We provide a motivated introduction to the theory of categorical actions groups and local geometric Langlands program. Along way, we emphasize applications, old new, usual representation reductive affine Lie algebras.

AbstractAbstractWe prove a number of structural and representation-theoretic results on linearly reductive quantum groups, i.e., objects dual to those cosemisimple Hopf algebras: (a) closed normal subgroup is automatically if its squared antipode leaves invariant each simple subcoalgebra the underlying algebra; (b) for embedding H⊴G there Clifford-style correspondence between two equivalence re...

we regard the shearlet group as a semidirect product group andshow that its standard representation is,typically, a quasiregu-lar representation. as a result we can characterize irreducibleas well as square-integrable subrepresentations of the shearletgroup.

Let F be a finite extension of ℚ p and let G connected reductive group over F. We assume that is large relatively to G. ′ an endoscopic Following Arthur, we have, roughly speaking, spectral transfer morphism, denoted by transfert, which, stable linear combination irreducible admissible representations (F), associates G(F). 0,G the Bernstein’s projector such that, for representation π G(F), have...

For a connected reductive algebraic group G defined over finite field F q , Kawanaka introduced the generalized Gelfand-Graev representations (GGGRs for short) of ( ) in case where is power good prime . This representation has been widely studied and used various contexts. Recently, Geck proposed conjecture, characterizing Lusztig's special unipotent classes terms weighted Dynkin diagrams. Base...

We prove quantitative versions of Borel and Harish-Chandra's theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds the lengths reduced integral vectors in any rational representation a reductive group. Secondly, construction fundamental sets subgroups groups, as latter vary real conjugacy class fixed Our results allow us to apply Pila--Zannier strategy Zilber-...

Let G be a reductive algebraic group over the local field k. The local Langlands conjecture predicts that the irreducible complex representations π of the locally compact group G(k) can be parametrized by objects of an arithmetic nature: homomorphisms φ from the Weil-Deligne group of k to the complex L-group of G, together with an irreducible representation ρ of the component group of the centr...

The purpose of this section is to describe a construction, due to Totaro [5], of an algebro-geometric substitute for the classifying space of topological group. For topological group G, the spaces EG and BG are infinite dimensional, it is hard to control in general. What Totaro produces for a reductive algebraic group is a directed system of G-bundles En → Bn with the following property: for an...

The goal of this article is to explain Howe’s correspondence to a reader who is not necessarily an expert on Representation Theory of Real Reductive Groups, but is familiar with general concepts of Harmonic Analysis. We recall Howe’s construction of the Oscillator Representation, the notion of a dual pair and a few basic and general facts concerning the correspondence.