Goddard, Russell (2017). Commuting varieties and nilpotent orbits. University of Birmingham. Ph.D.

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 previouslyknown 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): 


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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