Restricted root system

Intuition [Beginner]

For a non-compact symmetric space $G/K$ (like the hyperbolic plane or the space of symmetric positive-definite matrices), there is a special set of directions called the "Cartan subspace" --- a maximal flat subspace of tangent directions at the base point. The restricted roots are the eigenvalues of the Lie bracket acting on the full Lie algebra, but projected onto this flat subspace.

Think of the restricted roots as the "skeleton" of the symmetric space. Just as the roots of a Lie algebra organise its representations, the restricted roots organise the geometry of the symmetric space. They tell you how the curvature, geodesics, and harmonic functions are distributed.

A key feature: restricted roots can form a "non-reduced" root system, meaning a root and its double can both appear. This does not happen for ordinary root systems of complex semisimple Lie algebras, but it happens for symmetric spaces because the real structure can collapse some of the finer distinctions.

Why does this concept exist? The restricted root system is the tool that drives harmonic analysis on symmetric spaces. It determines the Plancherel measure, the spherical functions, and the spectral decomposition of invariant differential operators. It is the Lie-theoretic input to the Selberg trace formula.

Visual [Beginner]

A diagram showing the restricted root system of $\mathfrak{sl}(3,\mathbb{R})$, which is the root system $B{C}_1$ (or $A_2$ depending on convention). Drawn in the Cartan subspace $\mathfrak{a}\cong {\mathbb{R}}^2$, the roots appear as vectors emanating from the origin, with the Weyl group acting as reflections.

The restricted root system organises the directions in the Lie algebra according to their eigenvalues under the Cartan subspace.

Worked example [Beginner]

Consider $\mathfrak{sl}(2,\mathbb{R})$ with Cartan decomposition $\mathfrak{k}\oplus \mathfrak{p}$ from 07.04.03, where $\mathfrak{k}=\mathfrak{so}(2)$ and $\mathfrak{p}$ is the 2-dimensional space of symmetric traceless matrices. The Cartan subspace is $\mathfrak{a}=\mathrm{span}\{H\}$ where $H=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$, dimension 1.

Step 1. Compute the eigenvalues of $\mathrm{ad}(H)$ on $\mathfrak{sl}(2,\mathbb{R})$: $[H,H]=0$, $[H,E+F]=2(E-F)$, $[H,E-F]=2(E+F)$. Wait --- let me use the standard basis. $[H,E]=2E$, $[H,F]=-2F$. In terms of the Cartan decomposition: $E+F\in \mathfrak{p}$ and $E-F\in \mathfrak{k}$.

Step 2. The restricted root $\alpha (H)=2$ for the eigenspace containing $E$: $[H,E]=2E$, so $\alpha (H)=2$, giving $\alpha =2\in {\mathfrak{a}}^{\ast }$. Similarly $-\alpha$ for $F$. The restricted root space ${\mathfrak{g}}_{\alpha }$ has dimension that depends on the real form.

Step 3. The restricted roots are $\{\alpha ,-\alpha ,2\alpha ,-2\alpha \}$... actually for $\mathfrak{sl}(2,\mathbb{R})$ the restricted roots are $\{\pm \alpha \}$ with $\alpha (H)=2$, and the root spaces each have multiplicity 1. The root system is $A_1$ (type $B{C}_1$ does not appear for $\mathfrak{sl}(2)$).

What this tells us: even for this simplest case, the restricted root system captures the essential spectral information. The root $\alpha$ tells us the "spread" of eigenvalues of the adjoint action.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $M=G/K$ be a Riemannian symmetric space of non-compact type with Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ and orthogonal symmetric Lie algebra $(\mathfrak{g},\theta )$ 07.04.06.

Definition (Cartan subspace). A Cartan subspace $\mathfrak{a}$ of $(\mathfrak{g},\theta )$ is a maximal abelian subalgebra of $\mathfrak{p}$. All Cartan subspaces of a given pair $(\mathfrak{g},\theta )$ are conjugate under $\mathrm{Ad}(K)$.

Definition (Restricted root). A restricted root is a nonzero element $\lambda \in {\mathfrak{a}}^{\ast }$ for which the restricted root space

$${\mathfrak{g}}_{\lambda }=\{X\in \mathfrak{g}:[H,X]=\lambda (H)X\text{ for all }H\in \mathfrak{a}\}$$

is nonzero. The set of restricted roots is denoted $\Sigma =\Sigma (\mathfrak{g},\mathfrak{a})$.

