A multivariate version of the disk convolution
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Date
2015-04
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Abstract
We present an explicit product formula for the spherical functions of the compact
Gelfand pairs (G,K_1) = (SU(p + q), SU(p) × SU(q)) with p ≥ 2q, which can be considered
as the elementary spherical functions of one-dimensional K-type for the Hermitian
symmetric spaces G/K with K = S(U(p) × U(q)). Due to results of Heckman, they can
be expressed in terms of Heckman-Opdam Jacobi polynomials of type BC_q with specific
half-integer multiplicities. By analytic continuation with respect to the multiplicity
parameters we obtain positive product formulas for the extensions of these spherical functions
as well as associated compact and commutative hypergroup structures parametrized
by real p ∈]2q−1,∞[. We also obtain explicit product formulas for the involved continuous
two-parameter family of Heckman-Opdam Jacobi polynomials with regular, but not
necessarily positive multiplicities. The results of this paper extend well known results for
the disk convolutions for q = 1 to higher rank.
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Keywords
hypergeometric functions associated with root systems, Heckman-Opdam theory, Jacobi polynomials, disk hypergroups, positive product formulas, compact Grassmann manifolds, spherical functions