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

Let $F$ be a finite field or a local field of \emph{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 $x\in A$, is a locally compact group with the topology induced by the standard one on $F$ and the multiplication $(1+x)\cdot(1+y)=1+(x+y+xy)$. We prove a result conjectured by E.~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 $B\subset A$ is an $F$-subalgebra. In the case where $F$ is local and nonarchimedean we also establish an analogous result for \emph{smooth} irreducible representations of $1+A$ over $\bC$ and show that every such representation is admissible and carries an invariant Hermitian inner product.

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Published 20 May 2011