arXiv subjects

Tsao-Hsien Chen

Publications and source records attributed to Tsao-Hsien Chen.

At least 19 recordsLinked to original sources

arXiv2026

Singularities of orbit closures in loop spaces of symmetric varieties II: twisted cases

We study the singularities of orbit closures in loop symmetric varieties, extending our previous work https://arxiv.org/abs/2310.20006 to the twisted setting. We prove that the IC complexes of orbit closures are pointwise pure and satisfy a parity-vanishing property. We apply these geometric results to the study of twisted affine Lusztig--Vogan modules, establishing foundational results, including positivity properties of the twisted affine Kazhdan--Lusztig--Vogan polynomials. Along the way, we construct conical symplectic transversal slices in loop symmetric spaces. We provide applications to Langlands duality for real groups.

math.RT↗

arXiv2026

Loop spaces, twistor $\mathbb P^1$ and tempiric parameters

We establish bijections among spherical orbits on loop symmetric spaces, principal bundles on the twistor $\mathbb P^1$, and tempiric Langlands parameters. Using Vogan's theory of minimal $K$-types, we further establish bijections among irreducible equivariant local systems on loop symmetric spaces, irreducible local systems on principal bundles on the twistor $\mathbb P^1$, and irreducible representations of all strong inner forms of the dual symmetric subgroup. The latter bijections may be viewed as a combinatorial shadow of geometric Langlands duality for real reductive groups.

math.RT↗

arXiv2026

A relative Langlands dual realization of $T^*(G/K)$ and derived Satake

We show that the cotangent bundle $T^*(G/K)$ of a quasi-split symmetric space $G/K$ is isomorphic to the dual variety of the loop symmetric space for the Langlands dual group, providing instances of the relative Langlands duality for non-split groups. Then we establish a Langlands dual description of equivariant coherent sheaves on $T^*(G/K)$ in terms of constructible sheaves on the loop symmetric spaces, generalizing the derived Satake equivalence for reductive groups to quasi-split symmetric spaces. To this end, we prove the derived Satake equivalence for the twisted affine Grassmannians, study ring objects arising from loop symmetric spaces, and explore the formality and fully-faithfulness properties of $!$-pure objects. We deduce a version of Bezrukavnikov equivalence for quasi-split symmetric spaces and make connections to the geometric Langlands on the twistor $\mathbb P^1$.

math.RT↗

arXiv2026

Admissibility of Bernstein centers

We provide a criterion for determining when elements of the Bernstein center of a totally disconnected locally compact group are admissible invariant distributions in the sense of Harish-Chandra \cite{HC99}. As a consequence, we deduce the local integrability results for elements of bounded depth or Bernstein supports in the Bernstein centers of reductive $p$-adic groups. Our methods apply uniformly to both complex and mod-$\ell$ coefficients, generalizing results of Moy and Tadić \cite{MT02} in the complex case.

math.RT↗

arXiv2026

Geometric Langlands for Irregular Theta Connections and Epipelagic Representations

From a stable vector of a stable grading on a simple Lie algebra, Yun defined a rigid automorphic datum that encodes a epipelagic representation, and also an irregular connection on the projective line called $θ$-connection. We show that under geometric Langlands correspondence, $θ$-connection corresponds to the Hecke eigensheaf attached the rigid automorphic datum, assuming the stable grading is inner and its Kac coordinate $s_0$ is positive. We provide applications of the main result on cohomological rigidity of $θ$-connections, global oper structures, and a de Rham analog of Reeder-Yu's predictions on epipelagic Langlands parameters.

math.RT↗

arXiv2026

Matsuki duality for loop groups

We establish versions of Matsuki duality for loop groups. The main result is a bijection between symmetric loop group orbits and real polynomial loop group orbits on the affine Grassmannians or affine flag varieties. Along the way we obtain orbit parametrizations and make connections with vector bundles on real and twistor-$\mathbb P^1$ and Kottwitz sets .

math.RT↗

arXiv2025

A description of the integral depth-$r$ Bernstein center

In this paper we give a description of the depth-$r$ Bernstein center for non-negative integers $r$ of a reductive simply connected group $G$ over a non-archimedean local field as a limit of depth-$r$ standard parahoric Hecke algebras. Using the description, we construct maps from the algebra of stable functions on the $r$-th Moy-Prasad filtration quotient of hyperspecial parahorics to the depth-$r$ Bernstein center and use them to attach to each depth-$r$ irreducible representation $π$ an invariant $θ(π)$, called the depth-$r$ Deligne-Lusztig parameter of $π$. We show that $θ(π)$ is equal to the semi-simple part of minimal $K$-types of $π$.

