Double cosets $NgN$ of normalizers of maximal tori of simple algebraic groups and orbits of partial actions of Cremona subgroups (2112.06332v1)

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 $T\leq G$. 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 $\mathcal{N}\leq \operatorname{Cr}m(K)$ 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 $G = \operatorname{SL}_2(\mathbb{C})$.