Vermas, co-Vermas and simple modules

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: 9 February 2009

Vermas, co-Vermas and simple modules

Let $$\widehat{𝒪}=\left\{\text{isomorphism classes of irreducible finite dimensional }\mathbb{Z}\text{-graded }{𝔤}_0\text{-modules}\right\}$$

If $L_0\left(\lambda \right)\in \widehat{𝒪}$ then $L_0\left(\lambda \right)$ is concentrated in a single degree. Let $U=U\left(𝔤\right),U^{\ge 0}=U\left({𝔤}_{\ge 0}\right)$ and $U=U\left(𝔤\right),U^{\le 0}=U\left({𝔤}_{\le 0}\right),$ where $${𝔤}_{\ge 0}={\oplus }_{i\in {\mathbb{Z}}_{\ge 0}}{𝔤}_i$$ and $${𝔤}_{\le 0}={\oplus }_{i\in {\mathbb{Z}}_{\le 0}}{𝔤}_i.$$

The Verma module and the coVerma module are $$\Delta \left(\lambda \right)=U{\otimes }_{U^{\ge 0}}{L}_0\left(\lambda \right)$$ and $$\nabla \left(\lambda \right)={\mathrm{Hom}}_{U\le 0}\left(U,L_0\left(\lambda \right)\right).$$

The modules $$\begin{array}{rcl}L\left(\lambda \right)& =& \text{unique simple quotient of }\Delta \left(\lambda \right)\\ & =& \text{unique simple submodule of }\nabla \left(\lambda \right),\end{array}$$ for $\lambda \in \widehat{𝒪}$ are the simple objects in $𝒪.$

Vermas, co-Vermas and simple modules

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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