Geometric Models for the Spectra of Certain Gelfand Pairs Associated with Heisenberg Groups
نویسنده
چکیده
Let K be a compact Lie group acting on a finite dimensional Hermitian vector space V via some unitary representation. Now K acts by automorphisms on the associated Heisenberg group HV = V × R and we say that (K,HV ) is a Gelfand pair when the algebra LK(HV ) of integrable K-invariant functions on HV commutes under convolution. In this situation an application of the Orbit Method yields a injective mapping Ψ from the space ∆(K,HV ) of bounded K-spherical functions on HV to the space h ∗ V /K of K-orbits in the dual of the Lie algebra for HV . We prove that Ψ is a homeomorphism onto its image provided that the action of K on V is “well-behaved” in a sense made precise in this work. Our result encompasses a widely studied class of examples arising in connection with Hermitian symmetric spaces.
منابع مشابه
Gelfand pairs associated with finite Heisenberg groups
A topological group G together with a compact subgroup K are said to form a Gelfand pair if the set L1(K\G/K) of K-bi-invariant integrable functions on G is a commutative algebra under convolution. The situation where G and K are Lie groups has been the focus of extensive and ongoing investigation. Riemannian symmetric spaces G/K furnish the most widely studied and best understood examples. ([H...
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