Periodic Reporting for period 2 - RedLang (Modular representation theory of reductive algebraic groups and local Geometric Langlands duality)
Okres sprawozdawczy: 2023-03-01 do 2024-08-31
Representation Theory is the study of the main mathematical formalization of the notion of symmetry. The symmetries of an object form a structure called a group; depending on the structures of the object under consideration this group might have compatible additional structures (topological space, algebraic variety, etc.). The "representations" of the group of symmetries can be thought as the different ways these symmetries can appear in geometry. Usually one studies primarily the representations which are compatible with the additional structures of the group. The main subject of this project is the study of algebraic representations of algebraic groups over fields of positive characteristic; in other words one studies groups which have an additional structure of algebraic variety, and studies the representations which are compatible with this structure, in the case of base fields of positive charateristic.
The main objectives of the project RedLang is to develop new geometric tools for this study, with the aim of applying them to answer central unsolved questions in this area. One therefore aims at describing some categories of representations (or related structures) in terms of categories of geometric nature (mainly coherent sheaves on algebraic varieties or constructible sheaves on topological spaces) and then use some tools from geometry to solve the problems we are studying about representations. In particular this project proposes 2 important new constructions in this direction:
(1) The construction of a "ramified geometric Satake equivalence" for general coefficients. The geometric Satake equivalence is a celebrated construction (completed by Mirkovic-Vilonen) which has become a cornerstone in most geometric approaches to the Langlands program (a tentalizing web of theorems and conjectures relating Number Theory, Geometry and Representation Theory). We aim at developing a new variant of this construction (first considered by Zhu and Richarz), and explore some of the new directions of research it will open. (This part of the project is joint with P. Achar, J. Lourenço and T. Richarz.)
(2) The construction of an analogue of Bezrukavnikov's equivalence for positive-characteristic coefficients. Bezrukavnikov's equivalence is another geometric construction of central importance in the Langlands program, which has seen major applications in the recent years. It was initially developed for l-adic sheaves by Bezrukavnikov; we aim at constructing a version of this equivalence of categories for positive-characteristic coefficients, and explore its applications in Representation Theory and Geometry. (This part of the project is joint with R. Bezrukavnikov, and partly with L. Rider.)
In parallel to these main objectives, the members of the project will also study various questions and problems in Geometry and Representation Theory that arise from these studies.
The main objectives of the project RedLang is to develop new geometric tools for this study, with the aim of applying them to answer central unsolved questions in this area. One therefore aims at describing some categories of representations (or related structures) in terms of categories of geometric nature (mainly coherent sheaves on algebraic varieties or constructible sheaves on topological spaces) and then use some tools from geometry to solve the problems we are studying about representations. In particular this project proposes 2 important new constructions in this direction:
(1) The construction of a "ramified geometric Satake equivalence" for general coefficients. The geometric Satake equivalence is a celebrated construction (completed by Mirkovic-Vilonen) which has become a cornerstone in most geometric approaches to the Langlands program (a tentalizing web of theorems and conjectures relating Number Theory, Geometry and Representation Theory). We aim at developing a new variant of this construction (first considered by Zhu and Richarz), and explore some of the new directions of research it will open. (This part of the project is joint with P. Achar, J. Lourenço and T. Richarz.)
(2) The construction of an analogue of Bezrukavnikov's equivalence for positive-characteristic coefficients. Bezrukavnikov's equivalence is another geometric construction of central importance in the Langlands program, which has seen major applications in the recent years. It was initially developed for l-adic sheaves by Bezrukavnikov; we aim at constructing a version of this equivalence of categories for positive-characteristic coefficients, and explore its applications in Representation Theory and Geometry. (This part of the project is joint with R. Bezrukavnikov, and partly with L. Rider.)
In parallel to these main objectives, the members of the project will also study various questions and problems in Geometry and Representation Theory that arise from these studies.
During the first half of the project we have completed the 2 main constructions outlined above.
Regarding (1), with our collaborators we have undertaken a general study of fixed points of certain groups of automorphisms of reductive group schemes (over general bases), and used it to construct the wished-for "modular ramified geometric Satale equivalence". These studies gave rise to 2 preprints. In addition to providing new tools to study important problems in Representation Theory, it was also an occasion to revisit, generalize, and sometimes correct, some important geometric constructions related to the classical geometric Satake equivalence. We have started exploring possible concrete applications of our results, and we expect progress in this direction in the next periods.
Regarding (2), with our collaborators we have constructed the wished-for "modular Bezrukavnikov equivalence". This work gave rise to 3 preprints, totalizing more than 300 pages. Our constructions provide equivalences of categories relating 3 important geometric and representation-theoretic incarnations of the affine Hecke algebra, a central object in the Langlands program. We expect that these equivalences will have important applications; as an evidence of this claim, we have provided a proof of a 25 years old conjecture of Finkelberg and Mirkovic that enhances the geometric Satake equivalence to a description of the principal block of representations of a reductive group. We expect other applications of this construction in the next period.
