Related papers: Loop Spaces and Representations
We apply the technique of S^1-equivariant localization to sheaves on loop spaces in derived algebraic geometry, and obtain a fundamental link between two families of categories at the heart of geometric representation theory. Namely, we…
We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. Our focus is the Hecke category of Borel-equivariant D-modules on the flag variety of a complex reductive group…
In this article we study the representations of general linear groups which arise from their action on flag spaces. These representations can be decomposed into irreducibles by proving that the associated Hecke algebra is cellular. We give…
In this paper we provide, under some mild explicit assumptions, a geometric description of the category of representations of the centralizer of a regular unipotent element in a reductive algebraic group in terms of perverse sheaves on the…
We establish equivalences of derived categories of the following 3 categories: (1) Principal block of representations of the quantum at a root of 1; (2) G-equivariant coherent sheaves on the Springer resolution; (3) Perverse sheaves on the…
In this paper we use the quantization of fields based on Geometric Langlands Correspondence \cite{diep1} to realize the automorphic representations of some concrete series of groups: for the affine Heisenberg (loop) groups it is reduced to…
We examine the geometry of loop spaces in derived algebraic geometry and extend in several directions the well known connection between rotation of loops and the de Rham differential. Our main result, a categorification of the geometric…
With a view towards applications in the theory of infinite-dimensional representations of finite-dimensional Lie supergroups, we introduce a new category of supermanifolds. In this category, supermanifolds of `maps' and `fields' (fibre…
Kazhdan and Lusztig identified the affine Hecke algebra $\mathcal{H}$ with an equivariant $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of…
We (a) prove that continuous morphisms from locally compact groups to locally exponential (possibly infinite-dimensional) Lie groups factor through Lie quotients, recovering a result of Shtern's on factoring norm-continuous representations…
Actions of algebraic groups on DG categories provide a convenient, unifying framework in some parts of geometric representation theory, especially the representation theory of reductive Lie algebras. We extend this theory to loop groups and…
Let $G$ and $\tilde G$ be reductive groups over a local field $F$. Let $\eta : \tilde G \to G$ be a $F$-homomorphism with commutative kernel and commutative cokernel. We investigate the pullbacks of irreducible admissible…
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 graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and…
The Hecke category is at the heart of several fundamental questions in modular representation theory. We emphasise the role of the "philosophy of deformations" both as a conceptual and computational tool, and suggest possible connections to…
Let G be a connected reductive group over a non-archimedean local field. We say that an irreducible depth-zero (complex) G-representation is non-singular if its cuspidal support is non-singular. We establish a Local Langlands Correspondence…
We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$, the derived and the stable categories…
We consider four classes of classical groups over a non-archimedean local field F: symplectic, (special) orthogonal, general (s)pin and unitary. These groups need not be quasi-split over F. The main goal of the paper is to obtain a local…
We present a finite algorithm for computing the set of irreducible unitary representations of a real reductive group G. The Langlands classification, as formulated by Knapp and Zuckerman, exhibits any representation with an invariant…
In this paper we construct equivalences of monoidal categories relating three geometric or representation-theoretic categorical incarnations of the affine Hecke algebra of a connected reductive algebraic group $G$ over a field of positive…