## Quantization and representations of solvable Lie groups.(English)Zbl 0203.03302

Let $$G$$ be a connected, simply connected, solvable Lie group. Theorem 1 provides a necessary and sufficient condition for $$G$$ to be of Type I. Assuming that $$G$$ is of Type I, Theorem 2 provides a description of the irreducible representations of $$G$$ and Theorem 3 provides a construction of such representations (to within equivalence). The article is concerned solely with discussing and setting forth these theorems, and contains no proofs.
Reviewer: J. W. Baker

### MSC:

 2.2e+28 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
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### References:

 [1] Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. No. 62 (1966), 199. · Zbl 0204.14202 [2] P. Bernat, Sur les représentations unitaires des groups de Lie résolubles, Ann. Sci. École Norm. Sup. (3) 82 (1965), 37 – 99 (French). · Zbl 0138.07302 [3] A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57 – 110 (Russian). · Zbl 0090.09802 [4] Lajos Pukánszky, On the theory of exponential groups, Trans. Amer. Math. Soc. 126 (1967), 487 – 507. · Zbl 0207.33605
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