×

zbMATH — the first resource for mathematics

Spherical functions on affine Lie groups. (English) Zbl 0848.43010
A complex valued function \(f\) on a group \(K\) is said to be spherical if the vector space \(V_f\) spanned by the functions \(x \mapsto f(g^{-1} xg)\) is finite dimensional. If \(V\) is a finite dimensional representation of \(K\) the spherical function \(f\) is said to be of type \(V\) if \(V_f\) is isomorphic to \(V\). Further a function \(\Psi : K \to V^*\) (the dual representation of \(V)\) is said to be equivariant if \(g^{-1} \Psi (gxg^{-1}) = \Psi (x)\). Then, for \(v \in V\), the function \(f(x) = \langle v,\Psi (x) \rangle\) is spherical of type \(V\). In the first section one considers the case of a compact Lie group. Let \(V,W\) be finite dimensional representations of \(K\) and \(\Phi : W \to W \otimes V^*\) an intertwining operator, then \(\Psi (x) = Tr |_W (\Phi x)\) is an equivariant function. The restriction of \(\Psi\) to a maximal torus determines \(\Psi\), and, when \(W = N_\lambda\) is a highest weight representation, is the solution of a differential system \(R_V (Y) \Psi = \chi (Y) (\lambda + \rho) \Psi\), where \(R_V (Y)\) is an operator valued radial part associated to an element \(Y \in {\mathcal Z} (g)\), the center of the universal enveloping algebra of \(g = Lie (K)^C\). This theory is a vector valued generalization of the theory of characters (which corresponds to the special case \(V = C)\). It can be applied to the integrability of the Sutherland operator and to the study of the Jack polynomials. In the second part one extends the preceding results to the case of an affine Lie group: \(G\) is a complex simply connected simple Lie group, \(\widehat G\) is the universal extension of the loop group \(LG\), and \(\widetilde G\) is the semi-direct product \(C^* \times \widetilde G\). One proves that the space of equivaruant functions having a fixed homogeneity degree with respect to the action of the center is finite dimensional. One constructs an eigenbasis of the radial part of the second order Laplace operator. This is related to the affine Jack polynomials. One considers also higher order Laplace operators. In particular, for the affine Lie group \(\widetilde {SL}_2\), one obtains the classical Lamé operator.
Reviewer: J.Faraut (Paris)

