Highest Weight Modules and $b$-Functions of Semi-invariants

Gyoja, Akihiko

doi:10.2977/prims/1195165904

Cited by 15 publications

(12 citation statements)

References 22 publications

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“…The reader will see below that the construction of Ω j is guided by the requirement that J(Ω j ) = (τ j ), which thus comes for free. The difficulty is in establishing the conformal invariance of the resulting system for some s. Because of the connection between conformally invariant systems and reducibility of generalized Verma modules, our results can be tested against Jantzen's results [12] and against the expectations arising from a conjecture of Gyoja [8]. Recall that Jantzen [12] gave a necessary and sufficient condition for reducibility of a generalized Verma module.…”

Section: §1 Introductionmentioning

confidence: 99%

“…The highest root is γ = 2α 1 + 3α 2 + 4α 3 + 6α 4 + 5α 5 + 4α 6 + 3α 7 + 2α 8 , ρ(n) = (29/2)γ and we may take δ = α 8 to stand for the Ω 2 system constructed by the normal means, and Ω small 2 to stand for Ω 2 (Z 0 ). A dash indicates that no such system exists in that type.…”

Section: • −γmentioning

confidence: 99%

“…For Gyoja's original formulation, the reader is encouraged to consult [8]. For any parabolic subalgebra q = l ⊕ n we follow Gyoja in defining an L-quasi-invariant polynomial P ∈ C[n].…”

Section: S((vmentioning

confidence: 99%

“…We wish to determine In [8], Gyoja explains how to find a polynomial b(s) having the required property by using a sophisticated algorithm, and carries out the necessary calculations in some cases. We shall instead use a completely elementary approach that expresses a suitable polynomial b(s) directly in terms of the b-function of the relative invariant.…”

Section: §7 the B-function Of The Quasi-invariantmentioning

confidence: 99%

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Conformally Invariant Systems of Diﬀerential Equations and Prehomogeneous Vector Spaces of Heisenberg Parabolic Type

Barchini¹,

Kable²,

Zierau³

2008

Publ. Res. Inst. Math. Sci.

156

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Several systems of partial differential operators are associated to each complex simple Lie algebra of rank greater than one. Each system is conformally invariant under the given algebra. The systems so constructed yield explicit reducibility results for a family of scalar generalized Verma modules attached to the Heisenberg parabolic subalgebra of the given Lie algebra. Points of reducibility for such families lie in the union of several arithmetic progressions, possibly overlapping. For classical algebras, enough systems are constructed to account for the first point of reducibility in each progression. The relationship between these results and a conjecture of Akihiko Gyoja is explored. §1. Introduction To describe our results, it is first necessary to set the scene. In the body of the paper we shall work most of the time in an exclusively algebraic framework, but it will be useful here to take a more inclusive viewpoint, mixing the analytic and the algebraic. We shall first attempt to explain the significance of our results and place them in context. To some extent, these remarks may be taken as an introduction to a broader investigation of which this work and [2] are the first fruits. Then we shall draw a map to aid the reader in navigating on the admittedly lengthy journey through the proofs.

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Section: §1 Introductionmentioning

confidence: 99%

Section: • −γmentioning

confidence: 99%

“…For Gyoja's original formulation, the reader is encouraged to consult [8]. For any parabolic subalgebra q = l ⊕ n we follow Gyoja in defining an L-quasi-invariant polynomial P ∈ C[n].…”

Section: S((vmentioning

confidence: 99%

Section: §7 the B-function Of The Quasi-invariantmentioning

confidence: 99%

See 2 more Smart Citations

Conformally Invariant Systems of Diﬀerential Equations and Prehomogeneous Vector Spaces of Heisenberg Parabolic Type

Barchini¹,

Kable²,

Zierau³

2008

Publ. Res. Inst. Math. Sci.

156

View full text Add to dashboard Cite

show abstract

“…Moreover, using the commutative diagram (1.1) for ξ 0 and ψ 0 we give a new proof of the following criterion of the irreducibility of the generalized Verma module due to Suga [10], Gyoja [1], Wachi [13]:…”

Section: Theorem 11mentioning

confidence: 99%

The $b$-functions for Prehomogeneous Vector Spaces of Commutative Parabolic Type and Universal Generalized Verma Modules

Kamita

2005

Publ. Res. Inst. Math. Sci.

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We shall give a new elementary proof of the uniform expression for the b-functions of prehomogeneous vector spaces of commutative parabolic type obtained by Muller, Rubenthaler and Schiffmann [5] by using micro-local analysis. Our method is similar to Kashiwara's approach using the universal Verma modules. We shall also give a new proof for the criterion of the irreducibility of the generalized Verma module in terms of b-functions due to Suga [10], Gyoja [1], Wachi [13]. §1. IntroductionIn this paper we deal with the b-functions of the invariants on the flag manifolds G/P . In the case where P is a Borel subgroup, Kashiwara [3] determined the b-functions by using the universal Verma modules. For general parabolic subgroups P we show that b-functions are regarded as generators of ideals defined by universal generalized Verma modules. When the unipotent radical of P is commutative, we determine the generator.Let g be a simple Lie algebra over the complex number field C, and let G be a connected simply-connected simple algebraic group with Lie algebra g. Fix a parabolic subalgebra p of g. We denote the reductive part of p and the nilpotent part of p by l and n respectively. Let L be the subgroup of G

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Integral geometry for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {D}}$\end{document}‐modules on dual flag manifolds and generalized Verma modules

Marastoni

2013

Mathematische Nachrichten

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Highest Weight Modules and $b$-Functions of Semi-invariants

Cited by 15 publications

References 22 publications

Conformally Invariant Systems of Diﬀerential Equations and Prehomogeneous Vector Spaces of Heisenberg Parabolic Type

Conformally Invariant Systems of Diﬀerential Equations and Prehomogeneous Vector Spaces of Heisenberg Parabolic Type

The $b$-functions for Prehomogeneous Vector Spaces of Commutative Parabolic Type and Universal Generalized Verma Modules

Integral geometry for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {D}}$\end{document}‐modules on dual flag manifolds and generalized Verma modules

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