Variations on a Theme of Cline and Donkin |
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Authors: | Brian J Parshall Leonard L Scott |
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Institution: | 1. Department of Mathematics, University of Virginia, Charlottesville, VA, 22903, USA
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Abstract: | Let N be a normal subgroup of a group G. An N-module Q is called G-stable provided that Q is equivalent to the twist Q g of Q by g, for every g?∈?G. If the action of N on Q extends to an action of G on Q, then Q is obviously G-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups G asserts that the (obviously G-stable) projective indecomposable modules (PIMs) Q for the Frobenius kernels G r (r?≥?1) of G have a G-module structure. It is sometimes just as useful (for a general module Q) to know that a finite direct sum Q ?⊕?n of Q has a compatible G-module structure. In this paper, this property is called numerical stability. In recent work (Parshall and Scott, Adv Math 226:2065–2088, 2011), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of k-group schemes and a suitable version of G-stability, called strong G-stability. Among our results here is the determination of necessary and sufficient conditions for the existence of a compatible G-module structure on a strongly G-stable N-module, in the form of a cohomological obstruction which must be trivial precisely when the G-module structure exists. Our main result is achieved by giving an approach to killing the obstruction by tensoring with certain finite dimensional G/N-modules. |
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