单代数群极大环面正规化子的双陪集$NgN$与Cremona子群的部分作用轨道
Double cosets $NgN$ of normalizers of maximal tori of simple algebraic groups and orbits of partial actions of Cremona subgroups
摘要：
设$G$是代数闭域$K$上的单代数群，$N = N_G(T)$是固定极大环面$T\leq G$的正规化子.此外，令$U$是包含$T$的固定Borel子群$B$的幂单根，令$U^-$是相反Borel子群$B^-$的幂单根。Bruhat分解意味着分解$G = NU^-UN$。Zorski闭子集$U^-U\subset G$同构于仿射空间$A_K^m$，其中$m = \dim G -\dim T$是相应根系中的根的数目。在这里，我们构建了一个子群$\mathcal{N}\leq \operatorname{Cr}_m(K)$，“部分作用”$A_K^m\approx\mathcal{U}$和我们表明，有一个这样的部分行动和一组双陪集$\{NgN\}$的轨道之间的一一对应。在这里，我们也计算最简单情况下的集合$\{g_α\}_{α\in \mathfrak A}\subset \mathcal{U}$，即$G = \operatorname{SL}_2(\mathbb{C})$。
英文摘要：
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 $\{g_α\}_{α\in \mathfrak A}\subset \mathcal{U}$ in the simplest case $G = \operatorname{SL}_2(\mathbb{C})$.
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