Definition (Restricted root system). The collection $\Sigma \subset {\mathfrak{a}}^{\ast }\setminus \{0\}$ is the restricted root system of the symmetric space $G/K$.

Root space decomposition. The Lie algebra decomposes as:

$$\mathfrak{g}={\mathfrak{g}}_0\oplus \underset{\lambda \in \Sigma }{⨁}{\mathfrak{g}}_{\lambda }$$

where ${\mathfrak{g}}_0=\mathfrak{a}\oplus \mathfrak{m}$ with $\mathfrak{m}=\{X\in \mathfrak{k}:[H,X]=0\text{ for all }H\in \mathfrak{a}\}$ (the centraliser of $\mathfrak{a}$ in $\mathfrak{k}$).

Multiplicities. The dimension $m_{\lambda }=\dim {\mathfrak{g}}_{\lambda }$ is the multiplicity of the restricted root $\lambda$. For ordinary root systems of complex semisimple Lie algebras, all multiplicities are 1. For restricted root systems, multiplicities can be larger and can vary between roots.

Counterexamples to common slips

• The restricted root system need not be reduced. The type $B{C}_r$ contains roots $\pm e_i$, $\pm 2e_i$, and $\pm e_i\pm e_j$, which includes both $\alpha$ and $2\alpha$. This occurs for real forms where the restricted roots have multiplicity patterns that collapse two distinct roots of the complex algebra to a proportional pair.

• The restricted root system of a compact symmetric space is defined differently. For compact symmetric spaces $U/K$, the Cartan subspace lives in $\mathfrak{p}$ (the $-1$ eigenspace), and the restricted roots take purely imaginary values, requiring a different treatment. The standard theory focuses on the non-compact dual.

• Not every abelian subalgebra of $\mathfrak{p}$ is a Cartan subspace. It must be maximal: not properly contained in any other abelian subalgebra of $\mathfrak{p}$. The rank of the symmetric space is $\dim \mathfrak{a}$.

Key theorem with proof [Intermediate+]

Theorem (Restricted roots form a root system). Let $\mathfrak{g}$ be a real semisimple Lie algebra with Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ and Cartan subspace $\mathfrak{a}\subset \mathfrak{p}$. Then the set $\Sigma =\Sigma (\mathfrak{g},\mathfrak{a})$ of restricted roots, together with the inner product on ${\mathfrak{a}}^{\ast }$ induced by the Killing form, satisfies the axioms of a root system (possibly non-reduced).

Proof. We verify the four axioms of a (possibly non-reduced) root system.

Axiom 1 (Finiteness). The set $\Sigma$ is finite. Since $\mathfrak{g}$ is finite-dimensional and $\mathfrak{a}$ acts semisimply on $\mathfrak{g}$ (via the simultaneous diagonalisation of the commuting family $\{\mathrm{ad}(H):H\in \mathfrak{a}\}$), there are finitely many distinct eigenvalues, hence finitely many restricted roots.

Axiom 2 (Spans ${\mathfrak{a}}^{\ast }$). The restricted roots span ${\mathfrak{a}}^{\ast }$. Suppose $\lambda (H)=0$ for all $\lambda \in \Sigma$ and some $H\in \mathfrak{a}$. Then $[H,X]=0$ for all $X\in {\mathfrak{g}}_{\lambda }$ (all root spaces), and also $[H,{\mathfrak{g}}_0]=0$ (since $H\in \mathfrak{a}\subset {\mathfrak{g}}_0$). So $[H,\mathfrak{g}]=0$, meaning $H$ is in the center of $\mathfrak{g}$. But $\mathfrak{g}$ is semisimple, so the center is zero. Hence $H=0$.

Axiom 3 (Reflection invariance). For each $\alpha \in \Sigma$, the reflection $s_{\alpha }(\lambda )=\lambda -2\frac{(\lambda ,\alpha )}{(\alpha ,\alpha )}\alpha$ preserves $\Sigma$, where $(\cdot ,\cdot )$ is the inner product on ${\mathfrak{a}}^{\ast }$ from the Killing form. To see this, note that the reflection $s_{\alpha }$ corresponds to the Weyl group element $w_{\alpha }\in W(\mathfrak{g},\mathfrak{a})=N_K(\mathfrak{a})/Z_K(\mathfrak{a})$ (the normaliser modulo the centraliser of $\mathfrak{a}$ in $K$).

