Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

We consider smooth representations of the unit group G = A(x) of a finite-dimensional split basic algebra A. over a non-Archimedean local field. In particular, we prove a version of Gutkin's conjecture, namely, we prove that every irreducible smooth representation of G is compactly induced by a one-dimensional representation of the unit group of some subalgebra of A. We also discuss admissibility and unitarisability of smooth representations of G.