Spin\(^c\)-quantization and the \(K\)-multiplicities of the discrete series. (English) Zbl 1091.53059
Summary: In the 1970s, W. Schmid has shown that the representations of the discrete series of a real semi-simple Lie group \(G\) could be realized as the quantization of elliptic coadjoint orbits. We show that such orbits, equipped with the Hamiltonian action of a maximal compact subgroup \(K \subset G\), are non-compact examples where the philosophy of Guillemin-Sternberg – Quantization commutes with reduction – applies. If \(\mathcal H_{\mathcal O}\) is a representation of the discrete series of \(G\) associated to a coadjoint orbit \(\mathcal O\), we express the \(K\)-multiplicities of \(\mathcal H_{\mathcal O}\) in terms of Spin\(^c\)-index on symplectic reductions of \(\mathcal O\).
MSC:
| 53D50 | Geometric quantization |
| 22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
| 53D20 | Momentum maps; symplectic reduction |
| 58J20 | Index theory and related fixed-point theorems on manifolds |
References:
| [1] | Atiyah M.F. , Elliptic Operators and Compact Groups , Lecture Notes in Math. , vol. 401 , Springer-Verlag , Berlin/New York , 1974 . MR 482866 | Zbl 0297.58009 · Zbl 0297.58009 · doi:10.1007/BFb0057821 |
| [2] | Atiyah M.F. , Convexity and commuting Hamiltonians , Bull. London Math. Soc. 14 ( 1982 ) 1 - 15 . MR 642416 | Zbl 0482.58013 · Zbl 0482.58013 · doi:10.1112/blms/14.1.1 |
| [3] | Atiyah M.F. , Bott R. , Shapiro A. , Clifford modules , Topology 3 ( Suppl. 1 ) ( 1964 ) 3 - 38 . MR 167985 | Zbl 0146.19001 · Zbl 0146.19001 · doi:10.1016/0040-9383(64)90003-5 |
| [4] | Atiyah M.F. , Segal G.B. , The index of elliptic operators II , Ann. of Math. 87 ( 1968 ) 531 - 545 . MR 236951 | Zbl 0164.24201 · Zbl 0164.24201 · doi:10.2307/1970716 |
| [5] | Atiyah M.F. , Singer I.M. , The index of elliptic operators I , Ann. of Math. 87 ( 1968 ) 484 - 530 . MR 236950 | Zbl 0164.24001 · Zbl 0164.24001 · doi:10.2307/1970715 |
| [6] | Atiyah M.F. , Singer I.M. , The index of elliptic operators III , Ann. of Math. 87 ( 1968 ) 546 - 604 . MR 236952 | Zbl 0164.24301 · Zbl 0164.24301 · doi:10.2307/1970717 |
| [7] | Atiyah M.F. , Singer I.M. , The index of elliptic operators IV , Ann. of Math. 93 ( 1971 ) 139 - 141 . MR 279833 | Zbl 0212.28603 · Zbl 0212.28603 · doi:10.2307/1970756 |
| [8] | Berline N. , Getzler E. , Vergne M. , Heat Kernels and Dirac Operators , Grundlehren Math. Wiss. , vol. 298 , Springer-Verlag , Berlin , 1991 . MR 2273508 | Zbl 0744.58001 · Zbl 0744.58001 |
| [9] | Berline N. , Vergne M. , The Chern character of a transversally elliptic symbol and the equivariant index , Invent. Math. 124 ( 1996 ) 11 - 49 . MR 1369410 | Zbl 0847.46037 · Zbl 0847.46037 · doi:10.1007/s002220050045 |
| [10] | Berline N. , Vergne M. , L’indice équivariant des opérateurs transversalement elliptiques , Invent. Math. 124 ( 1996 ) 51 - 101 . MR 1369411 | Zbl 0883.58037 · Zbl 0883.58037 · doi:10.1007/s002220050046 |
| [11] | Cannas da Silva A. , Karshon Y. , Tolman S. , Quantization of presymplectic manifolds and circle actions , Trans. Amer. Math. Soc. 352 ( 2000 ) 525 - 552 . MR 1714519 | Zbl 0977.53080 · Zbl 0977.53080 · doi:10.1090/S0002-9947-99-02260-6 |
| [12] | Duflo M. , Représentations de carré intégrable des groupes semi-simples réels, in: Sém. Bourbaki (1977/78), Exp. n o 508 , Lecture Notes in Math. 710 ( 1979 ) 22 - 40 . Numdam | MR 554213 | Zbl 0411.22011 · Zbl 0411.22011 |
