Geodesic
Double cosets $Ng N$ of normalizers of maximal tori of simple algebraic groups and orbits of partial actions of Cremona subgroups
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 5, Tome 511 (2022), pp. 5-27 
N. L. Gordeev; E. A. Egorchenkova. Double cosets $Ng N$ of normalizers of maximal tori of simple algebraic groups and orbits of partial actions of Cremona subgroups. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 5, Tome 511 (2022), pp. 5-27. http://geodesic.mathdoc.fr/item/ZNSL_2022_511_a0/
@article{ZNSL_2022_511_a0,
     author = {N. L. Gordeev and E. A. Egorchenkova},
     title = {Double cosets $Ng N$ of normalizers of maximal tori of simple algebraic groups and orbits of partial actions of {Cremona} subgroups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--27},
     year = {2022},
     volume = {511},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_511_a0/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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 $G = \mathrm{SL}_2(\mathbb C)$.

[1] N. Gordeev, E. Egorchenkova, Double cosets $Ng N$ of normalizers of maximal tori of simple algebraic groups and orbits of partial actions of Cremona subgroups, arXiv: 2112.06332 [math.AG]

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