On the Poisson relation for compact Lie groups

Abstract

Intuition drawn from quantum mechanics and geometric optics raises the following long-standing question: Can the length spectrum of a closed Riemannian manifold be recovered from its Laplace spectrum? By demonstrating that the Poisson relation is an equality for a generic bi-invariant metric on a compact Lie group, we establish that the length spectrum of a generic bi-invariant metric on a compact Lie group can be recovered from its Laplace spectrum. Furthermore, we exhibit a substantial collection ${\mathscr{G}}$ of compact Lie groups—including those that are either tori, simple, simply connected, or products thereof—with the property that for each group $U \in {\mathscr{G}}$ the length spectrum of any bi-invariant metric g carried by U is encoded in the Laplace spectrum of g. The preceding statements are special cases of results concerning compact globally symmetric spaces for which the semi-simple part of the universal cover is split-rank. The manifolds considered herein join a short list of families of non-“bumpy” Riemannian manifolds for which the Poisson relation is known to be an equality.

Appendix A: Data on indecomposable root systems and simple Lie groups

We compute the co-root lattice, central lattice and lowest strongly dominant form associated to each of the indecomposable root systems. We also compute the integral lattice associated to each simple Lie group. This data is used in the proof of Theorem 1.11 in section 3. References for some aspects of this section include [21, Chap. 10.3], [7, Chap. V.6], [26, Chap. 7.3], [34, Chap. 2.14], [24, Chap. III] and [2].

1.1 A.1. Basic definitions

Let V be a finite dimensional vector space over ${\mathbb{R}}$ and let $V^{*}$ denote its dual space. A root system associated to V is a finite subset $R \subset V^{*}$ consisting of nonzero elements having the following properties:

• (1) R spans $V^*$;

• (2) There is a map $\lambda \mapsto \lambda \, \check{}$ from R to V such that for any $\alpha , \beta \in R$ we have

  • (a) $\alpha ( \alpha \,\check{}\, ) = 2$,

  • (b) $\beta (\alpha \,\check{}\,) \in {\mathbb{Z}}$, and

  • (c) $\beta - \beta (\alpha \,\check{}\,)\alpha \in R$.

From the second property, we deduce that $\alpha$ is a root in R if and only if $-\alpha$ is as well. In the event R is a root system with the property that for each root $\alpha \in R$ we have $c \alpha \in R$ if and only if $c = {\pm } 1$, then we say the root system is reduced. Otherwise, we say the root system is non-reduced and it can be seen that if $\alpha$ is a root in R and $c \alpha \in R$, then c is restricted to the values ${\pm } \frac{1}{2}, {\pm } 1, {\pm } 2$. The dimension of V is the rank of the root system. For each $\alpha \in R$, $\alpha \,\check{}$ is called the co-root of $\alpha$ and the set $R\,\check{} \subset V \simeq V^{**}$ consisting of the co-roots is a root system associated to $V^{*}$ known as the co-root system.

Associated to each $\alpha \in R$ we have the reflection through the root$\alpha$ which is defined to be the isomorphism $S_\alpha : V \rightarrow V$ given by $S_{\alpha }(X) = X - \alpha (X) \alpha \,\check{}$. The map $S_\alpha$ fixes the hyperplane $\ker (\alpha )$ and sends $\alpha \,\check{}$ to $-\alpha \,\check{}$. The Weyl group associated to R is the finite subgroup W of ${\text{Aut}}(V)$ generated by the $S_{\alpha }$’s. If for any $\lambda \in V^*$ we define $S_{\alpha } \cdot \lambda \equiv \lambda \circ S_\alpha ^{-1} = \lambda \circ S_\alpha$, then it follows from condition 2(c) that W acts on R via permutations. We also recall that if $\langle \cdot , \cdot \rangle$ is a W-invariant inner product on V and if for each $\lambda \in V^*$ we let $\lambda ^* \in V$ be the unique vector such that $\langle \lambda ^*, \cdot \rangle = \lambda ( \cdot )$, then $\alpha \check{} = \frac{2 \alpha ^*}{\langle \alpha ^*, \alpha ^* \rangle }$. It follows that for any $\beta , \alpha \in R$ we have $\beta (\alpha \check{} \,) = 0$ if and only if $\langle \beta ^*, \alpha ^* \rangle = 0$, we therefore agree to say that two roots $\alpha , \beta \in R$ are orthogonal if $\beta (\alpha \check{}\,) = 0$.

