Kazhdan-Lusztig polynomial

introduction

• https://www.math.ucdavis.edu/~vazirani/S05/KL.details.html
• KL polys have applications to the representation theory of semisimple algebraic groups, Verma modules, algebraic geometry and topology of Schubert varieties, canonical bases, immanant inequalities, etc.

appearance

• change of basis coeffs (more or less) from the standard basis to the KL basis in a Hecke algebra (see Humphreys)
• giving the dimensions of local intersection cohomology for Schubert varieties. (this interpretation proves their positivity and integrality, but NO combinatorial interpretation is known!!) (see Geometry papers below; proved by KL in "Schubert varieties and Poincare duality.")
• the multiplicity of standard modules in indecomposable tilting modules (Soergel) are given by parabolic, affine KL polys eval at q=1. [ $n_{\lambda + \rho, \mu + \rho}(q=1)$ ]
• the change of basis coeffs for the canonical basis in terms of the standard basis for the Fock space (basic representation) [$d_{\lambda, \mu}(q)$. note Goodman-Wenzl ; Varagnolo-Vasserot show $d_{\lambda, \mu}(q) = n_{\lambda + \rho, \mu + \rho}(q)$,]
• the multiplicity of simple modules in Specht modules for Hecke algebras at roots of unity over a field of characteristic 0are given by parabolic, affine KL polys eval at $q=1$. [ie decomposition numbers $d_{\lambda, \mu}= d_{\lambda, \mu}(q=1)$]
• the multiplicity of Verma modules (proved by Beilinson-Bernstein and Brylinski-Kashiwara) when eval at $q=1$
• the coefficient (+/-) of a Weyl module character (ie Schur function) in the expression of the character of an irreducible highest weight module ($\operatorname{ch} L(\lambda)$ ) of a quantum group at a root of unity , when evaluated at $q=1$. [this is Lusztig's conjecture, proved by KL, Kashiwara-Tanisaki]
• the multiplicities of standards in the Jantzen filtration on Vermas [Beilinson-Bernstein] and hence the multiplicity of irreducibles in the Jantzen filtration of standard modules [by a result of Suzuki by functoriality]. Note, as a consequence, when we evaluate at q=1 we get overall multiplicity (not w/ the grading of the filtration).
• there is more about geometry, perverse sheaves, Lagrangian subvarieties, Springer resolution ...