Square integrable representations of classical p-adic groups corresponding to segments

Abstract

Let S n S_n be either the group S p ( n ) Sp(n) or S O ( 2 n + 1 ) SO(2n+1) over a p p -adic field F F . Then Levi factors of maximal parabolic subgroups are (isomorphic to) direct products of G L ( k ) GL(k) and S n − k S_{n-k} , with 1 ≤ k ≤ n 1\leq k\leq n . The square integrable representations which we define and study in this paper (and prove their square integrability), are subquotients of reducible representations Ind P S n ( δ ⊗ σ ) , _P^{S_n}(\delta \otimes \sigma ), where δ \delta is an essentially square integrable representation of G L ( k ) GL(k) , and σ \sigma is a cuspidal representation of S n − k S_{n-k} . These square integrable representations play an important role in a construction of more general square integrable representations.