Let be a real vector space. A
σ-root system (R,\sigma) consists of a root system R\subset V that spans and a linear involution of that satisfies \sigma(R)=R. Let R_\bullet\subset R be the set of roots fixed by and let \Sigma := \left\{ \frac{\alpha-\sigma(\alpha)}{2}: \alpha\in R\setminus R_\bullet\right\}. is called the
restricted root system. The
Satake diagram of a σ-root system (R,\sigma) is obtained as follows: Let \alpha_1,\dots,\alpha_n be simple roots of such that \alpha_{n+1-p},\dots,\alpha_n are simple roots of R_\bullet. We can define an involution of \{1,\dots,n-p\} by having \sigma(\alpha_i) = \alpha_{\tau(i)} + \Z R_\bullet\qquad (i=1,\dots,n-p). The Satake diagram is then obtained from the Dynkin diagram describing by blackening the vertices corresponding to \alpha_{n+1-p},\dots,\alpha_n, and by drawing arrows between the white vertices that are interchanged by .
Satake diagram of a real semisimple Lie algebra Let \mathfrak{g}_\R be a real semisimple Lie algebra and let \mathfrak{g}=\mathfrak{g}_\R\otimes\C be its complexification. Define the map \sigma:\mathfrak{g}\to\mathfrak{g},\qquad X\otimes z\mapsto X\otimes\overline{z}. This is an anti-linear involutive automorphism of real Lie algebras and its fixed-point set is our original \mathfrak{g}_\R. Let \mathfrak{h}\le\mathfrak{g} be a
Cartan subalgebra that satisfies \sigma(\mathfrak{h})=\mathfrak{h} and is maximally split, i.e. when we split \mathfrak{h} into -eigenspaces, the -1-eigenspace has maximal dimension. induces an anti-linear involution on \mathfrak{h}^*: \sigma^*(\lambda)(v) = \overline{\lambda(\sigma(v))}\qquad (\lambda\in\mathfrak{h}^*,v\in\mathfrak{h}). If X\in\mathfrak{g}_\alpha is a root vector, one can show that \sigma(X)\in\mathfrak{g}_{\sigma^*(\alpha)}. Consequently, preserves the root system of \mathfrak{g}. We thus obtain a σ-root system (R,\sigma^*) whose Satake diagram is the Satake diagram of \mathfrak{g}_\R.
Satake diagram of a symmetric pair Let (\mathfrak{g},\mathfrak{k}) be a symmetric pair of complex Lie algebras where \mathfrak{g} is semisimple, i.e. let be an involutive Lie algebra automorphism of \mathfrak{g} and let \mathfrak{k} be its fixed-point set. It is shown in that these symmetric pairs (even for \mathfrak{g} an infinite-dimensional
Kac-Moody algebra), or equivalently these involutive automorphisms, can be classified using so-called admissible pairs. These admissible pairs describe again a σ-root system that can be obtained from the automorphism , and the Satake diagrams that arise this way are exactly the ones listed in and the Satake diagrams obtained by blackening all vertices.
Definition Given a Dynkin diagram with vertex set , an
admissible pair (I_\bullet, \tau) consists of a subset I_\bullet of finite type and a diagram automorphism satisfying • \tau^2=\operatorname{id} • \tau(I_\bullet)=I_\bullet • The permutation \tau|_{I_\bullet} coincides with -w_\bullet (where w_\bullet is the longest element of the Weyl group generated by the vertices in I_\bullet) • For j\in I\setminus I_\bullet with \tau(i)=i, we have \alpha_j(\rho^\vee_\bullet)\in\Z, where \rho^\vee_\bullet = \frac{1}{2}\sum_{\alpha\in R_\bullet^+} \alpha^\vee. Given an admissible pair (I_\bullet,\tau), we can define a σ-root system by equipping the root system of with the involution \sigma = -w_\bullet\circ\tau == Classification of Satake diagrams ==