Theta distinguished representations, inflation and the symmetric square L-function

A theta distinguished representation is a quotient of a tensor of exceptional representations, where "exceptional" is in the sense of Kazhdan and Patterson. We study relations between theta distinguished representations of and . In the case of (or ) exceptional (or small) representations were constructed by Bump, Friedberg and Ginzburg. We prove a Rodier-type hereditary property: a tempered representation is distinguished if and only if the representation induced to is distinguished, and the multiplicity of both quotients is at most one. If is supercuspidal and distinguished, then so is the Langlands quotient of . As a corollary, we characterize supercuspidal distinguished representations, in terms of the pole of the symmetric square L-function at s = 0.