Periodic Reporting for period 4 - ModRed (The geometry of modular representations of reductive algebraic groups)
Reporting period: 2020-09-01 to 2021-08-31
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 reductive 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 to 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 characteristic, 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.
These objects have been reached to a large extend as part of the work performed during this project. In particular we have explored three independent geometric approaches to the computation of characters of tilting modules, which each leads to a solution to this problem. (The main principles of two of these approaches were outlined in the proposal; the third one follows a different strategy which emerged from other works performed as part of this project.) In addition we have obtained a solution to the question of explicitly determining characters of simple modules using the known characters of tilting modules, and we have made important progress towards the solution of the "Humphreys conjecture" on support varieties of tilting modules (as envisaged in the proposal). Finally, we have explored further geometric and structural aspects of these "geometric models", and made the first steps towards building a new bridge that will unify all of these approaches.
As a conclusion, the work performed as part of this project has on the one hand clarified and explicitly constructed various expected geometric pictures for representations of reductive algebraic groups, and on the other hand used these pictures to give concrete solutions to some of the most important questions in this area.
The Representation Theory of reductive 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 to 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 characteristic, 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.
These objects have been reached to a large extend as part of the work performed during this project. In particular we have explored three independent geometric approaches to the computation of characters of tilting modules, which each leads to a solution to this problem. (The main principles of two of these approaches were outlined in the proposal; the third one follows a different strategy which emerged from other works performed as part of this project.) In addition we have obtained a solution to the question of explicitly determining characters of simple modules using the known characters of tilting modules, and we have made important progress towards the solution of the "Humphreys conjecture" on support varieties of tilting modules (as envisaged in the proposal). Finally, we have explored further geometric and structural aspects of these "geometric models", and made the first steps towards building a new bridge that will unify all of these approaches.
As a conclusion, the work performed as part of this project has on the one hand clarified and explicitly constructed various expected geometric pictures for representations of reductive algebraic groups, and on the other hand used these pictures to give concrete solutions to some of the most important questions in this area.
The three main objectives of the project have been reached to a very large extend.
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. After this paper was written we encountered some difficulties in trying to adapt these methods to general reductive groups, and therefore started developing alternative approaches. But more recently, in a joint paper with R. Bezrukavnikov ("Hecke action on the principal block") we found a way to overcome these difficulties using tools developed by other authors in the meantime, which allowed us to complete the program proposed with G. Williamson, and thus provide a proof of the character formula for tilting modules for all reductive groups.
Before we were able to complete this approach, we developed another approach to this question, which led to a first general solution. This alternative approach (whose principles were also outlined in the proposal) was explored 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 developed 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 give a first general proof of the tilting character conjecture, as soon as p is larger than the associated Coxeter number (i.e. in the expected generality).
Finally, in recent work with G. Williamson (see "Smith-Treumann theory and the linkage principle") we found an unexpected and simpler third approach to this question, based on earlier work performed as part of this project and a new perspective on some classical tools from algebraic topology. This approach, not envisaged in the proposal, in fact provides an even more general solution to our problem than what we were expecting, in that it provides character formulas for all reductive groups *in all characteristics*. This can be considered a considerable and unexpected breakthrough in the field.
In work with G. Williamson (see "A simple character formula") we haved used the tilting character formula so obtained to deduce an explicit character formula for simple modules in terms of the p-canonical basis, therefore giving an answer to the most important question in this field. (This answer cannot be considered a final solution to this problem, since the involved combinatorics is very difficult to manipulate, but it definitely is one first solution, which opens very interesting new directions of research.)
Taken together, these works provide a satisfactory solution to our goals (1) and (2) above. Regarding our third goal, the main other question to which our tools have been applied is the celebrated "Humphreys conjecture" on support varieties of tilting modules, which was largely open at this time. We have not been able to obtain a general proof of the conjecture, but in joint work with P. Achar and W. Hardesty ("On the Humphreys conjecture on support varieties of tilting modules") we have obtained a proof in large characteristics. The geometry involved in our approach turned out to be more subtle than we envisaged, and was further studied in later works with the same coauthors (in particular "Conjectures on tilting modules and antispherical p-cells" and "Integral exotic sheaves and the modular Lusztig-Vogan bijection"). These works have then led to further progress on the conjecture by P. Achar and W. Hardesty (without the collaboration of the PI). We expect our new perspective on these questions to be influential in the field. Another open question we have tried to attack is the "Donkin conjecture" on indecomposability of certain tilting modules. Our study has revealed new aspects of this conjecture (explained in "Dualité de Koszul formelle et théorie des représentations des groupes algébriques réductifs en caractéristique positive", with P. Achar), but our concrete progress in this direction is more modest.
