Distinction of some induced representations

Abstract

Abstract. Let K/F be a quadratic extension of p-adic fields, σ the nontrivial element of the Galois group of K over F , and ∆ a quasi-square-integrable representation of GL(n, K). Denoting by ∆ ∨ the smooth contragredient of ∆, and by ∆ σ the representation ∆ • σ, we show that representation of GL(2n, K) obtained by normalized parabolic induction of the representation ∆ ∨ ⊗ ∆ σ , is distinguished with respect to GL(2n, F ). This is a step towards the classification of distinguished generic representations of general linear groups over p-adic fields.

Periods of residual automorphic forms

“…When E(φ, λ) is a cuspidal Eisenstein series on G(A), Theorem 23 of [14] provides a formula for the regularized period P H (E(φ, λ)). Its local factor was used in the proof of [23,Theorem 4.2] to show the distinction of σ × θ(σ ∨ ) for all essentially square-integrable representation σ of G(F v ), where the representation θ(σ ∨ ) is defined by θ(σ ∨ )(g) = σ(θ( t g −1 )) for g ∈ G(F v ). Since its analytic properties are difficult to study, the proof was rather indirect.…”

“…The distinction in this case has been extensively studied both locally and globally. For various aspects of the theory, we refer to HarderLanglands-Rapoport [12], Flicker [6,7], Gelbart-Jacquet-Rogawski [9], Gan-Raghuram [10], Prasad [31], Hakim-Murnaghan [11], Kable [17], Matringe [22][23][24][25], Liu [20], Wei Zhang [35] and the references therein. In this case the group G is conjectured to be a quasisplit unitary group of n-variables by Flicker [7] and Rallis.…”

On -Distinguished Representations of the Quasi-Split Unitary Groups

“…• L( i ) is GL(F)-distinguished if w(i) = i. In fact, we will only apply the 'if' part of this statement, which is a consequence of [23,Theorem 4.2] and [8,Proposition 26].…”

The characterization of theta-distinguished representations of GL(n)

“…Let F 0 be a quadratic extension of F . Matringe [Mat10a,Mat10b,Mat11] studied (GL n (F ), η)distinguished representations of GL n (F 0 ). He proved, using different (but in some sense related) techniques, a combinatorial characterization similar to Theorem A.…”