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

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 $T\leq G$. 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 $G = NU^-UN$. 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 $\mathcal{N}\leq \mathrm{Cr}_m(K)$ 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 $G = \mathrm{SL}_2(\mathbb C)$.