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

Abstract

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 2n+1 . In the case of GSpin 2n+1 (or SO 2n+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 2n+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.

Distinguished theta representations for certain covering groups

“…Another significant direction of applications is the Rankin-Selberg integral representation for the symmetric square and cube L-functions (cf. [BG], [BGH], [Tak] and [Ka2]). Evidently, it should be mentioned that for distinguished theta representations, the theory of L-function could be developed as in the linear algebraic case, since the Casselman-Shalika formula is then scalar-valued.…”

The characterization of theta-distinguished representations of GL(n)

“…Let GSpin 2n+1 be the split odd general spin group of rank n + 1. A result similar to Theorem B, for exceptional representations Θ of GSpin 2n+1 (defined in [Kap14a] following the exposition in [BFG03] for SO 2n+1 ), was proved in [Kap14c] and used in a study of Θ ⊗Θ ′ . The definition of distinguished representations of GSpin 2n+1 is similar to that of GL n , an irreducible representation is distinguished if its contragradient is a quotient of Θ ⊗ Θ ′ .…”

“…× τ m , where each τ i is essentially square-integrable. Then τ is distinguished if and only if there is 0 ≤ m 0 ≤ ⌊m 2⌋ such that, perhaps after permuting the indices of the inducing data, τ 2i = τ ∨ 2i−1 for 1 ≤ i ≤ m 0 and τ i is distinguished for 2m 0 + 1 ≤ i ≤ m. The supercuspidal case has already been proved in [Kap14c]. The square-integrable case is handled in Theorem 4.12.…”

“…The definition of distinguished representations of GSpin 2n+1 is similar to that of GL n , an irreducible representation is distinguished if its contragradient is a quotient of Θ ⊗ Θ ′ . The results of [Kap14c] allow us to alternate between quotients of θ⊗θ ′ and Θ⊗Θ ′ , using parabolic induction.…”

“…One can also use this to deduce certain irreducibility results (see Remark 4.15). In more detail, Proposition 4.1 of [Kap14c] showed that for a tempered distinguished τ , a representation parabolically induced from ν 1 2 τ ⊗1 to GSpin 2n+1 will also be distinguished, but then it must be reducible, otherwise it is a generic quotient of Θ ⊗ Θ ′ .…”