A vector $v \in V$ is said to be regular if $\alpha (v) \ne 0$ for every $\alpha \in R$; otherwise, we will say the vector is singular. The Weyl chambers associated to the root system R are the connected components of $V - \cup _{\alpha \in R} \ker (\alpha )$ and a choice of Weyl chamber ${\mathcal {C}}$ allows us to partition R into two disjoint sets:

$$\begin{aligned} R^{+} = R^{+}({\mathcal {C}}) \equiv \{ \alpha \in R : \alpha \upharpoonright {\mathcal {C}} > 0 \} \end{aligned}$$

and

$$\begin{aligned} R^{-} = R^{-}({\mathcal {C}}) \equiv \{ \alpha \in R : \alpha \upharpoonright {\mathcal {C}} < 0 \}. \end{aligned}$$

The roots in $R^{+}$ (resp. $R^{-}$) are known as the positive (resp. negative) roots of R with respect to ${\mathcal {C}}$. An element $\alpha \in R^{+}$ is said to be decomposable if there are $\alpha _1, \alpha _2 \in R^{+}$ such that $\alpha = \alpha _1 + \alpha _2$; otherwise, we say that $\alpha \in R^{+}$ is indecomposable.

Definition A.1

Let R be a root system and ${\mathcal {C}}$ an associated Weyl chamber. A subset $B \subset R^{+} = R^{+}({\mathcal {C}})$ is said to be a (positive) basis for R if the following conditions are met:

• (1) B is a vector space basis of $V^*$;

• (2) The coordinates of each root $\alpha \in R$ with respect to the basis B are all nonnegative integers or all non-positive integers.

The elements of B are called simple roots.

Theorem A.2

[24, Sect. 10] Every root system has a positive basis. In fact, ifRis a root system and${\mathcal {C}}$is an associated Weyl chamber, the set$B^+({\mathcal {C}})$consisting of the indecomposable elements in$R^{+}({\mathcal {C}})$forms a positive basis and all positive bases ofRarise in this manner.

Corollary A.3

LetBbe a positive basis of the root systemR, then the set$B\,\check{} = \{ \alpha \,\check{} : \alpha \in B \}$is a positivebasis for$R\,\check{}$.

Associated to any root system $R \subset V^*$ there are two important lattices in V.

Definition A.4

Let V be a vector space and $R \subset V^*$ a root system with co-root system $R\,\check{} \subset V$.

• (1) The co-root lattice associated to R is the lattice $\Lambda _{R\,\check{}}$ generated by the co-roots.

• (2) The central lattice associated to R is the lattice $\Lambda _Z = \{ v \in V : \alpha (v) \in {\mathbb{Z}}\hbox{ for all } \alpha \in R \}.$

Clearly, $\Lambda _{R\,\check{}} \subseteq \Lambda _Z$ and in light of Corollary A.3 we have $\Lambda _{R\, \check{}} = \langle B\, \check{} \, \rangle$, for any positive basis B.

Given a root system $R \subset V^*$ of V with positive roots $R^+$ corresponding to some choice of Weyl chamber, the lowest strongly dominant form is the element $\rho \equiv \frac{1}{2} \sum _{\alpha \in R^+} \alpha$ (i.e., half the sum of the positive roots). This element enjoys the following well-known property which we will find useful in our proof of Proposition 1.8.

