Semisimple orbital integrals on the symplectic space for a real reductive dual pair

McKee, Mark; Pasquale, Angela; Przebinda, Tomasz

doi:10.1016/j.jfa.2014.10.002

Cited by 5 publications

(4 citation statements)

References 26 publications

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“…Here we use the notation introduced in [MPP20, Definition 4]. The Weyl integration formula on W (see [MPP15,Theorem 21] and [MPP20, (16), (23)]) shows that…”

Section: Resultsmentioning

confidence: 99%

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Symmetry breaking operators for dual pairs with one member compact

McKee,

Pasquale,

Przebinda

2021

Preprint

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We consider a dual pair (G, G ′ ), in the sense of Howe, with G compact acting on L 2 (R n ), for an appropriate n, via the Weil representation ω. Let G be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation Π of G, let Π ′ be the corresponding irreducible unitary representation of G ′ in the Howe duality. The orthogonal projection onto L 2 (R n ) Π , the Π-isotypic component, is the essentially unique symmetry breaking operator in HomWe study this operator by computing its Weyl symbol. Our results allow us to compute the wavefront set of Π ′ by elementary means. Contents 1. Notation and preliminaries. 4 2. The center of the metaplectic group. 5 3. Dual pairs as Lie supergroups. 7 4. Main results. 11 5. The integral over −G 0 as an integral over g. 16 6. The invariant integral over g as an integral over h. 17 7. An intertwining distribution in terms of orbital integrals on the symplectic space. 21 8. The special case for the pair O 2l , Sp 2l ′ (R). 32 9. The special case for the pair O 2l+1 , Sp 2l ′ (R), with 1 ≤ l ≤ l ′ . 35 10. The special case for the pair O 2l+1 , Sp 2l ′ (R) with l > l ′ . 37 11. The sketch of a computation of the wave front of Π ′ . 39 Appendix A. The Jacobian of the Cayley transform 40 Appendix B. Proof of formula (86) 41 Appendix C. The special functions P a,b and Q a,b 43 Appendix D. The covering G → G 49 References 52Let W be a finite dimensional vector space over R equipped with a non-degenerate symplectic form •, • and let Sp(W) denote the corresponding symplectic group. Write

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“…Here we use the notation introduced in [MPP20, Definition 4]. The Weyl integration formula on W (see [MPP15,Theorem 21] and [MPP20, (16), (23)]) shows that…”

Section: Resultsmentioning

confidence: 99%

“…To present the main results of this paper, we need the realization of dual pairs with one member compact as Lie supergroups. The content of this section is taken from [Prz06] and [MPP15]. We recall the relevant material for making our exposition self-contained.…”

Section: Dual Pairs As Lie Supergroupsmentioning

confidence: 99%

Symmetry breaking operators for dual pairs with one member compact

McKee,

Pasquale,

Przebinda

2021

Preprint

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“…The set of almost semisimple elements in s 1 coincides with the union of the S-orbits through the generalized Cartan subspaces h1 S h1 , [MPP15,(47)]. Since the set of the regular almost semisimple elements is dense in h1 S h1 , [MPP15, Theorem 19] implies that the set of the regular almost semisimple elements is dense in s 1 .…”

Section: Limits Of Orbital Integralsmentioning

confidence: 99%

“…To simplify the notation, when the Cartan subspace h 1 is fixed, we shall simply write µ O(w) instead of µ O(w),h 1 . These are well defined, tempered distribution on W, see [MPP15], which depend only on τ (w), or equivalently τ ′ (w) via the identification (57). Let µ W be the Lebesgue measure on W normalized as in the Introduction.…”

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confidence: 99%

Weyl calculus and dual pairs

McKee¹,

Pasquale²,

Przebinda³

2014

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We consider a dual pair (G, G ′ ), in the sense of Howe, with G compact acting on L 2 (R n ) for an appropriate n via the Weil Representation. Let G be the preimage of G in the metaplectic group. Given a genuine irreducible unitary representation Π of G we compute the Weyl symbol of orthogonal projection onto L 2 (R n ) Π , the Π-isotypic component. We apply the result to obtain an explicit formula for the character of the corresponding irreducible unitary representation Π ′ of G ′ and to compute of the wave front set of Π ′ by elementary means. Contents 1. Introduction. 1 2. The group G and the Weyl denominator. 7 3. A Theorem of G. W. Schwartz. 12 4. An almost semisimple orbital integral on the symplectic space. 13 5. Limits of orbital integrals. 28 6. Intertwining distributions. 40 7. The pair G = U l , G ′ = U l ′ , l ≤ l ′ . 59 8. Limits of orbital integrals in the stable range. 73 Appendix A: A few facts about nilpotent orbits 76 Appendix B: Pull-back of a distribution via a submersion 77 Appendix C: Some confluent hypergeometric polynomials 82 Appendix D: Wave front set of an asymptotically homogeneous distribution 90 Appendix E: A proof of a cocycle property 92 References 98 1. Introduction.Let W be a vector space of finite dimension 2n over R with a non-degenerate symplectic form •, • . Denote by Sp ⊆ GL(W) the symplectic group and let G, G ′ ⊆ Sp = Sp(W) be an irreducible dual pair. Denote by G, G ′ the preimages of G, G ′ in the metaplectic group

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