Weight elements of Chevalley groups

Abstract:

The paper is devoted to a detailed study of some remarkable semisimple elements of (extended) Chevalley groups that are diagonalizable over the ground field — the weight elements. These are the conjugates of certain semisimple elements $h_{\omega }(\varepsilon )$ of extended Chevalley groups $\overline {G}=\overline {G}(\Phi ,K)$, where $\omega$ is a weight of the dual root system $\Phi ^{\vee }$ and $\varepsilon \in K^*$. In the adjoint case the $h_{\omega }(\varepsilon )$’s were defined by Chevalley himself and in the simply connected case they were constructed by Berman and Moody. The conjugates of $h_{\omega }(\varepsilon )$ are called weight elements of type $\omega$. Various constructions of weight elements are discussed in the paper, in particular, their action in irreducible rational representations and weight elements induced on a regularly embedded Chevalley subgroup by the conjugation action of a larger Chevalley group. It is proved that for a given $x\in \overline {G}$ all elements $x(\varepsilon )=xh_{\omega }(\varepsilon )x^{-1}$, $\varepsilon \in K^*$, apart maybe from a finite number of them, lie in the same Bruhat coset $\overline {B}w\overline {B}$, where $w$ is an involution of the Weyl group $W=W(\Phi )$. The elements $h_{\omega }(\varepsilon )$ are particularly important when $\omega =\varpi _{i}$ is a microweight of $\Phi ^{\vee }$. The main result of the paper is a calculation of the factors of the Bruhat decomposition of microweight elements $x(\varepsilon )$ for the case where $\omega =\varpi _{i}$. It turns out that all nontrivial $x(\varepsilon )$’s lie in the same Bruhat coset $\overline {B}w\overline B$, where $w$ is a product of reflections in pairwise strictly orthogonal roots $\gamma _1,\ldots ,\gamma _{r+s}$. Moreover, if among these roots $r$ are long and $s$ are short, then $r+2s$ does not exceed the width of the unipotent radical of the $i$th maximal parabolic subgroup in $\overline G$. A version of this result was first announced in a paper by the author in Soviet Mathematics: Doklady in 1988. From a technical viewpoint, this amounts to the determination of Borel orbits of a Levi factor of a parabolic subgroup with Abelian unipotent radical and generalizes some results of Richardson, Röhrle, and Steinberg. These results are instrumental in the description of overgroups of a split maximal torus and in the recent papers by the author and V. Nesterov on the geometry of tori.