Two geometric character formulas for reductive Lie groups

Schmid, Wilfried; Vilonen, Kari

doi:10.1090/S0894-0347-98-00275-6

Two geometric character formulas for reductive Lie groups
HTML articles powered by AMS MathViewer

by Wilfried Schmid and Kari Vilonen

J. Amer. Math. Soc. 11 (1998), 799-867

DOI: https://doi.org/10.1090/S0894-0347-98-00275-6

PDF | Request permission

Abstract:

In this paper we prove two formulas for the characters of representations of reductive groups. Both express the character of a representation $\pi$ in terms of the same geometric data attached to $\pi$. When specialized to the case of a compact Lie group, one of them reduces to Kirillov’s character formula in the compact case, and the other, to an application of the Atiyah-Bott fixed point formula to the Borel-Weil realization of the representation $\pi$.

References

Michael Atiyah, Collected works. Vol. 1, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1988. Early papers: general papers. MR 951892
A. Beĭlinson, Localization of representations of reductive Lie algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 699–710. MR 804725
Alexandre Beĭlinson and Joseph Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
Alexander Beĭlinson and Joseph Bernstein, A generalization of Casselman’s submodule theorem, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 35–52. MR 733805
L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527, DOI 10.1007/BFb0073549
Edward Bierstone and Pierre D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5–42. MR 972342, DOI 10.1007/BF02699126
Nicole Berline and Michèle Vergne, Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 9, 539–541 (French, with English summary). MR 685019
Jean-Yves Charbonnel, Sur les semi-caractères des groupes de Lie résolubles connexes, J. Functional Analysis 41 (1981), no. 2, 175–203 (French, with English summary). MR 615160, DOI 10.1016/0022-1236(81)90086-0
M. Djuflo, Representations of the fundamental series of a semisimple Lie group, Funkcional. Anal. i Priložen. 4 (1970), no. 2, 38–42 (Russian). MR 0297923
Michel Duflo, Théorie de Mackey pour les groupes de Lie algébriques, Acta Math. 149 (1982), no. 3-4, 153–213 (French). MR 688348, DOI 10.1007/BF02392353
A. I. Fomin and N. N. Šapovalov, A certain property of the characters of irreducible representations of real semisimple Lie groups, Funkcional. Anal. i Priložen. 8 (1974), no. 3, 87–88 (Russian). MR 0367115
Mark Goresky and Robert MacPherson, Local contribution to the Lefschetz fixed point formula, Invent. Math. 111 (1993), no. 1, 1–33. MR 1193595, DOI 10.1007/BF01231277
Phillip Griffiths and Wilfried Schmid, Locally homogeneous complex manifolds, Acta Math. 123 (1969), 253–302. MR 259958, DOI 10.1007/BF02392390
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Harish-Chandra, Invariant differential operators and distributions on a semisimple Lie algebra, Amer. J. Math. 86 (1964), 534–564. MR 180628, DOI 10.2307/2373023
Harish-Chandra, Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions, Acta Math. 113 (1965), 241–318. MR 219665, DOI 10.1007/BF02391779
Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. MR 219666, DOI 10.1007/BF02392813
Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Functional Analysis 19 (1975), 104–204. MR 0399356, DOI 10.1016/0022-1236(75)90034-8
Henryk Hecht, The characters of some representations of Harish-Chandra, Math. Ann. 219 (1976), no. 3, 213–226. MR 427542, DOI 10.1007/BF01354284
Henryk Hecht, Dragan Miličić, Wilfried Schmid, and Joseph A. Wolf, Localization and standard modules for real semisimple Lie groups. I. The duality theorem, Invent. Math. 90 (1987), no. 2, 297–332. MR 910203, DOI 10.1007/BF01388707
H.Hecht, D.Miličić, W.Schmid and J.Wolf, Localisation and standard modules for real semisimple Lie groups II: Irreducibility, vanishing theorems and classification, preprint.
Henryk Hecht and Wilfried Schmid, Characters, asymptotics and ${\mathfrak {n}}$-homology of Harish-Chandra modules, Acta Math. 151 (1983), no. 1-2, 49–151. MR 716371, DOI 10.1007/BF02393204
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Takeshi Hirai, The characters of some induced representations of semi-simple Lie groups, J. Math. Kyoto Univ. 8 (1968), 313–363. MR 238994, DOI 10.1215/kjm/1250524058
Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
Heisuke Hironaka, Subanalytic sets, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453–493. MR 0377101
R. Hotta and M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984), no. 2, 327–358. MR 732550, DOI 10.1007/BF01388568
Masaki Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci. 20 (1984), no. 2, 319–365. MR 743382, DOI 10.2977/prims/1195181610
Masaki Kashiwara, Index theorem for constructible sheaves, Astérisque 130 (1985), 193–209. Differential systems and singularities (Luminy, 1983). MR 804053
M.Kashiwara, Open problems in group representation theory, Proceedings of Taniguchi Symposium held in 1986, RIMS preprint 569, Kyoto University, 1987.
Masaki Kashiwara, Character, character cycle, fixed point theorem and group representations, Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, Academic Press, Boston, MA, 1988, pp. 369–378. MR 1039844, DOI 10.2969/aspm/01410369
