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

2010/03/13 by Mitya Boyarchenko, Boyarchenko, Mitya
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1003.2742

openalex publication_date 2010/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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