Corwin-Greenleaf multiplicity functions for Hermitian symmetric spaces. (English) Zbl 1161.22008

Let \(G\) be a Lie group and \(H\) a closed subgroup of \(G\) with Lie algebras \(g\) and \(h\), respectively. There is a natural projection map \(pr:g^{*}\rightarrow h^{*}\), where \(g^{*}\) and \(h^{*}\) are the vector duals of \(g\) and \(h\), respectively. Given a co-adjoint orbit \({\mathcal O}^{G}\) of \(G\) and a co-adjoint orbit \({\mathcal O}^{H}\) of \(H\), the number \(n({\mathcal O}^{G},{\mathcal O}^{H})\) of \(H\)-orbits in the intersection \({\mathcal O}^{G}\cap pr^{-1}({\mathcal O}^{H})\) of \({\mathcal O}^{G}\) with the inverse image of \({\mathcal O}^{H}\) defines the so-called Corwin-Greenleaf multiplicity function.
In the paper under review, the author investigates relationships between this function, geometric properties of co-adjoint orbits and multiplicity formulas in the case where the group \(G\) is of Hermitian type. The main results proved by the author are a boundedness theorem and a non-vanishing criterion for Corwin-Greenleaf multiplicity functions. The proof is based on explicit formulas for the restriction of representations (branching laws) due to T. Kobayashi.
Reviewer: Salah Mehdi (Metz)


22E46 Semisimple Lie groups and their representations
22E60 Lie algebras of Lie groups
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
53C35 Differential geometry of symmetric spaces
81S10 Geometry and quantization, symplectic methods
Full Text: DOI Euclid


[1] L. Corwin and F. P. Greenleaf, Spectrum and multiplicities for restrictions of unitary representations in nilpotent Lie groups, Pacific J. Math. 135 (1988), no. 2, 233-267. · Zbl 0628.22007 · doi:10.2140/pjm.1988.135.233
[2] M. Duflo, G. Heckman and M. Vergne, Projection d’orbites, formule de Kirillov et formule de Blattner, Mém. Soc. Math. France (N.S.) No. 15 (1984), 65-128. · Zbl 0575.22014
[3] H. Fujiwara, Représentations monomiales des groupes de Lie résolubles exponentiels, in The orbit method in representation theory (Copenhagen, 1988) , 61-84, Progr. Math., 82, Birkhäuser, Boston, Boston, MA, 1990. · Zbl 0744.22010 · doi:10.1007/978-1-4612-4486-8_3
[4] F. P. Greenleaf, Harmonic analysis on nilpotent homogeneous spaces, in Representation theory and analysis on homogeneous spaces (New Brunswick, NJ, 1993) , 1-26, Contemp. Math., 177, Amer. Math. Soc., Providence, RI, 1994. · Zbl 0834.43006 · doi:10.1090/conm/177/01920
[5] L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains , Translated from the Russian by Leo Ebner and Adam Koranyi, Amer. Math. Soc., Providence, R.I, 1963. · Zbl 0112.07402
[6] H. P. Jakobsen and M. Vergne, Restrictions and expansions of holomorphic representations, J. Funct. Anal. 34 (1979), no. 1, 29-53. · Zbl 0433.22011 · doi:10.1016/0022-1236(79)90023-5
[7] A. A. Kirillov, Lectures on the orbit method , Amer. Math. Soc., Providence, RI, 2004. · Zbl 1229.22003
[8] T. Kobayashi, Discrete decomposability of the restriction of \(A_{q}(\lambda)\) with respect to reductive subgroups and its applications, Invent. Math. 117 (1994), no. 2, 181-205; Part II, Ann. of Math. 147 (1998), 1-21; Part III, Invent. Math. 131 (1998), 229-256. · Zbl 0826.22015 · doi:10.1007/BF01232239
[9] T. Kobayashi, Multiplicity-free branching laws for unitary highest weight modules, Proceedings of the Symposium on Representation Theory held at Saga, Kyushu 1997 (K. Mimachi, ed.), (1997), pp. 9-17. http://www.ms.u-tokyo.ac.jp/ toshi/pub/40.html T. Kobayashi, Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory, Amer. Math. Soc. Translation, Series II, 183 (1998), 1-31.
[10] T. Kobayashi, Geometry of multiplicity-free representations of \(\mathop{\mathrm{GL}}(n)\), visible actions on flag varieties, and triunity, Acta Appl. Math. 81 (2004), no. 1-3, 129-146. · Zbl 1050.22018 · doi:10.1023/B:ACAP.0000024198.46928.0c
[11] T. Kobayashi, Multiplicity-free representations and visible actions on complex manifolds, Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, 497-549. · Zbl 1085.22010 · doi:10.2977/prims/1145475221
[12] T. Kobayashi, Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs, in Representation theory and automorphic forms , 45-109, Progr. Math., 255, Birkhäuser, Boston, Boston, MA, 2007. · Zbl 1304.22013 · doi:10.1007/978-0-8176-4646-2_3
[13] T. Kobayashi, Visible actions on symmetric spaces, Transformation Groups, 12 , (2007), 671-694. · Zbl 1147.53041 · doi:10.1007/s00031-007-0057-4
[14] T. Kobayashi, A generalized Cartan decomposition for the double coset space \((U(n_{1}) \times U(n_{2}) \times U(n_{3})) \backslash U(n)/(U(p) \times U(q))\), J. Math. Soc. Japan, 59 (2007) 669-691. · Zbl 1124.22003 · doi:10.2969/jmsj/05930669
[15] T. Kobayashi, Propagation of multiplicity-free property for holomorphic vector bundles, math.RT/0607004 (Preprint). · Zbl 1284.32011
[16] T. Kobayashi and S. Nasrin, Multiplicity one theorem in the orbit method, in Lie groups and symmetric spaces , 161-169, Amer. Math. Soc. Transl. Ser. 2, 210, Amer. Math. Soc., Providence, RI, 2003. · Zbl 1048.22006
[17] T. Kobayashi and B. Ørsted, Conformal geometry and branching laws for unitary representations attached to minimal nilpotent orbits, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 8, 925-930. · Zbl 0910.22010 · doi:10.1016/S0764-4442(98)80115-8
[18] A. Korányi and J. A. Wolf, Realization of hermitian symmetric spaces as generalized half-planes, Ann. of Math. (2) 81 (1965), 265-288. · Zbl 0137.27402 · doi:10.2307/1970616
[19] R. L. Lipsman, Attributes and applications of the Corwin-Greenleaf multiplicity function, in Representation theory and analysis on homogeneous spaces (New Brunswick, NJ, 1993) , 27-46, Contemp. Math., 177, Amer. Math. Soc., Providence, RI, 1994. · Zbl 0836.22019 · doi:10.1090/conm/177/01925
[20] W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/1970), 61-80. · Zbl 0219.32013 · doi:10.1007/BF01389889
[21] D. A. Vogan, Jr., Unitary representations of reductive Lie groups , Ann. of Math. Stud., 118, Princeton Univ. Press, Princeton, NJ, 1987. · Zbl 0626.22011
[22] J. H. Xie, Restriction of discrete series of \(\mathrm{SU}(2,1)\) to \(\mathrm{S}(\mathrm{U}(1)\times\mathrm{U}(1,1))\), J. Funct. Anal. 122 (1994), no. 2, 478-518. · Zbl 0820.22009 · doi:10.1006/jfan.1994.1077
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.