On the Regularity of D-Modules Generated by Relative Characters

Li, Wen-Wei

doi:10.1007/s00031-020-09624-x

Cited by 6 publications

(23 citation statements)

References 30 publications

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“…Theorem E implies that the relative characters of π ∈ M Ξ (G) corresponding to Ξ-spherical subgroups are holonomic. For spherical subgroups, it is further shown in [Li20] that the relative characters are regular holonomic. We conjecture that this holds for Ξ-spherical subgroups as well, and hope to prove that at least for the case when Ξ is a closure of a Richardson orbit, using Theorem B and generalizing the proof in [Li20].…”

Section: Examplesmentioning

confidence: 98%

Finite multiplicities beyond spherical spaces

Aizenbud¹,

Gourevitch²

2021

Preprint

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Let G be a real reductive algebraic group, and let H ⊂ G be an algebraic subgroup. It is known that the action of G on the space of functions on G/H is "tame" if this space is spherical. In particular, the multiplicities of the space S(G/H) of Schwartz functions on G/H are finite in this case. In this paper we formulate and analyze a generalization of sphericity that implies finite multiplicities in S(G/H) for small enough irreducible representations of G.

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Section: Examplesmentioning

confidence: 98%

Finite multiplicities beyond spherical spaces

Aizenbud¹,

Gourevitch²

2021

Preprint

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“…A similar result of regularity is contained in [25] for localizations of Harish-Chandra (g, K)modules to X := H\G, where H, K ⊂ G are reductive spherical subgroups. By a Harish-Chandra (g, K)-module, we mean a (g, K)-module that is finitely generated over g and locally Z(g)-finite, where Z(g) is the center of U (g).…”

Section: Backgroundsmentioning

confidence: 68%

“…Let K ⊂ G and X be as in Theorem 5.3.4. Our proof of Theorem 5.3.4 is based on the result below from [25], which is a variant of Ginzburg's [13, Corollary 8.9.1]. Theorem 5.4.1.…”

Section: Criterion Of Regularitymentioning

confidence: 99%

“…We refer to Theorem 5.3.4 and the results in that section for complete statements. The proof is based on a criterion (Theorem 5.4.1) from [25]. One must show that the cohomologies of Loc X (V ) are (R1) finitely generated over D X , (R2) carry K-equivariant structures, and (R3) locally Z(g)-finite through…”

Section: H Nmentioning

confidence: 99%

“…Let m χ ⊂ Sym(s) be the corresponding maximal ideal; its image in D X := (π * D X ) S is central. Recall the following notion from [4, p.24] (see also [25]). Definition 6.4.1.…”

Section: Proposition 252 the Dg-functors From Lemma 251 Fit Into Adju...mentioning

confidence: 99%

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Higher localization and higher branching laws

Li¹

2022

Preprint

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For a connected reductive group G and an affine smooth G-variety X over the complex numbers, the localization functor takes g-modules to D X -modules. We extend this construction to an equivariant and derived setting using the formalism of h-complexes due to Beilinson-Ginzburg, and show that the localizations of Harish-Chandra (g, K)-modules onto X = H\G have regular holonomic cohomologies when H, K ⊂ G are both spherical reductive subgroups. The relative Lie algebra homologies and Ext-branching spaces for (g, K)-modules are interpreted geometrically in terms of equivariant derived localizations. As direct consequences, we show that they are finite-dimensional under the same assumptions, and relate Euler-Poincaré characteristics to local index theorem; this recovers parts of the recent results of M. Kitagawa. Examples and discussions on the relation to Schwartz homologies are also included.

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Annihilator varieties of distinguished modules of reductive Lie algebras

Gourevitch¹,

Sayag²,

Karshon³

2021

Forum of Mathematics, Sigma

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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.

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On the Regularity of D-Modules Generated by Relative Characters

Cited by 6 publications

References 30 publications

Finite multiplicities beyond spherical spaces

Finite multiplicities beyond spherical spaces

Higher localization and higher branching laws

Annihilator varieties of distinguished modules of reductive Lie algebras

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