Lemma A.5

$\rho$is integer valued on the co-root lattice. In fact, $\rho$assumes the value 1 on each element of$B\,\check{}$.

Proof

Let B be a basis for R. Then, since $B\, \check{}$ is a basis for $R\,\check{}$, we only need to check the value of $\rho$ on $B\,\check{}$. Now, we recall that for any $\alpha \in R$, the reflection $S_{\alpha }$ permutes the elements in $R^{+} -\{\alpha \}$ and sends $\alpha$ to $-\alpha$. Therefore, $S_\alpha \cdot \rho =\rho - \alpha$. On the other hand, by definition, $S_{\alpha } \cdot \rho = \rho - \rho (\alpha \,\check{} \,) \alpha$. Therefore, since B is a basis, we conclude that $\rho (\alpha \,\check{}\,) = 1$. $\square$

1.2 A.2. Various lattices and the lowest strongly dominant form

Given a vector space V, a root system $R \subset V^*$ is said to be decomposable if it can be written as the disjoint union of two non-empty, orthogonal sets $R_1$ and $R_2$; otherwise, we will say that R is indecomposable. In the event that R is decomposable with orthogonal decomposition $R = R_1 \cup R_2$, if we let $V_j^*$ be the span of $R_j$, for $j = 1, 2$, then $V^*= V_1^* \oplus V_2^*$, where $V_j^{*} = {\text{Span}}_{{\mathbb{R}}}(R_j)$ for $j = 1,2$, and one easily sees that $R_j$ is a root system of $V_j^*$, for $j=1,2$. The indecomposable root systems have been classified up to isomorphism. Since it will prove useful in section 3 and we were unable to find all of this information in one convenient location in the literature, we now give an explicit description of these root systems along with their co-root lattices, central lattices, center and lowest strongly dominant forms. We also compute the integral lattice of the corresponding simple Lie group. Throughout we will let $e_1, \ldots , e_n$ be the standard basis for ${\mathbb{R}}^n$ and $\epsilon _1, \ldots , \epsilon _n$ will denote the standard dual basis: $\epsilon _k (e_j) = \delta _{jk}$. We also let $\langle \cdot , \cdot \rangle$ denote the standard inner product on ${\mathbb{R}}^n$, so that $\epsilon _j^* = e_j$ for $j = 1, \ldots , n$. In all cases, our vector space V will be a subspace of the inner product space $({\mathbb{R}}^n, \langle \cdot , \cdot \rangle )$ and $\langle \cdot , \cdot \rangle$ will be invariant under the action of the Weyl group.

1.2.1 A.2.1. Type $A_n$

As a vector space, we have $V = \{ v \in {\mathbb{R}}^{n+1} : \sum _{j =1}^{n+1} \epsilon _j (v) = 0 \}$. The corresponding root system is $R = \{ (\epsilon _{\mu } - \epsilon _{\nu }) \upharpoonright V : 1 \le \mu \ne \nu \le n+1 \}$ with co-root system $R\,\check{} =\{e_\mu -e_\nu : 1 \le \mu \ne \nu \le n+1 \}$. For a Weyl chamber, we choose ${\mathcal {C}} = \{ v \in V : \epsilon _\nu (v) >\epsilon _{\nu +1}(v), 1\le \nu \le n \}$, which gives the corresponding positive roots $R^{+} = \{ (\epsilon _{\mu } -\epsilon _{\nu }) \upharpoonright V : 1 \le \mu < \nu \le n+1 \}$ and positive basis $B = \{ (\epsilon _{\nu } - \epsilon _{\nu +1}) \upharpoonright V : 1 \le \nu \le n \}$. The co-root lattice, central lattice, center and sum of the positive roots (i.e., $2\rho$) of this root system are given by:

• $\Lambda _{R\,\check{}} = \langle e_{\mu } - e_\nu : 1 \le \mu \ne \nu \le n+1 \rangle,$

