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.References
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Additional Information
- Rolf Farnsteiner
- Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany
- MR Author ID: 194225
- Email: rolf@math.uni-kiel.de
- 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’.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 1487-1531
- MSC (2010): Primary 14L15, 16G70; Secondary 16T05
- DOI: https://doi.org/10.1090/S0002-9947-2012-05672-8
- MathSciNet review: 3003272