Equivariant Sheaves for Classical Groups Acting on Grassmannians: Existence, Properties, and Applications to Springer Theory

Core Concepts

This paper introduces a new stratification of Grassmannians based on the action of classical groups and investigates the properties of parity sheaves within this framework, aiming to lay the groundwork for applications in Springer theory, particularly Mautner's cleanness conjecture.

Abstract

Bibliographic Information:

Achar, P. N., & Chatterjee, T. (2024). Equivariant Sheaves for Classical Groups Acting on Grassmannians. arXiv:2411.03158v1 [math.RT].

Research Objective:

This research paper delves into the study of parity sheaves on Grassmannians equipped with stratifications derived from the action of classical groups, aiming to establish their properties and explore potential applications in Springer theory.

Methodology:

The authors utilize tools from algebraic geometry and representation theory, including the classification of group orbits, analysis of equivariant fundamental groups, construction of resolutions of singularities, and the study of hypercohomology of sheaves.

Key Findings:

• The paper provides a classification of QB-orbits on the Grassmannian Grk(B), where QB is a specific group related to the classical group acting on the underlying vector space.
• It determines that the QB-equivariant fundamental group of each orbit is a product of copies of Z/2Z, implying the semisimplicity of the category of QB-equivariant k-local systems on each orbit when the characteristic of the field k is not 2.
• For each orbit and irreducible local system, the authors construct a unique indecomposable parity sheaf and prove a parity-vanishing property for its hypercohomology.
• The paper also details the construction of specific resolutions of singularities for orbit closures, crucial for the existence and properties of parity sheaves.

Main Conclusions:

The study establishes the existence and unique properties of parity sheaves on Grassmannians stratified by the action of classical groups. These findings are anticipated to be instrumental in future research on Springer theory, particularly in addressing Mautner's cleanness conjecture for classical groups.

Significance:

This research significantly contributes to the understanding of parity sheaves in a new geometric context and provides a theoretical framework for tackling open problems in Springer theory, a central area of representation theory with connections to other mathematical fields.

Limitations and Future Research:

The paper focuses on a specific type of stratification of Grassmannians. Exploring parity sheaves under different stratifications and extending the results to other algebraic varieties could be promising avenues for future research. Additionally, the application of these findings to explicitly address Mautner's cleanness conjecture is envisioned as a subsequent step.

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Equivariant sheaves for classical groups acting on Grassmannians

by Pramod N. Ac... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.03158.pdf

Deeper Inquiries

How can the properties of parity sheaves established in this paper be utilized to investigate other geometric objects or representation-theoretic problems beyond Springer theory?

The properties of parity sheaves established in the paper, particularly their even cohomology and their connection to equivariant local systems, can be utilized to investigate a variety of geometric objects and representation-theoretic problems beyond Springer theory. Here are some potential avenues:

Other stratified spaces: The techniques used in the paper, such as constructing even resolutions and universal covering submersions, can be adapted to study parity sheaves on other stratified spaces with group actions. Examples include:

Partial flag varieties: These are natural generalizations of Grassmannians, and the results of the paper could potentially be extended to study parity sheaves for suitable stratifications on partial flag varieties.
Toric varieties: These spaces admit torus actions and have a rich combinatorial structure. Parity sheaves in this context could provide insights into the interplay between geometry and combinatorics.
Singularities: Parity sheaves have been used to study singularities of Schubert varieties. The methods developed in the paper might be applicable to investigating singularities in other geometric contexts, especially those with group actions.

Modular representation theory: Parity sheaves play a crucial role in the study of modular representations of algebraic groups and related objects like Hecke algebras. The results of the paper could potentially be used to:

Construct and study modular representations: The parity sheaves constructed in the paper could lead to new constructions of modular representations of classical groups and their covers.
Analyze decomposition numbers: Parity sheaves are related to decomposition numbers, which measure the failure of modular representations to be semisimple. The paper's results could provide new tools for studying these numbers.

