Metaplectic Tensor Products for Automorphic Representation of (r)

Abstract

AbstractLet M = GLr1 ✗ … × GLrk ⊆ GLr be a Levi subgroup of GLr, where r = r1 + … +rk, and its metaplectic preimage in the n-fold metaplectic cover r1 of GLr. For automorphic representations π1, …, πk of r1 (), … ,rk (), we construct (under a certain technical assumption that is always satisfied when n = 2) an automorphic representation π of () that can be considered as the “tensor product” of the representations π1, … , πk. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place v, πv is equivalent to the local metaplectic tensor product of π1,v, … , πk,v defined by Mezo. Then we show that if all of the πi 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.