A nonabelian Fourier transform for tempered unipotent representations of p-adic groups

Abstract: In the representation theory of finite reductive groups, an essential role is played by Lusztig's nonabelian Fourier transform, an involution on the space of unipotent characters the group. This involution is the change of bases matrix between the basis of irreducible characters and the basis of `almost characters', certain class functions attached to character sheaves. For reductive p-adic groups, the unipotent local Langlands correspondence gives a natural parametrization of irreducible smooth representations with unipotent cuspidal support. However, many questions about the characters of these representations are still open. Motivated by the study of the characters on compact elements, we introduce in joint work with A.-M. Aubert and B. Romano (arXiv:2106.13969) an involution on the spaces of elliptic and compact tempered unipotent representations of pure inner twists of a split simple p-adic group. This generalizes a construction by Moeglin and Waldspurger (2003, 2016) for elliptic tempered representations of split orthogonal groups, and potentially gives another interpretation of a Fourier transform for p-adic groups introduced by Lusztig (2014). We conjecture (and give supporting evidence) that the restriction to reductive quotients of maximal compact open subgroups intertwines this involution with a disconnected version of Lusztig's nonabelian Fourier transform for finite reductive groups.

representation theory

Audience: researchers in the topic

MIT Lie groups seminar

Series comments: Description: Research seminar on Lie groups

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