In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial P y , w ( q ) {\displaystyle P_{y,w}(q)} is a member of a family of integral polynomials introduced by David Kazhdan and George 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.
Motivation and history In the spring of 1978 Kazhdan and Lusztig were studying Springer representations of the Weyl group of an algebraic group on ℓ {\displaystyle \ell } -adic cohomology groups related to conjugacy classes which are unipotent. 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 Jens Carsten Jantzen and Anthony Joseph relating primitive ideals of enveloping algebras to representations of Weyl groups, led to the Kazhdan–Lusztig conjectures.
Definition Fix a Coxeter group W with generating set S, and write ℓ ( w ) {\displaystyle \ell (w)} for the length of an element w (the smallest length of an expression for w as a product of elements of S). The Hecke algebra of W has a basis of elements T w {\displaystyle T_{w}} for w ∈ W {\displaystyle w\in W} over the ring Z [ q 1 / 2 , q − 1 / 2 ] {\displaystyle \mathbb {Z} [q^{1/2},q^{-1/2}]} , with multiplication defined by
T y T w = T y w , if ℓ ( y w ) = ℓ ( y ) + ℓ ( w ) ( T s + 1 ) ( T s − q ) = 0 , if s ∈ S . {\displaystyle {\begin{aligned}T_{y}T_{w}&=T_{yw},&&{\mbox{if }}\ell (yw)=\ell (y)+\ell (w)\\(T_{s}+1)(T_{s}-q)&=0,&&{\mbox{if }}s\in S.\end{aligned}}}
The quadratic second relation implies that each generator Ts is invertible in the Hecke algebra, with inverse Ts−1 = q−1Ts + q−1 − 1. These inverses satisfy the relation (Ts−1 + 1)(Ts−1 − q−1) = 0 (obtained by multiplying the quadratic relation for Ts by −Ts−2q−1), and also the braid relations. From this it follows that the Hecke algebra has an automorphism D that sends q1/2 to q−1/2 and each Ts to Ts−1. More generally one has D ( T w ) = T w − 1 − 1 {\displaystyle D(T_{w})=T_{w^{-1}}^{-1}} ; also D can be seen to be an involution. The Kazhdan–Lusztig polynomials Pyw(q) are indexed by a pair of elements y, w of W, and uniquely determined by the following properties.
They are 0 unless y ≤ w (in the Bruhat order of W), 1 if y = w, and for y < w their degree is at most (ℓ(w) − ℓ(y) − 1)/2. The elements
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