Research Interests/Projects

Harmonic analysis on homogeneous spaces and representations of Lie groups.

Explicit realization of singular representations of semisimple Lie groups and Lie algebras. Composition series and unitary structure of principle representations of Lie algebras. Holomorphic discrete series and their tensor products. Howe dual pairs. Special functions related to the decomposition of representations. Branching of holomorphic representations and coupling coefficients. Expansion of spherical functions. Symmetric and non-symmetric polynomials. Berezin quantization.

Bloch functions on bounded symmetric domains. Explicit realization (using Jordan triples) of invariant differential operators on line bundles over bounded symmetric domains. Relative discrete series of line bundles. Toeplitz and Hankel operators Hankel operators and Hankel forms on bounded symmetric domains. Relation to intertwining operators and invariant theory, for example, transvectant, Bol's lemma. Schatten - von Neumann properties of Hankel operators and forms. Dixmier trace and Toeplitz operators Hardy and Bergman spaces on bounded symmetric domains and their analytic continuation. Spaces of vector-valued analytic functions and harmonic $L^2$-forms on symmetric spaces. Expansion of reproducing matrix-valued reproducing kernel. Intertwining operator to relative discrete series. L^p-properties of the Bergman projection.


Some Related Recent Publications

Radon, cosine and sine transforms on Grassmannian manifolds preprint, 2006, pdf file,

Radon transform on symmetric matrix domains preprint, 2005/06, Trans. Amer. Math. Soc. to appear 2009(?) ps file,

Radon transform on real, complex and quaternionic Grassmannians Duke Math. J., 2007 ps file,

(with K. Koufany) Hua operators and Poisson transform on bounded symmetric domains J. Functional Analysis, to appear, 2006/7 ps file,

(with S. Sahi Bi-orthogonal expansions of the non-symmetric Jack functions MPIM-preprint 2007 ps file, to appear in SIGMA, Volume dedicated to Tom Branson.

(with L. Peng) Non-symmetric Jacobi and Wilson polynomials Intern. Math. Res. Notices, to appear, 2006. ps file,

Spherical transform and Jacobi type polynomials on root system of Type BC Intern. Math. Res. Notices, 2005.

(with M. Englis) On the Faraut-Koranyi hypergeometric functions in rank two Ann. Inst. Fourier, 2004, ps file,

Shimura invariant differential operators and their eigenvalues Math. Ann., 2001

Invariant Differential Operators on Bounded Symmetric Domains and Their Eigenvalues. preprint, 1999, dvi file Israel J. Math, 119(2000), 157-185.

Some recurrence formulas of spherical polynomials on tube domains, Trans. Amer. Math. Soc. , 347 (1995), 1725-1734

(with B. \Orsted) Laplacians on the quotients of Cauchy-Riemann complex and Szegö map for L^2-harmonic forms preprint, 2004, ps file,

Nearly holomorphic functions and realization of relative discrete series ESI preprint no 741, 1999, J. Math. Kyoto Univ., to appear

(with Jaak Peetre) Invariant Cauchy-Riemann operators and Realization of Relative Discrete Series of Line Bundle over the Unit Ball of $\mathbf C^n$}, Michigan J. Math, 45 (1998), no. 2, 387--397.

(with A. H. Dooley and Bent Orsted) Relative discrete series of line bundles over bounded symmetric domains, Annales de l'Institut Fourier, 46 (1996), 1011-1026.

Branching coefficients of holomorphic representations and Segal-Bargmann transform, J. Funct. Anal. 2003 ps file,

(with Lizhong Peng) Tensor product of holomorphic representations and bilinear differential operators ps file, JFA, to appear

(with Bent \Orsted ) Reproducing kernels and composition series for spaces of vector-valued holomorphic functions, Pacific J. Math. 171 (1995), no. 2, 493--510.

Berezin transform on real bounded symmetric domains Trans. Amer. Math. Soc., 2001,

Segal-Bargman transform and Laplace transform (with G. Olafsson, M. Davidson) to appear, J. Functional Analysis

Berezin transform on compact Hermitian symmetric spaces. Manuscripta Math. 97 (1998), no. 3, 371--388.

Tensor product of minimal holomorphic representations Representation Theory, 2001,

(with Kunyu Guo and Miraslav Englis) Toeplitz and Hankel operators and Dixmier traces on the unit ball of $\mathbb C^n$ Proc. Amer. Math. Soc. to appear, 2007/08, ps file,

Homogeneous tuple of multiplication operators on bounded symmetric domains (with Jonathan Arazy) J. Funct. Anal. 2003

A class of Hankel type operators related to complementary series (with M. Englis, S. Hille, J. Peetre, H. Rosengren) to appear, Arabic J. Math.

(with Bent \Orsted) Generalized principal series representations and tube domains, Duke Math. J., 78 (1995), 335-357

Jordan algebras and generalized principal series representations, Mathematische Annalen, 302(1995), 773-786.

(with A. H. Dooley) Generalized principal series representations of SL(1+n, C), Proc. Amer. Math. Soc. 126 (1998), no. 12, 3693--3699.

(with Bent \Orsted ) $L^2$-versions of the Howe Correspondence II, J. Math. Pures et Appl., 74 (1995), 165-183.

(with Bent \Orsted ) $L^2$-versions of the Howe Correspondence I, Math. Scand., 1995.

The asymptotics of spherical functions and the central limit theorem on symmetric cones, Annales de l'Institut Fourier, 45 (1995), 567-575

(with Dmitry Shklyarov) Berezin transform on the quantum unit ball J. Math. Phys., to appear, ps file,

Conferences/Workshops/Visiting

Representation Theory, Lie groups and conformal geometry, April 2009

Oberwolfach Meeting on Hilbert Modules, April 2009

IHES, Aug/Sept. 2009


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