Geometric Interpretation Abstract

Anthony W. Knapp

Abstract of "Geometric Interpretations of Two Branching Theorems of D. E. Littlewood"

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 \gt_{\gl} of U(n) has nonnegative highest weight \gl of depth \leq n/2. It says that the multiplicity in \gt_{\gl}|_{SO(n)} of an irreducible representation of SO(n) of highest weight \nu is the sum over \mu of the multiplicities of \gt_{\gl} in the U(n) tensor product \gt_{\mu}\otimes\gt_{\nu}, the allowable \mu'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 \gt_{\mu}\otimes\gt_{\nu} 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 \leq r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also.