Annihilator varieties of distinguished modules of reductive Lie algebras
| Autor: | Eitan Sayag, Dmitry Gourevitch, Ido Karshon |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Pure mathematics Automorphic form Unipotent 01 natural sciences Theoretical Computer Science 0103 physical sciences Lie algebra FOS: Mathematics Discrete Mathematics and Combinatorics 0101 mathematics Representation Theory (math.RT) Mathematics::Representation Theory Mathematical Physics Mathematics Algebra and Number Theory Functor 010102 general mathematics Orthogonal complement Reductive group 20G05 22E46 22E47 22E45 Annihilator Computational Mathematics 010307 mathematical physics Geometry and Topology Variety (universal algebra) Mathematics - Representation Theory Analysis |
| DOI: | 10.48550/arxiv.2001.11746 |
| Popis: | We provide a micro-local necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let $\bf G$ be a complex algebraic reductive group, and $\bf H\subset G$ be a spherical algebraic subgroup. Let $\mathfrak{g},\mathfrak{h}$ denote the Lie algebras of $\bf G$ and $\bf H$, and let $\mathfrak{h}^{\bot}$ denote the annihilator of $\mathfrak{h}$ in $\mathfrak{g}^*$. A $\mathfrak{g}$-module is called $\mathfrak{h}$-distinguished if it admits a non-zero $\mathfrak{h}$-invariant functional. We show that the maximal $\bf G$-orbit in the annihilator variety of any irreducible $\mathfrak{h}$-distinguished $\mathfrak{g}$-module intersects $\mathfrak{h}^{\bot}$. This generalizes a result of Vogan. 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 Prasad, if $H$ 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 models studied in the theory of automorphic forms involve an additive character on the unipotent radical of $\bf H$, and we 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 W-algebras. As an application 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 non-generic representations. We also prove more general results that ease the sphericity assumption on the subgroup, and apply them to local theta correspondence in type II and to degenerate Whittaker models. Comment: 33p. v3: added results for non-spherical subgroups, and applications to local theta correspondence in type II and to degenerate Whittaker models. Also added an appendix by Ido Karshon. v4: minor corrections in the proof of Theorem B and the generality of Theorem 9.2 |
| Databáze: | OpenAIRE |
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