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

[摘要] 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 tau(lambda) of U(n) has nonnegative highest weight lambda of depth less than or equal to n/2. It says that the multiplicity in tau(lambda)\(SO(n)) of an irreducible representation of SO(n) of highest weight v is the sum over mu of the multiplicities of tau(lambda) in the U (n) tensor product tau(mu) circle times tau(v), 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 tau(mu) circle times tau(v) 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 less than or equal to r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also. (C) 2003 Elsevier Inc. All rights reserved.

[发布日期] 2003-12-15

[关键词] branching theorem;classical group;representation