Infinite dimensional multiplicity free spaces. III: Matrix coefficients and regular functions. (English) Zbl 1207.22012
[Part I: Limits of compact commutative spaces, in: K.-H. Neeb and A. Pianzola (eds.), “Developments and trends in infinite dimensional Lie theory”, Birkhäuser (to appear); Part II in Contemporary Mathematics 491, 179–208 (2009; Zbl 1180.22021).]
Let \(G\) be a locally compact topological group and \(K\) a compact subgroup. Then \((G,K)\) is a Gelfand pair, i.e. \(L^1(K\setminus G/K)\) is commutative under convolution, if and only if the left regular representation of \(G\) on \(L^2(G/K)\) is multiplicity free. In the paper under review the author defines a ring \({\mathcal A}(G/K)\) of regular functions as a subalgebra of the inverse limit of finite dimensional Gelfand pairs in the usual sense. This is accompanied by a \(G\)-equivariant map \({\mathcal A}(G/K)\to L^2(G/K)\). The main result of this paper asserts that both for the direct limit of compact symmetric spaces and for the direct limit of commutative nilmanifolds \({\mathcal A}(G/K)\) injects to a dense subspace of \(L^2(G/K)\), so \(L^2(G/K)\) defines a \(G\)-invariant inner product on \({\mathcal A}(G/K)\), the regular representation of \(G\) on \({\mathcal A}(G/K)\) is unitarized, and \(L^2(G/K)\) can be interpreted as a Hilbert space completion of \({\mathcal A}(G/K)\).
Let \(G\) be a locally compact topological group and \(K\) a compact subgroup. Then \((G,K)\) is a Gelfand pair, i.e. \(L^1(K\setminus G/K)\) is commutative under convolution, if and only if the left regular representation of \(G\) on \(L^2(G/K)\) is multiplicity free. In the paper under review the author defines a ring \({\mathcal A}(G/K)\) of regular functions as a subalgebra of the inverse limit of finite dimensional Gelfand pairs in the usual sense. This is accompanied by a \(G\)-equivariant map \({\mathcal A}(G/K)\to L^2(G/K)\). The main result of this paper asserts that both for the direct limit of compact symmetric spaces and for the direct limit of commutative nilmanifolds \({\mathcal A}(G/K)\) injects to a dense subspace of \(L^2(G/K)\), so \(L^2(G/K)\) defines a \(G\)-invariant inner product on \({\mathcal A}(G/K)\), the regular representation of \(G\) on \({\mathcal A}(G/K)\) is unitarized, and \(L^2(G/K)\) can be interpreted as a Hilbert space completion of \({\mathcal A}(G/K)\).
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |
| 17B65 | Infinite-dimensional Lie (super)algebras |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 43A85 | Harmonic analysis on homogeneous spaces |
Keywords:
Gelfand pair; direct limit; symmetric space; regular representation; regular function; Hilbert spaceCitations:
Zbl 1180.22021References:
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