arXiv subjects
Publications and source records attributed to Tsao-Hsien Chen.
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.
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 \cite{CVX3}, we have established a Springer theory for the symmetric pair $(\operatorname{SL}(N),\operatorname{SO}(N))$. In this setting we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These representations arise from cohomology of families of certain (Hessenberg) varieties. In this paper we determine the Springer correspondence explicitly for IC sheaves supported on order 2 nilpotent orbits. In this process we encounter universal families of hyperelliptic curves. As an application we calculate the cohomolgy of Fano varieties of $k$-planes in the smooth intersection of two quadrics in an even dimensional projective space.
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-Kähler 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.
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 $ρ$-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 $G=\GL_n$.
In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [CVX2].
In this paper we compute the cohomology of the Fano varieties of $k$-planes in the smooth complete intersection of two quadrics in $\mathbb{P}^{2g+1}$, using Springer theory for symmetric spaces.
In this paper we establish Springer correspondence for the symmetric pair $(\mathrm{SL}(N),\mathrm{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaves construction due to Grinberg. As applications, we obtain results on cohomology of Hessenberg varieties and geometric constructions of irreducible representations of Hecke algebras of symmetric groups at $q=-1$.
In this article we prove a conjecture of Braverman and Kazhdan in \cite{BK1} on acyclicity of $ρ$-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.
In this paper, we explore the structure of the Hitchin morphism for higher dimensional varieties. We show that the Hitchin morphism factors through a closed subscheme of the Hitchin base, which is in general a non-linear subspace of lower dimension. We conjecture that the resulting morphism, which we call the spectral data morphism, is surjective. In the course of the proof, we establish connections between the Hitchin morphisms for higher dimensional varieties, the invariant theory of the commuting schemes, and Weyl's polarization theorem. We use the factorization of the Hitchin morphism to construct the spectral and cameral covers. In the case of general linear groups and algebraic surfaces, we show that spectral surfaces admit canonical finite Cohen-Macaulayfications, which we call the Cohen-Macaulay spectral surfaces, and we use them to obtain a description of the generic fibers of the Hitchin morphism similar to the case of curves. Finally, we study the Hitchin morphism for some class of algebraic surfaces.
We study the Casselman-Jacquet functor $J$, viewed as a functor from the (derived) category of $(\mathfrak{g},K)$-modules to the (derived) category of $(\mathfrak{g},N^-)$-modules, $N^-$ is the negative maximal unipotent. We give a functorial definition of $J$ as a certain right adjoint functor, and identify it as a composition of two averaging functors $\text{Av}^{N^-}_!\circ \text{Av}^N_*$. We show that it is also isomorphic to the composition $\text{Av}^{N^-}_*\circ \text{Av}^N_!$. Our key tool is the pseudo-identity functor that acts on the (derived) category of (twisted) $D$-modules on an algebraic stack.
In this paper, we explore the structure of the Hitchin map for higher dimensional varieties with emphasis on the case of algebraic surfaces.
Let $G$ be a simple complex group of adjoint type. In his unpublished work, Z. Yun associated to each $θ$-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.
Let (G,K) be a symmetric pair over the complex numbers, and let X=K\G be the corresponding symmetric space. In this paper we study a nearby cycles functor associated to a degeneration of X to MN\G, which we call the "wonderful degeneration". We show that on the category of character sheaves on X, this functor is isomorphic to a composition of two averaging functors (a parallel result, on the level of functions in the p-adic setting, was obtained in [BK, SV]). As an application, we obtain a formula for the geometric Jacquet functor of [ENV] and use this formula to give a geometric proof of the celebrated Casselman's submodule theorem and establish a second adjointness theorem for Harish-Chandra modules.
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.
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$, whose characteristic is positive and does not divide the order of the Weyl group of $G$, and let $\breve G$ be its Langlands dual group over $k$. Let $C$ be a smooth projective curve over $k$. Denote by $\Bun_G$ the moduli stack of $G$-bundles on $C$ and $ \Loc_{\breve G}$ the moduli stack of $\breve G$-local systems on $C$. Let $D_{\Bun_G}$ be the sheaf of crystalline differential operators on $\Bun_G$. In this paper we construct an equivalence between the bounded derived category $D^b(\on{QCoh}(\Loc_{\breve G}^0))$ of quasi-coherent sheaves on some open subset $\Loc_{\breve G}^0\subset\Loc_{\breve G}$ and bounded derived category $D^b(D_{\Bun_G}^0\on{-mod})$ of modules over some localization $D_{\Bun_G}^0$ of $D_{\Bun_G}$. This generalizes the work of Bezrukavnikov-Braverman in the $\GL_n$ case.
Let G be a reductive group over an algebraically closed field of positive characteristic. Let C be a smooth projective curve over k. We give a description of the moduli space of flat G-bundles in terms of the moduli space of G-Higgs bundles over the Frobenius twist C' of C. This description can be regarded as the non-abelian Hodge theory for curves in positive characteristic.
It is expected that, under mild conditions, local Langlands correspondence preserves depths of representations. In this article, we formulate a conjectural geometrisation of this expectation. We prove half of this conjecture by showing that the depth of a categorical representation of the loop group is less than or equal to the depth of its underlying geometric Langlands parameter. A key ingredient of our proof is a new definition of the slope of a meromorphic connection, a definition which uses opers. In the appendix, we consider a relationship between our conjecture and Zhu's conjecture on non-vanishing of the Hecke eigensheaves produced by Beilinson and Drinfeld's quantisation of Hitchin's fibration for non-constant groups.