# Commuting varieties and nilpotent orbits

Goddard, Russell (2017). Commuting varieties and nilpotent orbits. University of Birmingham. Ph.D.  Preview
Goddard17PhD.pdf
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## Abstract

Let $$G$$ be a reductive algebraic group over an algebraically closed field $$k$$ of good characteristic, let $$g$$=Lie($$G$$) be the Lie algebra of $$G$$, and let $$P$$ be a parabolic subgroup of $$G$$ with $$p$$=Lie($$P$$).
We consider the commuting variety $$C$$($$p$$) of $$p$$ and obtain two criteria for $$C$$($$p$$) to be irreducible. In particular we classify all cases when the commuting variety $$C$$($$b$$) is irreducible, for $$b$$ a Borel subalgebra of $$g$$.
We then let $$G$$ be a classical group and let $$O$$$$_1$$ and $$O$$$$_2$$ be nilpotent orbits of $$G$$ in $$g$$. We say that $$O$$$$_1$$ and $$O$$$$_2$$ commute if there exists a pair ($$X$$, $$Y$$) ∈ $$O$$$$_1$$×$$O$$$$_2$$ such that [$$X$$,$$Y$$]=0. For $$g$$=$$s$$$$p$$$$_2$$$$_m$$($$k$$) or $$g$$=$$s$$$$o$$$$_n$$($$k$$), we describe the orbits that commute with the regular orbit, and classify (with one exception) the orbits that commute with all other orbits in $$g$$. This extends previously-known results for $$g$$=$$g$$$$l$$$$_n$$($$k$$).
Finally let φ be a Springer isomorphism, that is, a $$G$$-equivariant isomorphism from the unipotent variety $$U$$ of $$G$$ to the nilpotent variety $$N$$ of $$g$$. We show that polynomial Springer isomorphisms exist when $$G$$ is of type G$$_2$$, but do not exist for types E$$_6$$ and E$$_7$$ for $$k$$ of small characteristic.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Goodwin, SimonUNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Mathematics
Funders: None/not applicable
Subjects: Q Science > QA Mathematics
URI: http://etheses.bham.ac.uk/id/eprint/7476

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