Rationality properties of complex representations of reductive p-adic groups

Abstract

For a reductive group G over a non-archimedean local field, we compare smooth representations over C with smooth representations over Qbar (an algebraic closure of Q). We show that an elliptic G-representation (in the sense of Arthur) can be realized over Qbar if and only if its central character takes values in Qbar. That applies in particular to all essentially square-integrable G-representations. We also study the action of the automorphism group of C/Q on complex G-representations. We prove that the sets of essentially square-integrable representations and of elliptic representations are stable under Gal(C/Q).

It turned out that several results from sections 2 and 3 of the first version had already been proven by Vignéras. In the second version we replaced those results by references, and we merged sections 2 and 3 into one section