Nilpotent orbits of height 2 and involutions in the affine Weyl group

Abstract

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Fully commutative elements and spherical nilpotent orbits

“…Indeed in this case α and β are strongly orthogonal, and it is easily computed that ht(e α + e β ) = 4 (see e.g. [7]). Thus G(e α + e β ) is not spherical by Panyushev's characterization.…”

“…Proof. Subsets of orthogonal roots Γ ⊂ Φ + giving rise to non-spherical nilpotent orbits were studied in [6], [7], and classified in [3,Proposition 3.7]. In particular, if Gx Γ is not spherical, since Φ is not of tye G 2 we only have the following possibilities:…”

Parametrization, structure and Bruhat order of certain spherical quotients

“…The problem of describing the inclusion relations between the orbit closures has been addressed in [BP19,BR12] in certain cases. To the best of our knowledge, there is no general approach to these problems, although recently [GMP20] associate to any nilpotent element of height 2 an involution in the affine Weyl group, and show that the orbit closures are described by restricting the Bruhat order on the affine Weyl group.…”