Complexity, periodicity and one-parameter subgroups

Farnsteiner, Rolf

doi:10.1090/S0002-9947-2012-05672-8

Complexity, periodicity and one-parameter subgroups
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Abstract:

Using the variety of infinitesimal one-parameter subgroups introduced by Suslin-Friedlander-Bendel, we define a numerical invariant for representations of an infinitesimal group scheme $\mathcal {G}$. For an indecomposable $\mathcal {G}$-module $M$ of complexity $1$, this number, which may also be interpreted as the height of a “vertex” $\mathcal {U}_M \subseteq \mathcal {G}$, is related to the period of $M$. In the context of the Frobenius category of $G_rT$-modules associated to a smooth reductive group $G$ and a maximal torus $T \subseteq G$, our methods give control over the behavior of the Heller operator of such modules, as well as precise values for the periodicity of their restrictions to $G_r$. Applications include the structure of stable Auslander-Reiten components of $G_rT$-modules as well as the distribution of baby Verma modules.

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