Use this URL to cite or link to this record in EThOS:  http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.715618 
Title:  Commuting varieties and nilpotent orbits  
Author:  Goddard, Russell 
ISNI:
0000 0004 6347 2558


Awarding Body:  University of Birmingham  
Current Institution:  University of Birmingham  
Date of Award:  2017  
Availability of Full Text: 


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.


Supervisor:  Not available  Sponsor:  Engineering and Physical Sciences Research Council  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.715618  DOI:  Not available  
Keywords:  QA Mathematics  
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