2012/09/06 by Simon Goodwin, Gerhard Roehrle, Goodwin, Simon +1
Mathematics · #FOS: Mathematics #Primary 20G15 #Representation Theory (math.RT) #Secondary 17B45 #math.RT #msc:17B45 #msc:20G15
paper · pdf · doi:10.48550/arxiv.1209.1289
10 pages
arxiv created 2012/09/06 · arxiv updated 2012/09/07
Let G be a connected reductive algebraic group defined over an algebraically closed field \mathbbm k of characteristic zero. We consider the commuting variety \mathcal C(\mathfrak u) of the nilradical \mathfrak u of the Lie algebra \mathfrak b of a Borel subgroup B of G. In case B acts on \mathfrak u with only a finite number of orbits, we verify that \mathcal C(\mathfrak u) is equidimensional and that the irreducible components are in correspondence with the \em distinguished B-orbits in \mathfrak u. We observe that in general \mathcal C(\mathfrak u) is not equidimensional, and determine the irreducible components of \mathcal C(\mathfrak u) in the minimal cases where there are infinitely many B-orbits in \mathfrak u.