Unipotent variety in the group compactification. (English) Zbl 1102.20032
Let \(G\) be a connected simple (hence adjoint) algebraic group with wonderful compactification \(\overline G\). The main result of this paper is the explicit determination of the closure \(\overline U\) in \(\overline G\) of the unipotent radical \(U\) of a Borel subgroup \(B\). It confirms a conjecture of Lusztig who expects to use it in his theory of “parabolic character sheaves” on \(\overline G\). Starting point is the description by Springer of \(B\times B\)-orbits in \(\overline G\). One also needs a study of Coxeter elements in Weyl groups. Some case by case combinatorics gives a lower bound on \(\overline U\). One shows this lower bound equals an upper bound. The unipotent radical is one of the Steinberg fibers. It turns out that the boundary \(\overline F-F\) is the same for every Steinberg fiber \(F\).
Reviewer: Wilberd van der Kallen (Utrecht)
MSC:
| 20G15 | Linear algebraic groups over arbitrary fields |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 20G05 | Representation theory for linear algebraic groups |
Keywords:
unipotent varieties; Steinberg fibers; group compactifications; connected simple algebraic groups; wonderful compactificationsReferences:
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