Modular invariant representations of infinite-dimensional Lie algebras and superalgebras. (English) Zbl 0652.17010

Kac, Victor G.; Wakimoto, Minoru

Proc. Natl. Acad. Sci. USA 85, No. 14, 4956-4960 (1988).

Let \({\mathfrak g}\) be the Kac-Moody Lie algebra associated to an arbitrary symmetrizable generalized Cartan matrix. Let L(\(\lambda)\) be a highest weight irreducible module for \({\mathfrak g}\). If \(\lambda\) is dominant integral, then the character of L(\(\lambda)\) is given by the famous Weyl- Kac character formula. In this paper the authors prove a character formula for a large class of highest weight representations of \({\mathfrak g}.\)
Explicitly, let \(R_{\lambda}\) denote the subset of real roots of \({\mathfrak g}\) such that (\(\lambda\),a)\(\in {\mathbb{Z}}\) for \(\alpha \in R_{\lambda}\), and let \(R^+_{\lambda}\) be the subset of positive real roots in \(R_{\lambda}\). Let \(W_{\lambda}\) be the subgroup of the Weyl group generated by the reflections \(\{s_{\alpha}:\) \(\alpha \in R_{\lambda}\}\). Then, if \(\lambda +\rho\) is in the Tits cone, and is dominant integral for \(R^+_{\lambda}\), the character of L(\(\lambda)\) is \(\sum_{w\in W_{\lambda}}\epsilon (w)ch M(w\lambda)\). As an immediate corollary, one has that the maximal submodule of M(\(\lambda)\) is generated by \(\{y_{\alpha}^{\lambda (h_{\alpha}+1)}:\) \(\alpha \in \Pi_{\lambda}\}\), where \(\Pi_{\lambda}\) is a set of simple roots for \(R_{\lambda}\) and \(y_{\alpha}\) is an element of the -\(\alpha\) root space of \({\mathfrak g}.\)
The authors study the case of affine Lie algebras, and prove that the class of representations described previously include certain modular invariant representations of \({\mathfrak g}\). The case \(A_ 1^{(1)}\) is worked out in detail and the results are used to determine all the modular invariant representations of the Virasoro algebra. The generalizations of these results to the case of Kac-Moody superalgebras are also considered.

Reviewer: V.Chari

MSC:

17B67	Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B70	Graded Lie (super)algebras
17B65	Infinite-dimensional Lie (super)algebras
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
11F03	Modular and automorphic functions

Keywords:

Kac-Moody Lie algebra; character formula; highest weight representations; affine Lie algebras; modular invariant representations; Virasoro algebra; Kac-Moody superalgebras