Abstract :
Several methods of evaluation are presented for a family {In,d,p
} of Selberg-like integrals that arise in the computation of the
algebraic–geometric degrees of a family of spherical nilpotent orbits associated to the symmetric space of a simple real Lie group.
Adapting the technique of Nishiyama, Ochiai and Zhu, we present an explicit evaluation in terms of certain iterated sums over
permutation groups. The resulting formula, however, is only valid when the integrand involves an even power of the Vandermonde
determinant. We then apply, to the general case, the theory of symmetric functions and obtain an evaluation of the integral In,d,p
as a product of polynomial of fixed degree times a particular product of gamma factors; thereby identifying the asymptotics of
the integrals with respect to their parameters. Lastly, we derive a recursive formula for evaluation of another general class of
Selberg-like integrals, by applying some of the technology of generalized hypergeometric functions.
© 2008 Elsevier Inc. All rights reserved
