Etingof, Pavel I.; Kirillov, Alexander A. jun. Representations of affine Lie algebras, parabolic differential equations, and Lamé functions. (English) Zbl 0811.17026 Duke Math. J. 74, No. 3, 585-614 (1994). Let \(\mathfrak g\) be a finite-dimensional complex simple Lie algebra and \({\mathfrak h} \subset {\mathfrak g}\) a Cartan subalgebra. Let \(\widehat {\mathfrak g}\) be the associated affine Lie algebra, and \(M_{\lambda, k}\) the Verma module for \(\widehat{\mathfrak g}\) with level \(k\) and highest weight \(\lambda \in {\mathfrak h}^*\) for \(\mathfrak g\). The authors consider the \(n\)- point correlation functions ‘on the torus’ \[ \text{trace}_{M_{\lambda_ 0,k}} (\Phi'(\tau_ 1) \dots \Phi^ n(\tau_ n) q^{-\partial} e^ h), \] where \(\Phi^ i(\tau_ i) : M_{\lambda_{i,k}} \to M_{\lambda_{i - 1,k}} \otimes V_ i\) are vertex operators, \(\tau_ 1, \dots,\tau_ n \in \mathbb{C}^ \times\), \(\lambda_ 0 = \lambda_ n\), \(V_ i\) is a finite-dimensional irreducible \(\mathfrak g\)-module, \(\partial\) is the grading operator on Verma modules, and \(h \in {\mathfrak h}^*_ \mathbb{R}\). In the ‘flat’ case, it is well known that the \(n\)-point functions satisfy the Knichnik- Zamolodchikov equation. The authors derive a differential equation satisfied by their functions and study the asymptotics of its solutions. When \({\mathfrak g} = \text{sl}_ 2\), \(V_ 0 = \cdots = V_ n\) and \(k \to -1\) (the critical level), they find that the asymptotics are closely related to Lamé functions. Reviewer: A.N.Pressley (London) Cited in 1 ReviewCited in 24 Documents MSC: 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 33E10 Lamé, Mathieu, and spheroidal wave functions 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 17B69 Vertex operators; vertex operator algebras and related structures Keywords:representations; affine Lie algebras; \(n\)-point correlation functions; vertex operators; Lamé functions PDFBibTeX XMLCite \textit{P. I. Etingof} and \textit{A. A. Kirillov jun.}, Duke Math. J. 74, No. 3, 585--614 (1994; Zbl 0811.17026) Full Text: DOI arXiv References: [1] H. Awata, A. Tsuchiya, and Y. Yamada, Integral formulas for the WZNW correlation functions , Nuclear Phys. B 365 (1991), no. 3, 680-696. [2] D. Bernard, On the Wess-Zumino-Witten models on the torus , Nuclear Phys. B 303 (1988), no. 1, 77-93. [3] D. Bernard and G. Felder, Fock representations and BRST cohomology in \(\mathrm SL(2)\) current algebra , Comm. Math. Phys. 127 (1990), no. 1, 145-168. · Zbl 0703.17013 [4] I. Cherednik, Integral solutions of trigonometric Knizhnik-Zamolodchikov equations and Kac-Moody algebras , Publ. Res. Inst. Math. Sci. 27 (1991), no. 5, 727-744. · Zbl 0753.17036 [5] E. T. Copson, Asymptotic expansions , Cambridge Tracts in Mathematics and Mathematical Physics, No. 55, Cambridge University Press, New York, 1965. · Zbl 0123.26001 [6] M. Crivelli, G. Felder, and C. Wieczerkowski, Generalized hypergeometric functions on the torus and adjoint representation of \(U_q(\mathfraksl_2)\) , · Zbl 0777.17007 [7] P. I. Etingof, Representations of affine Lie algebras, elliptic \(r\)-matrix systems, and special functions , to appear in Comm. Math. Phys. (hep-th 9303018). · Zbl 0803.17008 [8] B. L. Feigin and E. V. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds , Comm. Math. Phys. 128 (1990), no. 1, 161-189. · Zbl 0722.17019 [9] 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 [10] I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations , Russian Math. Surveys 32 (1977), 185-213. · Zbl 0386.35002 [11] V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions , Nuclear Phys. B 247 (1984), no. 1, 83-103. · Zbl 0661.17020 [12] M. A. Olshanetsky and A. M. Perelomov, Quantum integrable systems related to Lie algebras , Phys. Rep. 94 (1983), no. 6, 313-404. [13] A. Pressley and G. Segal, Loop groups , Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1986. · Zbl 0618.22011 [14] N. Yu. Reshetikhin and A. N. Varchenko, in preparation. [15] V. V. Schechtman and A. N. Varchenko, Arrangements of hyperplanes and Lie algebra homology , Invent. Math. 106 (1991), no. 1, 139-194. · Zbl 0754.17024 [16] 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) eds. M. Jimbo, T. Miwa, and A. Tsuchiya, Adv. Stud. Pure Math., vol. 16, Academic Press, Boston, MA, 1988, pp. 297-372. · Zbl 0661.17021 [17] A. N. Varchenko, Critical points of the product of powers of linear functions and families of bases of singular vectors , preprint, 1993. · Zbl 0798.33012 [18] E. T. Whittaker and G. N. Watson, Course of Modern Analysis , fourth edition, Cambridge Univ. Press, Cambridge, 1958. · JFM 45.0433.02 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.