Asymptotic dimension of invariant subspace in tensor product representation of compact Lie group

Abstract

We consider asymptotic behavior of the dimension of the invariant subspace in a tensor product of several irreducible representations of a compact Lie group $G$. It is equivalent to studying the symplectic volume of the symplectic quotient for a direct product of several coadjoint orbits of $G$. We obtain two formulas for the asymptotic dimension. The first formula takes the form of a finite sum over tuples of elements in the Weyl group of $G$. Each term is given as a multiple integral of a certain polynomial function. The second formula is expressed as an infinite series over dominant weights of $G$. This could be regarded as an analogue of Witten's volume formula in 2-dimensional gauge theory. Each term includes data such as special values of the characters of the irreducible representations of $G$ associated to the dominant weights.