is a member of a family of integral polynomials introduced by David Kazhdan and George Lusztig (1979).
They found a new construction of these representations over the complex numbers (Kazhdan & Lusztig 1980a).
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.
, with multiplication defined by The quadratic second relation implies that each generator Ts is invertible in the Hecke algebra, with inverse Ts−1 = q−1Ts + q−1 − 1.
The Kazhdan–Lusztig polynomials Pyw(q) are indexed by a pair of elements y, w of W, and uniquely determined by the following properties.
The Inventiones paper also put forth two equivalent conjectures, known now as Kazhdan–Lusztig conjectures, which related the values of their polynomials at 1 with representations of complex semisimple Lie groups and Lie algebras, addressing a long-standing problem in representation theory.
These conjectures were proved over characteristic 0 algebraically closed fields independently by Alexander Beilinson and Joseph Bernstein (1981) and by Jean-Luc Brylinski and Masaki Kashiwara (1981).
Thus, the Kazhdan–Lusztig conjectures describe the Jordan–Hölder multiplicities of Verma modules in any regular integral block of Bernstein–Gelfand–Gelfand category O.
A similar interpretation of all coefficients of Kazhdan–Lusztig polynomials follows from the Jantzen conjecture, which roughly says that individual coefficients of Py,w are multiplicities of Ly in certain subquotient of the Verma module determined by a canonical filtration, the Jantzen filtration.
The Jantzen conjecture in regular integral case was proved in a later paper of Beilinson and Bernstein (1993).
David Vogan showed as a consequence of the conjectures that and that Extj(My, Lw) vanishes if j + ℓ(w) + ℓ(y) is odd, so the dimensions of all such Ext groups in category O are determined in terms of coefficients of Kazhdan–Lusztig polynomials.
This result demonstrates that all coefficients of the Kazhdan–Lusztig polynomials of a finite Weyl group are non-negative integers.
By the Bruhat decomposition the space G/B of the algebraic group G with Weyl group W is a disjoint union of affine spaces Xw parameterized by elements w of W. The closures of these spaces Xw are called Schubert varieties, and Kazhdan and Lusztig, following a suggestion of Deligne, showed how to express Kazhdan–Lusztig polynomials in terms of intersection cohomology groups of Schubert varieties.
More precisely, the Kazhdan–Lusztig polynomial Py,w(q) is equal to where each term on the right means: take the complex IC of sheaves whose hyperhomology is the intersection homology of the Schubert variety of w (the closure of the cell Xw), take its cohomology of degree 2i, and then take the dimension of the stalk of this sheaf at any point of the cell Xy whose closure is the Schubert variety of y.
This gave the first proof that all coefficients of Kazhdan–Lusztig polynomials for finite Weyl groups are non-negative integers.
They are analogous to Kazhdan–Lusztig polynomials, but are tailored to representations of real semisimple Lie groups, and play major role in the conjectural description of their unitary duals.
The distinction, in the cases directly connection to representation theory, is explained on the level of double cosets; or in other terms of actions on analogues of complex flag manifolds G/B where G is a complex Lie group and B a Borel subgroup.
The original (K-L) case is then about the details of decomposing a classical theme of the Bruhat decomposition, and before that of Schubert cells in a Grassmannian.
Then the relevant object of study is In March 2007, a collaborative project, the "Atlas of Lie groups and representations", announced that the L–V polynomials had been calculated for the split form of E8.
Much of the later work of Lusztig explored analogues of Kazhdan–Lusztig polynomials in the context of other natural singular algebraic varieties arising in representation theory, in particular, closures of nilpotent orbits and quiver varieties.
They admit an elementary description, but the deeper properties of these polynomials necessary for representation theory follow from sophisticated techniques of modern algebraic geometry and homological algebra, such as the use of intersection cohomology, perverse sheaves and Beilinson–Bernstein–Deligne decomposition.
The coefficients of the Kazhdan–Lusztig polynomials are conjectured to be the dimensions of some homomorphism spaces in Soergel's bimodule category.
Combinatorial properties of Kazhdan–Lusztig polynomials and their generalizations are a topic of active current research.
Given their significance in representation theory and algebraic geometry, attempts have been undertaken to develop the theory of Kazhdan–Lusztig polynomials in purely combinatorial fashion, relying to some extent on geometry, but without reference to intersection cohomology and other advanced techniques.