The element $w_{\alpha }$ acts by $\mathrm{Ad}(k)$ on $\mathfrak{a}$ for a suitable $k\in N_K(\mathfrak{a})$, and $\mathrm{Ad}(k)$ preserves the root space decomposition (it is an automorphism of $\mathfrak{g}$). So $\mathrm{Ad}(k)({\mathfrak{g}}_{\lambda })={\mathfrak{g}}_{s_{\alpha }(\lambda )}$, and since $\mathrm{Ad}(k)$ is invertible, ${\mathfrak{g}}_{\lambda }\ne 0$ iff ${\mathfrak{g}}_{s_{\alpha }(\lambda )}\ne 0$, giving $s_{\alpha }(\Sigma )=\Sigma$.

Axiom 4 (Integrality). For $\alpha ,\lambda \in \Sigma$, the quantity $2(\lambda ,\alpha )/(\alpha ,\alpha )\in \mathbb{Z}$. This follows from the ${\mathfrak{sl}}_2$-subalgebra associated to each restricted root. For each $\alpha \in \Sigma$, choose $X_{\alpha }\in {\mathfrak{g}}_{\alpha }$ and $Y_{\alpha }\in {\mathfrak{g}}_{-\alpha }$ with $B(X_{\alpha },Y_{\alpha })=1$ (where $B$ is the Killing form). Then $H_{\alpha }=[X_{\alpha },Y_{\alpha }]\in \mathfrak{a}$ (since $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{-\alpha }]\subset {\mathfrak{g}}_0$, and the component in $\mathfrak{a}$ is nonzero). The triple $(Y_{\alpha },H_{\alpha },X_{\alpha })$ spans a subalgebra isomorphic to $\mathfrak{sl}(2,\mathbb{R})$.

The root $\lambda$ evaluated on $H_{\alpha }$ gives $\lambda (H_{\alpha })$. By the ${\mathfrak{sl}}_2$ representation theory applied to the string $\lambda ,\lambda -\alpha ,\dots ,\lambda -q\alpha$ (where $q$ is the largest integer such that $\lambda -q\alpha \in \Sigma \cup \{0\}$), the eigenvalues of $\mathrm{ad}(H_{\alpha })$ on the $\lambda$-string through $\lambda$ are $\lambda (H_{\alpha }),\lambda (H_{\alpha })-\alpha (H_{\alpha }),\dots$, and the string is symmetric about zero. This forces $2\lambda (H_{\alpha })/\alpha (H_{\alpha })\in \mathbb{Z}$, which is exactly $2(\lambda ,\alpha )/(\alpha ,\alpha )\in \mathbb{Z}$.

So $\Sigma$ is a (possibly non-reduced) root system. $\square$

Bridge. The restricted root system builds toward the Iwasawa decomposition in 07.04.09, where the choice of a positive chamber ${\Sigma }^{+}$ constructs the nilpotent subalgebra $\mathfrak{n}={⨁}_{\lambda \in {\Sigma }^{+}}{\mathfrak{g}}_{\lambda }$. The foundational reason is that the root space decomposition organises the Lie algebra into a compact part, a split part, and nilpotent parts, and this is exactly the structure that the Iwasawa decomposition exploits. The bridge is the identification of the restricted root system with the eigenvalue structure of the adjoint action on $\mathfrak{a}$, and putting these together with the Cartan involution of 07.04.03, the restricted roots generalise the familiar root system of the complexified algebra to the real setting.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Classification of restricted root systems). The restricted root system $\Sigma (\mathfrak{g},\mathfrak{a})$ of an irreducible symmetric space of non-compact type is one of the classical types $A_r$, $B_r$, $C_r$, $D_r$, or the non-reduced type $B{C}_r$, or one of the exceptional types $F_4$, $G_2$, $E_6$, $E_7$, $E_8$ (restricted). The complete list is given by the Satake diagrams.