math.RT↗

arXiv2025

A description of the depth-$r$ Bernstein center for rational depths

Let $G$ be a split connected reductive over a non-archimedean local field $k$. In this paper we give a description of the depth-$r$ Bernstein center of $G(k)$ for rational depths as a limit of depth-$r$ standard parahoric Hecke algebras, extending our previous work in the integral depths case (arXiv:2407.15128). Using this description, we construct maps from the space of stable functions on depth-$r$ Moy-Prasad quotients to the depth-$r$ center, and attach depth-$r$ Deligne-Lusztig parameters to smooth irreducible representations, with the assignment of parameters to irreducible representations shown to be consistent with restricted Langlands parameters for Moy-Prasad types described Chen-Debacker-Tsai (arXiv:2509.07780). As an application, we give a decomposition of the category of smooth representations into a product of full subcategories indexed by restricted depth-$r$ Langlands parameters.

math.RT↗

arXiv2025

Langlands parameters for Moy-Prasad types

Suppose $G$ is a tamely ramified $p$-adic reductive group. We construct a partial local Langlands correspondence between the set of irreducible smooth representations of $G$ having depth $r$ and a certain set of $G^\vee$-conjugacy classes of continuous homomorphisms $φ:I_F^r\rightarrow G^{\vee}$. Here $G^\vee$ is the dual group of $G$, and $I_F^r$ is the $r^{\text{th}}$ upper-numbering filtration subgroup of the inertia subgroup $I_F$.

math.RT↗

arXiv2025

Singularities of orbit closures in loop spaces of symmetric varieties

We study the singularities of closures of Iwahori orbits on loop spaces of symmetric varieties extending the celebrated work of Lusztig-Vogan to the affine setting. We show that the IC-complexes of orbit closures (with possible non-trivial coefficients) are pointwise pure and satisfy a parity vanishing property. We apply those geometric results to study the affine Lusztig-Vogan modules and obtain fundational results about them including the positivity properties of the affine Kazhdan-Lusztig-Vogan polynomials. Along the way, we construct conical transversal slices inside loop spaces of symmetric varieties generalizing the work of Mars-Springer in the finite dimensional setting. Our results answer a question of Lusztig. We deduce results for singularities of spherical orbit closures and provide applications to relative Langlands duality including the positivity for the relative Kostka-Foulkes polynomials and the formality conjecture.

math.RT↗

arXiv2024

Functorial transfer for reductive groups and central complexes

A class of Weyl group equivariant $\ell$-adic complexes on a torus, called the central complexes, was introduced and studied in our previous work on Braverman-Kazhdan conjecture. In this note we show that the category of central complexes admits functorial monoidal transfers with respect to morphisms between the dual groups. Combining with the work of Bezrukavnikov-Deshpande, we show that the $\ell$-adic bi-Whittaker categories (resp. the category of vanishing complexes, the category of stable complexes) on reductive groups admit functorial transfers. As an application, we give a new geometric proof of Laumon-Letellier's fromula for transfer maps of stable functions on finite reductive groups.

math.RT↗

arXiv2024

Lorentzian and Octonionic Satake equivalence

We establish a derived geometric Satake equivalence for the real group $G_{\mathbb R}=PSO(2n-1,1)$ (resp. $PE_6(F_4)$), to be called the Lorentzian Satake equivalence (resp. Octonionic Satake equivalence). By applying the real-symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the splitting rank symmetric variety $X=PSO_{2n}/SO_{2n-1}$ (resp. $PE_6/F_4$). As an application, we compute the stalks of the $\text{IC}$-complexes for spherical orbit closures in the real affine Grassmannian for $G_{\mathbb R}$ and the loop space of $X$. We show the stalks are given by the Kostka-Foulkes polynomials for $GL_2$ (resp. $GL_3$) but with $q$ replaced by $q^{n-1}$ (resp. $q^4$).

math.RT↗

arXiv2024

Real groups, symmetric varieties and Langlands duality

Let $G_\mathbb R$ be a connected real reductive group and let $X$ be the corresponding complex symmetric variety under the Cartan bijection. We construct a canonical equivalence between the relative Satake category of $G(\mathcal O)$-equivariant $\mathbb C$-constructible complexes on the loop space of $X$ and the real Satake category of $G_\mathbb R(\mathcal O_\mathbb R)$-equivariant $\mathbb C$-constructible complexes on the real affine Grassmannian. We show that the equivalence is $t$-exact with respect to the natural perverse $t$-structures and is compatible with the fusion products and Hecke actions. We further show that the relative Satake category is equivalent to the category of $\mathbb C$-constructible complexes on the moduli stack of $G_\mathbb R$-bundles on the real projective line $\mathbb P^1(\mathbb R)$ and hence provides a connection between the relative Langlands program and the geometric Langlands program for real groups. We provide numerous applications of the main theorems to real and relative Langlands duality including the formality and commutativity conjectures for the real and relative Satake categories and an identification of the dual groups for $G_\mathbb R$ and $X$.

math.RT↗

arXiv2023

Affine Matsuki correspondence for sheaves

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.

math.RT↗

arXiv2023

Quantization of Hitchin integrable system via positive characteristic

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.

math.RT↗

arXiv2023

Towards the depth zero stable Bernstein center conjecture

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.

math.RT↗

arXiv2023

Real and symmetric quasi-maps

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.

math.RT↗

arXiv2022

Quaternionic Satake equivalence

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.

math.RT↗