Geometric Langlands program: The geometric Langlands program seeks to relate representations of Galois groups to geometric objects on moduli spaces. Parity sheaves have emerged as important tools in this program. The paper's focus on classical groups and Grassmannians could potentially lead to new insights into the geometric Langlands correspondence for these groups.

Equivariant cohomology theories: The paper focuses on ordinary equivariant cohomology, but the techniques could potentially be adapted to study parity sheaves in other equivariant cohomology theories, such as K-theory or elliptic cohomology. This could lead to new connections between these theories and representation theory.

Could there be alternative constructions of resolutions of singularities for the orbit closures that provide different insights into the structure of parity sheaves or lead to different applications?

Yes, alternative constructions of resolutions of singularities for the orbit closures could certainly provide different insights into the structure of parity sheaves and lead to different applications. Here are some possibilities:

Bott-Samelson resolutions: These are well-known resolutions of Schubert varieties in flag varieties, and they could potentially be adapted to the setting of the paper. Bott-Samelson resolutions have a combinatorial structure that could shed light on the structure of parity sheaves.

De Concini-Procesi resolutions: These resolutions are defined for more general types of singularities, including those arising in the closures of nilpotent orbits. Adapting these resolutions to the Grassmannian setting could provide a different perspective on the geometry of the orbit closures.

Quiver varieties: In some cases, orbit closures in Grassmannians can be realized as quiver varieties. These varieties come with natural resolutions that could be used to study parity sheaves.

Symplectic resolutions: When the orbit closures have symplectic structures, one could seek symplectic resolutions, which are resolutions that preserve the symplectic form. These resolutions have special properties that could be relevant to the study of parity sheaves.
Different resolutions can lead to different insights because they emphasize different aspects of the geometry of the orbit closures. For example, some resolutions might be better suited for studying intersection theory, while others might be more useful for understanding the topology of the fibers.

What are the implications of the connection between the topology of the Grassmannian stratification and the structure of the classical groups acting on them, and how can this be further explored?

The connection between the topology of the Grassmannian stratification and the structure of the classical groups acting on them is a manifestation of a deep interplay between geometry and representation theory. Here are some implications and potential avenues for further exploration:

Representation-theoretic interpretations of geometric invariants: The paper shows that the equivariant cohomology of the strata is related to the structure of the stabilizers, which are subgroups of the classical groups. This suggests that other geometric invariants of the stratification, such as intersection cohomology or K-theoretic invariants, might also have representation-theoretic interpretations.

Geometric construction of representations: The parity sheaves themselves can be viewed as geometric realizations of certain representations of the classical groups. The stratification of the Grassmannian provides a framework for constructing and studying these representations geometrically.

Deeper understanding of the geometry of orbits: The structure of the classical groups dictates the possible orbit types and their closures. Further exploration could involve:

Studying singularities of orbit closures: The nature of the singularities is related to the stabilizers of points in the orbits.
Investigating the incidence relations between orbits: The closure relations between orbits are governed by the group action, and understanding these relations can provide insights into the geometry of the Grassmannian.

Generalizations to other groups and spaces: The connection between the topology of stratifications and the structure of groups acting on them is a general phenomenon. It would be interesting to explore this connection for other groups, such as exceptional groups, and other spaces, such as flag varieties or other homogeneous spaces.

Connections to other areas of mathematics: The interplay between geometry and representation theory explored in the paper has connections to other areas of mathematics, such as:

Combinatorics: The structure of the Grassmannian and its stratification can be described using combinatorial objects, such as partitions and Young diagrams.
Algebraic geometry: The study of orbit closures and their resolutions is a central topic in algebraic geometry.
Symplectic geometry: When the Grassmannian and the orbits have symplectic structures, the group action provides a rich source of examples and techniques for symplectic geometry.
By further exploring these connections, we can gain a deeper understanding of both the geometry of Grassmannians and the representation theory of classical groups.