| [13] | Duflo M. , Heckman G. , Vergne M. , Projection d’orbites, formule de Kirillov et formule de Blattner , Mém. Soc. Math. Fr. 15 ( 1984 ) 65 - 128 . Numdam | MR 789081 | Zbl 0575.22014 · Zbl 0575.22014 |
| [14] | Duistermaat J.J. , The Heat Lefschetz Fixed Point Formula for the Spin c -Dirac Operator , Progr. Nonlinear Differential Equation Appl. , vol. 18 , Birkhäuser , Boston , 1996 . MR 1365745 | Zbl 0858.58045 · Zbl 0858.58045 |
| [15] | Guillemin V. , Sternberg S. , Convexity properties of the moment mapping , Invent. Math. 67 ( 1982 ) 491 - 513 . MR 664117 | Zbl 0503.58017 · Zbl 0503.58017 · doi:10.1007/BF01398933 |
| [16] | Guillemin V. , Sternberg S. , Geometric quantization and multiplicities of group representations , Invent. Math. 67 ( 1982 ) 515 - 538 . MR 664118 | Zbl 0503.58018 · Zbl 0503.58018 · doi:10.1007/BF01398934 |
| [17] | Guillemin V. , Sternberg S. , A normal form for the moment map , in: Sternberg S. (Ed.), Differential Geometric Methods in Mathematical Physics , Reidel Publishing Company , Dordrecht , 1984 . MR 767835 | Zbl 0548.58011 · Zbl 0548.58011 |
| [18] | Jeffrey L. , Kirwan F. , Localization and quantization conjecture , Topology 36 ( 1997 ) 647 - 693 . MR 1422429 | Zbl 0876.55007 · Zbl 0876.55007 · doi:10.1016/S0040-9383(96)00015-8 |
| [19] | Harish-Chandra H. , Discrete series for semi-simple Lie group, I and II , Acta Math. 113 ( 1965 ) 242 - 318 , Acta Math. 116 ( 1966 ) 1 - 111 . MR 219666 | Zbl 0199.20102 · Zbl 0199.20102 · doi:10.1007/BF02392813 |
| [20] | Hecht H. , Schmid W. , A proof of Blattner’s conjecture , Invent. Math. 31 ( 1975 ) 129 - 154 . MR 396855 | Zbl 0319.22012 · Zbl 0319.22012 · doi:10.1007/BF01404112 |
| [21] | Kawasaki T. , The index of elliptic operators over V-manifolds , Nagoya Math. J. 84 ( 1981 ) 135 - 157 . Article | MR 641150 | Zbl 0437.58020 · Zbl 0437.58020 |
| [22] | Kirwan F. , Cohomology of Quotients in Symplectic and Algebraic Geometry , Princeton Univ. Press , Princeton , 1984 . MR 766741 | Zbl 0553.14020 · Zbl 0553.14020 |
| [23] | Kirwan F. , Convexity properties of the moment mapping III , Invent. Math. 77 ( 1984 ) 547 - 552 . MR 759257 | Zbl 0561.58016 · Zbl 0561.58016 · doi:10.1007/BF01388838 |
| [24] | Kostant B. , Quantization and unitary representations , in: Modern Analysis and Applications, Lecture Notes in Math. , vol. 170 , Springer-Verlag , Berlin/New York , 1970 , pp. 87 - 207 . MR 294568 | Zbl 0223.53028 · Zbl 0223.53028 |
| [25] | Lawson H. , Michelsohn M.-L. , Spin Geometry , Princeton Math. Ser. , vol. 38 , Princeton Univ. Press , Princeton , 1989 . MR 1031992 | Zbl 0688.57001 · Zbl 0688.57001 |
| [26] | Lerman E. , Meinrenken E. , Tolman S. , Woodward C. , Non-Abelian convexity by symplectic cuts , Topology 37 ( 1998 ) 245 - 259 . MR 1489203 | Zbl 0913.58023 · Zbl 0913.58023 · doi:10.1016/S0040-9383(97)00030-X |
| [27] | Meinrenken E. , On Riemann-Roch formulas for multiplicities , J. Amer. Math. Soc. 9 ( 1996 ) 373 - 389 . MR 1325798 | Zbl 0851.53020 · Zbl 0851.53020 · doi:10.1090/S0894-0347-96-00197-X |
| [28] | Meinrenken E. , Symplectic surgery and the Spin c -Dirac operator , Adv. Math. 134 ( 1998 ) 240 - 277 . MR 1617809 | Zbl 0929.53045 · Zbl 0929.53045 · doi:10.1006/aima.1997.1701 |
