Weight elements of Chevalley groups

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 Math. 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 description of overgroups of a split maximal torus and in the recent papers by the author and V. Nesterov on the geometry of tori.