The 𝑆𝐿(2)-type and base change

Abstract

The S L ( 2 ) SL(2) -type of any smooth, irreducible and unitarizable representation of G L n GL_n over a p 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 S L ( 2 ) SL(2) -type of a representation is preserved under the base change with respect to any finite extension. The Klyachko model of a smooth, irreducible and unitarizable representation π \pi of G L n GL_n depends only on the S L ( 2 ) SL(2) -type of π \pi . As a consequence we observe that the Klyachko model of π \pi and of its base change are of the same type.