Geometric interpretations of two branching theorems of D.E. Littlewood

Abstract

D.E. Littlewood proved two branching theorems for decomposing the restriction of an irreducible finite-dimensional representation of a unitary group to a symmetric subgroup. One is for restriction of a representation of U(n) to the rotation group SO(n) when the given representation τλ of U(n) has nonnegative highest weight λ of depth ⩽n/2. It says that the multiplicity in τλ|SO(n) of an irreducible representation of SO(n) of highest weight ν is the sum over μ of the multiplicities of τλ in the U(n) tensor product τμ⊗τν, the allowable μ’s being all even nonnegative highest weights for U(n). Littlewood’s proof is character-theoretic. The present paper gives a geometric interpretation of this theorem involving the tensor products τμ⊗τν explicitly. The geometric interpretation has an application to the construction of small infinite-dimensional unitary representations of indefinite orthogonal groups and, for each of these representations, to the determination of its restriction to a maximal compact subgroup. The other Littlewood branching theorem is for restriction from U(2r) to the rank-r quaternion unitary group Sp(r). It concerns nonnegative highest weights for U(2r) of depth ⩽r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also.