Irreducibility of the parabolic induction of essentially Speh representations and a representation of Arthur type over a p-adic field

2025/04/01 by Bošnjak, Barbara, Stadler, Alexander
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2504.01226

Abstract

Let F be a p-adic field. In this article, we consider representations of split special orthogonal groups SO_2n+1(F) and symplectic groups Sp_2n(F) of rank n. We denote by π₁ × … × π_r \rtimes π the normalized parabolically induced representation of either. Now let u_i be essentially Speh representations and π a representation of Arthur type. We prove that the parabolic induction u₁ × … × u_r \rtimes π is irreducible if and only if u_i × u_j, u_i × u_j^\vee and u_i \rtimes π are irreducible for all choices of i≠ j. If u_i are Speh representations, we determine the composition series of the above parabolically induced representation. Through this, we are able to produce a new collection of unitary representations.

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