The nonabelian Fourier transform for elliptic unipotent representations of exceptional $p$-adic groups

Abstract

We define an involution on the space of elliptic unipotent Langlands parameters of a reductive $p$-adic group $G$ and verify that when $G$ is split adjoint exceptional, the composition of this involution with the hyperspecial parahoric restriction map agrees with Lusztig's nonabelian Fourier transform for unipotent representations of the finite reductive quotient. This is inspired by recent works of Lusztig on the almost unipotent characters of $p$-adic groups and of Moeglin and Waldspurger on the elliptic Fourier transform of odd orthogonal groups.

27 pages; v2: I corrected a mistake in the example of the $A_2\times A_2^{\ast }$ parahoric restriction in $F_4$ (which was pointed out to me by J.L. Waldspurger). This correction supports now the expectation that Conjecture 1.3(2) should hold for all maximal parahorics in general

References in corpus (1)

• Induce/restrict matrices for exceptional Weyl groups

Cited by in corpus (1)

• A nonabelian Fourier transform for tempered unipotent representations