The structure of the space of coadjoint orbits of an exponential solvable Lie group

Author:
Bradley N. Currey

Journal:
Trans. Amer. Math. Soc. **332** (1992), 241-269

MSC:
Primary 22E25; Secondary 22E15, 22E27

DOI:
https://doi.org/10.1090/S0002-9947-1992-1046014-2

MathSciNet review:
1046014

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Abstract: In this paper we address the problem of describing in explicit algebraic terms the collective structure of the coadjoint orbits of a connected, simply connected exponential solvable Lie group . We construct a partition of the dual of the Lie algebra of into finitely many -invariant algebraic sets with the following properties. For each , there is a subset of which is a cross-section for the coadjoint orbits in and such that the natural mapping is bicontinuous. Each is the image of an analytic -invariant function on and is an algebraic subset of . The partition has a total ordering such that for each , is Zariski open. For each there is a cone , such that is naturally a fiber bundle over with fiber and projection . There is a covering of by finitely many Zariski open subsets such that in each , there is an explicit local trivialization . Finally, we show that if is the minimal element of (containing the generic orbits), then its cross-section is a differentiable submanifold of . It follows that there is a dense open subset of such that has the structure of a differentiable manifold and has Plancherel measure zero.

**[1]**I. Brown,*Dual topology of a nilpotent Lie group*, Ann. Sci. École Norm. Sup.**6**(1973), 407-411. MR**0352326 (50:4813)****[2]**L. Corwin, F. P. Greenleaf, and G. Grelaud,*Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups*, Trans. Amer. Math. Soc.**304**(1988), 549-583. MR**911085 (89b:22013)****[3]**B. Currey ,*On the dual of an exponential solvable Lie group*, Trans. Amer. Math. Soc.**309**(1988), 295-307. MR**957072 (89i:22015)****[4]**B. Currey and R. Penney,*The structure of the space of coadjoint orbits of a completely solvable Lie group*, Michigan Math. J.**36**(1989), 309-320. MR**1000533 (90f:22012)****[5]**M. Duflo and M. Rais,*Sur l'analyse harmonique sur les groupes de Lie resolubles*, Ann. Sci. École Norm. Sup.**9**(1976), 107-144. MR**0435294 (55:8254)****[6]**H. Fujiwara,*Representations monomiales des groupes de Lie resolubles exponentiels*, Progress in Math.**82**(1990), 61-84. MR**1095341 (92g:22021)****[7]**-,*Sur le dual d'un group de Lie resoluble exponentiel*, J. Math. Soc. Japan**36**(1984), 629-636. MR**759419 (87f:22008)****[8]**-,*Sur les restrictions des representations unitaires des groupes de Lie resolubles exponentiels*(to appear).**[9]**R. Lipsman,*Induced representations of completely solvable Lie groups*, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. (4)**17**(1990), 127-164. MR**1074629 (91j:22005)****[10]**-,*Orbital parameters for induced and restricted representations*, Trans. Amer. Math. Soc.**313**(1989), 433-473. MR**930083 (90a:22008)****[11]**-,*Restricting representations of completely solvable Lie groups*, Canad. J. Math.**42**(1990), 790-824. MR**1080997 (92h:22018)****[12]**N. V. Pedersen,*Geometric quantization and the universal enveloping algebra of nilpotent Lie groups*, Trans. Amer. Math. Soc.**315**(1989), 511-563. MR**967317 (90c:22026)****[13]**-,*On the characters of exponential solvable Lie groups*, Ann. Sci. École Norm. Sup.**17**(1984), 1-29. MR**744065 (85k:22022)****[14]**-,*On the infinitessimal kernel of irreducible representations of nilpotent Lie groups*, Bull. Soc. Math. France**112**(1984), 423-467. MR**802535 (87a:22018)****[15]**-,*Semicharacters and solvable Lie groups*, Math. Ann.**247**(1980), 191-244. MR**568989 (81j:22015)****[16]**L. Pukanszky,*On the characters and the Plancherel formula of nilpotent Lie groups*, J. Funct. Anal.**1**(1967), 255-280. MR**0228656 (37:4236)****[17]**-,*On the unitary representations of exponential groups*, J. Funct. Anal.**2**(1968), 73-113. MR**0228625 (37:4205)****[18]**-,*Unitary representations of solvable Lie groups*, Ann. Sci. École Norm. Sup.**4**(1971), 457-608. MR**0439985 (55:12866)**

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DOI:
https://doi.org/10.1090/S0002-9947-1992-1046014-2

Article copyright:
© Copyright 1992
American Mathematical Society