Abstract. In this paper, we consider the (partial) symmetric square L-function L S (s, π, Sym 2 ⊗ χ) of an irreducible cuspidal automorphic representation π of GLr(A) twisted by a Hecke character χ. In particular, we will show that the L-function L S (s, π, Sym 2 ⊗ χ) is holomorphic for the region Re(s) > 1 − 1 r with the exception that, if χ r ω 2 = 1, a pole might occur at s = 1, where ω is the central character of π. Our method of proof is essentially a (nontrivial) modification of the one by Bump and Ginzburg in which they considered the case χ = 1.
We give a proof of the Howe duality conjecture in the theory of local theta correspondence for symplectic-orthogonal or unitary dual pairs in arbitrary residual characteristic.
Abstract. Let M = GL r1 × · · · × GL r k ⊆ GL r be a Levi subgroup of GL r , where r = r 1 + · · · + r k , and M its metaplectic preimage in the n-fold metaplectic cover GL r of GL r . For automorphic representations π 1 , . . . , π k of GL r1 (A), . . . , GL r k (A), we construct (under a certain technical assumption, which is always satisfied when n = 2) an automorphic representation π of M(A) which can be considered as the "tensor product" of the representations π 1 , . . . , π k . This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place v, π v is equivalent to the local metaplectic tensor product of π 1,v , . . . , π k,v defined by Mezo. Then we show that if all of π i are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element, and show the compatibility with parabolic inductions.
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