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

2017 ◽  
Vol 120 (1) ◽  
pp. 87 ◽  
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.

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

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.

scholarly journals REGULAR FUNCTIONS ON THE SHILOV BOUNDARY

2005 ◽  
Vol 04 (06) ◽  
pp. 613-629 ◽  
Author(s):  
OLGA BERSHTEIN
Keyword(s):  
Symmetric Domain ◽  
Principal Series ◽  
Bounded Symmetric Domain ◽  
Bounded Symmetric Domains ◽  
Shilov Boundary ◽  
Generators And Relations ◽  
Regular Functions ◽  
Degenerate Principal Series

In this paper a *-algebra of regular functions on the Shilov boundary S(𝔻) of bounded symmetric domain 𝔻 is constructed. The algebras of regular functions on S(𝔻) are described in terms of generators and relations for two particular series of bounded symmetric domains. Also, the degenerate principal series of quantum Harish–Chandra modules related to S(𝔻) = Un is investigated.

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

1996 ◽  
Vol 11 (18) ◽  
pp. 3257-3295 ◽  
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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