Masaki Kashiwara, The invariant holonomic system on a semisimple Lie group, Algebraic analysis, Vol. I, Academic Press, Boston, MA, 1988, pp. 277–286. MR 992461
M.Kashiwara, $\mathcal {D}$-modules and Representation Theory of Lie Groups, RIMS preprint 940, Kyoto University, 1993.
M.Kashiwara and P.Schapira, Sheaves on manifolds, Springer, 1990.
M.Kashiwara and W.Schmid, Quasi-equivariant $\mathcal {D}$-modules, equivariant derived category, and representations of reductive Lie groups, Lie Theory and Geometry, in Honor of Bertram Kostant, Progress in Mathematics, Birkhäuser, 1994, pp. 457–488.
A. A. Kirillov, Characters of unitary representations of Lie groups, Funkcional. Anal. i Priložen. 2 (1968), no. 2, 40–55 (Russian). MR 0236318
A. A. Kirillov, Characters of unitary representations of Lie groups. Reduction theorems, Funkcional. Anal. i Priložen. 3 (1969), no. 1, 36–47 (Russian). MR 0248286
A. A. Kirillov, Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, Band 220, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt. MR 0412321, DOI 10.1007/978-3-642-66243-0
Toshihiko Matsuki, Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J. 12 (1982), no. 2, 307–320. MR 665498
Z. Mebkhout, Une équivalence de catégories, Compositio Math. 51 (1984), no. 1, 51–62 (French). MR 734784
D.Miličić, Localization and Representation Theory of Reductive Lie Groups, preliminary manuscript (available at www.math.utah.edu/˜milicic/), 1993.
I. Mirković, T. Uzawa, and K. Vilonen, Matsuki correspondence for sheaves, Invent. Math. 109 (1992), no. 2, 231–245. MR 1172690, DOI 10.1007/BF01232026
I. Mirković and K. Vilonen, Characteristic varieties of character sheaves, Invent. Math. 93 (1988), no. 2, 405–418. MR 948107, DOI 10.1007/BF01394339
Mason S. Osborne, Lefschetz formulas on nonelliptic complexes, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 171–222. Dissertation, Yale University, New Haven, CT, 1972. MR 1011898, DOI 10.1090/surv/031/04
Toshio Ōshima and Toshihiko Matsuki, A description of discrete series for semisimple symmetric spaces, Group representations and systems of differential equations (Tokyo, 1982) Adv. Stud. Pure Math., vol. 4, North-Holland, Amsterdam, 1984, pp. 331–390. MR 810636, DOI 10.2969/aspm/00410331
Wulf Rossmann, Kirillov’s character formula for reductive Lie groups, Invent. Math. 48 (1978), no. 3, 207–220. MR 508985, DOI 10.1007/BF01390244
W. Rossmann, Characters as contour integrals, Lie group representations, III (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1077, Springer, Berlin, 1984, pp. 375–388. MR 765560, DOI 10.1007/BFb0072345
W. Rossmann, Invariant eigendistributions on a semisimple Lie algebra and homology classes on the conormal variety. I. An integral formula, J. Funct. Anal. 96 (1991), no. 1, 130–154. MR 1093510, DOI 10.1016/0022-1236(91)90076-H
Wilfried Schmid, Homogeneous complex manifolds and representations of semisimple Lie groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 223–286. Dissertation, University of California, Berkeley, CA, 1967. MR 1011899, DOI 10.1090/surv/031/05
Wilfried Schmid, Some properties of square-integrable representations of semisimple Lie groups, Ann. of Math. (2) 102 (1975), no. 3, 535–564. MR 579165, DOI 10.2307/1971043
Wilfried Schmid, Two character identities for semisimple Lie groups, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1976), Lecture Notes in Math., Vol. 587, Springer, Berlin, 1977, pp. 196–225. MR 0507247
Wilfried Schmid, Construction and classification of irreducible Harish-Chandra modules, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 235–275. MR 1168487
W.Schmid, Character formulas and localization of integrals, Proceedings of the conference on symplectic geometry and mathematical physics, Ascona 1996, to appear.
W. Schmid and K. Vilonen, Characters, fixed points and Osborne’s conjecture, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 287–303. MR 1216196, DOI 10.1090/conm/145/1216196
W.Schmid and K.Vilonen, Weyl group actions on Lagrangian cycles and Rossmann’s formula, Proceedings of NATO Advanced Research Conference on Mathematical Physics and Group Theory, Kluwer Academic Press, 1994, pp. 242–250.
Wilfried Schmid and Kari Vilonen, Characters, characteristic cycles, and nilpotent orbits, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 329–340. MR 1358623, DOI 10.1016/S0934-8840(11)80140-4
Wilfried Schmid and Kari Vilonen, Characteristic cycles of constructible sheaves, Invent. Math. 124 (1996), no. 1-3, 451–502. MR 1369425, DOI 10.1007/s002220050060
Wilfried Schmid and Joseph A. Wolf, Geometric quantization and derived functor modules for semisimple Lie groups, J. Funct. Anal. 90 (1990), no. 1, 48–112. MR 1047577, DOI 10.1016/0022-1236(90)90080-5
David A. Vogan Jr., The algebraic structure of the representation of semisimple Lie groups. I, Ann. of Math. (2) 109 (1979), no. 1, 1–60. MR 519352, DOI 10.2307/1971266
Joseph A. Wolf, The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237. MR 251246, DOI 10.1090/S0002-9904-1969-12359-1