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
Awarding Body: University of Birmingham
Current Institution: University of Birmingham
Date of Award: 2017
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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.
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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