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 $Sp_{2n-2}$ in irreducible representations of $Sp_{2n}$. Our first theorem states that the map taking an element of $Sp_{2n}$ to its principal $n \times (n+1)$ submatrix induces an isomorphism of $\B$ to a different branching algebra $\B'$. The algebra $\B'$ encodes multiplicities of irreducible representations of $GL_{n-1}$ in certain irreducible representations of $GL_{n+1}$. Our second theorem is that each multiplicity space that arises in the restriction of an irreducible representation of $Sp_{2n}$ to $Sp_{2n-2}$ is canonically an irreducible module for the $n$-fold product of $SL_{2}$. In particular, this induces a canonical decomposition of the multiplicity spaces into one dimensional spaces, thereby resolving the multiplicities.