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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)



Complexity, periodicity and one-parameter subgroups

Author: Rolf Farnsteiner
Journal: Trans. Amer. Math. Soc. 365 (2013), 1487-1531
MSC (2010): Primary 14L15, 16G70; Secondary 16T05
Published electronically: October 4, 2012
MathSciNet review: 3003272
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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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Additional Information

Rolf Farnsteiner
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany

Received by editor(s): February 6, 2011
Received by editor(s) in revised form: July 15, 2011
Published electronically: October 4, 2012
Additional Notes: This work was supported by the D.F.G. priority program SPP1388 ‘Darstellungstheorie’.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.