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

Author： Dijk Gerrit  

Publisher： Springer Publishing Company

ISSN： 0025-5874

Source： Mathematische Zeitschrift, Vol.261, Iss.3, 2009-03, pp. : 525-529

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Abstract

Denote by G = U(p, q) the orthogonal group of the sesqui-linear quadratic form $${[x,, y]=x_1overline y_1 +cdots x_poverline y_p -x_{p+1}overline y_{p+1} -cdots - x_{p+q}overline y_{p+q}}$$ 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 $${chi_l(h)=chi_l (h_0h_1)=h_0^l (h_0in H_0,, h_1in H_1)}$$ where $${linmathbb Z}$$ . 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.

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