The non-degenerate spherical principal series of quantum Harish-Chandra modules is constructed. These modules appear in the theory of quantum bounded symmertic domains.
This work deals with function theory on quantum complex hyperbolic spaces. The principal notions are expounded. We obtain explicit formulas for invariant integrals on 'finite' functions on a quantum hyperbolic space and on the associated quantum isotropic cone. Also we establish principal series of U q su n,m -modules related to this cone. * Partially supported by the Stipend of the National Academy of Sciences of Ukraine under the action of quantum universal enveloping algebra U q su n,m . Finally, we introduce a quantum analog of the principal (unitary) series of U q su n,m -modules related to a quantum analog of the cone Ξ.These study were inspired and outlined by Leonid Vaksman some years ago. The authors are greatly indebted for him and D. Shklyarov for many helpful ideas towards this research.This project started out as joint work with Vaksman and Shklyarov. We are grateful to both of them for helpful discussions and drafts with preliminary definitions and computations.
PreliminariesLet q ∈ (0, 1). The Hopf algebra U q sl N is given by its generators K This * -algebra Pol(H n,m ) q will be called the algebra of regular functions on the quantum hyperbolic space.We are going to endow the * -algebra Pol(H n,m ) q with a structure of U q su n,m -module algebra [1]. For this purpose, we embed it into the U q su n,m -module * -algebra Pol X q of 'regular functions on the quantum principal homogeneous space' constructed in [11].Recall that Pol X q def = (C[SL N ] q , * ), with C[SL N ] q being the well-known algebra of regular functions on the quantum group SL N , and the involution * being defined byHere det q stands for the quantum determinant [1], and the matrix T ij is derived from the matrix T = (t kl ) by discarding its i's row and j's column. It follows from det q T = 1 that − n j=1
AbstractThis work deals with function theory on quantum complex hyperbolic spaces. The principal notions are expounded. We obtain explicit formulas for invariant integrals on 'finite' functions on a quantum hyperbolic space and on the associated quantum isotropic cone. Also we establish principal series of U q su n,m -modules related to this cone, and obtain the necessary conditions for those modules to be equivalent.
We obtain a q-analog of the well known result on a joint spectrum of invariant differential operators with polynomial coefficients on a prehomogeneous vector space of complex n × n-matrices. We are motivated by applications to the problems of harmonic analysis in the quantum matrix ball: our main theorem can be used while proving the Plancherel formula (to be published).Keywords: factorial Schur polynomials, Capelli identitites, quantum groups, quantum prehomogeneous vector spaces.MSC: 17B37, 20G42, 16S32.denote by z I J the minor of Z = (z i j ), which line numbers are from I and column numbers from J. Let ∂ I J be a similar minor of (∂ ij ). Puty k | C[Matn] λ is a scalar operator, since C[Mat n ] λ is a simple K-module and y k is K-invariant. There is an explicit formula for these scalars [11,7,6], [4, Proposition 3.3] (the so called Wallach-Okounkov formula):
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