• $\Lambda _Z = \langle L_j \equiv \frac{n}{n+1} e_j -\frac{1}{n+1} \sum _{{\mathop {k \ne j}\limits ^{k=1}}}^{n+1} e_k : 1 \le j \le n \rangle = \langle L_1, e_\mu - e_\nu : 1 \le \mu < \nu \le n+1 \rangle$,

• $Z = \Lambda _Z / \Lambda _{R\,\check{}} = \langle \overline{L}_1 \rangle \simeq {\mathbb{Z}}_{n+1}$,

• $2\rho \equiv \sum _{\alpha \in R^+} \alpha = \sum _{\mu =1}^{n} (2n-2\mu +2) \epsilon _\mu$, since $\epsilon _{n+1} = -\sum _{\mu =1}^{n} \epsilon _{j}$.

The corresponding simply connected compact Lie group is ${\text{SU}}(n+1)$ and all other Lie groups of type $A_n$ are of the form $U = {\text{SU}}(n+1) / \Gamma$, where $\Gamma \le Z({\text{SU}}(n+1)) \simeq {\mathbb{Z}}_{n+1}$. The integral lattice of U with respect to any bi-invariant metric is given by

$$\begin{aligned} \Lambda _{I}({\text{SU}}(n+1)/\Gamma ) = \langle k L_1\rangle + \Lambda _{R\,\check{}} \, , \end{aligned}$$

(A.1)

where $k = 0, 1, \ldots , n$ is the smallest generator of $\Gamma$.

1.2.2 A.2.2. Type $B_n$

As a vector space, we have $V = {\mathbb{R}}^n$. The corresponding root system is $R = \{ {\pm } \epsilon _\mu , {\pm } \epsilon _{\mu } {\pm } \epsilon _{\nu } : 1 \le \mu \ne \nu \le n, {\pm } \hbox{ independent} \}$ with co-root system $R\,\check{} = \{ {\pm } 2 e_\mu , {\pm } e_\mu {\pm } e_\nu : 1 \le \mu \ne \nu \le n , {\pm } \hbox{ independent} \}$. For a Weyl chamber, we choose ${\mathcal {C}} = \{ v \in V : \epsilon _\nu (v)>\epsilon _{\nu +1}(v), 1\le \nu \le n-1, \epsilon _n(v) > 0 \}$, which gives the corresponding positive roots $R^{+} = \{ \epsilon _\mu : 1 \le \mu \le n\} \cup \{ \epsilon _{\mu } {\pm } \epsilon _{\nu }: 1 \le \mu < \nu \le n\}$ and positive basis $B=\{ \epsilon _{\nu } - \epsilon _{\nu +1}, \epsilon _n: 1 \le \nu \le n-1 \}$. The co-root lattice, central lattice, center and sum of the positive roots (i.e., $2\rho$) of this root system are given by:

• $\Lambda _{R\,\check{}} = \langle 2e_\mu , e_{\mu } {\pm } e_\nu : 1 \le \mu \ne \nu \le n \rangle$,

• $\Lambda _Z = \langle e_1, \ldots , e_n \rangle = \langle e_1, e_1 - e_2, \ldots , e_1 - e_n \rangle$,

• $Z = \Lambda _Z / \Lambda _{R\,\check{}} = \langle \bar{e}_1 \rangle \simeq {\mathbb{Z}}_{2}$,

• $2\rho \equiv \sum _{\alpha \in R^+} \alpha = \sum _{\nu =1}^{n} (2n - 2\nu +1) \epsilon _\nu$.

