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Modular Reduction of Nilpotent Orbits
(Jay Taylor, Dec. 4, 2024)

The general linear group G=GLn(k) over a field k acts on the 
space gl=gln of (nxn)-matrices 
by conjugation. The set N(g) of nilpotent matrices 
is preserved by this action and G acts with finitely many orbits. 
The Jordan normal form gives a representative of each orbit that 
is contained in gln(Z). Importantly, the structure of 
the centralizer of this nilpotent matrix is independent of k.


It is natural to ask to what extent this statement extends to a
connected reductive algebraic group G acting on its Lie algebra 
g or the dual space g* via the (co-)adjoint 
representation. In general, the
structure of centralisers of nilpotent elements will depend on k but
one can hope that the centraliser dimension remains the same. In this
talk I will propose two variants of this idea, generalising the
classical situation above, and report on on-going joint work with Adam
Thomas (Warwick) to establish the existence of elements satisfying the
proposed properties.