Relative Discrete Series Representations for Two Quotients ofp-adic GL_n

Abstract

Abstract. We provide an explicit construction of representations in the discrete spectrum of two p-adic symmetric spaces. We consider GL n (F) × GL n (F) GL n (F) and GL n (F) GL n (E), where E is a quadratic Galois extension of a nonarchimedean local eld F of characteristic zero and odd residual characteristic. e proof of the main result involves an application of a symmetric space version of Casselman's Criterion for square integrability due to Kato and Takano.

Local unitary periods and relative discrete series

“…. , n 1 ) be a balanced partition of n. Now, we show that M is θ-elliptic if and only if (n) = (n/2, n/2) by applying [29,Lemma 3.8], which states that a θ-stable Levi subgroup M is θ-elliptic if and only if S M = S G . An element a = diag(a 1 , .…”

“…We are ultimately interested in the exponents of parabolically induced representations. For a proof of the following lemma, see [29,Lemma 4.15] Lemma 3.2. Let P = M N be a parabolic subgroup of G, let (ρ, V ρ ) be a finitely generated admissible representation of M and let π = ι G P ρ. .…”

“…(2). Suppose that P ∩ y L y L is a proper parabolic subgroup of y L. In particular, the Levi subgroup M ∩ y L of P ∩ y L is properly contained in y L. The following is the direct analogue of [29,Proposition 8.5]. The idea of the proof is to realize the exponents of π along P as restrictions of the exponents of τ along P ∩ y L (cf.…”

“…Theorem 5.11 is the direct analogue of [29,Theorem 6.3] and the overall method of proof is the same. The idea of the proof is to reduce the verification of the Relative Casselman's Criterion (Theorem 3.4) of Kato and Takano to the usual Casselman's Criterion for the inducing discrete series.…”

“…The idea of the proof is to reduce the verification of the Relative Casselman's Criterion (Theorem 3.4) of Kato and Takano to the usual Casselman's Criterion for the inducing discrete series. In contrast to the work in [29], here we can construct representations only by inducing from the maximal parabolic subgroup P (n,n) due to the nature of the symmetric space H\G and the (lack of) existence of θ-elliptic Levi subgroups (cf. Definition 4.4,Lemma 4.20).…”