The corresponding simply connected compact Lie group is ${\text{Spin}}(2n+1)$ and all Lie groups of type $B_n$ are of the form $U={\text{Spin}}(2n+1)/ \Gamma$, where $\Gamma \le Z({\text{Spin}}(2n+1)) = \langle \overline{e}_1 \rangle \simeq {\mathbb{Z}}_2$. The integral lattice of U with respect to any bi-invariant metric is given by

$$\begin{aligned} \Lambda _{I}({\text{Spin}}(2n+1)/ \Gamma ) = \left\{ \begin{array}{ll} \Lambda _{R\,\check{}} &{}\quad \hbox{if}\quad \Gamma \ \hbox{is trivial}, \\ \Lambda _Z &{}\quad \hbox{if}\quad \Gamma \simeq {\mathbb{Z}}_2. \end{array}\right. \end{aligned}$$

(A.2)

1.2.3 A.2.3. Type $C_n$

As a vector space we have $V = {\mathbb{R}}^n$. The corresponding root system is $R = \{ {\pm } 2 \epsilon _\mu , {\pm } \epsilon _{\mu } {\pm } \epsilon _{\nu } : 1 \le \mu \ne \nu \le n , {\pm } \hbox{ independent} \}$ with co-root system $R\,\check{} = \{ {\pm } e_\mu , {\pm } e_\mu {\pm } e_\nu : 1 \le \mu \ne \nu \le n , {\pm } \hbox{ independent} \}$. For a Weyl chamber, we choose ${\mathcal {C}} = \{ v \in V : \epsilon _1 (v)> \epsilon _{2}(v)> \cdots> \epsilon _n(v) > 0 \}$, which gives the corresponding positive roots $R^{+} = \{ 2 \epsilon _\mu : 1 \le \mu \le n\} \cup \{ \epsilon _{\mu } {\pm } \epsilon _{\nu }: 1 \le \mu < \nu \le n \}$ and positive basis $B=\{\epsilon _1 - \epsilon _2, \ldots , \epsilon _{n-1} - \epsilon _n, 2 \epsilon _n \}$. The co-root lattice, central lattice, center and sum of the positive roots (i.e., $2\rho$) of this root system are given by:

• $\Lambda _{R\,\check{}} = \langle e_1, \ldots , e_n \rangle$,

• $\Lambda _Z = \langle e_1, \ldots , e_n, F \equiv \frac{1}{2} \sum _{\mu = 1}^{n} e_\mu \rangle = \{(\frac{c_1}{2}, \ldots , \frac{c_n}{2}) : c_j \in {\mathbb{Z}}\hbox{ and } c_j \equiv c_i \mod 2 \}$,

• $Z = \Lambda _Z / \Lambda _{R\,\check{}} = \langle \overline{F} \rangle \simeq {\mathbb{Z}}_{2}$,

• $2\rho \equiv \sum _{\alpha \in R^+} \alpha = \sum _{\nu =1}^{n} 2(n-\nu +1) \epsilon _\nu$.

The corresponding simply\connected Lie group is ${\text{Sp}}(n)$ and all other Lie groups of type $C_n$ are of the form $U = {\text{Sp}}(n) / \Gamma$, where $\Gamma \le Z({\text{Sp}}(n)) = \langle \overline{F} \rangle \simeq {\mathbb{Z}}_2$. The integral lattice of U with respect to any bi-invariant metric is given by

$$\begin{aligned} \Lambda _{I}({\text{Sp}}(n)/ \Gamma ) = \left\{ \begin{array}{ll} \Lambda _{R\,\check{}} &{}\quad \hbox{if}\quad \Gamma \ \hbox{is trivial}, \\ \Lambda _Z &{}\quad \hbox{if}\quad \Gamma = \langle \overline{F} \rangle \simeq {\mathbb{Z}}_2. \end{array}\right. \end{aligned}$$

(A.3)

