Hua type integrals over unitary groups and over projective limits of unitary groups

Abstract

We discuss some natural maps from a unitary group U(n) to a smaller group U(n-m) (these maps are versions of the Livshic characteristic function). We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces (values of the integrals are products of Gamma-functions). These integrals generalize Hua Loo Keng integrals. We construct inverse limits of unitary groups equipped with analogues of the Haar measure and evaluate some integrals over these inverse limits.

20 pages; the final version; evaluation of some normalizing constants is added

References in corpus (5)

• Infinite random matrices and ergodic measures
• Distributions on partitions, point processes, and the hypergeometric kernel
• Plancherel formula for Berezin deformation of $L^2$ on Riemannian symmetric space
• Matrix analogs of B-functions and Plancherel formula for Berezin kernel representations
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Cited by in corpus (14)

• Harmonic analysis on the infinite symmetric group
• A few remarks on Colour-Flavour Transformations,truncations of random unitary matrices, Berezin reproducing kernels and Selberg type integrals
• On polynomial integrals over the orthogonal group
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• Spreading maps (polymorphisms), symmetries of Poisson processes and matching summation
• The space $L^2$ on semi-infinite Grassmannian over finite field
• Convergence and an explicit formula for the joint moments of the Circular Jacobi $\beta$-Ensemble characteristic polynomial
• Moments of Generalized Cauchy Random Matrices and continuous-Hahn Polynomials
• Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations
• On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions
• The Hilbert-Schmidt analyticity associated with infinite-dimensional unitary groups
• Hua type beta-integrals and projective systems of measures on flag spaces
• Paley-Wiener isomorphism over infinite-dimensional unitary groups
• Description of unitary representations of the group of infinite $p$-adic integer matrices