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 τλSO(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 μʹs being all even nonnegative highest weights for U(n). Littlewoodʹs 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
