We consider asymptotic behavior of the dimension of the invariant subspace in a tensor product of several irreducible representations of a compact Lie group . It is equivalent to studying the symplectic volume of the symplectic quotient for a direct product of several coadjoint orbits of . 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 . 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 . 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 associated to the dominant weights.
"Asymptotic dimension of invariant subspace in tensor product representation of compact Lie group." J. Math. Soc. Japan 61 (3) 921 - 969, July, 2009. https://doi.org/10.2969/jmsj/06130921