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Central subalgebras of the centralizer of a nilpotent element


Authors: George J. McNinch and Donna M. Testerman
Journal: Proc. Amer. Math. Soc. 144 (2016), 2383-2397
MSC (2010): Primary 20G15; Secondary 17B45, 17B05
Published electronically: October 21, 2015
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Abstract: Let $ G$ be a connected, semisimple algebraic group over a field $ k$ whose characteristic is very good for $ G$. In a canonical manner, one associates to a nilpotent element $ X \in \mathrm {Lie}(G)$ a parabolic subgroup $ P$ - in characteristic zero, $ P$ may be described using an $ \mathfrak{sl}_2$-triple containing $ X$; in general, $ P$ is the ``instability parabolic'' for $ X$ as in geometric invariant theory.

In this setting, we are concerned with the center $ Z(C)$ of the centralizer $ C$ of $ X$ in $ G$. Choose a Levi factor $ L$ of $ P$, and write $ d$ for the dimension of the center $ Z(L)$. Finally, assume that the nilpotent element $ X$ is even. In this case, we can deform $ \mathrm {Lie}(L)$ to $ \mathrm {Lie}(C)$, and our deformation produces a $ d$-dimensional subalgebra of $ \mathrm {Lie}(Z(C))$. Since $ Z(C)$ is a smooth group scheme, it follows that $ \dim Z(C) \ge d = \dim Z(L)$.

In fact, Lawther and Testerman have proved that $ \dim Z(C) = \dim Z(L)$. Despite only yielding a partial result, the interest in the method found in the present work is that it avoids the extensive case-checking carried out by Lawther and Testerman.


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Additional Information

George J. McNinch
Affiliation: Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, Massachusetts 02155
Email: george.mcninch@tufts.edu, mcninchg@member.ams.org

Donna M. Testerman
Affiliation: Institut de Géométrie, Algèbre et Topologie, Bâtiment BCH, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
Email: donna.testerman@epfl.ch

DOI: https://doi.org/10.1090/proc/12942
Received by editor(s): November 20, 2014
Received by editor(s) in revised form: July 30, 2015
Published electronically: October 21, 2015
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society