Double Cosets NgN 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 ≤ 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 ⊂ G is isomorphic to the affine space \({A}_{K}^{m},\) $A_K^m,$ where m = dim G − dim T is the number of roots in the corresponding root system. The goal of the paper is to construct a subgroup \(\mathcal{N}\) $\mathcal{N}$ ≤ Cr_m (K) that “acts partially” on \({A}_{K}^{m}\) $A_K^m$ ≈ \(\mathcal{U}\) $\mathcal{U}$ , and to show that there is one-to-one correspondence between the orbits of such a partial action and the set of double cosets \({\{N{g}_{\alpha }N\}}_{\alpha \in \mathfrak{A}}\) ${\{N{g}_{\alpha }N\}}_{\alpha \in \mathfrak{A}}$ . The set \({\{{g}_{\alpha }\}}_{\alpha \in \mathfrak{A}}\) ${\{g_{\alpha }\}}_{\alpha \in \mathfrak{A}}$ ⊂ \(\mathcal{U}\) $\mathcal{U}$ in the simplest case G = SL₂( \({\mathbb{C}}\) $\mathbb{C}$ ) is also calculated.