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Let $G$ be a split connected reductive group over a non-archimedan local field $F$. The depth zero stable Bernstein conjecture asserts that there is an algebra isomorphism between the depth zero stable Bernstein center of $G(F)$ and the ring of functions on the moduli of tame Langlands parameters. An approach to the depth zero stable Bernstein conjecture was proposed in the work of Bezrukavnikov-Kazhdan-Varshavsky \cite{BKV}. In this paper we generalize results and techniques in \cite{BKV} and apply them to give a geometric construction of elements in the depth zero Bernstein center. We conjecture that our construction produces all elements in the depth zero Bernstein center. An an illustration of the method, we give a construction of an algebra embedding from the (limit of) stable Bernstein centers for finite reductive groups to the depth zero Bernstein center and a family of elements in the depth zero Bernstein center coming from Deligne's epsilon factors. The paper is the first step toward the depth zero stable Bernstein center.

We construct a family of involutions on the space $\mathfrak{gl}_n'(\mathbb C)$ of $n\times n$ matrices with real eigenvalues interpolating the complex conjugation and the transpose. We deduce from it a stratified homeomorphism between the space of $n\times n$ real matrices with real eigenvalues and the space of $n\times n$ symmetric matrices with real eigenvalues, which restricts to a real analytic isomorphism between individual $\mathrm{GL}_n(\mathbb R)$-adjoint orbits and $\mathrm{O}_n(\mathbb C)$-adjoint orbits. We also establish similar results in more general settings of Lie algebras of classical types and quiver varieties. To this end, we prove a general result about involutions on hyper-K\"ahler quotients of linear spaces. We provide applications to the (generalized) Kostant-Sekiguchi correspondence, singularities of real and symmetric adjoint orbit closures, and Springer theory for real groups and symmetric spaces., Comment: 37 pages, minor expository changes

We establish a derived geometric Satake equivalence for the quaternionic general linear group GL_n(H). By applying the real-symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the symmetric variety GL_2n/Sp_2n. We explain how these equivalences fit into the general framework of a geometric Langlands correspondence for real groups and the relative Langlands duality conjecture. 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 GL_2n/Sp_2n. We show the stalks are given by the Kostka-Foulkes polynomials for GL_n but with all degrees doubled., Comment: 50 pages

Let G be a connected complex reductive group and let K be a symmetric subgroup of G. We prove a formula for the Drinfeld-Gaitsgory functor for the dg-category of K-equivariant sheaves on the flag manifold of G in terms of the Matsuki duality functor. As byproducts, we obtain a description of the Serre functor and the Deligne-Lusztig duality for (g,K)-modules., 11 pages

We prove the Chevalley restriction theorem for the commuting scheme of symplectic Lie algebras. The key step is the construction of the inverse map of the Chevalley restriction map called the spectral data map. Along the way, we establish a certain multiplicative property of the Pfaffian which is of independent interest.

In this article we prove a conjecture of Braverman and Kazhdan in \cite{BK1} on acyclicity of $\rho$-Bessel sheaves on reductive groups in both $\ell$-adic and de Rham settings. We do so by establishing a vanishing conjecture proposed in \cite{C1}. As a corollary, we obtain a geometric construction of the non-linear Fourier kernels for finite reductive groups as conjectured by Braverman and Kazhdan. The proof of the vanishing conjecture relies on the techniques developed in \cite{BFO} on Drinfeld center of Harish-Chandra bimodules and character D-modules, and a construction of a class of character sheaves in mixed-characteristic., Comment: 36 pages. New introduction

In this article we propose a vanishing conjecture for a certain class of $\ell$-adic complexes on a reductive group $G$, which can be regraded as a generalization of the acyclicity of the Artin-Schreier sheaf. We show that the vanishing conjecture contains, as a special case, a conjecture of Braverman and Kazhdan on the acyclicity of $\rho$-Bessel sheaves \cite{BK1}. Along the way, we introduce a certain class of Weyl group equivariant $\ell$-adic complexes on a maximal torus called \emph{central complexes} and relate the category of central complexes to the Whittaker category on $G$. We prove the vanishing conjecture in the case when $G=\GL_n$., Comment: 22 pages. Minor corrections

