Related papers: Categorification of local relative Langlands duali…

We first compute the denominator formulas for quantum affine algebras of all exceptional types. Then we prove the isomorphisms among Grothendieck rings of categories $C_Q^{(t)}$ $(t=1,2,3)$, $\mathscr{C}_{\mathscr{Q}}^{(1)}$ and…

We introduce and motivate -- based on ongoing joint work with Germ\'an Stefanich -- the notion of potent categorical representations of a complex reductive group $G$, specifically a conjectural Langlands correspondence identifying potent…

We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of Donagi-Pantev, and inspired by the work of…

We prove a descent criterion for certain families of smooth representations of GL_n(F) (F a p-adic field) in terms of the gamma factors of pairs constructed in previous work of the second author. We then use this descent criterion, together…

Let $F$ be a non Archimedean local field, and $G$ be the $F$-points of a connected quasi-split reductive group defined over $F$. In this note we propose a converse theorem statement for generic Langlands parameters of $G$ when the Langlands…

Kato and Fukaya conjectured existence of certain elements in certain ($p$-adic) Iwasawa algebras which are related to Deligne-Langland's ($l$-adic) local epsilon constants. We prove the existence of these elements in the $l \neq p$ case.

In this article, we will prove that the formal degree conjecture is compatible with the Deligne-Kazhdan correspondence for quasi-split groups, assuming that the local Langlands correspondence is compatible with the Deligne-Kazhdan…

We revisit the problem of local normality of Kraus-Polley-Reents infravacuum representations and provide a straightforward proof based on the Araki-Yamagami criterion. We apply this result to the theory of superselection sectors. Namely, we…

We give a parametrization of the simple Bernstein components of inner forms of a general linear group over a local field by invariants constructed from type theory, and explicitly describe its behaviour under the Jacquet-Langlands…

We prove a conjecture of Frenkel-Gaitsgory-Kazhdan-Vilonen on some exponential sums related to the geometric Langlands correspondence. Our main ingredients are the resolution of Lusztig scheme of lattices introduced by Laumon and the…

Lusztig \cite{L5,L6} gave a parametrization for $\rm{Irr}(G^F)$, where $G$ is a reductive algebraic group defined over $\mathbb{F}_q$, with Frobenius map $F$. This parametrization is known as Lusztig's Jordan decomposition or Lusztig…

Let G be any reductive p-adic group. We discuss several conjectures, some of them new, that involve the representation theory and the geometry of G. At the heart of these conjectures are statements about the geometric structure of Bernstein…

We propose a generalization of the level-rank dualities arising from Uglov's work on higher-level Fock spaces. The statements use Hecke algebras defined by Brou\'{e}-Malle, which conjecturally describe the endomorphisms of Lusztig induction…

Recent work of Mao, Wan and Zhang \cite{MWZ} has provided a complete list of strongly tempered hyperspherical varieties and they proposed some new period integrals. In this paper, I will present new period integrals of distinguished…

We formulate a global conjecture for the automorphic period integral associated to the symmetric pairs defined by unitary groups over number fields, generalizing a theorem of Waldspurger's toric period for $\mathrm{GL}(2)$. We introduce a…

We establish an Esakia duality for the categories of temporal Heyting algebras and temporal Esakia spaces. This includes a proof of contravariant equivalence and a congruence/filter/closed-upset correspondence. We then study two notions of…

We prove Kudla-Rallis conjecture on first occurrences of local theta correspondence, for all type I irreducible dual pairs and all local fields of characteristic zero.

We prove a refinement of the global Gan-Gross-Prasad conjecture proposed by Ichino-Ikeda and N. Harris for unitary groups under some local conditions. We need to assume some expected properties of L-packets and some part of the local…

The analytic Langlands correspondence was developed by Etingof, Frenkel and Kazhdan in arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243, arXiv:2311.03743. For a curve $X$ and a group $G$ over a local field $F$, in the tamely ramified…

The stack of local Langlands parameters for a torus is a Picard stack. In this article, we explicitly determine its Picard dual and show that the Fourier-Mukai transform gives rise to the integral categorical local Langlands correspondence…