Abstract

In this talk, I will consider two nonabelian Fourier transforms related to elliptic unipotent representations of semisimple p-adic groups. The elliptic representation theory concerns the study of characters modulo the proper parabolically induced ones. The unipotent category of representations was defined by Lusztig and it can be thought of as being the smallest subcategory of smooth representations that is closed under the formation of L-packets and such that it contains the Iwahori representations. The first Fourier transform is defined on the p-adic group side in terms of the pseudocoefficients of these representations and Lusztig’s nonabelian Fourier transform for characters of finite groups of Lie type. The second one is defined ``on the dual side’’ in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group. I will present a conjectural relation between them, and exemplify this conjecture in some cases that are known, the most notable case being that of split special orthogonal groups, by the work of Moeglin and Waldspurger. I will also try to explain the relevance of this picture to the verification of the properties of unipotent L-packets and to a geometric interpretation of formal degrees of square integrable representations. The talk is based on joint work with Eric Opdam.