Toward proof of Lusztig's conjecture concerning negative level representations of affine Lie algebras (Q1322063)

Let \({\mathfrak g}\) be an affine Lie algebra of type \(A^{(1)}_ \ell\), \(D^{(1)}_ \ell\), \(E^{(1)}_ \ell\), \(A^{(2)}_{2\ell -1}\), \(D^{(2)}_{\ell +1}\), \(E^{(2)}_ 6\) or \(D^{(3)}_ 4\), and let \(h\) be the dual Coxeter number of \({\mathfrak g}\). \textit{G. Lusztig} [J. Algebra 131, 466-475 (1990; Zbl 0698.16007)] made a conjecture determining the decomposition (in an appropriate Grothendieck group) of irreducible highest weight \({\mathfrak g}\)-modules \(L(\lambda)\) of level \(-l- h\) for any integer \(l\geqslant 1\) in terms of Verma modules \(M(\mu)\) of the same level. The aim of the paper under review is to show that the conjecture of \textit{D. Kazhdan} and \textit{G. Lusztig} [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] which determines the decomposition of \(L(-w \rho- \rho)\) in terms of \(M(-v\rho -\rho)\) for the affine Lie algebra \({\mathfrak g}\) implies the validity of Lusztig's conjecture for every \(l\geqslant 1\). The main ingredient of the proof is an extension of Jantzen's translation functor for the negative level representations of \({\mathfrak g}\) obtained by tensoring the representation with an integrable highest weight \({\mathfrak g}\)-module and then projecting onto a component of certain specific type.