Abstract
AbstractWe provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs.Let${\mathbf {G}}$be a complex algebraic reductive group and${\mathbf {H}}\subset {\mathbf {G}}$be a spherical algebraic subgroup. Let${\mathfrak {g}},{\mathfrak {h}}$denote the Lie algebras of${\mathbf {G}}$and${\mathbf {H}}$, and let${\mathfrak {h}}^{\bot }$denote the orthogonal complement to${\mathfrak {h}}$in${\mathfrak {g}}^*$. A${\mathfrak {g}}$-module is called${\mathfrak {h}}$-distinguished if it admits a nonzero${\mathfrak {h}}$-invariant functional. We show that the maximal${\mathbf {G}}$-orbit in the annihilator variety of any irreducible${\mathfrak {h}}$-distinguished${\mathfrak {g}}$-module intersects${\mathfrak {h}}^{\bot }$. This generalises a result of Vogan [Vog91].We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – whenHis a symmetric subgroup of a real reductive groupG, the existence of a temperedH-distinguished representation ofGimplies the existence of a genericH-distinguished representation ofG.Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup$\bf H$, and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules overW-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20].Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models.
Highlights
The study of periods of automorphic forms and of distinguished representations has received a lot of attention
The work of Sakellaridis and Venkatesh [SV17] regarding harmonic analysis on spherical varieties led to a conjectural parameterisation of classes of distinguished representations, and [Wan] pushed the conjectures of Gan, Gross and Prasad beyond classical groups to a more general setup of certain spherical pairs
The question becomes: given a nilpotent orbit O ⊂, what nilpotent orbits in can we hit by adding elements of ? The maximal among these orbits is the induced orbit, and the minimal is the saturation of O, but already in the case of = we will see from Lemma 7.2 that not every orbit in between is possible
Summary
The study of periods of automorphic forms and of distinguished representations has received a lot of attention. Denote by Irr( ) the collection of simple -modules that have a nonzero -invariant functional. By Theorem 1.1 and the Casselman–Wallach equivalence of categories (see Section 5), An V( ) is the closure of a unique coadjoint nilpotent orbit O( ) for any ∈ Irr( ). The case = 1 corresponds to branching problems for the restrictions from GL +1 (R) to GL (R), restrictions from SO( , + 1) and SO( , ) and analogous restrictions for complex groups and unitary groups This theorem partially confirms the nontempered Gan–Gross–Prasad conjectures [GGP20] in the Archimedean case.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
