Abstract. Let E/F be a quadratic extension of p-adic fields. If π is an admissible representation of GLn(E) that is parabolically induced from discrete series representations, then we prove that the space of Pn(F )-invariant linear functionals on π has dimension one, where Pn(F ) is the mirabolic subgroup. As a corollary, it is deduced that if π is distinguished by GLn(F ), then the twisted tensor L-function associated to π has a pole at s = 0. It follows that if π is a discrete series representation, then at most one of the representations π and π ⊗ χ is distinguished, where χ is an extension of the local class field theory character associated to E/F . This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.
Abstract. Let E/F be a quadratic extension of p-adic fields. We compute the multiplicity of the space of SL 2 (F )-invariant linear forms on a representation of SL 2 (E). This multiplicity varies inside an L-packet similar in spirit to the multiplicity formula for automorphic representations due to Labesse and Langlands.
Abstract. Let E/F be a quadratic extension of non-archimedean local fields of characteristic zero. An irreducible admissible representation π of GL(n, E) is said to be distinguished with respect to GL(n, F) if it admits a non-trivial linear form that is invariant under the action of GL(n, F). It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the F-points of the mirabolic subgroup when π is unitary and generic. In this paper, we prove that the essential vector of [JPSS81] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local L-value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case.
Let F/F o be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and σ be its non-trivial automorphism. We show that any σ-self-dual cuspidal representation of GL n (F) contains a σ-self-dual Bushnell-Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai L-function of a GL n (F o )-distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands-Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation. NotationLet F/F o be a quadratic extension of locally compact non-archimedean fields of residual characteristic p = 2. Write σ for the non-trivial F o -automorphism of F. For any finite extension E of F o , we denote by O E its ring of integers, by p E the unique maximal ideal of O E and by k E its residue field. We abbreviate O F to O and O Fo to O o , and define similarly p, p o , k, k o . The involution σ induces a k o -automorphism of k, still denoted σ.
Let \mathrm{F}/\mathrm{F}_{\mathsf{o}} be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and \sigma be its non-trivial automorphism. We show that any \sigma -self-dual cuspidal representation of \operatorname{GL}_n(\mathrm{F}) contains a \sigma -self-dual Bushnell–Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai L-function of a \operatorname{GL}_n(\mathrm{F}_\mathsf{o}) -distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands–Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation.
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