V^{\prime},*}$.

There is a twisted action $\operatorname{D-mod}_{-1/2}(G_{\mathcal{K}})\circlearrowright{\mathcal{W}}{\operatorname{-mod}}$ that gives rise to an action ${\operatorname{D-mod}_{-1/2}^{G_{{\mathcal{O}}}}}({\operatorname{Gr}}_{G})\circlearrowright({\mathcal{W}}{\operatorname{-mod}})^{G_{\mathcal{O}}}$, see [R, §10].

One of the main results of [DLYZ] is a construction of a monoidal equivalence $\Phi\colon D^{b}_{G_{\mathcal{O}}}({\widetilde{\operatorname{Gr}}}_{G})\mathbin{\vphantom{j^{X^{2}}}\smash{\overset{\sim}{\vphantom{\rule{0.0pt}{1.99997pt}}\smash{\longrightarrow}}}}D^{G}_{\mathop{\operatorname{\rm perf}}}(\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{g}}[-2]))$ (dg-category of perfect complexes of dg-modules over the dg-algebra $\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{g}}[-2])$ equipped with a trivial differential). It extends to a monoidal equivalence of Ind-completions $\Phi\colon D_{G_{\mathcal{O}}}({\widetilde{\operatorname{Gr}}}_{G})\mathbin{\vphantom{j^{X^{2}}}\smash{\overset{\sim}{\vphantom{\rule{0.0pt}{1.99997pt}}\smash{\longrightarrow}}}}D^{G}(\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{g}}[-2]))$.

Here is one of the key properties of the twisted derived Satake equivalence $\Phi$. We choose a pair of opposite maximal unipotent subgroups $U_{G},U_{G}^{-}\subset G$, their regular characters $\psi,\psi^{-}$, and denote by $\varkappa\colon D^{G}(\mathop{\operatorname{\rm Sym}}^{\bullet}({\mathfrak{g}}[-2]))\to D({\mathbb{C}}[\Xi_{\mathfrak{g}}])$ the functor of Kostant-Whittaker reduction with respect to $(U_{G}^{-},\psi^{-})$ (see e.g. [BeF, §2]). Here $\Xi_{\mathfrak{g}}$ with grading disregarded is the tangent bundle $T\Sigma_{\mathfrak{g}}$ of the Kostant slice $\Sigma_{\mathfrak{g}}\subset{\mathfrak{g}}^{*}$. Let us write $\kappa$ for the Ad-invariant bilinear form on ${\mathfrak{g}}$, i.e., level, corresponding to our central charge of $-1/2$. Explicitly, if we write $\kappa_{b}$ for the basic level giving the short coroots of ${\mathfrak{g}}$ squared length two, and $\kappa_{c}$ for the critical level, then $\kappa$ is defined by

If we consider the Langlands dual Lie algebra ${\mathfrak{g}}^{\!\scriptscriptstyle\vee}\simeq{\mathfrak{so}}_{2n+1}$, the form $\kappa$ gives rise to identifications $\Sigma_{\mathfrak{g}}\cong\Sigma_{{\mathfrak{g}}^{\!\scriptscriptstyle\vee}}$ and $\Xi_{\mathfrak{g}}\cong\Xi_{{\mathfrak{g}}^{\!\scriptscriptstyle\vee}}$. Also, we have a canonical isomorphism $H^{\bullet}_{G_{\mathcal{O}}}({\operatorname{Gr}}_{G})\cong{\mathbb{C}}[\Xi_{{\mathfrak{g}}^{\!\scriptscriptstyle\vee}}]\cong{\mathbb{C}}[\Xi_{\mathfrak{g}}]$. This is a theorem of V. Ginzburg [G] (for a published account see e.g. [BeF, Theorem 1]).

Now given ${\mathcal{F}}\in D^{b}_{G_{\mathcal{O}}}({\widetilde{\operatorname{Gr}}})$ we consider the tensor product ${\mathcal{F}}\overset{!}{\otimes}\operatorname{RT}(\Theta)$ (notation of §2.1). Since the monodromies of the factors cancel out, it canonically descends to an object of $D_{G_{\mathcal{O}}}({\operatorname{Gr}}_{G}).$ The aforementioned key property is a canonical isomorphism

of $H^{\bullet}_{G_{\mathcal{O}}}({\operatorname{Gr}}_{G})\cong{\mathbb{C}}[\Xi_{\mathfrak{g}}]$-modules.

To introduce the universal ring object and show its relation to the $\varTheta$-sheaf, we recall the following general construction of internal Hom objects.

Let $\mathcal{C}$ be a module category over $\operatorname{D-mod}_{-1/2}(G_{\mathcal{K}})$. Given a subgroup $H$ of $G_{\mathcal{K}}$ and an $H$-equivariant object $\xi$ of $\mathcal{C}$, convolution with it yields a $\operatorname{D-mod}_{-1/2}(G_{\mathcal{K}})$-equivariant functor $(\operatorname{D-mod}_{-1/2}(G_{\mathcal{K}})_{*})_{H}\rightarrow\mathcal{C}$, and upon restriction to spherical vectors a $\operatorname{D-mod}_{-1/2}({\operatorname{Gr}}_{G})^{G_{\mathcal{O}}}$-equivariant functor $\operatorname{D-mod}_{-1/2}(G_{\mathcal{O}}\backslash G_{\mathcal{K}})_{H}\rightarrow(\mathcal{C})^{G_{\mathcal{O}}}.$ If both $\mathcal{C}$ and $(\operatorname{D-mod}_{-1/2}(G_{\mathcal{O}}\backslash G_{\mathcal{K}})_{*})_{H}$ are dualizable as abstract dg-categories, we obtain the dual $\operatorname{D-mod}_{-1/2}({\operatorname{Gr}}_{G})^{G_{\mathcal{O}}}$-equivariant functor

We apply this as follows. First, taking $\mathcal{C}={\mathcal{W}}\operatorname{-mod}$, $H=G_{{\mathcal{O}}}$, and $\xi=\mathbb{C}[{\mathbf{M}}_{{\mathcal{O}}}]$, we obtain a functor

Setting $M=\mathbb{C}[{\mathbf{M}}_{\mathcal{O}}]$, we obtain the internal Hom ring object

Second, taking $\mathcal{C}={\mathcal{W}}\operatorname{-mod}$, $H=G_{\mathbb{C}[t^{-1}]}$, and $\xi={\boldsymbol{\omega}}_{t^{-1}{\mathbf{M}}_{\mathbb{C}[t^{-1}]}}$, i.e., the colimit of the dualizing sheaves ${\boldsymbol{\omega}}_{V}$ over finite dimensional subspaces $V\subset t^{-1}{\mathbf{M}}_{\mathbb{C}[t^{-1}]}$, we obtain a functor

Recall the Radon transform (2.1.1). We keep the same notation for its $D$-module version ${\mathop{\rm RT}}\colon\operatorname{D-mod}_{-1/2}({\mathbf{Gr}}_{G})_{!}^{G_{\mathcal{O}}}\to\operatorname{D-mod}_{-1/2}({\operatorname{Gr}}_{G})^{G_{\mathcal{O}}}$. See the Appendix starting from §A.5, where it is denoted ${\mathop{\rm RT}}_{!}^{-1}$.