Smooth representations of involutive algebra groups over non-archimedean local fields

2024/01/17 by André, Carlos A. M., Dias, João
#20G25 #22D12 #22D30 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2401.09302

Abstract

An algebra group over a field F is a group of the form G = 1+J where J is a finite-dimensional nilpotent associative F-algebra. A theorem of M. Boyarchenko asserts that, in the case where F is a non-archimedean local field, every irreducible smooth representation of G is admissible and smoothly induced by a one-dimensional smooth representation of some algebra subgroup of G. If J is a nilpotent algebra endowed with an involution σ:J→ J, then σ naturally defines a group automorphism of G, and we may consider the fixed point subgroup C_G(σ). Assuming that F has characteristic different from 2, we extend Boyarchenko's result and show that every irreducible smooth representation of C_G(σ) is admissible and smoothly induced by a one-dimensional smooth representation of a subgroup of the form C_H(σ) where H is an σ-invariant algebra subgroup of G. As a particular case, the result holds for maximal unipotent subgroups of the classical Chevalley groups defined over F.

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