Dacha seminar


Somewhere 147km from Moscow, 15-20 July, 2019.



Preseminar notes kinly provided by Vanya Losev and Vasya Krylov.

Monday: Representations of Reductive Algebraic Groups

Representations of reductive algebraic groups 1(Riche)
Induction functors, Chevalley theorem, Weyl character formula, Steinberg tensor product theorem, p-restricted weights, examples.

Representations of reductive algebraic groups 2 (Williamson)
Linkage principle, affine Weyl group, translation functors, principal block.

Character formulas 1 (Riche)
Hecke algebra and Kazhdan-Lusztig basis. Lusztig’s conjecture. Discussion of the bound. Examples.

Character formulas 2 (Williamson)
Curtis’ theorem. G_1T-modules. Linkage principle revisited. Periodic module. Jantzen’s generic decomposition patterns. Lusztig’s conjecture in periodic form.

Tuesday: Constructible sheaves and the Hecke category

Brief review of constructible and perverse sheaves (Mautner)
Stratifications, constructible sheaves, perverse sheaves, IC sheaves, Decomposition theorem.

Parity sheaves (Juteau)
Definition, uniqueness and existence, introduction to the equivariant derived category.

Hecke category via parity sheaves (Juteau)
Hecke category via parity sheaves. Stalks over Q are given by Kazhdan-Lusztig polynomials.

Hecke category via Soergel bimodules (Williamson)
Soergel bimodules (classical definition). Examples. Equivalence with geometric definition.

Wednesday: Decomposition Theorem and Torsion Explosion

Intersection forms and the Decomposition Theorem (Juteau)
Role of the intersection form in studying the Decomposition Theorem. Examples: surface singularities; contraction of zero section.
Explanation why, for a semi-small map, intersection forms being non-degenerate is equivalent to DT (as long as local systems are semi-simple).

Modular category O (Mautner)
Review of category O. Definition. Lusztig’s conjecture around the Steinberg weight. Relation to constructible sheaves. (Basically, a review of Soergel’s “on the relation…”)
Note: On page 4 the translation functor should go from St to p ρ. Also, in the actual lecture, modular category O was denoted pO to avoid confusion with classical category O.

Torsion explosion (Williamson)
Basically a review of “On torsion in the intersection cohomology”: miracle situation, Schubert calculus, some amusing number theory.


Diagrammatic Hecke category (Juteau)
How to draw monoidal categories. Frobenius objects and one-colour calculus. Two colour calculus. Jones-Wenzl projectors. Light leaves basis.

Thursday: Tilting modules and the anti-spherical module

Tilting modules for G (Williamson)
Classification. Tilting tensor theorem. Schur functor. Relevance of tilting modules to representation theory of the symmetric group. Example of SL(2).

Anti-spherical module and RW-conjecture (Riche)
Translation functors and tilting modules. Anti-spherical module. Conjecture. Philosophy of “higher representation theory”.

Koszul duality 1 (Riche)
Classical Koszul duality. Koszul rings etc. Composition with Ringel duality.

Two realizations (Bezrukavnikov)
Guest talk by Bezrukavnikov on “On two realizations of the affine Hecke algebra”.
Comments by Bezrukavnikov following his talk.


Friday: Reductive groups, the loop Grassmannian and the Springer resolution

The mixed derived category (Juteau)
Definition and relation to Soergel modules. (Following Achar-Riche.)
Note: On page 1 it was forgotten to mention that i* and j* preserve parity, hence induce functors on the mixed modular derived category.

Koszul duality 2 (Williamson)
Monoidal Koszul duality. Outline of Bezrukavnikov-Yun equivalence. Koszul duality between spherical and anti-spherical modules.

The Finkelberg-Mirkovic conjecture (Riche)
Brief review of the geometric Satake equivalence. Statement of parity = tilting theorem. Statement of Finkelberg-Mirkovic conjecture. Relation to Lusztig’s conjecture. Graded version.

The exotic t-structure (Mautner)
Exceptional collections and corresponding t-structures. Equivariant coherent sheaves.The exotic t-structure on the Springer resolution. Equivalence with the Iwahori constructible sheaves.

Optional extra talk: Iwahori-Whittaker model (Riche)
Review the results of the recent beautiful paper of BGMRW.

Saturday: Putting it all together

Achar-Riche theorem (Riche)
Statement and outline of proof of the formality and induction theorem.

Tilting characters (Williamson)
Deduction of numerical RW conjecture from AR theorem and monoidal Koszul duality.

Simple characters (Williamson)
Embedding spherical into anti-spherical. Deduction of the simple character formula from the tilting character formula.

**Banquet!**
Last modified: Mon Aug 12 16:13:51 AEST 2019