Spherical functions on affine lie groups

Duke Mathematical Journal

Volumn 80, Issue 1, 1995, Pages 59-90

  Etingof, Pavel I ^a   Frenkel, Igor B ^b   Kirillov, Alexander A ^c  

^a HARVARD UNIVERSITY   (United States)

^b YALE UNIVERSITY   (United States)

^c MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0003076446     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-95-08003-X     Document Type: Article

References (44)

1. D. BERNARD, On the Wess-Zumino-Witten models on the torus, Nuclear Phys. B 303 (1988), 77-93.
2. I. N. BERNSTEIN AND O. W. SCHWARTZMAN, Chevalley’s theorem for complex crystallographic Coxeter groups, Functional Anal. Appl. 12 (1980), 308-310.
3. F. CALOGERO, Solution of the one-dimensional n-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436.
4. V. CHARI AND A. PRESSLEY, New unitary representations of loop groups, Math. Ann. 275 (1986), 87-104.
5. P. I. ETINGOF, Quantum integrable systems and representations of Lie algebras, J. Math. Phys. 36 (1995), 2636-2651.
6. P. I. ETINGOF AND I. B. FRENKEL, Central extensions of current groups in two dimensions, Comm. Math. Phys. 165 (1994), 429-444.
7. P. I. ETINGOF AND A. A. KIRILLOV, JR., Representations of affine Lie algebras, parabolic differential equations, and Lamé functions, Duke Math. J. 74 (1994), 585-614.
8. P. I. ETINGOF AND A. A. KIRILLOV, JR., A unified representation-theoretic approach to special functions, Functional Anal. Appl. 28 (1994), 73-76.
9. P. I. ETINGOF AND A. A. KIRILLOV, JR., On the affine analogue of Jack and Macdonald polynomials, Duke Math. J. 78 (1995), 229-256.
10. P. I. ETINGOF AND A. A. KIRILLOV, JR., Macdonald’s polynomials and representations of quantum groups, Math. Res. Let. 1 1994, 279-296.
11. P. ETINGOF AND K. STYRKAS, Algebraic integrability of Schrödinger operators and representations of Lie algebras, to appear in Compositio Math.
12. F. FALCETO AND K. GAWEDZKI, Chern-Simons states in genus 1, Comm. Math. Phys. 159 (1994), 471-503.
13. B. L. FEIGIN AND E. V. FRENKEL, Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras, Internat. J. Modern Phys. A 7 (1992), Suppl 1A, 197-215.
14. I. B. FRENKEL AND N. YU. RESHETIKHIN, Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992), 1-60.
15. I. B. FRENKEL AND Y. ZHU, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123-168.
16. H. GARLAND, The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480-551.
17. A. GORSKY AND N. NEKRASOV, Relativistic Calogero-Moser Model as gauged "Wess-Zumino-Witten" theory, preprint, 1994.
18. A. GORSKY AND N. NEKRASOV, Elliptic Calogero-Moser system from two-dimensional current algebra, preprint, 1994.
19. R. GOODMAN AND N. WALLACH, Structure of unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle, J. Reine Angew. Math. 347 (1984), 69133.
20. R. GOODMAN AND N. WALLACH, Higher-order Sugawara operators for affine Lie algebras, Trans. Amer. Math. Soc. 315 (1989), 1-55.
21. T. HAYASHI, Sugawara operators and Kac-Kazhdan conjecture, Invent. Math. 94 (1988), 13-52.
22. HARISH-CHANDRA, Collected Papers, Springer, New York, 1984.
23. G. J. HECKMAN, Root systems and hypergeometric functions II, Compositio Math. 64 (1987), 353-373.
24. G. J. HECKMAN AND E. M. OPDAM, Root systems and hypergeometric functions I, Compositio Math. 64 (1987), 329-352.
25. S. HELGASON, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.
26. V. G. KAC, Infinite-dimensional Lie Algebras, 3rd edition, Cambridge Univ. Press, Cambridge, 1990.
27. S. HELGASON, Laplace operators of infinite-dimensional Lie algebras and theta functions, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 645-647.
28. D. KAZHDAN, B. KOSTANT, AND S. STERNBERG, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appi. Math. 31 (1978), 481-507.
29. E. LOOIJENGA, Root systems and elliptic curves, Invent. Math. 33 (1976), 17-32.
30. I. G. MACDONALD, A new class of symmetric functions, Pubi. Inst. Rech. Math. Av. 49 (1988), 131-171.
31. F. G. MALIKOV, Special vectors in Verma modules over affine Lie alqebras, Functional Anal. Appi. 23 (1989), 66-67.
32. G. MOORE AND N. SEIBERG, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), 177-254.
33. M. NOUMI, Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, preprint, Dept, of Math. Sciences, Univ. of Tokyo, Japan, Oct. 1993, to appear in Adv. Math.
34. M. S. NARASIMHAN AND C. S. SESHADRI, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540-567.
35. M. A. OLSHANETSKY AND A. M. PERELOMOV, Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
36. E. M. OPDAM, Root systems and hypergeometric functions III, Compositio Math. 67 (1988), 21-49.
37. E. M. OPDAM, Root systems and hypergeometric functions IV, Compositio Math. 67 (1988), 191-207.
38. A. PRESSLEY AND G. SEGAL, Loop Groups, Clarendon Press, Oxford, 1986.
39. A. RAMANATHAN, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 1975, 129-152.
40. G. SEGAL, Conformal field theory, proceedings of the international conference on mathematical physics, Swansea (1988).
41. B. SUTHERLAND, Exact results for a quantum many-body problem in one dimension, Phys. Rev. A 5 (1972), 1372-1376.
42. 1 and monodromy representations of braid group, Adv. Stud. Pure Math. 16, Academic Press, Boston, 1988, 297-372.
43. G. WARNER, Harmonic Analysis On Semi-simple Lie Groups, Springer-Verlag, Berlin, 1972.
44. E. T. WHITTAKER AND G. N. WATSON, Course of Modern Analysis, 4th edition, Cambridge Univ. Press, Cambridge, 1958.