Double cosets $NgN$ of normalizers of maximal tori of simple algebraic groups and orbits of partial actions of Cremona subgroups

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $K$ and let $N=N_G(T)$ be the normalizer of a fixed maximal torus $T\le G$. Further, let $U$ be the unipotent radical of a fixed Borel subgroup $B$ that contains $T$ and let $U^{-}$ be the unipotent radical of the opposite Borel subgroup $B^{-}$. The Bruhat decomposition implies the decomposition $G=N{U}^{-}UN$. The Zariski closed subset $U^{-}U\subset G$ is isomorphic to the affine space $A_K^m$ where $m=\dim G-\dim T$ is the number of roots in the corresponding root system. Here we construct a subgroup $\mathcal{N}\le {\mathrm{Cr}}_m(K)$ that ``acts partially'' on $A_K^m\approx \mathcal{U}$ and we show that there is one-to-one correspondence between the orbits of such a partial action and the set of double cosets $\{NgN\}$. Here we also calculate the set $\{g_{\alpha }{\}}_{\alpha \in \mathfrak{A}}\subset \mathcal{U}$ in the simplest case $G={\mathrm{SL}}_2(\mathbb{C})$.

arXiv comment18 pages