| [29] | Meinrenken E. , Sjamaar S. , Singular reduction and quantization , Topology 38 ( 1999 ) 699 - 762 . MR 1679797 | Zbl 0928.37013 · Zbl 0928.37013 · doi:10.1016/S0040-9383(98)00012-3 |
| [30] | Paradan P.-É. , Formules de localisation en cohomologie équivariante , Compositio Math. 117 ( 1999 ) 243 - 293 . MR 1702424 | Zbl 0934.55006 · Zbl 0934.55006 · doi:10.1023/A:1000602914188 |
| [31] | Paradan P.-É. , The moment map and equivariant cohomology with generalized coefficient , Topology 39 ( 2000 ) 401 - 444 . MR 1722000 | Zbl 0941.37050 · Zbl 0941.37050 · doi:10.1016/S0040-9383(99)00028-2 |
| [32] | Paradan P.-É. , The Fourier transform of semi-simple coadjoint orbits , J. Funct. Anal. 163 ( 1999 ) 152 - 179 . MR 1682831 | Zbl 0915.22008 · Zbl 0915.22008 · doi:10.1006/jfan.1998.3381 |
| [33] | Paradan P.-É. , Localization of the Riemann-Roch character , J. Funct. Anal. 187 ( 2001 ) 442 - 509 . MR 1875155 | Zbl 1001.53062 · Zbl 1001.53062 · doi:10.1006/jfan.2001.3825 |
| [34] | Schmid W. , On a conjecture of Langlands , Ann. of Math. 93 ( 1971 ) 1 - 42 . MR 286942 | Zbl 0291.43013 · Zbl 0291.43013 · doi:10.2307/1970751 |
| [35] | Schmid W. , L 2 -cohomology and the discrete series , Ann. of Math. 103 ( 1976 ) 375 - 394 . MR 396856 | Zbl 0333.22009 · Zbl 0333.22009 · doi:10.2307/1970944 |
| [36] | Schmid W. , Discrete series , in: Proc. Symp. Pure Math. , vol. 61 , 1997 , pp. 83 - 113 . MR 1476494 | Zbl 0936.22009 · Zbl 0936.22009 |
| [37] | Sjamaar R. , Symplectic reduction and Riemann-Roch formulas for multiplicities , Bull. Amer. Math. Soc. 33 ( 1996 ) 327 - 338 . MR 1364017 | Zbl 0857.58021 · Zbl 0857.58021 · doi:10.1090/S0273-0979-96-00661-1 |
| [38] | Sjamaar R. , Convexity properties of the moment mapping re-examined , Adv. Math. 138 ( 1998 ) 46 - 91 . MR 1645052 | Zbl 0915.58036 · Zbl 0915.58036 · doi:10.1006/aima.1998.1739 |
| [39] | Segal G. , Equivariant K-theory , Inst. Hautes Études Sci. Publ. Math. 34 ( 1968 ) 129 - 151 . Numdam | MR 234452 | Zbl 0199.26202 · Zbl 0199.26202 · doi:10.1007/BF02684593 |
| [40] | Tian Y. , Zhang W. , An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg , Invent. Math. 132 ( 1998 ) 229 - 259 . MR 1621428 | Zbl 0944.53047 · Zbl 0944.53047 · doi:10.1007/s002220050223 |
| [41] | Vergne M. , Geometric quantization and equivariant cohomology , in: First European Congress in Mathematics, vol. 1 , Progr. Math. , vol. 119 , Birkhäuser , Boston , 1994 , pp. 249 - 295 . MR 1341826 | Zbl 0827.58020 · Zbl 0827.58020 |
| [42] | Vergne M. , Multiplicity formula for geometric quantization, Part I, Part II, and Part III , Duke Math. J. 82 ( 1996 ) 143 - 179 , 181-194, 637-652. Article | MR 1387225 | Zbl 0855.58033 · Zbl 0855.58033 · doi:10.1215/S0012-7094-96-08206-X |
| [43] | Vergne M. , Quantification géométrique et réduction symplectique , Astérisque 282 ( 2002 ) 249 - 278 . Numdam | MR 1975181 | Zbl 1037.53062 · Zbl 1037.53062 |
| [44] | Witten E. , Two dimensional gauge theories revisited , J. Geom. Phys. 9 ( 1992 ) 303 - 368 . MR 1185834 | Zbl 0768.53042 · Zbl 0768.53042 · doi:10.1016/0393-0440(92)90034-X |
| [45] | Woodhouse N.M.J. , Geometric Quantization , Oxford Math. Monogr. , Clarendon , Oxford , 1997 . Zbl 0747.58004 · Zbl 0747.58004 |
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.
