The SL(2)-type and Base Change

Abstract

The SL(2)-type of any smooth, irreducible and unitarizable representation of GL(n) over a p-adic field was defined by Venkatesh. We provide a natural way to extend the definition to all smooth and irreducible representations. For unitarizable representations we show that the SL(2)-type of a representation is preserved under base change with respect to any finite extension. The Klyachko model of a smooth, irreducible and unitarizable representation \pi of GL(n) depends only on the SL(2)-type of \pi. As a consequence we observe that the Klyachko model of \pi and of its base-change are of the same type.