A Harish-Chandra Homomorphism for Reductive Group Actions

Knop, Friedrich

doi:10.2307/2118600

Cited by 84 publications

(116 citation statements)

References 23 publications

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“…This will also establish Theorem 1.2 since the two results are equivalent in view of Theorem 3.1. Our proof uses some results from [15] and [18] concerning the derived action of the Lie algebra of K on the space C[V ]. We begin by relating this action to the moment map.…”

Section: Geometric Criteriamentioning

confidence: 99%

A geometric criterion for Gelfand pairs associated with the Heisenberg group

Benson¹,

Jenkins²,

Lipsman³

et al. 1997

Pacific J. Math.

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Let K be a closed subgroup of U (n) acting on the (2n + 1)-dimensional Heisenberg group H n by automorphisms. One calls (K, H n ) a Gelfand pair when the integrable K-invariant functions on H n form a commutative algebra under convolution. We prove that this is the case if and only if the coadjoint orbits for G := K H n which meet the annihilator k ⊥ of the Lie algebra k of K do so in single K-orbits. Equivalently, the representation of K on the polynomial algebra over C n is multiplicity free if and only if the moment map from C n to k * is one-to-one on K-orbits.It is also natural to conjecture that the spectrum of the quasi-regular representation of G on L 2 (G/K) corresponds precisely to the integral coadjoint orbits that meet k ⊥ . We prove that the representations occurring in the quasi-regular representation are all given by integral coadjoint orbits that meet k ⊥ . Such orbits can, however, also give rise to representations that do not appear in L 2

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Section: Geometric Criteriamentioning

confidence: 99%

A geometric criterion for Gelfand pairs associated with the Heisenberg group

Benson¹,

Jenkins²,

Lipsman³

et al. 1997

Pacific J. Math.

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“…пп. 6.3, 6.5 в [4]). Автор благодарит Э. Б. Винберга и Д. А. Тимашева за помощь и обсуждение этих результатов в процессе их появления, а также многочисленные замечания по предварительному тексту статьи.…”

Section: список основных обозначенийunclassified

“…Результаты настоящей работы тесно связаны с общими результатами Кнопа [3], [4] об эквивариантной симплектической геометрии кокасательных расслоений гладких -многообразий (подробнее см. п.…”

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Гармонический Анализ На Некотором Классе Сферических Однородных Пространств

Горфинкель¹

2011

Mat. Zametki

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Том 90 выпуск 5 ноябрь 2011 УДК 512.745+512.816.4 Гармонический анализ на некотором классе сферических однородных пространств Н. Е. ГорфинкельИзучается спектр представлений полупростой алгебраической группы в про-странствах сечений однородных линейных расслоений на некотором классе сфе-рических однородных пространств; описывается алгебра инвариантных функ-ций на кокасательных расслоениях пространств этого класса, а также инвари-антные дифференциальные операторы.Библиография: 11 названий.

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“…Remember that in general Z(U(g) K ) Z(g) ⊗ Z(k) (Knop's Theorem [9]). Given h 0 and t 0 Cartan subalgebras of g 0 and k 0 respectively, we will denote γ G h and γ K t the Harish-Chandra homomorphisms of Z(g) and Z(k) with respect to the subalgebras h and t: Then we have…”

Section: Definitionmentioning

confidence: 99%

Fundamental solutions of invariant differential operators on a semisimple Lie group II

Ames¹

2002

Pacific J. Math.

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A Harish-Chandra Homomorphism for Reductive Group Actions

Cited by 84 publications

References 23 publications

A geometric criterion for Gelfand pairs associated with the Heisenberg group

A geometric criterion for Gelfand pairs associated with the Heisenberg group

Гармонический Анализ На Некотором Классе Сферических Однородных Пространств

Fundamental solutions of invariant differential operators on a semisimple Lie group II

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