Satake diagram

Assume the Axiom of Choice (The Axiom of Choice).

Let ${\mathfrak{g}}_0$ be a finite-dimensional real semisimple Lie algebra with Cartan involution $\theta$ and Cartan decomposition ${\mathfrak{g}}_0={\mathfrak{k}}_0\oplus {\mathfrak{p}}_0$ (Theta-stable Cartan subalgebras and their compact and split parts). Let ${\mathfrak{h}}_0={\mathfrak{t}}_0\oplus {\mathfrak{a}}_0$ be a $\theta$-stable Cartan subalgebra of ${\mathfrak{g}}_0$ whose split part ${\mathfrak{a}}_0$ is a maximal abelian subspace of ${\mathfrak{p}}_0$; such an ${\mathfrak{h}}_0$ is maximally split, or maximally noncompact (Theta-stable Cartan subalgebras and their compact and split parts, Maximal split abelian subspace and real rank). Write $\mathfrak{h}={\mathfrak{h}}_0\oplus i{\mathfrak{h}}_0$ for the complexification, $\mathfrak{t}={\mathfrak{t}}_0\oplus i{\mathfrak{t}}_0$ and $\mathfrak{a}={\mathfrak{a}}_0\oplus i{\mathfrak{a}}_0$, so that $\mathfrak{h}=\mathfrak{t}\oplus \mathfrak{a}$, and let $\mathrm{\Phi }=\mathrm{\Phi }(\mathfrak{g},\mathfrak{h})$ be the root system of $(\mathfrak{g},\mathfrak{h})$, with root spaces ${\mathfrak{g}}_{\alpha }$; the $\mathbb{C}$-linear extension of $\theta$ is again written $\theta$.

Root types. By Cayley transform of a theta-stable Cartan subalgebra every root $\alpha \in \mathrm{\Phi }$ satisfies $\alpha ({\mathfrak{t}}_0)\subseteq i\mathbb{R}$ and $\alpha ({\mathfrak{a}}_0)\subseteq \mathbb{R}$, and $\alpha$ is $\text{real if }\alpha {\mathrm{\mid }}_{{\mathfrak{t}}_0}=0,\text{imaginary if }\alpha {\mathrm{\mid }}_{{\mathfrak{a}}_0}=0,\text{complex otherwise};$ equivalently $\theta \alpha =-\alpha$, $\theta \alpha =\alpha$, or neither, where $\theta \alpha =\alpha \circ {\theta }^{-1}$. In particular an imaginary root is exactly a root whose restriction to ${\mathfrak{a}}_0$ vanishes, and $\theta$ permutes the three classes of roots. An imaginary root has $\theta$-stable root space ${\mathfrak{g}}_{\alpha }$, which is one-dimensional (Root spaces of a complex semisimple Lie algebra are one-dimensional), so it lies either in the $+1$-eigenspace $\mathfrak{k}$ of $\theta$ or in the $-1$-eigenspace $\mathfrak{p}$; the imaginary root is compact in the first case and noncompact in the second. Because ${\mathfrak{h}}_0$ is maximally split, no noncompact imaginary root of $(\mathfrak{g},\mathfrak{h})$ exists: a noncompact imaginary root admits a noncompact-imaginary Cayley transform, which produces a $\theta$-stable Cartan subalgebra whose noncompact dimension is larger by one (Cayley transforms connect theta-stable Cartans in the classification, assertion 2).

