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

Kitagawa, Masatoshi

doi:10.48550/arxiv.2109.05556

Cited by 1 publication

(9 citation statements)

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“…The property (a') has been proved in our previous paper [31,Theorem 7.18] (Fact 4.7). The property (b') is proved in Section 5 in a general setting.…”

Section: Introductionmentioning

confidence: 82%

“…In Section 3, we recall several fundamental notions about (g, K)-modules and Casselman-Wallach representations. In Section 4, we recall several results in our previous paper [31] related to uniformly bounded families, and prove the upper bound of (1.0.4). Section 5 is devoted to prove the lower bound of (1.0.4) in a general setting.…”

Section: The Assumption Annmentioning

confidence: 99%

“…Since the number of two-sided ideals of U(g ′ ) with a fixed infinitesimal character is bounded by a constant independent of the infinitesimal character [31,Proposition 7.7], n is bounded by a constant depending only on the codimension of I ∩ Z(g) in Z(g).…”

Section: Fact 312 ([21]mentioning

confidence: 99%

“…We have introduced the notion of uniformly bounded families of g-modules in [31]. We recall the notion and related results.…”

Section: Uniformly Bounded Family and Applicationsmentioning

confidence: 99%

“…We omit the details because we do not use the definition itself of uniformly bounded families in this paper. See [31,Definition 7.2] for the definition.…”

Section: Uniformly Bounded Familymentioning

confidence: 99%

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Uniformly bounded multiplicities, polynomial identities and coisotropic actions

Kitagawa

2021

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Let G R be a real reductive Lie group and G ′ R a reductive subgroup of G R such that g ′ is algebraic in g. In this paper, we consider restrictions of irreducible representations of G R to G ′ R and induced representations of irreducible representations of G ′ R to G R . Our main concern is when such a representation has uniformly bounded multiplicities, i.e. the multiplicities in the representation are (essentially) bounded. We give characterizations of the uniform boundedness by polynomial identities and coisotropic actions.For the restriction of (cohomologically) parabolically induced representations, we find a sufficient condition for the uniform boundedness by spherical actions and some fiber condition. This result gives an affirmative answer to a conjecture by T. Kobayashi.Our results can be applied to (g, K)-modules, Casselman-Wallach representations, unitary representations and objects in the BGG category O. We also treat with an upper bound of cohomological multiplicities.

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“…The property (a') has been proved in our previous paper [31,Theorem 7.18] (Fact 4.7). The property (b') is proved in Section 5 in a general setting.…”

Section: Introductionmentioning

confidence: 82%

Section: The Assumption Annmentioning

confidence: 99%

Section: Fact 312 ([21]mentioning

confidence: 99%

“…We have introduced the notion of uniformly bounded families of g-modules in [31]. We recall the notion and related results.…”

Section: Uniformly Bounded Family and Applicationsmentioning

confidence: 99%

“…We omit the details because we do not use the definition itself of uniformly bounded families in this paper. See [31,Definition 7.2] for the definition.…”

Section: Uniformly Bounded Familymentioning

confidence: 99%

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