1.2.4 A.2.4. Type $D_n$

As a vector space we have $V = {\mathbb{R}}^n$. The corresponding root system is $R = \{ {\pm } \epsilon _{\mu } {\pm } \epsilon _{\nu } : 1 \le \mu \ne \nu \le n , {\pm } \hbox{ independent} \}$ with co-root system $R\,\check{} = \{ {\pm } e_\mu {\pm } e_\nu : 1 \le \mu \ne \nu \le n , {\pm } \hbox{ independent} \}$. For a Weyl chamber we choose ${\mathcal {C}} = \{ v \in V : \epsilon _1 (v)> \epsilon _{2}(v)>\cdots> \epsilon _{n-1}(v) > |\epsilon _{n}(v)| \}$, which gives the corresponding positive roots $R^{+} = \{ \epsilon _{\mu } {\pm } \epsilon _{\nu }: 1 \le \mu < \nu \le n \}$ and positive basis $B=\{\epsilon _1 - \epsilon _2, \ldots , \epsilon _{n-1} - \epsilon _n, \epsilon _{n-1} + \epsilon _n\}$. The co-root lattice, central lattice, center and sum of the positive roots (i.e., $2\rho$) of this root system are given by:

• $\Lambda _{R\,\check{}} = \langle e_{\mu } {\pm } e_\nu : 1 \le \mu < \nu \le n \rangle$,

• $\Lambda _Z = \langle e_1, e_1 - e_2, \ldots , e_1 - e_n, F \equiv \frac{1}{2} \sum _{\mu = 1}^{n} e_\mu \rangle = \langle e_1, \ldots , e_n , F \rangle$,

• $$\begin{aligned} Z = \Lambda _Z / \Lambda _{R\,\check{}} = \langle \bar{e}_1, \overline{F} \rangle \simeq \left\{ \begin{array}{ll} {\mathbb{Z}}_4 &{} \quad \hbox{if}\quad n\ \hbox{is odd},\\ {\mathbb{Z}}_2 \oplus {\mathbb{Z}}_{2} &{}\quad \hbox{if}\quad n\ \hbox{is even}, \end{array}\right. \end{aligned}$$

• $2\rho \equiv \sum _{\alpha \in R^+} \alpha = \sum _{\nu =1}^{n} 2(n-\nu ) \epsilon _\nu$.

The corresponding simply connected Lie group is ${\text{Spin}}(2n)$ and all other groups of type $D_n$ are of the form $U = {\text{Spin}}(2n) /\Gamma$, where

$$\begin{aligned} \Gamma \le Z({\text{Spin}}(2n)) \simeq \left\{ \begin{array}{ll} {\mathbb{Z}}_4 &{} \quad \hbox{if}\quad n\ \hbox{is odd}, \\ {\mathbb{Z}}_2 \oplus {\mathbb{Z}}_{2} &{} \quad \hbox{if}\quad n\ \hbox{is even}. \end{array}\right. \end{aligned}$$

In the event that $2n \equiv 2 \mod 4$, the integral lattice of U with respect to a bi-invariant metric is

$$\begin{aligned} \Lambda _I({\text{Spin}}(2n)/\Gamma ) = \left\{ \begin{array}{ll} \Lambda _{R\,\check{}} &{} \quad \hbox{if}\quad \Gamma \ \hbox{is trivial}, \\ \langle 2F \rangle + \Lambda _{R\,\check{}} &{} \quad \hbox{if}\quad \Gamma = \langle 2\overline{F} \rangle \simeq {\mathbb{Z}}_2, \\ \langle F \rangle + \Lambda _{R\,\check{}} &{} \quad \hbox{if}\quad \Gamma = \langle \overline{F} \rangle \simeq {\mathbb{Z}}_4. \end{array}\right. \end{aligned}$$

(A.4)

For $2n \equiv 0 \mod 4$, the integral lattice of U with respect to a bi-invariant metric is