Finally, in the last years of the project we have started developing a very ambitious program to construct a "tamely ramified local Langlands equivalence" relating equivariant coherent sheaves on the Steinberg variety of a reductive algebraic group and perverse sheaves on the affine flag variety of the Langlands dual group, for coefficients of positive characteristic. We expect that this construction will at the same time unify the various geometric approaches to representations of reductive groups we have developed so far, provide further tools to attack the questions that remain, and finally open the way to new interactions with Number Theory (in particular through the study of the geometry of Shimura varieties.) A first step towards this goal has been achieved during this project in joint work with R. Bezrukavnikov and L. Rider (see "Modular affine Hecke category and regular unipotent centralizer, I"), and a second (more important) step is close to being completed (joint with R. Bezrukavnikov).
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. After this paper was written we encountered some difficulties in trying to adapt these methods to general reductive groups, and therefore started developing alternative approaches. But more recently, in a joint paper with R. Bezrukavnikov ("Hecke action on the principal block") we found a way to overcome these difficulties using tools developed by other authors in the meantime, which allowed us to complete the program proposed with G. Williamson, and thus provide a proof of the character formula for tilting modules for all reductive groups.
Before we were able to complete this approach, we developed another approach to this question, which led to a first general solution. This alternative approach (whose principles were also outlined in the proposal) was explored 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 developed 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 give a first general proof of the tilting character conjecture, as soon as p is larger than the associated Coxeter number (i.e. in the expected generality).
Finally, in recent work with G. Williamson (see "Smith-Treumann theory and the linkage principle") we found an unexpected and simpler third approach to this question, based on earlier work performed as part of this project and a new perspective on some classical tools from algebraic topology. This approach, not envisaged in the proposal, in fact provides an even more general solution to our problem than what we were expecting, in that it provides character formulas for all reductive groups *in all characteristics*. This can be considered a considerable and unexpected breakthrough in the field.
In work with G. Williamson (see "A simple character formula") we haved used the tilting character formula so obtained to deduce an explicit character formula for simple modules in terms of the p-canonical basis, therefore giving an answer to the most important question in this field. (This answer cannot be considered a final solution to this problem, since the involved combinatorics is very difficult to manipulate, but it definitely is one first solution, which opens very interesting new directions of research.)
Taken together, these works provide a satisfactory solution to our goals (1) and (2) above. Regarding our third goal, the main other question to which our tools have been applied is the celebrated "Humphreys conjecture" on support varieties of tilting modules, which was largely open at this time. We have not been able to obtain a general proof of the conjecture, but in joint work with P. Achar and W. Hardesty ("On the Humphreys conjecture on support varieties of tilting modules") we have obtained a proof in large characteristics. The geometry involved in our approach turned out to be more subtle than we envisaged, and was further studied in later works with the same coauthors (in particular "Conjectures on tilting modules and antispherical p-cells" and "Integral exotic sheaves and the modular Lusztig-Vogan bijection"). These works have then led to further progress on the conjecture by P. Achar and W. Hardesty (without the collaboration of the PI). We expect our new perspective on these questions to be influential in the field. Another open question we have tried to attack is the "Donkin conjecture" on indecomposability of certain tilting modules. Our study has revealed new aspects of this conjecture (explained in "Dualité de Koszul formelle et théorie des représentations des groupes algébriques réductifs en caractéristique positive", with P. Achar), but our concrete progress in this direction is more modest.
Finally, in the last years of the project we have started developing a very ambitious program to construct a "tamely ramified local Langlands equivalence" relating equivariant coherent sheaves on the Steinberg variety of a reductive algebraic group and perverse sheaves on the affine flag variety of the Langlands dual group, for coefficients of positive characteristic. We expect that this construction will at the same time unify the various geometric approaches to representations of reductive groups we have developed so far, provide further tools to attack the questions that remain, and finally open the way to new interactions with Number Theory (in particular through the study of the geometry of Shimura varieties.) A first step towards this goal has been achieved during this project in joint work with R. Bezrukavnikov and L. Rider (see "Modular affine Hecke category and regular unipotent centralizer, I"), and a second (more important) step is close to being completed (joint with R. Bezrukavnikov).
As explained above, the work performed as part of this project is a breakthrough in its field, since it provides character formulas for tilting and simple representations of reductive algebraic groups, which were considered out of reach a few years ago, and which give a solution to the most central question in this area. The third approach to this question goes even further than what was envisaged in the proposal, since it provides characters for characteristics below the Coxeter number. We have also made important contributions to some other important conjectures on representations of reductive groups (in particular the Humphreys conjecture), explored related geometric questions (leading in particular to new conjectures which should have an impact on future research), and started developing new connections between several "geometric models" involved in our study, which will pave the way to new and exciting developments which might go beyond our initial field of research.