In parallel to these works, the other members of the project have also contributed to the research in this area, in particular by a very general study of attractors and fixed points for actions of diagonalizable group schemes on algebraic spaces (work of A. Mayeux) and important new developments in the motivic approach to the Langlands program (work of T. van den Hove with a group of collaborators).
Regarding (1), with our collaborators we have undertaken a general study of fixed points of certain groups of automorphisms of reductive group schemes (over general bases), and used it to construct the wished-for "modular ramified geometric Satale equivalence". These studies gave rise to 2 preprints. In addition to providing new tools to study important problems in Representation Theory, it was also an occasion to revisit, generalize, and sometimes correct, some important geometric constructions related to the classical geometric Satake equivalence. We have started exploring possible concrete applications of our results, and we expect progress in this direction in the next periods.
Regarding (2), with our collaborators we have constructed the wished-for "modular Bezrukavnikov equivalence". This work gave rise to 3 preprints, totalizing more than 300 pages. Our constructions provide equivalences of categories relating 3 important geometric and representation-theoretic incarnations of the affine Hecke algebra, a central object in the Langlands program. We expect that these equivalences will have important applications; as an evidence of this claim, we have provided a proof of a 25 years old conjecture of Finkelberg and Mirkovic that enhances the geometric Satake equivalence to a description of the principal block of representations of a reductive group. We expect other applications of this construction in the next period.
In parallel to these works, the other members of the project have also contributed to the research in this area, in particular by a very general study of attractors and fixed points for actions of diagonalizable group schemes on algebraic spaces (work of A. Mayeux) and important new developments in the motivic approach to the Langlands program (work of T. van den Hove with a group of collaborators).
The results obtained so far clearly go beyond the state of the art in Geometric Representation Theory, and open the way to exciting new directions of research.
Geometric Representation Theory has initially began (in particular in work of Lusztig and Kashiwara in the late 1970's) with the use of geometric tools to answer questions of representation theory involving objects with coefficients in characteristic 0. It rapidly obtained spectacular successes, but it was only in the late 1990's that these methods started to be explored for questions involving coefficients in positive characteristic. The PI has contributed to these developments, and obtained important results in this direction. This project aims at pushing these methods even further. This involves many new ideas and manipulation of difficult and powerful geometric methods to overcome the new difficulties arising in this context. The new constructions developed as part of this project provide essential tools that will open the way to the solution of long standing questions in this area.
Beyond the new geometric constructions that have already been achieved, the PI has obtained (in joint work with Bezrukavnikov) a proof of a highly influential conjecture of Finkelberg-Mirkovic on the geometric realization of the principal block of representations of a reductive group. During the next periods we expect progress on the following questions:
- study of restrictions of tilting representations to some fixed-points subgroups (theme initiated by Brundan, and studied in joint work with Achar, Lourenço and Richarz);
- study of tilting ideals for reductive groups, in connection with cohomology of higher Frobenius kernels and p-cells (joint work with Achar, Bezrukavnikov and Boixeda-Alvarez);
- development of the motivic approach to the geometric Langlands program (work of T. van den Hove).
We also expect to initiate the use of methods of infinity-categories (which have become a central tool in the geometric Langlands program) for questions in geometry and representation theory involving positive-characteristic coefficients (joint work with Achar and Dhillon).
Geometric Representation Theory has initially began (in particular in work of Lusztig and Kashiwara in the late 1970's) with the use of geometric tools to answer questions of representation theory involving objects with coefficients in characteristic 0. It rapidly obtained spectacular successes, but it was only in the late 1990's that these methods started to be explored for questions involving coefficients in positive characteristic. The PI has contributed to these developments, and obtained important results in this direction. This project aims at pushing these methods even further. This involves many new ideas and manipulation of difficult and powerful geometric methods to overcome the new difficulties arising in this context. The new constructions developed as part of this project provide essential tools that will open the way to the solution of long standing questions in this area.
Beyond the new geometric constructions that have already been achieved, the PI has obtained (in joint work with Bezrukavnikov) a proof of a highly influential conjecture of Finkelberg-Mirkovic on the geometric realization of the principal block of representations of a reductive group. During the next periods we expect progress on the following questions:
- study of restrictions of tilting representations to some fixed-points subgroups (theme initiated by Brundan, and studied in joint work with Achar, Lourenço and Richarz);
- study of tilting ideals for reductive groups, in connection with cohomology of higher Frobenius kernels and p-cells (joint work with Achar, Bezrukavnikov and Boixeda-Alvarez);
- development of the motivic approach to the geometric Langlands program (work of T. van den Hove).
We also expect to initiate the use of methods of infinity-categories (which have become a central tool in the geometric Langlands program) for questions in geometry and representation theory involving positive-characteristic coefficients (joint work with Achar and Dhillon).