Representations of unipotent groups over local fields and Gutkin's conjecture

TL;DR

This paper proves Gutkin's conjecture that all irreducible unitary representations of groups formed from nilpotent algebras over local or finite fields can be induced from one-dimensional characters, extending to smooth representations in the local case.

Contribution

It confirms Gutkin's 1973 conjecture and extends the result to smooth irreducible representations over local fields, showing they are admissible and have invariant Hermitian forms.

Findings

All unitary irreducible representations are induced from 1-dimensional characters.

In the local nonarchimedean case, all smooth irreducible representations are admissible.

Such representations carry an invariant Hermitian inner product.

Abstract

Let F be a finite field or a local field of any characteristic. If A is a finite dimensional associative nilpotent algebra over F, the set 1+A of all formal expressions of the form 1+x, where x ranges over the elements of A, is a locally compact group with the topology induced by the standard one on F and the multiplication given by (1+x)(1+y)=1+(x+y+xy). We prove a result conjectured by Eugene Gutkin in 1973: every unitary irreducible representation of 1+A can be obtained by unitary induction from a 1-dimensional unitary character of a subgroup of the form 1+B, where B is an F-subalgebra of A. In the case where F is local and nonarchimedean we also establish an analogous result for smooth irreducible representations of 1+A over the field of complex numbers and show that every such representation is admissible and carries an invariant Hermitian inner product.