An Analysis of the Multiplicity Spaces in Branching of Symplectic Groups

Branching of symplectic groups is not multiplicity-free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra B. The algebra B is a graded algebra whose components encode the multiplicities of irreducible representations of Sp2n−2 in irreducible representations of Sp2n. Our rst theorem states that the map taking an element of Sp2n to its principal n× (n+ 1) submatrix induces an isomorphism of B to a di erent branching algebra B ′. The algebra B ′ encodes multiplicities of irreducible representations of GLn−1 in certain irreducible representations of GLn+1. Our second theorem is that each multiplicity space that arises in the restriction of an irreducible representation of Sp2n to Sp2n−2 is canonically an irreducible module for the n-fold product of SL2. In particular, this induces a canonical decomposition of the multiplicity spaces into one dimensional spaces, thereby resolving the multiplicities.