The generalized linear periods

Abstract

Let F be a local field of characteristic zero. Let μ be a good character of GLp(F)×GLp+1(F). We study the generalized linear period problem for the pair (G,Hp,p+1)=(GL2p+1(F),GLp(F)×GLp+1(F)) and we prove that any bi-(Hp,p+1,μ)-equivariant tempered generalized function on G is invariant under the matrix transpose. We also show that any P∩Hp,p+1-invariant linear functional on an Hp,p+1-distinguished irreducible smooth representation of G is also Hp,p+1-invariant if F is nonarchimedean, where P is the standard mirabolic subgroup of G consisting of matrices with last row vector (0,⋯,0,1).