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.

category O

• We start by considering Category O, which is the setting of the original Kazhdan-Lusztig polynomials.
• We change notation and G be a connected, complex reductive group, and B a Borel subgroup.
• Then B has a finite number of orbits on G/B, parametrized by the Weyl group W.
• Fix a regular integral infinitesimal character.
• For any element $w$ in $W$ there is a Verma module $L(w)$ (this is like the standard module above), with the given infinitesimal character, containing a unique irreducible submodule $\pi(w)$.
• There is a decomposition $L(w)=\sum_{y} m(y,w)\pi(y)$.
• Again this can be inverted, to give $\pi(w)=\sum_{y}M(y,w)I(y)$.
• The integers $m(y,w)$ are given by Kazhdan-Lusztig polynomials.
• These are defined in terms of the flag variety $G/B$, and are related to singularities of, and closure relations between, the orbits of $B$ on $G/B$.
• If $w,y$ are elements of $W$, then the Kazhdan-Lusztig polynomial $P_{x,y}$ is a polynomial in $q$, defined in terms of the orbits corresponding to $x$ and $y$.
• Then $M(x,y)=P_{x,y}(1)$ up to (an explicitly computed) sign.

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 ...

periodic KL polynomial

• In 1980, Lusztig introduced the periodic Kazhdan-Lusztig polynomials, which are conjectured to have important information about the characters of irreducible modules of a reductive group over a field of positive characteristic, and also about those of an affine Kac-Moody algebra at the critical level. The periodic Kazhdan-Lusztig polynomials can be computed by using another family of polynomials, called the periodic $R$-polynomials
• Affine Lie algebras at the critical level

questions

• http://mathoverflow.net/questions/80635/what-information-is-contained-in-the-kazhdan-lusztig-polynomials

articles

• Hideya Watanabe, Satoshi Naito, A combinatorial formula expressing periodic $R$-polynomials, http://arxiv.org/abs/1603.02778v1
• Adams, Jeffrey, and David A. Voga Jr. ‘Parameters for Twisted Representations’. arXiv:1502.03304 [math], 11 February 2015. http://arxiv.org/abs/1502.03304.
• Lusztig, G., and D. A. Vogan Jr. ‘Quasisplit Hecke Algebras and Symmetric Spaces’. Duke Mathematical Journal 163, no. 5 (April 2014): 983–1034. doi:10.1215/00127094-2644684.
• Fan, Neil J. Y., Peter L. Guo, and Grace L. D. Zhang. “On Parabolic Kazhdan-Lusztig R-Polynomials for the Symmetric Group.” arXiv:1501.04275 [math], January 18, 2015. http://arxiv.org/abs/1501.04275.
• Elias, Ben, Nicholas Proudfoot, and Max Wakefield. “The Kazhdan-Lusztig Polynomial of a Matroid.” arXiv:1412.7408 [math], December 23, 2014. http://arxiv.org/abs/1412.7408.