Moment Graphs in Geometry, Combinatorics, and Representation Theory

Moment graphs are combinatorial gadgets which allow to attack problems in geometry, representation theory and algebraic combinatorics (and probably many more fields!). On top of it, they are even interesting to be investigated as combinatorial objects themselves.

The main goal of the course is to get people convinced that moment graphs (and sheaves on them) are useful!!

Among the applications of moment graphs, there are:

- combinatorial description of equivariant cohomology of an equivariantly formal variety, such as smooth varieties, flag and Schubert varieties, ... (see, for example, this paper of Julianna Tymoczko)

- combinatorial description of intersection cohomology of Schubert varieties (after Braden and MacPherson's paper)

- categorical approach to the study of Kazhdan-Lusztig polynomials (have a look at my survey)

- realisation in terms of moment graph sheaves of relevant representation theoretical categories (see, for example, this paper, where also the geometric view point is discussed)

Since most of the above applications involve objects which are not supposed to be known by the audience, we will choose some of the topics, based on the research interests of the students and provide the necessary background.

The application of combinatorial (and categorical) techniques to describe intersection cohomology of Schubert varieties and hence the use of such a description to attack problems in algebraic combinatorics and representation theory has helped Geordie Williamson to become plenary speaker at the next ICM!

Tentative programme of the course: the course will consist of three main parts:

• 1. Equivariant cohomology of flag and Schubert varieties: definition and first proprties of equivariant cohomology; statement of the localisation theorem; flag varieties and their Schubert varieties; Bruaht graphs; application of the localisation theorem to flag and Schubert varieties;

  2. Coxeter groups and Kazhdan-Lusztig combinatorics: basics on Coxeter groups and Hecke algebras; categorical lifting of Hecke the algebra via sheaves on moment graphs;

  3. The Kazhdan-Lusztig conjecture on simple characters of finite dimensional complex simple Lie algebras: category O, statement of the Kazhdan-Lusztig conjecture; sheaves on moment graph approach to the conjecture.

Main references: For parts 1.and 3. we will mainly follow Jantzen's lecture notes "Moment graphs and Representations", while for the second part our the classical theory can be found in Humphrey's book on "Reflection groups and Coxeter groups" and for the moment graph approach I will follow my FPSAC paper. For basics on flag and Schubert varieties look also at Brion's lecture notes: "Lectures on the geometry of flag varieties".

Practical information: The course will be held between January and March 2018 (for a total of 25 hours). To get a grade you have to submit solutions to some of the exercises I will propose during the lectures.

All the lectures will take place in Aula D'Antoni from 10:30 to 12:30, unless otherwise specified.

Diary of the lectures:

• • Dec 18, 2017: preliminary meeting.

  • Jan 26, 2018: Principal bundles and universal principal bundles; examples.

  • Jan 30, 2018: Definition of equivariant cohomology and first properties & examples; torus equivariant cohomology of a point; sketch of the P^1C-case; statement of GKM-localisation Theorem.

  • Feb 1, 2018: definition of moment graph; P^1C-case revisited; Grassmann varieties (definition, realisation as a homogeneous space, T-fixed points and 1-dimensional orbits); flag varieties (definition, realisation as a homogeneous space).

  • Feb 6, 2018: RESCHEDULED -> 29th of March

  • Feb 13, 2018: moment graph for the maximal torus action on flag varieties; Schubert cells in the variety of full flags (coset description and description as flags in a given relative position wrt the standard flag, linear algebraic description); Bruhat decomposition; Schubert varieties (definition, linear algebraic description).

  • Feb 15, 2018: Bruhat order on the symmetric group; moment graph for the maximal torus action on a Schubert variety; the case of partial flag varieties; the case of generalised flag varieties G/B, for G a connected reductive complex algebraic group; Bruhat graph; Schubert classes in the moment graph setting.

  • Feb 27, 2018: RESCHEDULED -> 5th of April

  • Mar 2, 2018: Coxeter groups (definition and first examples); length function; root system and geometric representation of a Coxeter group.

  • Mar 16, 2018, h. 2-3pm +3:30-4:30pm: (this lecture will be held by Prof. Kirill Zainoulline, from the University of Ottawa) Titles and abstracts of his talks can be found here

  • Mar 20, 2018:(in Aula Dal Passo) roots amd reflections; parabolic subgroups; Bruhat graphs; Bruhat order; some properties of the Bruhat order; the longest element of a Coxeter group; Hecke algebras; the bar-involution; KL-polynomials.

• Mar 23, 2018, h. 2-4pm : sheaves on moment graphs; modules of sections; the structure sheaf and the structure algebra; flabby sheaves; Braden-MacPherson sheaves; the positivity conjecture

  • Mar 27, 2018: (in Aula Dal Passo) Weak categorification of the Hecke algebra via moment graph theory.

  • Mar 29, 2018: (in Aula Dal Passo) Basics on representation theory of finite dimensional complex Lie algebras: Verma modules, simple modules; Kazhdan-Lusztig's conjecture on multiplicities of simples in Jordan-Hoelder series of Vermas; deformation algebras, deformed Verma modules, deformed weight modules, locally finiteness for deformed modules; deformed category O.

  • Apr 5, 2018: (in Aula Dal Passo) Projective covers in deformed category O; block structure of category O; the centre of an abelian category; the moment graoh associated with a block; the centre of deformed category O and the structure algebra; a functor into combinatorics: from a deformed block of category O to the category of modules over the structure algebra of the associated moment graph (after base change); BMP sheaves and Verma multiplicities in indecomposable projectives (i.e. Fiebig's approach to the Kazhdan-Lusztig conjecture).