ON RANK VARIETIES FOR VERMA MODULES OF HIGHER FROBENIUS KERNELS

GERHARD RÖHRLE

Abstract

This is a report on a joint project with Rolf Farnsteiner.

In this talk we shall discuss new results on rank varieties of Verma modules of Frobenius kernels of reductive algebraic groups. We indicate how these geometric results can be used to determine the structure of the component of the stable Auslander-Reiten quiver containing the isoclass of a Verma module.

More specifically, let G_{r} be the r-th Frobenius kernel of a connected, reductive group G and let Z_{r}(λ) be the Verma module of G_{r} with highest weight λ. By Θ_{r}(λ) we denote the component of the stable Auslander-Reiten quiver containing the isoclass of Z_{r}(λ). It was shown in [1] that Θ_{1}(λ) is of tree class A_{∞} and that Z_{1}(λ) is located at an end of Θ_{1}(λ). The goal of this project is to extend this result to higher Frobenius kernels. We shall explain how new results on the rank varieties of Z_{r}(λ) for r ≥ 2 allow us to reduce this study of the Auslander-Reiten components Θ_{r}(λ) to the second Frobenius kernel of SL(2).

References

[1] R. Farnsteiner, G. Röhrle. Almost split sequences for Verma modules. Math. Ann. 322 (2002), 701–743.

School of Mathematics and Statistics, The University of Birmingham, Birmingham B15 2TT, UK

   E-mail address: ger@for.mat.bham.ac.uk