Central subalgebras of the centralizer of a nilpotent element
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- by George J. McNinch and Donna M. Testerman
- Proc. Amer. Math. Soc. 144 (2016), 2383-2397
- DOI: https://doi.org/10.1090/proc/12942
- 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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Bibliographic Information
- George J. McNinch
- Affiliation: Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, Massachusetts 02155
- MR Author ID: 625671
- 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
- MR Author ID: 265736
- Email: donna.testerman@epfl.ch
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2383-2397
- MSC (2010): Primary 20G15; Secondary 17B45, 17B05
- DOI: https://doi.org/10.1090/proc/12942
- MathSciNet review: 3477055