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(GL(n + 1, ℝ), GL(n, ℝ)) is a generalized Gelfand pair

Research paper by G. van Dijk

Indexed on: 07 Dec '08Published on: 07 Dec '08Published in: Russian Journal of Mathematical Physics



Abstract

Denote by G = GL(n + 1, ℝ) the group of invertible (n + 1) × (n + 1) matrices with real entries, acting on ℝn+1 in the usual way, and let H1 = GL(n, ℝ) be the stabilizer of the first unit vector e0. Let H0 = GL(1, ℝ) and set H = H0 × H1. It is known that the pair (G,H) is a generalized Gelfand pair. Define a character χ of H by χ(h) = χ(h0h1) = χ0(h0) where χ0 is a unitary character of H0 (h0 ∈ H0, h1 ∈ H1). Let σ be the anti-involution on G given by σ(g) = tg. In this note, we show that any distribution T on G satisfying T(h1gh2) = χ(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.