An analysis of the multiplicity spaces in branching of symplectic groups, Selecta Math N.S., Volume 16, Issue 4, (2010) E-mail address : skim@maths.uq.edu.au School of Mathematics and

Abstract. 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 first theorem states that the map taking an element of Sp2n to its principal n × (n+ 1) submatrix induces an isomorphism of B to a different 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 on