Free-monodromic mixed tilting sheaves on flag varieties

Pramod N. Achar; Shotaro Makisumi; Simon Riche; Geordie Williamson

Home/Archive/arXiv

arXiv · 1703.05843

Free-monodromic mixed tilting sheaves on flag varieties

Pramod N. Achar·Shotaro Makisumi·Simon Riche·Geordie Williamson

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we propose a construction of a monoidal category of "free-monodromic" tilting perverse sheaves on (Kac-Moody) flag varieties in the setting of the "mixed modular derived category" introduced by the first and third authors. This category shares most of the properties of their counterpart in characteristic 0, defined by Bezrukavnikov-Yun using certain pro-objects in triangulated categories. This construction is the main new ingredient in the construction of a "modular Koszul duality" equivalence for constructible sheaves on flag varieties, see [P. Achar, S. Makisumi, S. Riche, and G. Williamson, "Koszul duality for Kac-Moody groups and characters of tilting modules", J. Amer. Math. Soc. 32 (2019)].

Explore related subjects

math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Pramod N. Achar, Shotaro Makisumi, Simon Riche, Geordie Williamson. 2022-11-14. Free-monodromic mixed tilting sheaves on flag varieties. https://arxiv.org/abs/1703.05843

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

KEEP EXPLORING

Related papers

arXiv2026

Minuscule Relations in Quantum $K$-Theory of Flag Varieties

We study the quantum $K$-theory of the flag variety $G/B$. For each minuscule fundamental weight $\varpi$, we construct an explicit relation in the torus-equivariant quantum $K$-theory $QK_T(G/B)$. The relation can be regarded as a quantum deformation of the character of the irreducible representation with highest weight $\varpi$.

math.RT↗

arXiv2026

A Gelfand model for the Okada algebra

In this paper, we construct a Gelfand model for the Okada algebra $O_n(X,Y)$ with generic parameters $X$ and $Y$, on the space of symmetric Okada arc diagrams using a conjugation-type action. The model is constructed inductively by identifying the Okada algebra as a diagram algebra and using the Jones basic construction to obtain a tower of algebras that are themselves Okada algebras at lower levels. We use the model to obtain all the irreducible representations of $O_n(X,Y)$, indexed by the elements of rank $n$ of the Young--Fibonacci lattice, and identify them with the cell modules of $O_n(X,Y)$.

math.RT↗

arXiv2026

Categorical Lie-Rinehart modules and Shen-Larsson functors

We develop a categorical framework for Lie-Rinehart monoids and their weak modules in a symmetric monoidal category. Using crossed homomorphisms, we construct a natural action of the monoidal category of modules over a Lie monoid on the category of weak Lie-Rinehart modules, thereby obtaining categorical versions of the Shen-Larsson functors. We further characterize the conditions under which the category of weak modules admits a monoidal structure and identify the corresponding condition for the associated functors to be strict monoidal. A dual theory for Lie- Rinehart comonoids and weak comodules is developed using cocrossed homomorphisms. Combining the module and comodule constructions, we obtain a bimodule category structure on the category of weak modules. Finally, we specialize the general framework to the symmetric monoidal category of super vector spaces, recovering Lie-Rinehart superalgebras and their associated Shen-Larsson-type constructions.

math.RT↗