MSC:
43A90 Harmonic analysis and spherical functions
22E67 Loop groups and related constructions, group-theoretic treatment
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] D. Bernard, On the Wess-Zumino-Witten models on the torus , Nuclear Phys. B 303 (1988), no. 1, 77-93. · doi:10.1016/0550-3213(88)90217-9
[2] I. N. Bernstein and O. W. Schwartzman, Chevalley’s theorem for complex crystallographic Coxeter groups , Functional Anal. Appl. 12 (1980), 308-310. · Zbl 0458.32017 · doi:10.1007/BF01076385
[3] F. Calogero, Solution of the one-dimensional \(N\)-body problems with quadratic and/or inversely quadratic pair potentials , J. Mathematical Phys. 12 (1971), 419-436. · Zbl 1002.70558 · doi:10.1063/1.531804
[4] V. Chari and A. Pressley, New unitary representations of loop groups , Math. Ann. 275 (1986), no. 1, 87-104. · Zbl 0603.17012 · doi:10.1007/BF01458586 · eudml:164136
[5] P. I. Etingof and I. B. Frenkel, Central extensions of current groups in two dimensions , Comm. Math. Phys. 165 (1994), no. 3, 429-444. · Zbl 0822.22014 · doi:10.1007/BF02099419
[6] P. I. Etingof and A. A. Kirillov, Jr., Representations of affine Lie algebras, parabolic differential equations, and Lamé functions , Duke Math. J. 74 (1994), no. 3, 585-614. · Zbl 0811.17026 · doi:10.1215/S0012-7094-94-07421-8
[7] P. I. Etingof and A. A. Kirillov, Jr., A unified representation-theoretic approach to special functions , Functional Anal. Appl. 28 (1994), 73-76. · Zbl 0868.33010 · doi:10.1007/BF01079011
[8] P. I. Etingof and A. A. Kirillov, Jr., On the affine analogue of Jack and Macdonald polynomials , Duke Math. J. 78 (1995), no. 2, 229-256. · Zbl 0873.33011 · doi:10.1215/S0012-7094-95-07810-7
[9] P. I. Etingof and A. A. Kirillov, Jr., Macdonald’s polynomials and representations of quantum groups , Math. Res. Lett. 1 (1994), no. 3, 279-296. · Zbl 0833.17007 · doi:10.4310/MRL.1994.v1.n3.a1
[10] P. Etingof and K. Styrkas, Algebraic integrability of Schrödinger operators and representations of Lie algebras , to appear in Compositio Math. · Zbl 0861.17003 · numdam:CM_1995__98_1_91_0 · eudml:90396
[11] F. Falceto and K. Gawedzki, Chern-Simons states at genus one , Comm. Math. Phys. 159 (1994), no. 3, 549-579. · Zbl 0797.57011 · doi:10.1007/BF02099984
[12] 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), 197-215, Suppl 1 A. · Zbl 0925.17022 · doi:10.1142/S0217751X92003781
[13] I. B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations , Comm. Math. Phys. 146 (1992), no. 1, 1-60. · Zbl 0760.17006 · doi:10.1007/BF02099206
[14] I. B. Frenkel and Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras , Duke Math. J. 66 (1992), no. 1, 123-168. · Zbl 0848.17032 · doi:10.1215/S0012-7094-92-06604-X
[15] H. Garland, The arithmetic theory of loop algebras , J. Algebra 53 (1978), no. 2, 480-551. · Zbl 0383.17012 · doi:10.1016/0021-8693(78)90294-6
[16] A. Gorsky and N. Nekrasov, Relativistic Calogero-Moser Model as gauged “Wess-Zumino-Witten” theory , preprint, 1994. · Zbl 1007.81547 · doi:10.1016/0550-3213(94)90429-4
[17] A. Gorsky and N. Nekrasov, Elliptic Calogero-Moser system from two-dimensional current algebra , preprint, 1994. · Zbl 1007.81547 · doi:10.1016/0550-3213(94)90429-4
[18] R. Goodman and N. R. Wallach, Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle , J. Reine Angew. Math. 347 (1984), 69-133. · Zbl 0514.22012 · doi:10.1515/crll.1984.347.69 · crelle:GDZPPN002201127 · eudml:152598
[19] R. Goodman and N. R. Wallach, Higher-order Sugawara operators for affine Lie algebras , Trans. Amer. Math. Soc. 315 (1989), no. 1, 1-55. · Zbl 0676.17013 · doi:10.2307/2001371
[20] T. Hayashi, Sugawara operators and Kac-Kazhdan conjecture , Invent. Math. 94 (1988), no. 1, 13-52. · Zbl 0674.17005 · doi:10.1007/BF01394343 · eudml:143616
[21] Harish-Chandra, Collected Papers , Springer, New York, 1984. · Zbl 0527.01019
[22] G. J. Heckman, Root systems and hypergeometric functions. II , Compositio Math. 64 (1987), no. 3, 353-373. · Zbl 0656.17007 · numdam:CM_1987__64_3_353_0 · eudml:89880
[23] G. J. Heckman and E. M. Opdam, Root systems and hypergeometric functions. I , Compositio Math. 64 (1987), no. 3, 329-352. · Zbl 0656.17006 · numdam:CM_1987__64_3_329_0 · eudml:89879
[24] S. Helgason, Differential geometry, Lie groups, and symmetric spaces , Pure and Applied Mathematics, vol. 80, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978. · Zbl 0451.53038
[25] V. G. Kac, Infinite-dimensional Lie algebras , Cambridge University Press, Cambridge, 1990. · Zbl 0716.17022
[26] V. G. Kac, Laplace operators of infinite-dimensional Lie algebras and theta functions , Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 2, Phys. Sci., 645-647. JSTOR: · Zbl 0532.17008 · doi:10.1073/pnas.81.2.645 · links.jstor.org
[27] D. Kazhdan, B. Kostant, and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type , Comm. Pure Appl. Math. 31 (1978), no. 4, 481-507. · Zbl 0368.58008 · doi:10.1002/cpa.3160310405
[28] E. Looijenga, Root systems and elliptic curves , Invent. Math. 38 (1976/77), no. 1, 17-32. · Zbl 0358.17016 · doi:10.1007/BF01390167 · eudml:142440
[29] I. G. Macdonald, A new class of symmetric functions , Publ. Inst. Rech. Math. Av. 49 (1988), 131-171. · Zbl 0962.05507
[30] F. G. Malikov, Special vectors in Verma modules over affine Lie algebras , Functional Anal. Appl. 23 (1989), 66-67. · Zbl 0676.17012 · doi:10.1007/BF01078582
[31] G. Moore and N. Seiberg, Classical and quantum conformal field theory , Comm. Math. Phys. 123 (1989), no. 2, 177-254. · Zbl 0694.53074 · doi:10.1007/BF01238857
[32] 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., to appear in Adv. Math., 1993. · Zbl 0874.33011 · doi:10.1006/aima.1996.0066
[33] 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. JSTOR: · Zbl 0171.04803 · doi:10.2307/1970710 · links.jstor.org
[34] M. A. Olshanetsky and A. M. Perelomov, Quantum integrable systems related to Lie algebras , Phys. Rep. 94 (1983), no. 6, 313-404. · doi:10.1016/0370-1573(83)90018-2
[35] E. M. Opdam, Root systems and hypergeometric functions. III , Compositio Math. 67 (1988), no. 1, 21-49. · Zbl 0669.33007 · numdam:CM_1988__67_1_21_0 · eudml:89908
[36] E. M. Opdam, Root systems and hypergeometric functions. IV , Compositio Math. 67 (1988), no. 2, 191-209. · Zbl 0669.33008 · numdam:CM_1988__67_2_191_0 · eudml:89916
[37] A. Pressley and G. Segal, Loop groups , Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1986. · Zbl 0618.22011
[38] A. Ramanathan, Stable principal bundles on a compact Riemann surface , Math. Ann. 213 (1975), 129-152. · Zbl 0284.32019 · doi:10.1007/BF01343949 · eudml:162664
[39] G. Segal, Conformal field theory , Swansea, 1988, proceedings of the international conference on mathematical physics. · Zbl 0657.53060
[40] B. Sutherland, Exact results for a quantum many-body problem in one dimension , Phys. Rev. A 5 (1972), 1372-1376.
[41] A. Tsuchiya and Y. Kanie, Vertex operators in conformal field theory on \(\mathbf P^ 1\) and monodromy representations of braid group , Conformal field theory and solvable lattice models (Kyoto, 1986), Adv. Stud. Pure Math., vol. 16, Academic Press, Boston, MA, 1988, pp. 297-372. · Zbl 0661.17021
[42] G. Warner, Harmonic Analysis On Semi-simple Lie Groups , Springer-Verlag, Berlin, 1972. · Zbl 0265.22020
[43] E. T. Whittaker and G. N. Watson, A course of modern analysis , Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. · Zbl 0951.30002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.