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 GL _n and GSpin _{2 n + 1}. In the case of GSpin _{2 n + 1} (or SO _{2 n + 1}) exceptional (or small) representations were constructed by Bump, Friedberg and Ginzburg. We prove a Rodier-type hereditary property: a tempered representation τ is distinguished if and only if the representation I (τ) induced to GSpin _{2 n + 1} is distinguished, and the multiplicity of both quotients is at most one. If τ is supercuspidal and distinguished, then so is the Langlands quotient of I (τ). As a corollary, we characterize supercuspidal distinguished representations, in terms of the pole of the symmetric square L-function at s= 0.