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Spectra for Gelfand pairs associated with the Heisenberg group

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Benson C., Jenkins J., Ratcliff G., Worku T.

Colloquium Mathematicum

1996

tom 71

nr 2

305-328

Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group $H_n$. We say that $(K,H_n)$ is a Gelfand pair when the set $L^1_K(H_n)$ of integrable K-invariant functions on $H_n$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for $L^1_K(H_n)$ can be identified with the set $Δ(K,H_n)$ of bounded K-spherical functions on $H_n$. In this paper, we study the natural topology on $Δ(K,H_n)$ given by uniform convergence on compact subsets in $H_n$. We show that $Δ(K,H_n)$ is a complete metric space and that the 'type 1' K-spherical functions are dense in $Δ(K,H_n)$. Our main result shows that one can embed $Δ(K,H_n)$ quite explicitly in a Euclidean space by mapping a spherical function to its eigenvalues with respect to a certain finite set of ($K ⋉ H_n$)-invariant differential operators on $H_n$. This viewpoint on the spectrum for $Δ(K,H_n)$ was previously known for K=U(n) and is referred to as 'the Heisenberg fan'.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Benson C.

1 Jenkins J.

1 Ratcliff G.

1 Worku T.

1 1996

Wyniki wyszukiwania

Spectra for Gelfand pairs associated with the Heisenberg group