The full automorphism group of $\overline{T}$

TL;DR

This paper determines the automorphism group of the closure of a maximal torus in the wonderful compactification of a simple affine algebraic group, revealing a structure involving the normalizer and Dynkin diagram automorphisms.

Contribution

It explicitly describes the automorphism group of the torus closure in the wonderful compactification, extending understanding of symmetries in algebraic group compactifications.

Findings

Automorphism group is the semi-direct product of the normalizer and Dynkin diagram automorphisms.

Special case for G=PSL(2,C), automorphism group is PSL(2,C).

Provides a complete description for all simple affine algebraic groups except PSL(2,C).

Abstract

Let $\overline G$ be the wonderful compactification of a simple affine algebraic group $G$ of adjoint type defined over $\mathbb C.$ 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 $N_G(T)\rtimes D,$ where $N_G(T)$ is the normalizer of $T$ in $G$ and $D$ is the group of all automorphisms of the Dynkin diagram, if $G\not= {\rm PSL}(2,\mathbb{C})$. Note that if $G = {\rm PSL}(2,\mathbb{C})$, then $\overline{T} = {\mathbb C}{\mathbb P}^1$ and so in this case $\text{Aut}(\overline T)= {\rm PSL}(2,\mathbb{C})$.

Equations

$$R\subset X(T)$$

$$S=\{{\alpha }_1,\cdots ,{\alpha }_n\}$$

$$\rho :T⟶{\mathrm{Aut}}^0(\overline{T})$$

$$A=\mathrm{Aut}(\overline{T}).$$

$$\tilde{\Delta }\subset {\mathfrak{t}}_{\mathbb{R}}$$

$$N_G(T)⋊D\subset A.$$

$$g=l_{t_0}h,$$

$$N_G(T)/T⋊D=W⋊D$$

Full text

Résumé. Le groupe complet des automorphismes de $\overline{T}$.  Soit $\overline{G}$ la compactification magnifique d’un groupe algébrique affine simple $G$ de type adjoint défini sur $\mathbb{C}$. Soit ${\overline{T}}\,\subset\,{\overline{G}}$ la clôture d’un tore maximal $T\,\subset\,G$. Si $G\not\,=\,{\rm PSL}(2,{\mathbb{C}})$, nous montrons que le groupe de tous les automorphismes de la variété $\overline{T}$ est le produit semi-direct $N_{G}(T)\rtimes D$, 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 $G\,=\,{\rm PSL}(2,\mathbb{C})$, alors $\overline{T}\,=\,{\mathbb{C}}{\mathbb{P}}^{1}$ et donc dans ce cas $\text{Aut}(\overline{T})\,=\,{\rm PSL}(2,\mathbb{C})$.

1. Introduction

Let $G$ be a simple affine algebraic group of adjoint type defined over the field of complex numbers. De Concini and Procesi constructed a very important compactification of $G$ [DP, p. 14, 3.1, THEOREM]; it is known as the wonderful compactification. The wonderful compactification of $G$ will be denoted by $\overline{G}$. Fix a maximal torus $T$ of $G$, and denote by $\overline{T}$ the closure of the variety $T$ in the wonderful compactification $\overline{G}$ [BJ, § 1]. Let ${\rm Aut}(\overline{T})$ denote the group of all holomorphic automorphisms of $\overline{T}$. For $G\,\not=\,{\rm PSL}(2,\mathbb{C}),$ the connected component of ${\rm Aut}(\overline{T})$ containing the identity element coincides with $T$ acting on $\overline{T}$ by translations [BKN, Theorem 3.1]. Our aim here is to compute the full automorphism group ${\rm Aut}(\overline{T})$.

It may be noted that $\overline{T}$ is stable under the conjugation of the normalizer $N_{G}(T)$ of $T$ in $G$. This indicates that ${\rm Aut}(\overline{T})$ need not be connected.

For $G$ different from ${\rm PSL}(2,\mathbb{C})$, we prove that ${\rm Aut}(\overline{T})$ is the semi-direct product $N_{G}(T)\rtimes D,$ where $N_{G}(T)$ is the normalizer of $T$ in $G,$ and $D$ is the group of all automorphisms of the Dynkin diagram (see Theorem 3.1).

2. Lie algebra and algebraic groups

We recall the set-up of [BKN]. Throughout $G$ will denote an affine algebraic group over $\mathbb{C}$ such that $G$ is simple and of adjoint type (equivalently, the center of the simple group is trivial). We will always assume that $G\,\not=\,{\rm PSL}(2,\mathbb{C})$.

Fix a maximal torus $T$ of $G$. The group of all characters of $T$ will be denoted by $X(T)$. The Weyl group of $G$ with respect to $T$ is defined to be $W\ :=\,N_{G}(T)/T$, where $N_{G}(T)$ is the normalizer of $T$ in $G$. Let

[TABLE]

be the root system of $G$ with respect to $T$. For a Borel subgroup $B$ of $G$ containing the maximal torus $T$, let $R^{+}(B)$ denote the set of positive roots determined by $T$ and $B$. Let

[TABLE]

be the set of simple roots in $R^{+}(B)$, where $n$ is the rank of $G$. Let $B^{-}$ denote the opposite Borel subgroup of $G$ determined by $B$ and $T.$ So in particular $B\bigcap B^{-}\,=\,T$. For any $\alpha\,\in\,R^{+}(B)$, let $s_{\alpha}\,\in\,W$ be the reflection corresponding to $\alpha$.

