Merits and demerits of the orbit method

Kirillov, A.

doi:10.1090/S0273-0979-99-00849-6

Merits and demerits of the orbit method
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Bull. Amer. Math. Soc. 36 (1999), 433-488 Request permission

Abstract:

This survey is the expanded version of my talk at the AMS meeting in April 1997. I explain to non-experts how to use the orbit method, discuss its strong and weak points and advertise some open problems.

References

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
V. I. Arnol′d and A. B. Givental′, Symplectic geometry, Current problems in mathematics. Fundamental directions, Vol. 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 5–139, 291 (Russian). MR 842908
L. Auslander and B. Kostant, Polarization and unitary representations of solvable Lie groups, Invent. Math. 14 (1971), 255–354. MR 293012, DOI 10.1007/BF01389744
A. Alekseev and S. Shatashvili, Path integral quantization of the coadjoint orbits of the Virasoro group and $2$-d gravity, Nuclear Phys. B 323 (1989), no. 3, 719–733. MR 1014632, DOI 10.1016/0550-3213(89)90130-2
Ian D. Brown, Dual topology of a nilpotent Lie group, Ann. Sci. École Norm. Sup. (4) 6 (1973), 407–411. MR 352326, DOI 10.24033/asens.1253
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
Busyatskaya I.K., Representations of exponential Lie groups, Funct. Anal. and Appl. 7, No 2 (1973), 151-152.
P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Raïs, P. Renouard, and M. Vergne, Représentations des groupes de Lie résolubles, Monographies de la Société Mathématique de France, No. 4, Dunod, Paris, 1972. MR 0444836
Ranee Brylinski and Bertram Kostant, Minimal representations, geometric quantization, and unitarity, Proc. Nat. Acad. Sci. U.S.A. 91 (1994), no. 13, 6026–6029. MR 1278630, DOI 10.1073/pnas.91.13.6026
Jacques Dixmier, Sur les représentations unitaries des groupes de Lie nilpotents. III, Canadian J. Math. 10 (1958), 321–348. MR 95427, DOI 10.4153/CJM-1958-033-5
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
Robert L. Devaney, An introduction to chaotic dynamical systems, 2nd ed., Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. MR 1046376
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, Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 2, 265–288 (French, with English summary). MR 444841, DOI 10.24033/asens.1327
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
J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), no. 2, 259–268. MR 674406, DOI 10.1007/BF01399506
B. A. Dubrovin, Igor Moiseevich Krichever, and S. P. Novikov, Integrable systems. I, Current problems in mathematics. Fundamental directions, Vol. 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 179–284, 291 (Russian). MR 842910
Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov Jr., Spherical functions on affine Lie groups, Duke Math. J. 80 (1995), no. 1, 59–90. MR 1360611, DOI 10.1215/S0012-7094-95-08003-X
A. A. Kirillov Jr. and P. I. Ètingof, On a unified representation-theoretic approach to the theory of special functions, Funktsional. Anal. i Prilozhen. 28 (1994), no. 1, 91–94 (Russian); English transl., Funct. Anal. Appl. 28 (1994), no. 1, 73–76. MR 1275729, DOI 10.1007/BF01079011
J. M. G. Fell, Weak containment and induced representations of groups, Canadian J. Math. 14 (1962), 237–268. MR 150241, DOI 10.4153/CJM-1962-016-6
I. B. Frenkel, Orbital theory for affine Lie algebras, Invent. Math. 77 (1984), no. 2, 301–352. MR 752823, DOI 10.1007/BF01388449
V. V. Fock and A. A. Rosly, Flat connections and polyubles, Teoret. Mat. Fiz. 95 (1993), no. 2, 228–238 (English, with English and Russian summaries); English transl., Theoret. and Math. Phys. 95 (1993), no. 2, 526–534. MR 1243250, DOI 10.1007/BF01017138
