Branching Theorems for Compact Symmetric Spaces 405 [

نویسنده

  • Anthony W. Knapp
چکیده

A compact symmetric space, for purposes of this article, is a quotient G=K, where G is a compact connected Lie group and K is the identity component of the subgroup of xed points of an involution. A branching theorem describes how an irreducible representation decomposes upon restriction to a subgroup. The article deals with branching theorems for the passage from G to K 2 K 1 , where G=(K 2 K 1) is any of U(n + m)=(U(n) U(m)), SO(n + m)=(SO(n) SO(m)), or Sp(n + m)=(Sp(n) Sp(m)), with n m. For each of these compact symmetric spaces, one associates another compact symmetric space G 0 =K 2 with the following property: To each irreducible representation (; V) of G whose space V K 1 of K 1-xed vectors is nonzero, there corresponds a canonical irreducible representation (0 ;V 0) of G 0 such that the representations (j K 2 ; V K 1) and (0 ;V 0) are equivalent. For the situations under study, G 0 =K 2 is equal respectively to (U(n) U(n))=diag(U(n)), U(n)=SO(n), and U(2n)=Sp(n), independently of m. Hints of the kind of \duality" that is suggested by this result date back to a 1974 paper by S. Gelbart. 1. Branching Theorems Branching theorems tell how an irreducible representation of a group decomposes when restricted to a subgroup. The rst such theorem historically for a compact connected Lie group is due to Hermann Weyl. It already appeared in the 1931 book W] and described how a representation of the unitary group U(n) decomposes when restricted to the subgroup U(n?1) embedded in the upper left n?1 entries. With respect to standard choices, the highest weight of the given representation may be written in the modern form a 1 e 1 + + a n e n , where a 1 a n are integers, or in the more traditional form (a 1 ; : : :; a n). Weyl's theorem is that the representation of U(n) with highest weight (a 1 ; : : :; a n) decomposes with multiplicity one under U(n ? 1), and the representations of U(n ? 1) that appear are exactly those with highest weights (c 1 ; : : :; c n?1) such that a 1 c 1 a 2 a n?1 c n?1 a n : (1.1) Similar results for rotation groups are due to Murnaghan and appeared in his 1938 …

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تاریخ انتشار 2001