  In this paper we give an asymptotic formula for a matrix integral which plays
a crucial role in the approach of Diaconis et al. to random matrix eigenvalues.
The choice of parameter for the asymptotic analysis is motivated by an
invariant theoretic interpretation of this type of integral. For arbitrary
regular irreducible representations of arbitrary connected semisimple compact
Lie groups, we obtain an asymptotic formula for the trace of certain operators
on the space of tensor invariants, thus extending a result of Biane on the
dimension of these spaces.
