The moment map for a multiplicity free action
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- by Chal Benson, Joe Jenkins, Ronald L. Lipsman and Gail Ratcliff PDF
- Bull. Amer. Math. Soc. 31 (1994), 185-190 Request permission
Abstract:
Let K be a compact connected Lie group acting unitarily on a finite-dimensional complex vector space V. One calls this a multiplicity-free action whenever the K-isotypic components of $\mathbb {C}{\text {[}}V]$ are K-irreducible. We have shown that this is the case if and only if the moment map $\tau :V \to {\mathfrak {k}^{\ast } }$ for the action is finite-to-one on K-orbits. This is equivalent to a result concerning Gelfand pairs associated with Heisenberg groups that is motivated by the Orbit Method. Further details of this work will be published elsewhere.References
- Chal Benson, Joe Jenkins, Ronald L. Lipsman, and Gail Ratcliff, A geometric criterion for Gelfand pairs associated with the Heisenberg group, Pacific J. Math. 178 (1997), no. 1, 1–36. MR 1447402, DOI 10.2140/pjm.1997.178.1
- Chal Benson, Joe Jenkins, and Gail Ratcliff, On Gel′fand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc. 321 (1990), no. 1, 85–116. MR 1000329, DOI 10.1090/S0002-9947-1990-1000329-0
- Giovanna Carcano, A commutativity condition for algebras of invariant functions, Boll. Un. Mat. Ital. B (7) 1 (1987), no. 4, 1091–1105 (English, with Italian summary). MR 923441
- Lawrence Corwin and Frederick P. Greenleaf, Spectrum and multiplicities for restrictions of unitary representations in nilpotent Lie groups, Pacific J. Math. 135 (1988), no. 2, 233–267. MR 968611, DOI 10.2140/pjm.1988.135.233
- I. M. Gel′fand, Spherical functions in symmetric Riemann spaces, Doklady Akad. Nauk SSSR (N.S.) 70 (1950), 5–8 (Russian). MR 0033832
- V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538. MR 664118, DOI 10.1007/BF01398934
- Victor Guillemin and Shlomo Sternberg, Multiplicity-free spaces, J. Differential Geom. 19 (1984), no. 1, 31–56. MR 739781
- G. J. Heckman, Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups, Invent. Math. 67 (1982), no. 2, 333–356. MR 665160, DOI 10.1007/BF01393821
- Roger Howe and T\B{o}ru Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), no. 3, 565–619. MR 1116239, DOI 10.1007/BF01459261
- V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), no. 1, 190–213. MR 575790, DOI 10.1016/0021-8693(80)90141-6
- Friedrich Knop, A Harish-Chandra homomorphism for reductive group actions, Ann. of Math. (2) 140 (1994), no. 2, 253–288. MR 1298713, DOI 10.2307/2118600
- Ronald L. Lipsman, Orbit theory and harmonic analysis on Lie groups with co-compact nilradical, J. Math. Pures Appl. (9) 59 (1980), no. 3, 337–374. MR 604474
- Ronald L. Lipsman, Orbital parameters for induced and restricted representations, Trans. Amer. Math. Soc. 313 (1989), no. 2, 433–473. MR 930083, DOI 10.1090/S0002-9947-1989-0930083-1
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 31 (1994), 185-190
- MSC: Primary 22C05; Secondary 22E30
- DOI: https://doi.org/10.1090/S0273-0979-1994-00514-2
- MathSciNet review: 1260517