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

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.