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Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. II: Nonintegral case. (English) Zbl 0929.17027

In an earlier paper [cf. Part I, Duke Math. J. 77, 21-62 (1995; Zbl 0829.17020)] the authors proved a Kazhdan-Lusztig type character formula for the irreducible highest weight modules over affine Lie algebras with negative level for the integral highest weight case. In this paper, they extend this result to the rational highest weight case.
Let \({\mathfrak g}\) be an affine Lie algebra, \({\mathfrak h}\) its Cartan subalgebra, \(c\) the canonical central element, \(\{h_i\}_{i\in I}\) the set of simple coroots and \(W\) be the Weyl group. For \(\lambda\in {\mathfrak h}^*\), let \(M(\lambda)\) (resp. \(L(\lambda)\)) denote the Verma module (resp. irreducible module) with highest weight \(\lambda\) and let \(W(\lambda)\) denote the subgroup of \(W\) generated by the reflections with respect to the real coroots \(h\) satisfying \(\langle \lambda, h\rangle\in\mathbb{Z}\). Then \(W(\lambda)\) is a Coxeter group. Let \(\ell^\lambda\) (resp. \(\leq^\lambda\)) denote the length function (resp. Bruhat order) in \(W(\lambda)\). Choose \(\rho\in {\mathfrak h}^*\) such that \(\langle \rho, h_i\rangle=1\) for all \(i\in I\). Let \(w\) be an element in \(W\) such that its length is smallest among the elements in the coset \(wW(\lambda)\). For \(y,x\in W(\lambda)\), let \(P_{y,x}(q)\) denote the Kazhdan-Lusztig polynomial for the Coxeter group \(W(\lambda)\). The main result in this paper is the following character formula for the irreducible \({\mathfrak g}\) module \(L(w(\lambda+ \rho)-\rho)\).
Let \(w\in W\) as above. For \(\lambda\in {\mathfrak h}^*\) and \(x\in W(\lambda)\) satisfying: \(\langle \lambda+\rho, h_i\rangle\in \mathbb{Q}_{\leq 0}\) for all \(i\in I\), \(\langle \lambda+\rho, c\rangle<0\) and \(w'(\lambda+\rho)\neq wx(\lambda+\rho)\) for any \(w'\in W\) with \(w'< wx\); we have \[ \text{ch} (L(wx(\lambda+\rho)- \rho)= \sum_{y\leq^\lambda x}(-1)^{\ell^\lambda(x)- \ell^\lambda(y)} P_{y,x}(1) \text{ ch } M(wy(\lambda+ \rho)-\rho), \] where ch denotes the character.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)

Citations:

Zbl 0829.17020
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References:

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