We lift the affine Matsuki correspondence between real and symmetric loop group orbits in affine Grassmannians to an equivalence of derived categories of sheaves. In analogy with the finite-dimensional setting, our arguments depend upon the Morse theory of energy functions obtained from symmetrizations of coadjoint orbits. The additional fusion structures of the affine setting lead to further equivalences with Schubert constructible derived categories of sheaves on real affine Grassmannians.

Let $G_\mathbb R$ be a real reductive group and let $X$ be the corresponding complex symmetric variety under the Cartan bijection. We construct a stratified homeomorphism between the based polynomial arc group of $G_\mathbb R$ and the based polynomial arc space of $X$. We also prove a multi-point version where we replace arcs by moduli spaces of quasi-maps from the projective line $\mathbb P^1$ to $G_\mathbb R$ and $X$. The key ingredients in the proof include: (i) a multi-point generalization of the ``Gram-Schmidt" factorization of loop groups, and (ii) a nodal degeneration of moduli spaces of quasi-maps. As an application, we show that for the closures of real spherical orbits in the real affine Grassmannian, their singularities near the base point are locally homeomorphic to complex algebraic varieties., Comment: complete revision (including title) with some material moved to arXiv:2006.10279; 22 pages, 1 figure

In this paper, we explore the structure of the Hitchin map for higher dimensional varieties with emphasis on the case of algebraic surfaces., 25 pages, revised version

In this article we prove a conjecture of Braverman-Kazhdan in \cite{BK} on the acyclicity of gamma sheaves in the de Rham setting. The proof relies on the techniques developed in \cite{BFO} on Drinfeld center of Harish-Chandra bimodules and character D-modules. As an application, we show that the functors of convolution with gamma D-modules, which can be viewed as a version of non-linear Fourier transforms, commute with induction functors and are exact on the category of admissible D-modules on a reductive group., Comment: 25 pages

In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable system by showing that the global sections of critically twisted differential operators on the moduli stack of G-bundles on an algebraic curve is identified with the ring of regular functions on the space of G-opers; they deduce existence of an automorphic D-module corresponding to a local system carrying a structure of an oper. In this note we show for G=GL(n) that those results admit a short proof by reduction to positive characteristic, where they are deduced from generic Langlands duality established earlier by the first author and A. Braverman. The appendix contains a proof of some properties of the p-curvature map restricted to the space of opers., Comment: paper by Roman Bezrukavnikov and Roman Travkin with an appendix by Roman Bezrukavnikov, Tsao-Hsien Chen and Xinwen Zhu. The new version contains minor updates to some proofs and to the introduction and a new remark on potential relation to a conjecture of Etingof, Frenkel and Kazhdan

Let $G$ be a simple complex group of adjoint type. In his unpublished work, Z. Yun associated to each $��$-group $(G_0, \mathfrak g_1)$ and a vector $X\in\mathfrak g_1$ a flat $G$-connection $\nabla ^X$ on $\mathbb P^1-\{0,\infty\}$, generalizing the construction of Frenkel and Gross in [FG]. In this paper we study the local monodromy of those flat $G$-connections and compute the de Rham cohomology of $\nabla^X$ with values in the adjoint representations of $G$. In particular, we show that in many cases the connection $\nabla^X$ is cohomologically rigid., Rewrite introduction and typos corrected. To appear in IMRN

For each integral dominant weight $\lambda$, we construct a twisted global section functor $\Gamma^{\lambda}$ from the category of critical twisted $D$-modules on affine Grassmannian to the category of $\lambda$-regular modules of affine Lie algebra at critical level. We proved that $\Gamma^{\lambda}$ is exact and faithful. This generalized the work of Frenkel and Gaitsgory in the case when $\lambda=0$., Comment: 19 pages, preliminary version