Reductive Dual Pair Correspondence for Complex Groups

Shtukas for Reductive Groups and Langlands Correspondence for Functions Fields

This text gives an introduction to the Langlands correspondence for function fields and in particular to some recent works in this subject. We begin with a short historical account (all notions used below are recalled in the text). The Langlands correspondence [Lan70] is a conjecture of utmost importance, concerning global fields, i.e. number fields and function fields. Many excellent surveys a...

Gerbes on Complex Reductive Lie Groups

Let K be a compact Lie group with complexification G. We construct by geometric methods a conjugation invariant gerbe on G; this then gives by restriction an invariant gerbe on K. Our construction works for any choice of level. When K is simple and simply-connected, the level is just an integer as usual. For general K, the level is a bilinear form b on a Cartan subalgebra where b satisfies a qu...

Howe ' s Rank and Dual Pair Correspondence in Semistable Range

ABSRACT Let G be a classical group of type I. For an irreducible unitary representation, Howe defined the notion of rank in analytic terms. On the algebraic side, there is the theory of primitive ideals and associated variety. In the first part of this thesis, we relate Howe's rank with the associated variety. In the second part, We study the Bargmann-Segal model of the oscillator representatio...

Reductive Groups

Semisimple orbital integrals on the symplectic space for a real reductive dual pair

We prove a Weyl Harish-Chandra integration formula for the action of a reductive dual pair on the corresponding symplectic space W. As an intermediate step, we introduce a notion of a Cartan subspace and a notion of an almost semisimple element in the symplectic space W. We prove that the almost semisimple elements are dense in W. Finally, we provide estimates for the orbital integrals associat...