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.