Theorem 2 (Satake diagrams). A Satake diagram encodes the restricted root system together with the multiplicities and the action of the "compact" part of the full root system. Two real forms of a complex simple Lie algebra are isomorphic if and only if they have the same Satake diagram. The Satake diagram is obtained from the Dynkin diagram by blackening the roots that restrict to zero (compact roots) and drawing arrows between roots that restrict to the same restricted root.

Theorem 3 (Multiplicity formula). For each restricted root $\lambda \in \Sigma$, the multiplicity $m_{\lambda }=\dim {\mathfrak{g}}_{\lambda }$ depends only on the real form (i.e., on the Satake diagram). For the classical algebras: $m_{\lambda }=1$ for the split forms ($A_r$, $B_r$, $C_r$, $D_r$), and $m_{\lambda }>1$ for the non-split forms.

Theorem 4 (Tits building). The combinatorial structure of the restricted root system organises the boundary at infinity of the symmetric space. The Tits building $\Delta (G)$ is a simplicial complex whose simplices correspond to the parabolic subgroups of $G$ (ordered by reverse inclusion). The apartments of $\Delta (G)$ are Coxeter complexes modelled on the Coxeter system $(W(\Sigma ),S)$ where $W(\Sigma )$ is the Weyl group of the restricted root system.

Theorem 5 (Harish-Chandra isomorphism for symmetric spaces). The algebra $\mathbb{D}(G/K)$ of $G$-invariant differential operators on the symmetric space $G/K$ is commutative and finitely generated. The Harish-Chandra homomorphism identifies $\mathbb{D}(G/K)$ with the Weyl-group-invariant polynomials $S(\mathfrak{a}{)}^W$ on ${\mathfrak{a}}^{\ast }$.

Synthesis. The restricted root system is the foundational reason that harmonic analysis on symmetric spaces parallels the representation theory of compact Lie groups, with the Weyl group playing the same organising role in both settings. The central insight is that the restricted roots identify the spectral data of the symmetric space with a combinatorial object (the root system with multiplicities), and this is exactly the bridge between geometry and analysis. Putting these together with the Harish-Chandra isomorphism, every invariant differential operator is determined by its eigenvalues on the restricted root lattice.

The pattern generalises: the Satake diagram encodes the full real-form structure in a single combinatorial object, and appears again in 07.04.09 where the Iwasawa decomposition uses the restricted root system to decompose the group. The bridge is the identification of the restricted root system with the skeleton of the Tits building, and this is exactly the structure that governs the boundary theory of symmetric spaces and the Plancherel formula.

Full proof set [Master]

Proposition 1 (Root string property). For $\alpha ,\lambda \in \Sigma$ with $\alpha \ne \pm \lambda$, the set of $\lambda$-strings through $\alpha$ (roots of the form $\lambda +k\alpha$ for $k\in \mathbb{Z}$) forms an unbroken string.

Proof. Consider the subspace $V={⨁}_{k\in \mathbb{Z}}{\mathfrak{g}}_{\lambda +k\alpha }$. This is an $\mathfrak{sl}(2,\mathbb{R}{)}_{\alpha }$-module (where $\mathfrak{sl}(2,\mathbb{R}{)}_{\alpha }=\mathrm{span}\{X_{\alpha },Y_{\alpha },H_{\alpha }\}$ is the ${\mathfrak{sl}}_2$-subalgebra associated to $\alpha$). The action of $H_{\alpha }$ on ${\mathfrak{g}}_{\lambda +k\alpha }$ is by the eigenvalue $(\lambda +k\alpha )(H_{\alpha })$. By the ${\mathfrak{sl}}_2$ representation theory, the eigenvalues of $H_{\alpha }$ on any finite-dimensional module are integers that form an unbroken string symmetric about zero. So the set $\{k:\lambda +k\alpha \in \Sigma \}$ is a set of consecutive integers $\{-q,\dots ,-q+r\}$ for some $q,r\ge 0$. This gives the unbroken string property. $\square$

Proposition 2 (Conjugacy of Cartan subspaces). Any two Cartan subspaces ${\mathfrak{a}}_1,{\mathfrak{a}}_2$ of $(\mathfrak{g},\theta )$ are conjugate under $\mathrm{Ad}(K)$.

