The full automorphism group of $ \stackrel{‾}{T}$

Biswas, Indranil; Kannan, Subramaniam Senthamarai; Nagaraj, Donihakalu Shankar

doi:10.1016/j.crma.2017.02.008

Algebraic geometry

The full automorphism group of

\overline{T}

[Le groupe complet des automorphismes de

\overline{T}

]

Présenté par : Claire Voisin

Biswas, Indranil ¹ ; Kannan, Subramaniam Senthamarai ² ; Nagaraj, Donihakalu Shankar ³

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
² Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam 603103, India
³ The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India

Comptes Rendus. Mathématique, Tome 355 (2017) no. 4, pp. 452-454.

Résumé
Abstract

Soit $\overline{G}$ la compactification magnifique d'un groupe algébrique affine simple G de type adjoint défini sur $C$ . Soit $\overline{T} \subset \overline{G}$ la clôture d'un tore maximal $T \subset G$ . Si $G \neq PSL (2, C)$ , nous montrons que le groupe de tous les automorphismes de la variété $\overline{T}$ est le produit semi-direct $N_{G} (T) ⋊ D$ , où $N_{G} (T)$ est le normalisateur de T dans G et D est le groupe de tous les automorphismes du diagramme de Dynkin. Remarquez que si $G = PSL (2, C)$ , alors $\overline{T} = C P^{1}$ et donc dans ce cas $Aut (\overline{T}) = PSL (2, C)$ .

Let $\overline{G}$ be the wonderful compactification of a simple affine algebraic group G of adjoint type defined over $C$ . Let $\overline{T} \subset \overline{G}$ be the closure of a maximal torus $T \subset G$ . We prove that the group of all automorphisms of the variety $\overline{T}$ is the semi-direct product $N_{G} (T) ⋊ D$ , where $N_{G} (T)$ is the normalizer of T in G and D is the group of all automorphisms of the Dynkin diagram, if $G \neq PSL (2, C)$ . Note that if $G = PSL (2, C)$ , then $\overline{T} = C P^{1}$ and so in this case $Aut (\overline{T}) = PSL (2, C)$ .

Reçu le : 2016-11-25
Accepté le : 2017-02-27
Publié le : 2017-03-11

DOI : 10.1016/j.crma.2017.02.008

Affiliations des auteurs :

Biswas, Indranil ¹ ; Kannan, Subramaniam Senthamarai ² ; Nagaraj, Donihakalu Shankar ³

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     title = {The full automorphism group of $ \stackrel{‾}{T}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {452--454},
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Biswas, Indranil; Kannan, Subramaniam Senthamarai; Nagaraj, Donihakalu Shankar. The full automorphism group of $ \stackrel{‾}{T}$. Comptes Rendus. Mathématique, Tome 355 (2017) no. 4, pp. 452-454. doi : 10.1016/j.crma.2017.02.008. http://www.numdam.org/articles/10.1016/j.crma.2017.02.008/

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