# Quantization and harmonic analysis on nilpotent Lie groups

## Abstract

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.

- Authors:

- Publication Date:

- Research Org.:
- Yale Univ., New Haven, CT (USA)

- OSTI Identifier:
- 5464732

- Resource Type:
- Thesis/Dissertation

- Resource Relation:
- Other Information: Thesis (Ph. D.)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; LIE GROUPS; FOURIER TRANSFORMATION; QUANTIZATION; ALGEBRA; HILBERT SPACE; UNITARITY; BANACH SPACE; INTEGRAL TRANSFORMATIONS; MATHEMATICAL SPACE; MATHEMATICS; SPACE; SYMMETRY GROUPS; TRANSFORMATIONS; 658000* - Mathematical Physics- (-1987)

### Citation Formats

```
Wildberger, N J.
```*Quantization and harmonic analysis on nilpotent Lie groups*. United States: N. p., 1983.
Web.

```
Wildberger, N J.
```*Quantization and harmonic analysis on nilpotent Lie groups*. United States.

```
Wildberger, N J. Sat .
"Quantization and harmonic analysis on nilpotent Lie groups". United States.
```

```
@article{osti_5464732,
```

title = {Quantization and harmonic analysis on nilpotent Lie groups},

author = {Wildberger, N J},

abstractNote = {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.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1983},

month = {1}

}