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-Hoelder 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.
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 , 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.
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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)
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.
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 , Adv. Math. 119 (1996), no. 2, 207–281, MR97k:17033, doi
A. V. Zelevinskiĭ, The -adic analogue of the Kazhdan-Lusztig conjecture, Funktsional. Anal. i Prilozhen. 15 (1981), no. 2, 9–21, 96.