Admissible representations, multiplicity-free representations and visible actions on non-tube type Hermitian symmetric spaces

Abstract. Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity-free. If ƛ denotes the isomorphism class of an irreducible representation of K, let ρƛ : K → GL(Vƛ) denote the corresponding irreducible representation and Sƛ the ƛ-isotypic component of S. Write Sƛ ・ Sμ for the subspace of S spanned by products of Sƛ and Sμ. If Vν occurs as an irreducible constituent of Vƛ ⊗ Vμ, is it true that Sν ⊆ Sƛ ・ Sμ? In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring S to the usual Littlewood–Richardson rule.

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Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.111 ◽

2021 ◽

pp. 1-50

Author(s):

BRUNO DUCHESNE ◽

JEAN LÉCUREUX ◽

MARIA BEATRICE POZZETTI

Keyword(s):

Symmetric Spaces ◽

Hermitian Symmetric Spaces ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Boundary Map ◽

Density Assumption ◽

Maximal Representations ◽

Tube Type ◽

Maximal Representation ◽

Hyperbolic Lattices

Abstract We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

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Hermitian symmetric spaces of tube type and multivariate Meixner-Pollaczek polynomials

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-25506 ◽

2017 ◽

Vol 120 (1) ◽

pp. 87 ◽

Cited By ~ 1

Author(s):

Jacques Faraut ◽

Masato Wakayama

Keyword(s):

Symmetric Spaces ◽

Symmetric Domain ◽

Type Equation ◽

Bounded Symmetric Domain ◽

Cayley Transform ◽

Hermitian Symmetric Spaces ◽

Spherical Fourier Transform ◽

Pollaczek Polynomials ◽

Tube Type ◽

The Difference

Harmonic analysis on Hermitian symmetric spaces of tube type is a natural framework for introducing multivariate Meixner-Pollaczek polynomials. Their main properties are established in this setting: orthogonality, generating and determinantal formulae, difference equations. For proving these properties we use the composition of the following transformations: Cayley transform, Laplace transform, and spherical Fourier transform associated to Hermitian symmetric spaces of tube type. In particular the difference equation for the multivariate Meixner-Pollaczek polynomials is obtained from an Euler type equation on a bounded symmetric domain.

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Lp-Poisson integral representations of solutions of the Hua system on Hermitian symmetric spaces of tube type

Journal of Functional Analysis ◽

10.1016/j.jfa.2005.11.005 ◽

2006 ◽

Vol 235 (2) ◽

pp. 413-429 ◽

Cited By ~ 5

Author(s):

Abdelhamid Boussejra

Keyword(s):

Symmetric Spaces ◽

Integral Representations ◽

Poisson Integral ◽

Hermitian Symmetric Spaces ◽

Tube Type

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CORWIN–GREENLEAF MULTIPLICITY FUNCTIONS FOR HERMITIAN SYMMETRIC SPACES AND MULTIPLICITY-ONE THEOREM IN THE ORBIT METHOD

International Journal of Mathematics ◽

10.1142/s0129167x10006021 ◽

2010 ◽

Vol 21 (03) ◽

pp. 279-296 ◽

Cited By ~ 6

Author(s):

SALMA NASRIN

Keyword(s):

Classical Limit ◽

Symmetric Spaces ◽

Discrete Series ◽

Coadjoint Orbit ◽

Symmetric Pair ◽

Orbit Method ◽

Hermitian Symmetric Spaces ◽

Branching Laws ◽

Scalar Type ◽

Multiplicity Free

Kobayashi's multiplicity-free theorem asserts that irreducible unitary highest-weight representations π are multiplicity-free when restricted to every symmetric pair if π is of scalar type. The aim of this paper is to find the "classical limit" of this multiplicity-free theorem in terms of the geometry of two coadjoint orbits, for which the correspondence is predicted by the Kirillov–Kostant–Duflo orbit method.For this, we study the Corwin–Greenleaf multiplicity function [Formula: see text] for Hermitian symmetric spaces G/K. First, we prove that [Formula: see text] for any G-coadjoint orbit [Formula: see text] and any K-coadjoint orbit [Formula: see text] if [Formula: see text]. Here, 𝔤 = 𝔨 + 𝔭 is the Cartan decomposition of the Lie algebra 𝔤 of G.Second, we find a necessary and sufficient condition for [Formula: see text] by means of strongly orthogonal roots. This criterion may be regarded as the "classical limit" of a special case of the Hua–Kostant–Schmid–Kobayashi branching laws of holomorphic discrete series representations with respect to symmetric pairs.

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ON MATRIX KP AND SUPER-KP HIERARCHIES IN THE HOMOGENEOUS GRADING

International Journal of Modern Physics A ◽

10.1142/s0217751x96001565 ◽

1996 ◽

Vol 11 (18) ◽

pp. 3257-3295 ◽

Cited By ~ 5

Author(s):

F. TOPPAN

Keyword(s):

Power Series ◽

Symmetric Spaces ◽

Unexpected Result ◽

Hermitian Symmetric Spaces ◽

Regular Elements ◽

Symmetry Properties ◽

Alternating Sums ◽

Spectral Parameter Power Series ◽

Special Limit ◽

Free Parameters

Constrained KP and super-KP hierarchies of integrable equations (generalized NLS hierarchies) are systematically produced through a Lie-algebraic AKS matrix framework associated with the homogeneous grading. The role played by different regular elements in defining the corresponding hierarchies is analyzed, as well as the symmetry properties under the Weyl group transformations. The coset structure of higher order Hamiltonian densities is proven. For a generic Lie algebra the hierarchies considered here are integrable and essentially dependent on continuous free parameters. The bosonic hierarchies studied in Refs. 1 and 2 are obtained as special limit restrictions on Hermitian symmetric spaces. In the supersymmetric case the homogeneous grading is introduced consistently by using alternating sums of bosons and fermions in the spectral parameter power series. The bosonic hierarchies obtained from [Formula: see text] and the supersymmetric ones derived from the N=1 affinization of sl (2), sl (3) and osp (1|2) are explicitly constructed. An unexpected result is found: only a restricted subclass of the sl (3) bosonic hierarchies can be supersymmetrically extended while preserving integrability.

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