Quantization and harmonic analysis on nilpotent Lie groups
Weyl Quantization is a procedure for associating a function on which the canonical commutation relations are realized. If G is a simply-connected, connected nilpotent Lie group with Lie algebra g and dual g/sup */, it is shown how to inductively construct symplectic isomorphisms between every co-adjoint orbit O and the bundle in Hilbert Space for some m. Weyl Quantization can then be used to associate to each orbit O a unitary representation rho/sub 0/ of G, recovering the classification of the unitary dual by Kirillov. It is used to define a geometric Fourier transform, F : L/sup 1/(G) ..-->.. functions on g/sup */, and it is shown that the usual operator-valued Fourier transform can be recovered from F, characters are inverse Fourier transforms of invariant measures on orbits, and matrix coefficients are inverse Fourier transforms of non-invariant measures supported on orbits. Realizations of the representations rho/sub 0/ in subspaces of L/sup 2/(O) are obtained.. Finally, the kernel function is computed for the upper triangular unipotent group and one other example.
- Research Organization:
- Yale Univ., New Haven, CT (USA)
- OSTI ID:
- 5464732
- Resource Relation:
- Other Information: Thesis (Ph. D.)
- Country of Publication:
- United States
- Language:
- English
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