Compatible positive systems. A positive system ${\mathrm{\Phi }}^{+}$ of $\mathrm{\Phi }$ and its base $\mathrm{\Delta }$ are as in Positive systems and simple roots. Let $\mathrm{\Sigma }=\mathrm{\Sigma }({\mathfrak{g}}_0,{\mathfrak{a}}_0)$ be the finite set of restricted roots (Restricted root and restricted root space, Restricted root space decomposition). A restricted positive system means ${\mathrm{\Sigma }}^{+}=\{\lambda \in \mathrm{\Sigma }:\lambda (H)>0\}$ for a regular $H\in {\mathfrak{a}}_0$, with $\lambda (H)\mathrm{\ne }0$ for all $\lambda \in \mathrm{\Sigma }$. This definition also applies when the restricted root system is nonreduced. The nonzero restrictions of complex roots are exactly $\mathrm{\Sigma }$: restriction of the complex root decomposition groups its weight spaces according to $\alpha {\mathrm{\mid }}_{{\mathfrak{a}}_0}$, while complexification of a real restricted-weight space gives the same joint eigenspace, since its eigenvalue equations have real coefficients (Root-space decomposition, Restricted root and restricted root space).

A positive system ${\mathrm{\Phi }}^{+}$ is compatible with ${\mathrm{\Sigma }}^{+}$ if $\alpha \in {\mathrm{\Phi }}^{+},\alpha {\mathrm{\mid }}_{{\mathfrak{a}}_0}\mathrm{\ne }0⟹\alpha {\mathrm{\mid }}_{{\mathfrak{a}}_0}\in {\mathrm{\Sigma }}^{+}\mathrm{.}$ Given the specified ${\mathrm{\Sigma }}^{+}$, take a regular $H$ defining it. Choose $T\in {\mathfrak{t}}_0$ with $\alpha (T)\mathrm{\ne }0$ for every imaginary root; the finite-union lemma gives such a $T$ because those restrictions are nonzero (A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces). Use lexicographic positivity of $(\alpha (H),\mathrm{Im}\alpha (T))$. To see that this is a positive system in the stated Euclidean sense, choose $ϵ>0$ so small that $\mathrm{\mid }ϵ\mathrm{Im}\alpha (T)\mathrm{\mid }<\mathrm{\mid }\alpha (H)\mathrm{\mid }$ for every root with $\alpha (H)\mathrm{\ne }0$. There are only finitely many such inequalities, so they can all hold. The regular vector $H-iϵT\in {\mathfrak{a}}_0\oplus i{\mathfrak{t}}_0$ then gives exactly those signs, since $\alpha (H-iϵT)=\alpha (H)+ϵ\mathrm{Im}\alpha (T)$. The real positive Killing space here is justified in step 2.1 of Cayley transforms connect theta-stable Cartans in the classification. If a relevant root family is empty, choose the corresponding vector to be zero and omit its inequalities.

The Satake diagram. Let ${\mathrm{\Phi }}^{+}$ be a positive system of $\mathrm{\Phi }$ compatible with a positive system ${\mathrm{\Sigma }}^{+}$ of $\mathrm{\Sigma }$, with base $\mathrm{\Delta }$. The Satake diagram of the quadruple $({\mathfrak{g}}_0,{\mathfrak{h}}_0,{\mathrm{\Sigma }}^{+},{\mathrm{\Phi }}^{+})$ is the Dynkin diagram of $\mathrm{\Phi }$ relative to $\mathrm{\Delta }$ (Dynkin diagram with edge multiplicity and arrow convention) together with the following two decorations of its vertices:

• Colouring. A vertex $\alpha \in \mathrm{\Delta }$ is black if it is an imaginary simple root, that is, if $\alpha {\mathrm{\mid }}_{{\mathfrak{a}}_0}=0$, and white otherwise, that is, if $\alpha {\mathrm{\mid }}_{{\mathfrak{a}}_0}\mathrm{\ne }0$. Since ${\mathfrak{h}}_0$ is maximally split, every imaginary root of $(\mathfrak{g},\mathfrak{h})$ is compact, so the black vertices are exactly the compact imaginary simple roots.
• Arrows. Two distinct white vertices $\alpha ,\beta \in \mathrm{\Delta }$ are joined by a Satake arrow when their restrictions to ${\mathfrak{a}}_0$ agree: $\alpha {\mathrm{\mid }}_{{\mathfrak{a}}_0}=\beta {\mathrm{\mid }}_{{\mathfrak{a}}_0}(\mathrm{\ne }0)\mathrm{.}$ In the standard drawing a Satake arrow is curved or drawn in a distinct style, so that it is not confused with the arrows attached to multiple edges of the underlying Dynkin diagram.

