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\section*{Kazhdan-Lusztig theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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-H\"o{}lder 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-module]]s and [[perverse sheaves]].

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[David Kazhdan]], [[George Lusztig]], \emph{Representations of Coxeter groups and Hecke algebras}, Invent. Math. \textbf{53} (1979), no. 2, 165--184, \href{http://www.ams.org/mathscinet-getitem?mr=560412}{MR81j:20066}, \href{http://dx.doi.org/10.1007/BF01390031}{doi}


\item D. Kazhdan, G. Lusztig, \emph{Schubert varieties and Poincar\'e{} duality, in: Geometry of the Laplace operator, 185--203, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc. 1980.}


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


\item [[Wolfgang Soergel]], \emph{Kazhdan-Lusztig-Polynome und eine Kombinatorik f\"u{}r Kipp-Moduln}, Represent. Theory 1 (1997) 37-68, \href{http://www.ams.org/ert/1997-001-04/S1088-4165-97-00006-X/S1088-4165-97-00006-X.pdf}{pdf}; engl. version Kazhdan-Lusztig polynomials and a combinatoric for tilting modules. Represent. Theory 1 (1997) 83-114, \href{http://www.ams.org/ert/1997-001-06/S1088-4165-97-00021-6/S1088-4165-97-00021-6.pdf}{pdf}.


\item R. Hotta, K. Takeuchi, T. Tanisaki, \emph{D-modules, perverse sheaves, and representation theory}, Progress in Mathematics \textbf{236}, Birkh\"a{}user, Boston 2008.


\item Jean-Luc Brylinski, [[Masaki Kashiwara]], \emph{D\'e{}monstration de la conjecture de Kazhdan-Lusztig sur les modules de Verma}, C. R. Acad. Sci. Paris S\'e{}r. A-B \textbf{291} (1980), no. 6, A373--A376, \href{http://www.ams.org/mathscinet-getitem?mr=596075}{MR81k:17004}


\item Walter Borho, Jean-Luc Brylinski, \emph{Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules.} Invent. Math. \textbf{69} (1982), no. 3, 437--476, \href{http://www.ams.org/mathscinet-getitem?mr=679767}{MR84b:17007}, \href{http://ams.mpim-bonn.mpg.de/leavingmsn?url=http://dx.doi.org/10.1007/BF01389364}{doi}; \emph{II. Relative enveloping algebras.}, Bull. Soc. Math. France \textbf{117} (1989), no. 2, 167--210, \href{http://www.ams.org/mathscinet-getitem?mr=1015807}{MR90j:17023}, \href{http://www.numdam.org/item?id=BSMF_1989__117_2_167_0}{numdam}


\item N. Chriss, [[V. Ginzburg]], \emph{Representation theory and complex geometry}, Birkh\"a{}user 1997. x+495 pp.


\item Vinay V. Deodhar, \emph{On a construction of representations and a problem of Enright}, Invent. Math. \textbf{57} (1980), no. 2, 101--118, \href{http://www.ams.org/mathscinet-getitem?mr=567193}{MR81f:17004}, \href{http://dx.doi.org/10.1007/BF01390091}{doi}


\item Vinay V. Deodhar, \emph{On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells}, Invent. Math. \textbf{79} (1985), no. 3, 499--511, \href{http://www.ams.org/mathscinet-getitem?mr=782232}{MR86f:20045}, \href{http://dx.doi.org/10.1007/BF01388520}{doi}; \emph{II. The parabolic analogue of Kazhdan-Lusztig polynomials}, \href{http://www.ams.org/mathscinet-getitem?mr=916182}{MR89a:20054}, \href{http://dx.doi.org/10.1016/0021-8693(87}{doi}90232-8)


\item [[Anthony Joseph]], \emph{The Enright functor on the Bernstein-Gelfand-Gelfand category $\mathcal{O}$}, Invent. Math. \textbf{67} (1982), no. 3, 423--445, \href{http://www.ams.org/mathscinet-getitem?mr=664114}{MR84j:17005}, \href{http://dx.doi.org/10.1007/BF01398930}{doi}


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


\item V. Deodhar, \emph{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. \textbf{56}, Part 1, Amer. Math. Soc. 1994.


\item Vinay V. Deodhar, [[Ofer Gabber]], [[Victor Kac]], \emph{Structure of some categories of representations of infinite-dimensional Lie algebras}, Adv. in Math. \textbf{45} (1982), no. 1, 92--116. \href{http://www.ams.org/mathscinet-getitem?mr=663417}{MR83i:17012}, \href{http://dx.doi.org/10.1016/S0001-8708(82}{doi}80014-5)


\item O. Gabber, A Joseph, \emph{Towards the Kazhdan-Lusztig conjecture}, Ann. Sci. \'E{}cole Norm. Sup. (4) 14 (1981), no. 3, 261--302, \href{http://www.ams.org/mathscinet-getitem?mr=644519}{MR83e:17009}, \href{http://www.numdam.org/item?id=ASENS_1981_4_14_3_261_0}{numdam}



\end{itemize}
The following article proves a conjecture from above article of Deodhar, Gabber and Kac:

\begin{itemize}%
\item Luis Casian, \emph{Proof of the Kazhdan-Lusztig conjecture for Kac-Moody algebras (the characters} $\mathrm{ch}\,L_{\omega\rho-\rho})$, Adv. Math. \textbf{119} (1996), no. 2, 207--281, \href{http://www.ams.org/mathscinet-getitem?mr=1390798}{MR97k:17033}, doi


\item A. V. Zelevinski, \emph{The $p$-adic analogue of the Kazhdan-Lusztig conjecture}, Funktsional. Anal. i Prilozhen. \textbf{15} (1981), no. 2, 9--21, 96.


\item Luis G. Casian, David H. Collingwood, \emph{The Kazhdan-Lusztig conjecture for generalized Verma modules}, Math. Z. \textbf{195} (1987), no. 4, 581--600, \href{http://www.ams.org/mathscinet-getitem?mr=900346}{MR88i:17008}, \href{http://dx.doi.org/10.1007/BF01166705}{doi}



\end{itemize}
[[!redirects Kazhdan-Lusztig polynomial]]



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