Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On modular categories O for quantized symplectic resolutions (1712.07726v2)

Published 20 Dec 2017 in math.RT

Abstract: In this paper we study highest weight and standardly stratified structures on modular analogs of categories $\mathcal{O}$ over quantizations of symplectic resolutions and show how to recover the usual categories $\mathcal{O}$ (reduced mod $p\gg 0$) from our modular categories. More precisely, we consider a conical symplectic resolution that is defined over a finite localization of $\mathbb{Z}$ and is equipped with a Hamiltonian action of a torus $T$ that has finitely many fixed points. We consider algebras $\mathcal{A}\lambda$ of global sections of a quantization in characterstic $p\gg 0$, where $\lambda$ is a parameter. Then we consider a category $\tilde{\mathcal{O}}\lambda$ consisting of all finite dimensional $T$-equivariant $\mathcal{A}\lambda$-modules. We show that for $\lambda$ lying in a {\it p-alcove} $\,p!A$, the category $\tilde{\mathcal{O}}\lambda$ is highest weight (in some generalized sense). Moreover, we show that every face of $\,p!A$ that survives in $\,p!A/p$ when $p\rightarrow \infty$ defines a standardly stratified structure on $\tilde{\mathcal{O}}_\lambda$. We identify the associated graded categories for these standardly stratified structures with reductions mod $p$ of the usual categories $\mathcal{O}$ in characteristic $0$. Applications of our construction include computations of wall-crossing bijections in characteristic $p$ and the existence of gradings on categories $\mathcal{O}$ in characteristic $0$.

Summary

We haven't generated a summary for this paper yet.