The Lie algebras of $G$, $T$ and $B$ will be denoted by $\mathfrak{g}$, $\mathfrak{t}$ and $\mathfrak{b}$ respectively. The dual of the real form $\mathfrak{t}_{\mathbb{R}}$ of $\mathfrak{t}$ is $X(T)\otimes\mathbb{R}\,=\,{\rm Hom}_{\mathbb{R}}(\mathfrak{t}_{\mathbb{R}},\,\mathbb{R})$.

Now, let $\sigma$ be the involution of $G\times G$ defined by $\sigma(x\,,y)\,=\,(y\,,x)$. We note that the diagonal subgroup $\Delta(G)$ of $G\times G$ is the subgroup of fixed points of $\sigma$. The subgroup $T\times T\,\subset\,G\times G$ is a $\sigma$–stable maximal torus of $G\times G$, while $B\times B^{-}$ is a Borel subgroup of $G\times G$; this Borel subgroup $B\times B^{-}$ has the property that $\sigma(\alpha)\,\in\,-R^{+}(B\times B^{-})$ for every $\alpha\,\in\,R^{+}(B\times B^{-}).$

The group $G$ is identified with the symmetric space $(G\times G)/\Delta(G)$. Let $\overline{G}$ denote the corresponding wonderful compactification of $G$ (see [DP, p. 14, 3.1, THEOREM]). In particular $G\times G$ acts on $\overline{G}$. Let $\overline{T}$ be the closure of $T$ in $\overline{G}.$ The action of the subgroup $N_{G}(T)\,\subset\,G\,=\,\Delta(G)$ on $\overline{G}$ preserves $\overline{T}$.

3. The automorphism group of $\overline{T}$

Let ${\rm Aut}(\overline{T})$ denote the group of all holomorphic automorphisms of $\overline{T}$; any holomorphic automorphism is algebraic. Let ${\rm Aut}^{0}(\overline{T})\,\subset\,{\rm Aut}(\overline{T})$ be the connected component containing the identity element. The translation action of $T$ on itself produces an isomorphism

[TABLE]

[BKN, p. 786, Theorem 3.1].

Theorem 3.1**.**

The automorphism group ${\rm Aut}(\overline{T})$ is the semi-direct product $N_{G}(T)\rtimes D,$ where $N_{G}(T)$ is the normalizer of $T$ in $G$, and $D$ is the group of all automorphisms of the Dynkin diagram of $G$.

Proof.

For notational convenience denote

[TABLE]

Note that $\overline{T}$ is stable under the conjugation action of $N_{G}(T)$ on $\overline{G}.$ Let

[TABLE]

be the fan of the toric variety $\overline{T}$. This $\widetilde{\Delta}$ consists of cones associated to the Weyl chambers (see [BK, p. 187, 6.1.6, Lemma]). Note that any automorphism $\sigma$ of the Dynkin Diagram associated to set $S\subset R$ of simple roots with respect to $(T,\,B)$ preserves the fan $\widetilde{\Delta}.$ Therefore, we have [Co, p. 47]

[TABLE]

Next we will show that $N_{G}(T)\rtimes D\,=\,A$.

Since $\rho$ in (2) is an isomorphism, it follows immediately that $T$ is a normal subgroup of $A$. Therefore, the intersection $T\bigcap g(T)$ is a $T$ stable open dense subset of $\overline{T}$ for every element $g\,\in\,A$. Consequently, the open subset $T\,\subset\,\overline{T}$ is preserved by the natural action of $A$ on $\overline{T}$. Consequently, every automorphism $g\,\in\,A$ can be expressed as

[TABLE]

where $l_{t_{0}}$ is the left translation by some $t_{0}\,\in\,T$, and $h\,\in\,A$ satisfies the condition that $h(1)\,=\,1,$ with $1$ being the identity element of $T$.

By a result of Rosenlicht, the action of the $h$ (in (4)) on $T$ is by group automorphism (see [MR, p. 986, Theorem 3]). Therefore, $h$ gives an automorphism of $X(T)$, and hence $h$ gives an automorphism of $\mathfrak{t}_{\mathbb{R}}$. Since $T$ is left invariant under the action of $h$ the toric variety data of $\overline{T}$ is preserved by $h.$ Hence we see that the automorphism of $\mathfrak{t}_{\mathbb{R}}$ given by $h$ preserves the fan $\widetilde{\Delta}$ in (3). Since $\widetilde{\Delta}$ is given by the Weyl chambers and its faces, we see that the induced action of $h$ on $X(T)$ leaves the root system $R$ of $G$ in (1) invariant. Consequently, $h$ produces an automorphism of the root system $R$.

On the other hand, the automorphism group ${\rm Aut}(R)$ of the root system $R$ is precisely

[TABLE]

(see [HJ, p. 231, (A.8)]). ∎

Corollary 3.2**.**

The quotient group ${\rm Aut}(\overline{T})/{\rm Aut}^{0}(\overline{T})$ is isomorphic to ${\rm Aut}(R)=W\rtimes D.$

Remark 3.3*.*

The automorphism group $D$ is trivial except for the types $A_{\ell}$ with $\ell\,\geq\,2$, $D_{\ell}$ and $E_{6}$ (see [HJ, p. 231, (A.8)]).

Remark 3.4*.*

We note that the structure of the automorphism group of a complete simplicial toric variety is described by D. A. Cox (see [Co, p. 48, Corallary 4.7]).

Acknowledgements

We thank the referee for very helpful comments. The first–named author thanks the Institute of Mathematical Sciences for hospitality while this work was carried out. He also acknowledges the support of the J. C. Bose Fellowship. The second named author would like to thank the Infosys Foundation for the partial support.