Branching Abstract

Anthony W. Knapp

Abstract of "Branching theorems for compact symmetric spaces"

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 fixed 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 x K_1, where G/(K_2 x K_1) is any of the three quotients U(n+m)/(U(n) x U(m)), SO(n+m)/(SO(n) x SO(m)), or Sp(n+m)/(Sp(n) x Sp(m)), with n \leq m. For each of these compact symmetric spaces, one associates another compact symmetric space G'/K_2 with the following property: To each irreducible representation (\sigma,V) of G whose space V^{K_1} of K_1-fixed vectors is nonzero, there corresponds a canonical irreducible representation (\sigma',V') of G' such that the representations (\sigma|_{K_2},V^{K_1}) and (\sigma',V') are equivalent. For the situations under study, G'/K_2 is equal respectively to (U(n) x 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.