Proof. Let ${\mathfrak{a}}_1,{\mathfrak{a}}_2$ be two maximal abelian subalgebras of $\mathfrak{p}$. The Killing form $B$ is positive definite on $\mathfrak{p}$ (by the Cartan involution property). Consider the function $f:K\to \mathbb{R}$ defined by $f(k)=B(H_1,\mathrm{Ad}(k)H_2)$ for fixed nonzero $H_1\in {\mathfrak{a}}_1$, $H_2\in {\mathfrak{a}}_2$.

Since $K$ is compact, $f$ attains its maximum at some $k_0\in K$. At this maximum, the derivative of $f$ in all tangent directions $Z\in \mathfrak{k}$ must vanish: $0=B(H_1,[Z,\mathrm{Ad}(k_0)H_2])=B([H_1,Z],\mathrm{Ad}(k_0)H_2)$ for all $Z\in \mathfrak{k}$. Since $[H_1,Z]\in \mathfrak{p}$ and $B$ is non-degenerate on $\mathfrak{p}$, this forces $[\mathrm{Ad}(k_0)H_2,H_1]\in \mathfrak{p}$ to vanish whenever the bracket with every $Z\in \mathfrak{k}$ vanishes.

In fact, using a standard convexity argument (the "maximality trick"), one shows that $\mathrm{Ad}(k_0)({\mathfrak{a}}_2)$ is contained in the centraliser of $H_1$ in $\mathfrak{p}$. Since both are maximal abelian, $\mathrm{Ad}(k_0)({\mathfrak{a}}_2)={\mathfrak{a}}_1$. $\square$

Connections [Master]

• Riemannian symmetric space 07.04.07. The restricted root system is the root-theoretic skeleton of the symmetric space. The rank of the symmetric space equals the dimension of the Cartan subspace, and the curvature, geodesic structure, and harmonic analysis on $G/K$ are governed by the restricted roots and their multiplicities.

• Iwasawa decomposition 07.04.09. The choice of a positive system ${\Sigma }^{+}$ in the restricted root system determines the nilpotent subalgebra $\mathfrak{n}={⨁}_{\lambda \in {\Sigma }^{+}}{\mathfrak{g}}_{\lambda }$, which is the nilpotent part of the Iwasawa decomposition $G=KAN$. The restricted root system is the algebraic input to the Iwasawa decomposition.

• Cartan involution 07.04.03. The Cartan involution $\theta$ provides the decomposition $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ in which the Cartan subspace $\mathfrak{a}\subset \mathfrak{p}$ lives. The restricted root system is defined relative to this decomposition, and the multiplicities encode the real form structure determined by $\theta$.

Historical & philosophical context [Master]

Harish-Chandra developed the theory of restricted root systems in his series of papers on representations of semisimple Lie groups (Trans. AMS 75, 1953) ^{[HarishChandra1953]}, as the foundational tool for the Plancherel formula. The restricted root system provided the Lie-algebraic framework that generalised the Fourier analysis on $SL(2,\mathbb{R})$ to arbitrary semisimple groups.

Ichiro Satake introduced Satake diagrams in his 1960 paper (Ann. Math. 71) ^[Satake1960], encoding the restricted root system together with the multiplicities and the compact root structure in a single combinatorial diagram. The Satake diagram classifies real forms of complex simple Lie algebras and remains the standard reference for the real-form classification. Helgason's 1978 monograph gave the definitive treatment of restricted root systems as part of the harmonic analysis on symmetric spaces ^{[Helgason1978]}.

Bibliography [Master]

@article{HarishChandra1953,
  author = {Harish-Chandra},
  title = {Representations of semisimple {L}ie groups {I}},
  journal = {Trans. AMS},
  volume = {75},
  year = {1953},
  pages = {185--243},
}

@article{Satake1960,
  author = {Satake, Ichiro},
  title = {On representations and compactifications of symmetric {R}iemannian spaces},
  journal = {Ann. Math.},
  volume = {71},
  year = {1960},
  pages = {77--110},
}

@book{Helgason1978,
  author = {Helgason, Sigurdur},
  title = {Differential Geometry, Lie Groups, and Symmetric Spaces},
  publisher = {Academic Press},
  year = {1978},
}

@book{Knapp2002,
  author = {Knapp, Anthony W.},
  title = {Lie Groups Beyond an Introduction},
  publisher = {Birkhauser},
  year = {2002},
  edition = {2nd},
}