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

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

2010-03-13

arxiv.org/abs/1003.2742v1

Mathematics

Representation Theory

20 pages, LaTeX

Scientific paper

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.

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