# 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 ^{p}O 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