MULTIPLICITY-FREE DECOMPOSITIONS OF THE MINIMAL REPRESENTATION OF THE INDEFINITE ORTHOGONAL GROUP

摘要

Kazhdan, Kostant, Binegar-Zierau and Kobayashi-Orsted constructed a distinguished infinite-dimensional irreducible unitary representation p of the indefinite orthogonal group G = O(2p, 2q) for p, q >= 1 with p + q > 2, which has the smallest Gelfand-Kirillov dimension 2p + 2q - 3 among all infinite-dimensional irreducible unitary representations of G and hence is called the minimal representation. We consider, for which subgroup G' of G, the restriction pi vertical bar G' is multiplicity-free. We prove that the restriction of pi to any subgroup containing the direct product group U(p(1)) x U(p(2)) x U(q) for p(1), p(2) >= 1 with p(1) + p(2) = p is multiplicity-free, whereas the restriction to U(p(1)) x U(p(2)) x U(q(1)) x U(q(2)) for q(1), q(2) >= 1 with q(1) + q(2) = q has infinite multiplicities.