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Teaser, summary, work performed and final results

Periodic Reporting for period 2 - ModRed (The geometry of modular representations of reductive algebraic groups)

Teaser

\"The main goal of the present project is to explore the Representation Theory of reductive algebraic groups over fields of positive characteristic through Geometry. The Representation Theory of redutive algebraic groups is a subject of important study since the 1960\'s. One of...

Summary

\"The main goal of the present project is to explore the Representation Theory of reductive algebraic groups over fields of positive characteristic through Geometry.

The Representation Theory of redutive algebraic groups is a subject of important study since the 1960\'s. One of the major open questions in the field is the determination of character formulas for simple modules (i.e. understand the \"\"size\"\" of the \"\"building blocks\"\" of the theory). This question is the subject of a conjecture due to G. Lusztig in 1980, which was then partially proved in the 1990\'s. However, a recent breakthrough due tu G. Williamson (one of the main collaborators of the PI on this project) has shown that this conjecture does *not* provide a correct answer in the generality that Lusztig (and most specialists after him) expected. The starting point of this project was a program (including collaborations with P. Achar and G. Williamson, among others) providing a new approach to this question. We proposed to look for a character formula for indecomposable tilting modules (another important family of representations, parametrized in a way similar to simple modules), whose solution is known to provide (in theory) a solution to the simple character formula question.

This study is important for the mathematical community because algebraic groups form a basic structure which appears in many different fields, and because this question lies at the intersection of fundamental problems in various fields (in particular Representation Theory of finite groups of Lie type in defining characteristics, Number Theory via various versions of the Langlands program, Geometry). Moreover, a solution to this problem would provide tools for attacking many other problems in Representation Theory, and we expect our methods to lead to progress on these questions too.

The main objectives of this project are:
1. to build a \"\"geometric model\"\" for Representations of reductive algebraic groups over fields of positive characteristic via coherent sheaves on the Springer resolution and perverse sheaves on affine flag varieties;
2. to use this model to obtain character formulas for simple and tilting representations;
3. to explore how these results can help solving various other open questions in the area.\"

Work performed

\"During the first period of this project we have essentially reached the first two objectives outlined above.

The first main achievement of this project was a paper joint with G. Williamson (entitled \"\"Tilting modules and the p-canonical basis\"\") where we explain a new paradigm in Representation Theory over fields of characteristic p>0, stating that the combinatorial data people are interested in in this area should be expressed in terms of the \"\"p-canonical basis\"\" of various Hecke algebras, as introduced recently by Williamson. We illustrate this idea by stating a conjecture giving a character formula for indecomposable tilting modules over reductive algebraic groups, which we expect to be valid as soon as p is bigger than the associated Coxeter number (i.e. in the biggest \"\"reasonable\"\" generality for this problem). In this publication we prove this conjecture in the (very important) special case of the groups GL(n), via diagrammatic techniques.

The next main achievement was a proof of this character formula for all reductive groups, in two steps. First, in a joint work with P. Achar (entitled \"\"Reductive groups, the loop Grassmannian, and the Springer resolution\"\"), and building on earlier work with C. Mautner (entitled \"\"Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković-Vilonen conjecture\"\") we develop a \"\"geometric model\"\" for the Representation Theory of reductive algebraic groups, involving coherent sheaves on the Springer resolution and perverse sheaves on affine flag varieties. Then, as a second step, in collaboration with P. Achar, S. Makisumi and G. Williamson we developed a \"\"Koszul duality\"\" for constructible sheaves on Kac-Moody flag varieties (see \"\"Koszul duality for Kac-Moody groups and characters of tilting modules\"\"). Combining these two tools we were able to prove the tilting character conjecture with Williamson, as soon as p is bigger than the associated Coxeter number.

In work in progress with G. Williamson we use this tilting character formula to deduce an explicit character formula for simple modules in terms of the p-canonical basis.

Finally, we also started exploring our 3rd main goal, and in particular obtained important progress on the celebrated \"\"Humphreys conjecture\"\" on support varieties of tilting modules.\"

Final results

\"As explained above, the work performed so far is already a breakthrough in the subject, since it provides character formulas for tilting and simple representations of reductive algebraic groups, which were considered out of reach a few years ago.

Our main goals for the rest of the project are:
- Trying to make our character formulas as explicit as possible, and study their combinatorics.
- Exploring how these new character formulas can help solving various other questions in the field (in particular, the Humphreys conjecture on character varieties of tilting modules).
- Developing positive-characteristic versions of some \"\"Geometric Langlands\"\" equivalences of categories.\"