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Annihilator varieties of distinguished modules of reductive Lie algebras

Part of: Linear algebraic groups and related topics Lie groups

Published online by Cambridge University Press:  26 July 2021

Dmitry Gourevitch Open the ORCID record for Dmitry Gourevitch [Opens in a new window] ,
Eitan Sayag Open the ORCID record for Eitan Sayag [Opens in a new window]  and
Ido Karshon
Dmitry Gourevitch
Affiliation:
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel E-mail: dimagur@weizmann.ac.il
Eitan Sayag
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, POB 653, Be’er Sheva84105, Israel E-mail: eitan.sayag@gmail.com
Ido Karshon
Affiliation:
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel E-mail: ido.karshon@gmail.com

Abstract

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We 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] – when H is a symmetric subgroup of a real reductive group G, the existence of a tempered H-distinguished representation of G implies the existence of a generic H-distinguished representation of G.

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 over W-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.

MSC classification

Primary: 20G05: Representation theory
Secondary: 22E46: Semisimple Lie groups and their representations 22E47: Representations of Lie and real algebraic groups 22E45: Representations of Lie and linear algebraic groups over real fields
Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e52
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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