On the modality of parabolic subgroups of linear algebraic groups (Q1279842)

The author generalizes to positive characteristic the ``monotonicity'' results of \textit{V. L. Popov} and \textit{G. Röhrle} [in: Algebraic groups and Lie groups, Aust. Math. Soc. Lect. Ser. 9, 297-320 (1997; Zbl 0887.14020)]. Namely, let \(G\) be a linear algebraic group over an algebraically closed field \(k\) and \(P\) be a parabolic subgroup of \(G\). It is proved that: (1) If \(P\) is stable with respect to a semisimple automorphism \(\Theta\) of \(G\), then \(\text{mod }P^\Theta\leq\text{mod }P\) (here and further \(\text{mod }Q\) stands for the modality of \(Q\), see the above mentioned paper). (2) If \(G\) is reductive, the characteristic of \(k\) is a good prime for \(G\), and \(H\) is a closed reductive subgroup of \(G\) normalized by a maximal torus of \(P\), then \(\text{mod}(P\cap H)\leq\text{mod }P\).