Algebraic Methods in the Theory of Generalized Harish-Chandra Modules

Penkov, Ivan; Zuckerman, Gregg J.

doi:10.1007/978-3-319-09804-3_15

Cited by 5 publications

(5 citation statements)

References 14 publications

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“…Such a module is called a generalized Harish-Chandra module by I. Penkov and G. Zuckerman (see e.g. [40]). We write C χ (g, k) for the full subcategory of C(g, k) whose object has the infinitesimal character χ.…”

Section: Family Of Modules With a Boundedness Property 331mentioning

confidence: 99%

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Untitled

2023

Represent. Theory

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In the representation theory of real reductive Lie groups, many objects have finiteness properties. For example, the lengths of Verma modules and principal series representations are finite, and more precisely, they are bounded. In this paper, we introduce a notion of uniformly bounded families of holonomic D-modules to explain and find such boundedness properties.A uniform bounded family has good properties. For instance, the lengths of modules in the family are bounded and the uniform boundedness is preserved by direct images and inverse images. By the Beilinson-Bernstein correspondence, we deduce several boundedness results about the representation theory of complex reductive Lie algebras from corresponding results of uniformly bounded families of D-modules. In this paper, we concentrate on proving fundamental properties of uniformly bounded families, and preparing abstract results for applications to the branching problem and harmonic analysis.

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Section: Family Of Modules With a Boundedness Property 331mentioning

confidence: 99%

“…I. Penkov and G. Zuckerman call a g-module M a generalized Harish-Chandra module if M is locally finite, completely reducible and admissible as a k-module (see e.g. [40]). A relation between generalized Harish-Chandra modules and supports of D-modules on G/B is studied by A. V. Petukhov [41].…”

Section: Introductionmentioning

confidence: 99%

Untitled

2023

Represent. Theory

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show abstract

“…Such a module is called a generalized Harish-Chandra module by I. Penkov and G. Zuckerman (see e.g. [39]). We write C χ (g, k) for the full subcategory of C(g, k) whose object has the infinitesimal character χ.…”

Section: Finite Orbits and Uniformly Bounded Familymentioning

confidence: 99%

“…I. Penkov and G. Zuckerman call a g-module M a generalized Harish-Chandra module if M is locally finite, completely reducible and admissible as a k-module (see e.g. [39]). A relation between generalized Harish-Chandra modules and supports of D-modules on G/B is studied by A. V. Petukhov [40].…”

Section: Introductionmentioning

confidence: 99%

Family of $\mathscr{D}$-modules and representations with a boundedness property

Kitagawa¹

2021

Preprint

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In the representation theory of real reductive Lie groups, many objects have finiteness properties. For example, the lengths of Verma modules and principal series representations are finite, and more precisely, they are bounded. In this paper, we introduce a notion of uniformly bounded families of holonomic D-modules to explain and find such boundedness properties.A uniform bounded family has good properties. For instance, the lengths of modules in the family are bounded and the uniform boundedness is preserved by direct images and inverse images. By the Beilinson-Bernstein correspondence, we can deduce several boundedness results about the representation theory of complex reductive Lie algebras from corresponding results of uniformly bounded families of D-modules. In this paper, we concentrate on proving fundamental properties of uniformly bounded families, and preparing abstract results for applications to the branching problem and harmonic analysis.

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“…In this note, we draw a corollary of our earlier work [4]. In the subsequent works [5], [6], [7], [8] we have built foundations of an algebraic theory of generalized Harish-Chandra modules.…”

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confidence: 99%