Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. II: Nonintegral case
type: Item
instance of: scholarly article
title: Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. II: Nonintegral case
zbMATH Open document ID: 0929.17027
DOI: 10.1215/S0012-7094-96-08424-0
author: Masaki Kashiwara
author: Toshiyuki Tanisaki
published in: Duke Mathematical Journal
publication date: 1997-03-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 22E47
zbMATH DE Number: 943725
zbMATH Keywords: \(D\)-modules
zbMATH Keywords: Kazhdan-Lusztig type character formula
zbMATH Keywords: affine Lie algebras
zbMATH Keywords: rational highest weight
zbMATH Keywords: Kazhdan-Lusztig polynomial
zbMATH Keywords: Coxeter group
reviewed by: Kailash C. Misra
Wikidata QID: Q123347975
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 943725