Character Formulas of Kazhdan-Lusztig Type

摘要

We give a survey of character formulas for irreducible modules over symmetrizable Kac-Moody Lie algebras, quantized enveloping algebras at roots of unity, and reductive algebraic groups in large positive characteristics. All of them are described by the Kazhdan-Lusztig polynomials. We also include a brief sketch of the proof in the case of symmetrizable Kac-Moody Lie algebras and a reformulation of the formula for affine Lie algebras in terms of the alcove geometry.