Existence of affine pavings for varieties of partial flags associated to nilpotent elements (Q2797798)

Let \(G\) be a connected reductive group over \(\mathbb{C}\), \(e\) be a nilpotent element in the Lie algebra \(\mathrm{Lie} (G)\), \(P\) be a parabolic subgroup of \(G\), and \(\mathfrak{i}\) be a \(P\)-stable subspace in \(\mathrm{Lie} (P)\). Define a subvariety of \(G/P\) by NEWLINE\[NEWLINE\mathcal{P}_{e,\mathfrak{i}}=\{gP\in G/P: g^{-1}e\in \mathfrak{i} \}.NEWLINE\]NEWLINE The author proved that if the minimal Levi subalgebra of \(\mathrm{Lie} (G)\) containing \(e\) has no non-regular component of exceptional type (see the definition on p. 422), then \(\mathcal{P}_{e,\mathfrak{i}}\) admits an affine paving, which means that it is covered by a finite subset of closed subvarieties, each of them isomorphic to an affine space. This generalizes a theorem of \textit{C. de Concini} et al. [J. Am. Math. Soc. 1, No. 1, 15--34 (1988; Zbl 0646.14034)], which asserts the statement when \(P\) is a Borel subgroup and \(\mathfrak{i}=\mathrm{Lie}(P)\), i.e., when \(\mathcal{P}_{e,\mathfrak{i}}\) is the Springer fiber over \(e\). This affine-paving property is desirable because it guarantees good cohomological properties for the subvariety.NEWLINENEWLINEWe give a few words about the proof. After recalling some basic facts in the first two sections about parabolic orbits of partial flag varieties and smooth pavings, the author reduces the proof to the case when \(G\) is of classical type and \(e\) is distinguished. During the reduction process, we apply a useful theorem of Bialynicki-Birula, which was also used in [loc. cit.]. In the last three sections, he proceeds with explicit calculations using root spaces of \(\mathrm{Lie} (G)\).