Alphabeta MathCategory O and Geometric Representation Theory
Category O and geometric representation theory relate highest-weight modules to Hecke combinatorics, bimodules and geometric constructions. This collection begins with Kazhdan–Lusztig bases, polynomials and cells, and develops Soergel intersection forms and Hodge theory. Projective functors and the Soergel functor connect the coinvariant algebra and wall crossing to category O. The character and multiplicity theorem then leads to singular and parabolic Kazhdan–Lusztig theory.
Parabolic category O supplies the standard, simple, projective and tilting bases used in the Fock-space dictionary. Arakawa–Suzuki functors and higher-level Schur–Weyl duality connect this branch to the degenerate Hecke categorification route. Proper cohomological descent and constructible hypercovers provide geometric foundations, while equivariant Borel–Moore convolution and constructible Ext develop the correspondence formalism. Basic category O remains in Lie Theory, and braid-group Soergel applications remain in Braid Groups. Hecke Algebras and Categorification contains the resulting Ariki and KLR applications. Each topic retains its examples companion and publication status.
Pathway
The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.
Part 1 · Category O and Geometric Representation Theory
9 pagesKazhdan–Lusztig bases and Soergel Hodge theory lead to projective functors, character formulas and singular or parabolic multiplicities. Parabolic category O and Arakawa–Suzuki functors develop the Fock-space and higher-level Schur–Weyl interfaces. The geometric branch supplies proper cohomological descent, constructible hypercovers and equivariant convolution. Their KLR and Hecke applications are linked from Hecke Algebras and Categorification.
This page builds the convolution/Ext interface used by the geometric realizations of quiver Hecke algebras and their canonical bases.
1 definition, 2 lemmas, 2 theoremsExamples & counterexamples →This page develops cohomological descent for proper morphisms of finite-type complex algebraic varieties in the analytic topology, with bounded algebraically constructible…
2 lemmas, 1 theoremExamples & counterexamples →This page develops the Hodge theory of Soergel bimodules for type A in the real category of The real type-A Soergel category by scalar extension: intersection forms on…
7 definitions, 15 lemmas, 7 theorems, 1 corollaryExamples & counterexamples →This page records the full Kazhdan–Lusztig basis theorem and its coefficient positivity, followed by polynomial recursion, inversion and cell theory.
3 definitions, 4 lemmas, 1 proposition, 5 theoremsExamples & counterexamples →This page develops projective functors on a regular integral category-O block and Soergel's functor V=Hom O(P(w 0·λ),-) with values in the coinvariant algebra.
6 definitions, 9 lemmas, 2 propositions, 8 theoremsExamples & counterexamples →This page proves the Kazhdan–Lusztig character and multiplicity theorem in a regular integral block of type A, through the Soergel functor rather than as an imported conjecture.
1 definition, 1 lemma, 1 proposition, 3 theorems, 1 remarkExamples & counterexamples →This page develops the two parabolic Hecke modules attached to a standard parabolic subgroup and the two categories in which their canonical bases appear.
5 definitions, 2 lemmas, 7 theoremsExamples & counterexamples →This page builds the type-A dictionary between genuine parabolic category O and the higher-level wedge-tensor Fock module: filtered and tilting objects in a finite…
7 definitions, 7 lemmas, 10 theoremsExamples & counterexamples →This page develops the Arakawa–Suzuki action of the degenerate affine Hecke algebra on tensor functors and the higher-level Schur–Weyl duality of Brundan–Kleshchev: the…
11 definitions, 6 lemmas, 11 theoremsExamples & counterexamples →