Mathematics – Representation Theory
Scientific paper
2004-07-27
Mathematics
Representation Theory
39 pages
Scientific paper
We study branching laws for a classical group $G$ and a symmetric subgroup $H$. Our approach is through the {\it branching algebra}, the algebra of covariants for $H$ in the regular functions on the natural torus bundle over the flag manifold for $G$. We give concrete descriptions of (natural subalgebras of) the branching algebra using classical invariant theory. In this context, it turns out that the ten classes of classical symmetric pairs $(G,H)$ are associated in pairs, $(G,H)$ and $(H',G')$, and that the (partial) branching algebra for $(G,H)$ also describes a branching law from $H'$ to $G'$. (However, the second branching law may involve certain infinite-dimensional highest weight modules for $H'$.) To highlight the fact that these algebras describe two branching laws simultaneously, we call them {\it reciprocity algebras}. Our description of the reciprocity algebras reveals that they all are related to the tensor product algebra for $GL_n$. This relation is especially strong in the {\it stable range}. We give quite explicit descriptions of reciprocity algebras in the stable range in terms of the tensor product algebra for $GL_n$. This is the structure lying behind formulas for branching multiplicities in terms of Littlewood-Richardson coefficients.
Howe Roger E.
Tan Eng Chye
Willenbring Jeb F.
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