Kazhdan–Lusztig conjecture

Kazhdan–Lusztig conjecture
Name	Kazhdan–Lusztig conjecture
Field	Representation theory
Proposed	1979
Proponents	David Kazhdan, George Lusztig
Status	Theorem (proven)

Kazhdan–Lusztig conjecture The Kazhdan–Lusztig conjecture proposed a precise description of characters of certain representations of complex semisimple Lie algebras and connected profound ideas across David Kazhdan, George Lusztig, Bertram Kostant, Harish-Chandra, Élie Cartan, Hermann Weyl and later work by James Arthur, Robert Langlands, Jean-Pierre Serre. Originally conjectured in 1979, it connected combinatorial invariants from Hecke algebra theory with geometric phenomena in flag varietys and singularities appearing in the work of Bernstein–Gelfand–Gelfand and Joseph Bernstein, leading to major advances involving researchers such as Alexander Beilinson, Joseph Bernstein, Stephen Gelbart, Vladimir Drinfeld and Edward Frenkel.

History and statement

The conjecture arose from interactions among researchers at institutions such as Harvard University, Massachusetts Institute of Technology, Princeton University, University of Cambridge, and Institute for Advanced Study where Kazhdan and Lusztig published their seminal paper relating Hecke algebra polynomials to representation-theoretic multiplicities; contemporaneous work by David Vogan, Anthony Joseph, Bertram Kostant, Irving Kaplansky and Roger Howe framed the context of highest-weight modules, Verma modules and primitive ideals. The original statement predicted that coefficients of the newly defined Kazhdan–Lusztig polynomials give multiplicities of simple modules in Verma modules for a complex semisimple Lie algebra with respect to the action of a corresponding Weyl group; subsequent geometric reinterpretations involved the cohomology of intersection complexes on Schubert varietys and perverse sheaves developed by Alexander Beilinson and Joseph Bernstein.

Representation-theoretic background

The conjecture sits at the intersection of algebraic structures studied by Weyl group theory, algebraic groups like GL_n, SL_n, Sp_n, and SO_n, and module categories such as the Bernstein–Gelfand–Gelfand category O introduced by David Bernstein, Israel Gelfand, Sergei Gelfand and I. N. Bernstein. Fundamental figures whose work supplies background include Élie Cartan for classification, Hermann Weyl for character formulae, Harish-Chandra for harmonic analysis on reductive groups, Igor Frenkel for connections to affine algebras, and Robert MacPherson for topological tools. Key algebraic objects appearing in the background are Verma modules, simple modules, highest weight theory, Harish-Chandra modules, primitive ideals and the structure of the universal enveloping algebra.

Formulation and equivalent versions

Kazhdan and Lusztig formulated polynomials indexed by pairs of elements in a Coxeter group and proved symmetry properties conjectured to control composition multiplicities in category O; the formulation used the basis change in the Iwahori–Hecke algebra associated to a Weyl group and produced polynomials now bearing their names. Equivalent versions were later expressed geometrically using intersection cohomology and perverse sheaves on flag varietys and Schubert varietys by George Lusztig, Robert MacPherson, Mark Goresky, R. K. Brylinski, Wilfried Schmid and Beilinson–Bernstein, and algebraically via translation functors and Jantzen filtration studied by J. C. Jantzen, Anthony Joseph and David Vogan.

Proofs and developments

The original conjecture was proven independently by a geometric method of Beilinson–Bernstein using localization theory on flag varieties linking D-modules and representation theory, and by an approach of Brylinksi–Kashiwara and Masaki Kashiwara using holonomic systems and Riemann–Hilbert correspondence techniques; these proofs invoked results of Pierre Deligne on perverse sheaves and of Michel Demazure on flag variety geometry. Subsequent extensions and refinements involved work of Soergel, who related the polynomials to bimodules now called Soergel bimodules, and of Beilinson–Ginzburg–Soergel, Stavros Argyrios, Ben Webster and Ivan Mirković on categorification and linkages with categorification programs in Mathematical Institute, Oxford and University of California, Berkeley. Later generalizations addressed affine and quantum analogues via researchers such as I. Grojnowski, H. H. Andersen, V. Kac, George Lusztig and Kashiwara.

Examples and computations

Concrete computations began with classical types A, B, C, D where Kazhdan–Lusztig polynomials can be computed using explicit reduced expressions in the symmetric group and its cousins; these computations were pursued by André Joyal, Richard Stanley, Arnold Schönhage and others in combinatorial representation theory. Algorithms implemented by groups at École Normale Supérieure, Massachusetts Institute of Technology, and University of Toronto used Coxeter group combinatorics, Hecke algebra recursions, and geometry of Schubert cells to compute tables that illuminated exceptional Lie types such as E_6, E_7, E_8, F_4 and G_2, with contributions from Friedrich Knop, Hanspeter Kraft and Nicolas Bourbaki-influenced classification efforts.

Applications and consequences

The theorem reshaped representation theory, influencing the formulation of the Langlands program, geometric representation theory, and connections to knot theory and low-dimensional topology via link homology theories developed by Mikhail Khovanov, Jacob Rasmussen and Edward Witten. It provided tools for computing primitive ideals in universal enveloping algebras used by Anthony Joseph and influenced categorical and geometric methods used in work of Maxim Kontsevich, Alexander Beilinson, Edward Frenkel and Nigel Higson. Applications extend to study of canonical bases in quantum groups initiated by George Lusztig and Vladimir Drinfeld, to modular representation theory explored by J. L. Alperin, Geoffrey Robinson and Henning Haahr Andersen, and to interplay with Schubert calculus studied by William Fulton and Leonard Gross.

Category:Representation theory