Singular localization and intertwining functors for reductive Lie algebras in prime characteristic

Roman Bezrukavnikov; Ivan Mirkovic; Dmitry Rumynin

Home/Archive/arXiv

arXiv · math/0602075

Singular localization and intertwining functors for reductive Lie algebras in prime characteristic

Roman Bezrukavnikov·Ivan Mirkovic·Dmitry Rumynin

View original source ↗Read source document ↗Export metadata ↓

Abstract

In math.RT/0205144 we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters. The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character as sheaves on the partial flag variety corresponding to the singularity of the character. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators, but is actually larger. We discuss translation functors and intertwining functors. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of D-modules and coherent sheaves.

Explore related subjects

math.RT math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Roman Bezrukavnikov, Ivan Mirkovic, Dmitry Rumynin. 2006-11-27. Singular localization and intertwining functors for reductive Lie algebras in prime characteristic. https://arxiv.org/abs/math/0602075

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Archimedean period relations for Rankin-Selberg convolutions

We formulate and prove the archimedean period relations for Rankin-Selberg convolutions of $\text{GL}(n)\times \text{GL}(n)$ and $\text{GL}(n)\times \text{GL}(n-1)$, for all generic cohomological representations. As a consequence, we prove the non-vanishing of the archimedean modular symbols. This extends the earlier results in [LLS24] for essentially tempered representations of $\text{GL}(n)\times\text{GL}(n-1)$.

math.RT↗

arXiv2026

Rational structures on quivers and a generalization of Gelfand's equivalence

We introduce the notion of rational structure on a quiver and associated representations to establish a coherent framework for studying quiver representations in separable field extensions. This notion is linked to a refinement of the notion of $K$-species, which we term étale $K$-species. We establish a categorical anti-equivalence between $K$-rational quivers and étale $K$-species, which extends to an equivalence of their respective representation categories. For $K$-rational quivers there is a canonical notion of base change, which suggests a corresponding notion of base change for (étale) $K$-species which we elaborate. As a primary application, we generalize Gelfand's celebrated equivalence between certain blocks of Harish-Chandra modules for $\mathrm{SL}_2(\mathbb{R})$ and representations of the Gelfand quiver to a rational setting. To this end, we define a $\mathbb{Q}$-rational structure on the Gelfand quiver and its representations. A key technical tool, which we call unipotent stabilization, is developed to construct the functor from rational Harish-Chandra modules to nilpotent rational quiver representations. We prove that this functor is an equivalence. A similar result is established for the cyclic quiver. Both results hold over every field $E$ of characteristic $0$, for suitable $E$-rational structures on the two quivers; for $E=\mathbb{C}$ they recover Gelfand's equivalence. In this rational framework the defining relation of the Gelfand quiver acquires a descent-theoretic meaning: its Galois orbit is stable, it descends to a single relation on the associated $\mathbb{Q}$-species, and a $\mathbb{Q}$-rational representation satisfies it if and only if the composite of the two arrows through the vertex $-$ is a $\mathbb{Q}$-rational endomorphism of the space at the central vertex. This allows us to recast our results in the language of $\mathbb{Q}$-species.

math.RT↗

arXiv2026

Perverse filtration on the cohomology of affine Springer fibers

We show that the top cohomology of any affine Springer fiber, as a Weyl group representation, contains a large part of the total cohomology of certain Springer fibers. The main ingredient of the proof is the construction of a ``perverse filtration'' on the pure part of the cohomology of affine Springer fibers.

math.RT↗