nLab Kazhdan-Lusztig theory

Idea

Kazhdan-Lusztig theory is about special recursive combinatorics which appears in several setups in mathematics, most notably in representation theory where it concerns the Jordan-Hölder coefficients of certain modules. As a phenomenon it has been discovered by David Kazhdan and George Lusztig, and some partial aspects independently by Deodhar. A central result is the Kazhdan-Lusztig conjecture, proved by Borho-Brylinski and by Masaki Kashiwara using D-modules and perverse sheaves.

References

• David Kazhdan, George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184, MR81j:20066, doi

• D. Kazhdan, G. Lusztig, Schubert varieties and Poincaré duality, in: Geometry of the Laplace operator, 185–203, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc. 1980.

• Kazhdan-Lusztig theory, chapter 8 in James E. Humphreys, Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$, Graduate Studies in Mathematics 94, Amer. Math. Soc. 2008. xvi+289 pp.

• Wolfgang Soergel, Kazhdan-Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln, Represent. Theory 1 (1997) 37-68, pdf; engl. version Kazhdan-Lusztig polynomials and a combinatoric for tilting modules. Represent. Theory 1 (1997) 83-114, pdf.

• R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory, Progress in Mathematics 236, Birkhäuser, Boston 2008.

• Jean-Luc Brylinski, Masaki Kashiwara, Démonstration de la conjecture de Kazhdan-Lusztig sur les modules de Verma, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 6, A373–A376, MR81k:17004

• Walter Borho, Jean-Luc Brylinski, Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules. Invent. Math. 69 (1982), no. 3, 437–476, MR84b:17007, doi; II. Relative enveloping algebras., Bull. Soc. Math. France 117 (1989), no. 2, 167–210, MR90j:17023, numdam

• N. Chriss, V. Ginzburg, Representation theory and complex geometry, Birkhäuser 1997. x+495 pp.

• Vinay V. Deodhar, On a construction of representations and a problem of Enright, Invent. Math. 57 (1980), no. 2, 101–118, MR81f:17004, doi

• Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499–511, MR86f:20045, doi; II. The parabolic analogue of Kazhdan-Lusztig polynomials, MR89a:20054, doi90232-8)

• Anthony Joseph, The Enright functor on the Bernstein-Gelʹfand-Gelʹfand category $\mathcal{O}$, Invent. Math. 67 (1982), no. 3, 423–445, MR84j:17005, doi

• Kazhdan-Lusztig theory and related topics, Proc. of the AMS Special Session at Loyola Univ., Chicago 1989. Edited by Vinay Deodhar. Contemporary Mathematics 139, Amer. Math. Soc. 1992.

• V. Deodhar, A brief survey of Kazhdan-Lusztig theory and related topics, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), 105–124, Proc. Sympos. Pure Math. 56, Part 1, Amer. Math. Soc. 1994.

• Vinay V. Deodhar, Ofer Gabber, Victor Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. in Math. 45 (1982), no. 1, 92–116. MR83i:17012, doi80014-5)

• O. Gabber, A Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 261–302, MR83e:17009, numdam

The following article proves a conjecture from above article of Deodhar, Gabber and Kac:

• Luis Casian, Proof of the Kazhdan-Lusztig conjecture for Kac-Moody algebras (the characters $\mathrm{ch}\,L_{\omega\rho-\rho})$, Adv. Math. 119 (1996), no. 2, 207–281, MR97k:17033, doi

• A. V. Zelevinskiĭ, The $p$-adic analogue of the Kazhdan-Lusztig conjecture, Funktsional. Anal. i Prilozhen. 15 (1981), no. 2, 9–21, 96.

• Luis G. Casian, David H. Collingwood, The Kazhdan-Lusztig conjecture for generalized Verma modules, Math. Z. 195 (1987), no. 4, 581–600, MR88i:17008, doi