This paper gives a Langlands classification of constituents of ramified principal series representations for split p-adic groups with connected center.

The Langlands classification theorem describes all admissible representations of a reductive group G in terms of the tempered representations of Levi subgroups of G. I will describe work with Susana Salamanca-Riba that provides (conjecturally) a similar description of the unitary representations of G in terms of certain very special unitary representations of Levi subgroups.

SupposeG to be a quasi-split reductive group defined over F . To it is associated a connected complex group G∨ whose root datum is dual to that ofG, and the structure ofG determines an algebraic action ofG(F/F ) on G∨. Define G to be the semidirect productG∨⋊WF . Langlands has conjectured a partition of a large part of the set of irreducible representations ofG(F ) into finite packets (calledL ...

This paper begins the project of adapting the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups, to padic groups, continuing in the direction charted by Vogan in his 1993 paper on the Langlands correspondence. This paper presents three theorems in that direction. The first theorem shows how Lusztig’s work on perverse sheaves aris...

We carry out (with technical modifications) some cases of a procedure proposed by R. Langlands in Beyond Endoscopy. This gives a new proof of the classification of “dihedral forms” on GL(2), avoiding endoscopic methods: instead, it uses an infinite limiting process in the trace formula. We also explain the relationship of Langlands’ idea to the results of [10] that count automorphic forms assoc...

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...

After explaining the concepts of Langlands dual and miniscule representations, we define an analog of the Gauss sum for any compact, simple Lie group with a simply laced Lie algebra. We then show a reciprocity property when a Lie group is exchanged with its Langlands dual. We also explore the relation with theta functions and modular transformations. In the non-simply laced case, we construct a...

In this paper we give an overview over the basic features of the continuous representation theory of p-adic Lie groups as it has emerged during the last five years. The main motivation for developing such a theory is a possible extension of the local Langlands program to p-adic Galois representations. This is still very much in its infancy. But in the last section we will describe a first appro...

‎In the spirit of the Langlands proposal on  Beyond Endoscopy ‎we discuss the explicit relation between the Langlands functorial transfers and automorphic $L$-functions‎. ‎It is well-known that the poles of the $L$-functions have deep impact to the Langlands functoriality‎. ‎Our discussion also‎ ‎includes the meaning of the central value of the tensor product $L$-functions in terms of the Langl...

We introduce a new approach to the representation theory of reductive p-adic groups G, based on the Geometric Invariant Theory (GIT) of Moy-Prasad quotients. Stable functionals on these quotients are used to give a new construction of supercuspidal representations of G having small positive depth, called epipelagic. With some restrictions on p, we classify the stable and semistable functionals ...