Representations of Unipotent Groups over Local Fields and Gutkin’s Conjecture

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 ∈ A, is a locally compact group with the topology induced by the standard one on F and the multiplication (1 + x) · (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 ⊂ A is an F -subalgebra. In the case where F is local and nonarchimedean we also establish an analogous result for smooth irreducible representations of 1 + A over C and show that every such representation is admissible and carries an invariant Hermitian inner product.