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Hua-type integrals over unitary groups and over projective limits of unitary groups. (English) Zbl 1019.43008

The author studies natural maps from U(n) to U(n-1) for which the image of the normalized Haar measure of U(n) is that of U(n-1). This makes it possible to define probability measures on infinite dimensional unitary groups. A family of integrals relative to these measures are evaluated. Let us state more precisely some of the main results of the paper. For K=R, C, or H, U 0 (n,K) is the group SO(n), U(n) or Sp(n). The author considers the map

γ m :U 0 (n,K)U 0 (n-m,K),
g=PQRTT-R(1+P) -1 Q,

for an (m+(n-m))×(m+(n-m)) block representation of g. It is proved that the image under the map γ m of the normalized Haar measure σ n of U 0 (n,K) is the Haar measure σ n-m . Further the author considers the map

ξ m :U 0 (n,K)U 0 (n-m,K)×B m ,
gγ m (g) , [g] m ,

where [g] m is the upper left m×m block of the matrix g, and B m is the unit ball in the space of m×m matrices over K and proves that the image under ξ m of σ n is

σ n-m (g)×Const.det(1-Z * Z) r-1 dZ

with τ=1 2(n-2m+1)dim R K and dZ the Lebesgue measure. As an application the integrals

U 0 (n,K) k=1 n det1 + [g] n-k+1 λ k -λ k-1 dσ n (g)

are evaluated in terms of the gamma function.


MSC:
43A85Analysis on homogeneous spaces
53C35Symmetric spaces (differential geometry)
15A52Random matrices (MSC2000)