The Category $𝒪$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 5 February 2010

The category $𝒪$

Let $$ℱ=\underset{i\in \mathbb{Z}}{⨁}{ℱ}_i\text{be a}\mathbb{Z}\text{-graded algebra}$$ such that

1. $\dim \left({ℱ}_i\right)\le \infty$,
2. ${ℱ}_0$ is reductive, and
3. $ℱ$ is semisimple as a ${ℱ}_0$-module (under adjoint action).

The category $𝒪$ for $ℱ=\underset{i\in \mathbb{Z}}{⨁}{ℱ}_i$ is the category of $$\mathbb{Z}\text{-graded}ℱ\text{-modules}M=\underset{i\in \mathbb{Z}}{⨁}{M}_i$$ such that

1. $M$ is semisimple as a ${ℱ}_0$-module, and
2. if $m\in M$ then $\dim \left(U^{\ge 0}m\right)<\infty$,

where $U^{\ge 0}=U\left({ℱ}_{\ge 0}\right)$ with ${ℱ}_{\ge 0}=\underset{i\in {\mathbb{Z}}_{\ge 0}}{⨁}{ℱ}_i$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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