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(U(p, q), U(p − 1, q)) is a generalized Gelfand pair

Research paper by Gerrit van Dijk

Indexed on: 02 Sep '08Published on: 02 Sep '08Published in: Mathematische Zeitschrift



Abstract

Denote by G = U(p, q) the orthogonal group of the sesqui-linear quadratic form \({[x,\, y]=x_1\overline y_1 +\cdots x_p\overline y_p -x_{p+1}\overline y_{p+1} -\cdots - x_{p+q}\overline y_{p+q}}\) on \({\mathbb C^{p+q}}\) and let H1 = U(p − 1, q) be the stabilizer of the first unit vector e1. Let H0 = U(1) and set H = H0 × H1. Define the character χl of H by \({\chi_l(h)=\chi_l (h_0h_1)=h_0^l\ (h_0\in H_0,\, h_1\in H_1)}\)where \({l\in\mathbb Z}\) . Define the anti-involution σ on G by \({\sigma (g)=\overline g^{-1}}\) . In this note we show that any distribution T on G satisfying T(h1gh2) = χl(h1h2) T(g) (g ∈ G; h1, h2 ∈ H) is invariant under the anti-involution σ. This result implies that (G,  H1) is a generalized Gelfand pair.