Inner Product of Irreducible Finite-dimensional Representations of Classical Groups

In this paper, we study the inner product of states corresponding to weights of irreducible finite-dimensional representations of classical groups. We propose a recursion algorithm for calculating the inner product effectively. As applications, we discuss the unitary of the representations space. We construct the norms of a special kind of states. We also determine the inner product of states in the minuscule representations. These results can be used to study the construction of the solutions to Kapustin-Witten equations basing on the fundamental solutions of Toda systems.