Izrail M. Gelfand, Collected papers. Vol. I, Springer-Verlag, Berlin, 1987. With remarks by V. W. Guillemin and S. Sternberg. MR 929821, DOI 10.1007/978-3-642-61705-8
V. A. Ginzburg, The orbit method in the theory of representations of complex Lie groups, Funktsional. Anal. i Prilozhen. 15 (1981), no. 1, 23–37, 96 (Russian). MR 609792
Victor Guillemin, Eugene Lerman, and Shlomo Sternberg, Symplectic fibrations and multiplicity diagrams, Cambridge University Press, Cambridge, 1996. MR 1414677, DOI 10.1017/CBO9780511574788
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
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, On Frobenius reciprocity for unipotent algebraic groups over $Q$, Amer. J. Math. 93 (1971), 163–172. MR 281842, DOI 10.2307/2373455
Roger Howe, Reciprocity laws in the theory of dual pairs, Representation theory of reductive groups (Park City, Utah, 1982) Progr. Math., vol. 40, Birkhäuser Boston, Boston, MA, 1983, pp. 159–175. MR 733812
I. M. Isaacs and Dikran Karagueuzian, Erratum: “Conjugacy in groups of upper triangular matrices” [J. Algebra 202 (1998), no. 2, 704–711; MR1617655 (99b:20011)], J. Algebra 208 (1998), no. 2, 722. MR 1655475, DOI 10.1006/jabr.1998.7430
R. S. Ismagilov, Representations of infinite-dimensional groups, Translations of Mathematical Monographs, vol. 152, American Mathematical Society, Providence, RI, 1996. Translated from the Russian manuscript by D. Deart. MR 1393939, DOI 10.1090/mmono/152
A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001
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
Kirillov A.A., Introduction to the theory of representations and noncommutative harmonic analysis, Encyclopaedia of Mathematical Sciences, vol. 22 Representation theory and noncommutative harmonic analysis I, Springer, 1994, pp. 1-156.
Aleksandr Aleksandrovich Kirillov, The orbit method. II. Infinite-dimensional Lie groups and Lie algebras, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 33–63. MR 1216180, DOI 10.1090/conm/145/1216180
A. A. Kirillov, Geometric quantization, Current problems in mathematics. Fundamental directions, Vol. 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 141–178, 291 (Russian). MR 842909
A. A. Kirillov, Variations on the triangular theme, Lie groups and Lie algebras: E. B. Dynkin’s Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 169, Amer. Math. Soc., Providence, RI, 1995, pp. 43–73. MR 1364453, DOI 10.1090/trans2/169/05
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, Geometric approach to discrete series of unirreps for Vir, J. Math. Pures Appl. (9) 77 (1998), no. 8, 735–746 (English, with English and French summaries). MR 1646792, DOI 10.1016/S0021-7824(98)80007-X
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
B. Kostant, Quantization and unitary representations, Uspehi Mat. Nauk 28 (1973), no. 1(169), 163–225 (Russian). Translated from the English (Lectures in Modern Analysis and Applications, III, pp. 87–208, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970) by A. A. Kirillov. MR 0385022
Bertram Kostant, On the existence and irreducibility of certain series of representations, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 231–329. MR 0399361
Bertram Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math. 34 (1979), no. 3, 195–338. MR 550790, DOI 10.1016/0001-8708(79)90057-4
A. A. Kirillov and M. L. Kontsevich, The growth of the Lie algebra generated by two generic vector fields on the line, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4 (1983), 15–20 (Russian, with English summary). MR 713969
A. A. Kirillov, M. L. Kontsevich, and A. I. Molev, Algebras of intermediate growth, Akad. Nauk SSSR Inst. Prikl. Mat. Preprint 39 (1983), 19 (Russian, with English summary). Translated in Selecta Math. Soviet. 9 (1990), no. 2, 137–153. MR 753873
