In the context of the local Langlands philosopy for R, Adams, Barbasch and Vogan describe a bijection between the simple Harish-Chandra modules for a real reductive group G(R) and the space of “complete geometric parameters”—a space of equivariant local systems on a variety on which the Langlands-dual of G(R) acts. By a conjecture of Soergel, this bijection can be enhanced to an equivalence of ...

This is the fourth in a series of papers (the earlier ones are [Casselman:1989], [Casselman:1993] and [Casselman:1999]) intended to present eventually a new way of proving, among other things, the well known results of Chapter 7 of [Langlands:1977] on the completeness of the spectrum arising from cusp forms and Eisenstein series. That Langlands’ results have lasted for nearly 40 years without m...

The local Langlands conjecture for GL(n) over a p-adic field, n < p.

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in seminal work of Riemann Euler–Gauss function and has blossomed into an active field connections to many areas mathematics. In this paper, we construct Hecke eigensheaves whose eigenvalues irreducible systems, thus confirming a central conjecture geo...

In this paper we study the theta correspondence for Unitary groups of the same size over local and global fields. This correspondence has been studied in many cases by several authors. We are able to unify and generalise all these known results in terms of two conjectures, one local and the other global. These conjectures are in terms of the parametrisation of irreducible admissible representat...

The Local Langlands Conjecture posits the existence of a bijection between certain classes of irreducible representations of a reductive group G, such as GLnpKq, and certain ndimensional representations of an additive extension of the Weil group WK (closely related to the Galois group). Here we focus solely on the case G “ GLn and K is a p-adic field (such as Qp). The Local Langlands conjecture...

We initiate the study of affine Deligne–Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. prove that $${{\,\mathrm{GL}\,}}_n$$ and its inner forms, Lusztig’s semi-infinite construction is isomorphic to an variety at infinite level. their homology give geometric realizations Langlands Jacquet–Langlands correspondences in setting Weil paramete...

We compute the universal deformations of cuspidal representations π of GL2(F ) over Fl, where F is a local field of residue characteristic p and l is an odd prime different from p. When π is supercuspidal there is an irreducible, two dimensional representation ρ of GF that corresponds to π via the mod l local Langlands correspondence of [Vi2]; we show that there is a natural isomorphism between...

The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of G2 over a p-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp6 orPGL3. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of G2 and other representations of PGSp6 and PGL3. This corresponden...

The general problem we discuss in this paper is how to prove stability properties for a linear combination of characters of irreductible discrete series of p-adic groups. Here we give ideas on how to reduce the case where the Langlands parameter is trivial on the wild ramification group to the case where this Langlands parameter factorizes through the Frobenius; we handle only the case of an od...