Here the equal-restriction relation really is a pairing. We give the linear-algebra verification. Put ${\mathrm{\Delta }}_0=\{\delta \in \mathrm{\Delta }:\delta {\mathrm{\mid }}_{{\mathfrak{a}}_0}=0\}$, $V_0={\mathrm{span}}_{\mathbb{R}}{\mathrm{\Delta }}_0$, and $s=-\theta$ on the real root space. The map $s$ is a root-system involution, acts as minus identity on $V_0$, and preserves restrictions to ${\mathfrak{a}}_0$. For a white simple root $\alpha$, $s\alpha$ is positive, since its nonzero restriction is the same positive restricted root as that of $\alpha$. The simple-root expansion theorem (Simple roots form a signed integral basis) therefore makes the induced matrix of $s$ on the quotient by $V_0$ nonnegative integral in the basis of white simple-root classes. Its inverse is itself and is also nonnegative. Such an invertible matrix permutes the extreme rays of the nonnegative coordinate cone: those rays are exactly the coordinate axes, since a vector with two positive coordinates splits into two nonproportional nonnegative vectors. Thus the matrix permutes axes up to positive scalars. The matrix and its inverse are integral, so each scalar and its reciprocal are positive integers, forcing the scalar to be one. Consequently there is an involution $\alpha \mapsto {\alpha }^{\mathrm{\prime }}$ of the white simple roots such that $s\alpha -{\alpha }^{\mathrm{\prime }}\in V_0$.

It follows that ${\alpha }^{\mathrm{\prime }}$ and $\alpha$ have equal restriction. Conversely, if white $\alpha ,\beta$ have equal restriction, then $\alpha +s\alpha =\beta +s\beta$: half of either sum is the corresponding functional on ${\mathfrak{a}}_0$, extended by zero on $i{\mathfrak{t}}_0$. Modulo $V_0$, equality reads $[\alpha ]+[{\alpha }^{\mathrm{\prime }}]=[\beta ]+[{\beta }^{\mathrm{\prime }}]$ in the basis of white classes. This forces the two unordered orbits, with repetitions for a fixed point, to be the same. Hence a nonzero restriction is shared by at most two white simple roots, and the distinct pair is precisely a two-element orbit of this involution. This proves the assertion behind the arrow drawing, without identifying the pairing with the possibly base-nonpreserving map $-\theta$ itself.

Isomorphism and equivalence. Two Satake diagrams are equivalent if they are isomorphic as decorated Dynkin diagrams: a vertex bijection preserves edge multiplicities, edge arrows, black and white colors, and Satake pairs. In based-root-system language the isomorphism preserves roots and Cartan integers; arbitrary absolute length scales on separate components are not part of these data.

A formally colored and paired finite-type Dynkin diagram is called admissible Satake data if it occurs from a quadruple as above; the word Satake diagram is reserved here for these admissible data. This definition does not assert that arbitrary black subsets and pairings are realizable. In particular the admissibility condition for Satake data differs from the unrestricted fixed-vertex painting in an abstract Vogan diagram (Vogan diagram).

A change of compatible positive system on a realized quadruple requires recomputing colors and pairs using the actual restriction map. It is not an operation specified by an arbitrary abstract decoration. Independence of the resulting equivalence class from the choices, and its relation to the Vogan classification, are the assertions of Vogan and Satake diagrams give equivalent real form classifications. They are not assumed in this definition. For ${\mathfrak{a}}_0=0$ all vertices are black and no Satake arrows occur; for the zero algebra the whole diagram is empty.