Kazhdan–Lusztig conjectures

In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial P_y,w(q) is a member of a family of integral polynomials introduced by Kazhdan and Lusztig (1979). They are indexed by pairs of elements y, w of a Coxeter group W, which can in particular be the Weyl group of a Lie group.

In the spring of 1978 Kazhdan and Lusztig were studying Springer representations of the Weyl group of an algebraic group on l-adic cohomology groups related to unipotent conjugacy classes. They found a new construction of these representations over the complex numbers (Kazhdan & Lusztig 1980a). The representation had two natural bases, and the transition matrix between these two bases is essentially given by the Kazhdan–Lusztig polynomials. The actual Kazhdan–Lusztig construction of their polynomials is more elementary. Kazhdan and Lusztig used this to construct a canonical basis in the Hecke algebra of the Coxeter group and its representations.

In their first paper Kazhdan and Lusztig mentioned that their polynomials were related to the failure of local Poincaré duality for Schubert varieties. In Kazhdan & Lusztig (1980b) they reinterpreted this in terms of the intersection cohomology of Mark Goresky and Robert MacPherson, and gave another definition of such a basis in terms of the dimensions of certain intersection cohomology groups.

The two bases for the Springer representation reminded Kazhdan and Lusztig of the two bases for the Grothendieck group of certain infinite dimensional representations of semisimple Lie algebras, given by Verma modules and simple modules. This analogy, and the work of Jantzen and Joseph relating primitive ideals of enveloping algebras to representations of Weyl groups, led to the Kazhdan–Lusztig conjectures.

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