# Varieties Related to the Problem of Lifting \(G_r\)-Modules to *G*

## Abstract

Let *G* be a simple simply connected algebraic group over an algebraically closed field *k* of characteristic *p*, with *r*th Frobenius kernel \(G_r\). Let *M* be a \(G_r\)-module and *V* a rational *G*-module. We put a variety structure on the set of all \(G_r\)-summands of *V* that are isomorphic to *M*, and study basic properties of these varieties. This is primarily to set the stage for later work that will bring techniques from geometric invariant theory to bear on the problem of lifting \(G_r\)-modules to *G*. However, we do give a few applications of the work in this paper to the representation theory of *G*, in particular noting that the truth of Donkin’s tilting module conjecture is equivalent to the linearizability of *G*-actions on certain affine spaces.

## 2010 Mathematics Subject Classification

17B10 (primary ) 20G05 (secondary)## Notes

### Acknowledgements

We wish to thank Chris Drupieski and Eric Friedlander for helpful comments on an earlier version of this paper. We thank Hanspeter Kraft for pointing us to work on reductive groups acting on affine spaces in positive characteristic. This work was partially supported by the Research Training Grant, DMS-1344994, from the NSF.

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