Dual pairs and Kostant-Sekiguchi correspondence. II. Classification of nilpotent elements

We classify the homogeneous nilpotent orbits in certain Lie color algebras and specialize the results to the setting of a real reductive dual pair. For any member of a dual pair, we prove the bijectivity of the two Kostant-Sekiguchi maps by straightforward argument. For a dual pair we determine the correspondence of the real orbits, the correspondence of the complex orbits and explain how these two relations behave under the Kostant-Sekiguchi maps. In particular we prove that for a dual pair in the stable range there is a Kostant-Sekiguchi map such that the conjecture formulated in [6] holds. We also show that the conjecture cannot be true in general.     

The Oscillator Character Formula,for isometry groups of split forms in deep stable range

Let G˜ and G˜ ^′ be a reductive dual pair of the type mentioned in the title, with G˜ the smaller member. Let Π and Π′ be unitary representations of G˜,G˜ ^′ which occur in Howe’s correspondence. We express the distribution character of Π′ in terms of the character of Π via an explicit integral kernel operator. Oblatum 4-I-1995     

The character and the wave front set correspondence in the stable range

We relate the distribution characters and the wave front sets of unitary representation for real reductive dual pairs of type I in the stable range.     

The Oscillator Character Formula,for isometry groups of split forms in deep stable range

Let and be a reductive dual pair of the type mentioned in the title, with the smaller member. Let and be unitary representations of , which occur in Howe's correspondence. We express the distribution character of in terms of the character of via an explicit integral kernel operator.Oblatum 4-I-1995Research partially supported by the¹UMK Grant 514-M, and the²NSF Grant DMS 9204488.     

A Capelli Harish-Chandra homomorphism

For a real reductive dual pair the Capelli identities define a homomorphism from the center of the universal enveloping algebra of the larger group to the center of the universal enveloping algebra of the smaller group. In terms of the Harish-Chandra isomorphism, this map involves a -shift. We view a dual pair as a Lie supergroup and offer a construction of the homomorphism based solely on the Harish-Chandra's radial component maps. Thus we provide a geometric interpretation of the -shift.

     

Howe correspondence and Springer correspondence for real reductive dual pairs

We consider a real reductive dual pair (G′, G) of type I, with rank ${({\rm G}^{\prime}) \leq {\rm rank(G)}}$ . Given a nilpotent coadjoint orbit ${\mathcal{O}^{\prime} \subseteq \mathfrak{g}^{{\prime}{*}}}$ , let ${\mathcal{O}^{\prime}_\mathbb{C} \subseteq \mathfrak{g}^{{\prime}{*}}_\mathbb{C}}$ denote the complex orbit containing ${\mathcal{O}^{\prime}}$ . Under some condition on the partition λ′ parametrizing ${\mathcal{O}^{\prime}}$ , we prove that, if λ is the partition obtained from λ by adding a column on the very left, and ${\mathcal{O}}$ is the nilpotent coadjoint orbit parametrized by λ, then ${\mathcal{O}_\mathbb{C}= \tau (\tau^{\prime -1}(\mathcal{O}_\mathbb{C}^{\prime}))}$ , where ${\tau, \tau^{\prime}}$ are the moment maps. Moreover, if ${chc(\hat\mu_{\mathcal{O}^{\prime}}) \neq 0}$ , where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to ${\mu_{\mathcal{O}^{\prime}}}$ with corresponds to ${\mathcal{O}_\mathbb{C}}$ via the Springer correspondence.     

Local geometry of orbits for an ordinary classical lie supergroup

In this paper we identify a real reductive dual pair of Roger Howe with an Ordinary Classical Lie supergroup. In these terms we describe the semisimple orbits of the dual pair in the symplectic space, a slice through a semisimple element of the symplectic space, an analog of a Cartan subalgebra, the corresponding Weyl group and the corresponding Weyl integration formula.     

Platonic Orthonormal Wavelets

We classify all orthonormal wavelets which occur in theL²space of the faces of a platonic solid.