[Paper Review] Modular affine Hecke category and regular unipotent centralizer

Roman Bezrukavnikov, Simon Riche+1 more|arXiv (Cornell University)|May 12, 2020

Advanced Algebra and Geometry43 references4 citations

TL;DR

This paper establishes a geometric equivalence between the category of representations of the centralizer of a regular unipotent element in a reductive group over a field of positive characteristic and a certain quotient of the Iwahori-equivariant derived category of perverse sheaves on the affine flag variety of the Langlands dual group. The result is a modular analogue of Soergel's Endomorphismensatz, constructed via a Serre quotient of perverse sheaves and tied to the geometry of the nilpotent cone and centralizer structures.

ABSTRACT

In this paper we provide, under some mild explicit assumptions, a geometric description of the category of representations of the centralizer of a regular unipotent element in a reductive algebraic group in terms of perverse sheaves on the Langlands dual affine flag variety. This equivalence is suggested and motivated by the "geometric Langlands" philosophy, and is used in later work to construct equivalences of categories relating various geometric incarnations of the affine Hecke algebra of the given reductive group.

Motivation & Objective

• To establish a modular variant of the geometric Langlands equivalence for affine Hecke algebras in positive characteristic.
• To provide a geometric description of the category of representations of the centralizer of a regular unipotent element in terms of perverse sheaves on the Langlands dual affine flag variety.
• To lay the foundation for a future equivalence between derived categories of constructible sheaves and coherent sheaves, analogous to Soergel's Struktursatz.
• To generalize the geometric Satake equivalence and Soergel's Endomorphismensatz to the modular setting using Iwahori-equivariant perverse sheaves.

Proposed method

• Construct a Serre quotient category $\mathsf{P}_{\mathrm{Iw}}^{0}$ of Iwahori-equivariant perverse sheaves on the affine flag variety $\mathrm{Fl}_G$, by quotienting out simple objects with positive-dimensional support.
• Define a monoidal structure $\star^{0}_{\mathrm{Iw}}$ on $\mathsf{P}_{\mathrm{Iw}}^{0}$ via the zeroth perverse cohomology of convolution.
• Use the geometric Satake equivalence to relate $G^\vee_{\Bbbk}$-modules to $\mathrm{Gr}_G$-sheaves and identify the centralizer $Z_{G^\vee_{\Bbbk}}(\mathsf{u})$ with a subgroup containing the center $Z(G^\vee_{\Bbbk})$.
• Establish an algebra isomorphism $\operatorname{End}_{\mathsf{P}^{0}_{\mathrm{Iw}}}(\mathscr{Z}^{0}(V)) \cong \operatorname{End}_{Z_{G^\vee_{\Bbbk}}(\mathsf{u})}(V)$, showing that the endomorphism algebra of the image of a tilting module is isomorphic to the endomorphism algebra of its $Z_{G^\vee_{\Bbbk}}(\mathsf{u})$-invariants.
• Use the grading on the coordinate ring $\mathscr{O}(\mathcal{N}_{G^\vee_{\Bbbk}})$ induced by the $\mathbb{G}_m$-dilation action to show that $\operatorname{End}_{Z_{G^\vee_{\Bbbk}}(\mathsf{u})}(V)$ is local when $V$ is indecomposable, implying indecomposability of $\mathscr{Z}^{0}(V)$.

Experimental results

Research questions

• RQ1 Can the geometric Langlands equivalence for affine Hecke algebras be extended to positive characteristic coefficients?
• RQ2 Is there a geometric realization of the category of representations of the centralizer of a regular unipotent element in terms of perverse sheaves on the Langlands dual affine flag variety?
• RQ3 Does the endomorphism algebra of the image of a tilting module under the proposed functor remain local when the module is indecomposable?
• RQ4 How does the action of the center $Z(G^\vee_{\Bbb{k}})$ relate to the structure of the centralizer $Z_{G^\vee_{\Bbb{k}}}(\mathsf{u})$?
• RQ5 Can the structure of the category $\mathsf{P}_{\mathrm{Iw}}^{0}$ be used to construct a full equivalence between derived categories of constructible and coherent sheaves in the modular setting?

Key findings

• The category $\mathsf{P}_{\mathrm{Iw}}^{0}$ of Iwahori-equivariant perverse sheaves modulo positive-dimensional support admits a well-defined monoidal structure $\star^{0}_{\mathrm{Iw}}$ via perverse cohomology of convolution.
• There is a monoidal equivalence between $\mathsf{P}_{\mathrm{Iw}}^{0}$ and the category $\mathrm{Rep}(Z_{G^\vee_{\Bbb{k}}}(\mathsf{u}))$ of finite-dimensional representations of the centralizer of a regular unipotent element in the Langlands dual group.
• The endomorphism algebra $\operatorname{End}_{\mathsf{P}^{0}_{\mathrm{Iw}}}(\mathscr{Z}^{0}(V))$ is isomorphic to $\operatorname{End}_{Z_{G^\vee_{\Bbb{k}}}(\mathsf{u})}(V)$, establishing a key link between geometric and representation-theoretic structures.
• For any finite-dimensional tilting $G^\vee_{\Bbb{k}}$-module $V$, the dimension of the $Z_{G^\vee_{\Bbb{k}}}(\mathsf{u})$-invariant subspace equals the dimension of the zero-weight space $V_0$, implying $V^{Z_{G^\vee_{\Bbb{k}}}(\mathsf{u})} = V^H$ where $H$ contains $Z(G^\vee_{\Bbb{k}})$.
• If $V$ is an indecomposable tilting $G^\vee_{\Bbb{k}}$-module, then $\mathscr{Z}^{0}(V)$ is indecomposable, as the endomorphism algebra $\operatorname{End}_{Z_{G^\vee_{\Bbb{k}}}(\mathsf{u})}(V)$ is local due to the $\mathbb{G}_m$-grading on $\mathscr{O}(\mathcal{N}_{G^\vee_{\Bbb{k}}})$.