An analysis of the multiplicity spaces in branching of symplectic groups

作者：Yacobi Oded^*

来源：Selecta Mathematica, New Series, 2010, 16(4): 819-855.

DOI：10.1007/s00029-010-0033-z

• Symplectic group
• Branching algebra
• Resolution of multiplicities

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 Sp2n to its principal n x (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.

• 出版日期2010-12