Kirillov A.A. and Melnikov A., On a remarkable sequence of polynomials, Proceedings of the Franco-Belge Colloquium, Reims 1995, Collection S.M.F., 1997, pp. 35-42.
Maxim Kontsevich, Formality conjecture, Deformation theory and symplectic geometry (Ascona, 1996) Math. Phys. Stud., vol. 20, Kluwer Acad. Publ., Dordrecht, 1997, pp. 139–156. MR 1480721
Masaki Kashiwara and Michèle Vergne, The Campbell-Hausdorff formula and invariant hyperfunctions, Invent. Math. 47 (1978), no. 3, 249–272. MR 492078, DOI 10.1007/BF01579213
A. A. Kirillov and D. V. Yur′ev, Kähler geometry of the infinite-dimensional homogeneous space $M=\textrm {Diff}_+(S^1)/\textrm {Rot}(S^1)$, Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 35–46, 96 (Russian). MR 925071
Horst Leptin and Jean Ludwig, Unitary representation theory of exponential Lie groups, De Gruyter Expositions in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1307383, DOI 10.1515/9783110874235
Jiang-Hua Lu and Alan Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501–526. MR 1037412
Calvin C. Moore, Decomposition of unitary representations defined by discrete subgroups of nilpotent groups, Ann. of Math. (2) 82 (1965), 146–182. MR 181701, DOI 10.2307/1970567
Yu. A. Neretin, Categories of symmetries and infinite-dimensional groups, London Mathematical Society Monographs. New Series, vol. 16, The Clarendon Press, Oxford University Press, New York, 1996. Translated from the Russian by G. G. Gould; Oxford Science Publications. MR 1418863
Lajos Pukánszky, On the theory of exponential groups, Trans. Amer. Math. Soc. 126 (1967), 487–507. MR 209403, DOI 10.1090/S0002-9947-1967-0209403-7
L. Pukanszky, Unitary representations of solvable Lie groups, Ann. Sci. École Norm. Sup. (4) 4 (1971), 457–608. MR 439985, DOI 10.24033/asens.1218
Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 900587
Wulf Rossmann, Kirillov’s character formula for reductive Lie groups, Invent. Math. 48 (1978), no. 3, 207–220. MR 508985, DOI 10.1007/BF01390244
S. P. Novikov (ed.), Dynamical systems. VII, Encyclopaedia of Mathematical Sciences, vol. 16, Springer-Verlag, Berlin, 1994. Integrable systems, nonholonomic dynamical systems; A translation of Current problems in mathematics. Fundamental directions, Vol. 16 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987 [ MR0922069 (88g:58004)]; Translation by A. G. Reyman [A. G. Reĭman] and M. A. Semenov-Tian-Shansky [M. A. Semenov-Tyan-Shanskiĭ]; Translation edited by V. I. Arnol′d and S. P. Novikov. MR 1256257, DOI 10.1007/978-3-642-57884-7
J.-M. Souriau, Structure des systèmes dynamiques, Dunod, Paris, 1970 (French). Maîtrises de mathématiques. MR 0260238
I. M. Ščepočkina, Orbit method in restriction and induction problems for a normal subgroup of a solvable Lie group, C. R. Acad. Bulgare Sci. 33 (1980), no. 8, 1039–1042 (Russian). MR 620833
Yan Soibelman, Orbit method for the algebras of functions on quantum groups and coherent states. I, Internat. Math. Res. Notices 6 (1993), 151–163. MR 1224113, DOI 10.1155/S1073792893000169
David A. Vogan Jr., Associated varieties and unipotent representations, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 315–388. MR 1168491
Edward Witten, Nonabelian bosonization in two dimensions, Comm. Math. Phys. 92 (1984), no. 4, 455–472. MR 736403, DOI 10.1007/BF01215276
Edward Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), no. 1, 153–209. MR 1133264, DOI 10.1007/BF02100009
N. J. Wildberger, On a relationship between adjoint orbits and conjugacy classes of a Lie group, Canad. Math. Bull. 33 (1990), no. 3, 297–304. MR 1077099, DOI 10.4153/CMB-1990-048-4
Andrey V. Zelevinsky, Representations of finite classical groups, Lecture Notes in Mathematics, vol. 869, Springer-Verlag, Berlin-New York, 1981. A Hopf algebra approach. MR 643482, DOI 10.1007/BFb0090287
Don Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 497–512. MR 1341859