$$\begin{aligned} \Lambda _I({\text{Spin}}(2n)/\Gamma ) = \left\{ \begin{array}{ll} \Lambda _{R\,\check{}} &{} \quad \hbox{if}\quad \Gamma \ \hbox{is trivial}, \\ \langle e_1 \rangle + \Lambda _{R\,\check{}} &{} \quad \hbox{if}\quad \Gamma = \langle \overline{e}_1 \rangle \simeq {\mathbb{Z}}_{2} \oplus 1,\\ \langle F \rangle + \Lambda _{R\,\check{}} &{} \quad \hbox{if}\quad \Gamma = \langle \overline{F} \rangle \simeq 1 \oplus {\mathbb{Z}}_2, \\ \langle e_1 + F \rangle + \Lambda _{R\,\check{}} &{} \quad \hbox{if}\quad \Gamma = \langle \overline{e_1 + F} \rangle \simeq {\mathbb{Z}}_2,\\ \Lambda _Z &{} \quad \hbox{if}\quad \Gamma = \langle \overline{e}_1 , \overline{F} \rangle \simeq {\mathbb{Z}}_2 \oplus {\mathbb{Z}}_2. \end{array}\right. \end{aligned}$$

(A.5)

1.2.5 A.2.5. Type $BC_n$

As a vector space, we have $V = {\mathbb{R}}^n$. The corresponding root system is a non-reduced root system which is the union of the root systems of type $B_n$ and $C_n$: $R = \{ {\pm } \epsilon _\mu , {\pm } 2 \epsilon _\mu , {\pm } \epsilon _{\mu } {\pm } \epsilon _{\nu } : 1 \le \mu \ne \nu \le n \}$ with co-root system $R\,\check{} = \{ {\pm } 2 \epsilon , {\pm } e_\mu , {\pm } e_\mu {\pm } e_\nu : 1 \le \mu \ne \nu \le n \}$. For a Weyl chamber, we choose ${\mathcal {C}} = \{ v \in V : \epsilon _1(v)> \epsilon _2(v)> \cdots \epsilon _n(v) >0 \}$, which gives the corresponding positive roots $R^{+} = \{ \epsilon _j, 2 \epsilon _j: 1 \le j \le n\} \cup \{ \epsilon _i {\pm } \epsilon _j : 1 \le i < j \le n\}$ and positive basis $B = \{ \epsilon _1 -\epsilon _2, \ldots , \epsilon _{n-1} - \epsilon _n, 2 \epsilon _n \}$. The co-root lattice, central lattice, center and sum of the positive roots are given by:

• $\Lambda _{R\,\check{}} = \Lambda _{R\,\check{}}^{C_n} = \langle e_1, \ldots , e_n \rangle$,

• $\Lambda _Z = \Lambda _Z^{B_n} = \langle e_1, \ldots , e_n\rangle$,

• $Z = \Lambda _Z / \Lambda _{R\,\check{}} = 1$,

• $2 \rho \equiv \sum _{\alpha \in R^+} \alpha = \sum _{j = 1}^{n} (2(n-j) +3) \epsilon _j$.

1.2.6 A.2.6. Type $F_4$

As a vector space, we have $V = {\mathbb{R}}^4$. The corresponding root system contains the roots coming from $B_4$: $R = \{ {\pm } \epsilon _\mu , {\pm } \epsilon _{\mu } {\pm } \epsilon _{\nu } : 1 \le \mu \ne \nu \le 4 , {\pm } \hbox{ independent} \} \cup \{ \frac{1}{2} \sum _{\mu =1}^{4} {\pm } \epsilon _\mu : {\pm } \hbox{ independent}\}$ with co-root system $R\,\check{} = \{ {\pm } 2 e_\mu , {\pm } e_{\mu } {\pm } e_{\nu } : 1 \le \mu \ne \nu \le 4 , {\pm } \hbox{ independent} \} \cup \{ \frac{1}{2} \sum _{\mu =1}^{4} {\pm } e_\mu : {\pm } \hbox{ independent} \}$. For a Weyl chamber ${\mathcal {C}}$ we choose the component of $V-\cup _{\alpha \in R} \ker (\alpha )$ containing the regular vector (8, 3, 2, 1). Then, the positive roots are $R^{+} = \{ \epsilon _\mu : 1 \le \mu \le n\} \cup \{ \epsilon _{\mu } {\pm } \epsilon _{\nu } : 1 \le \mu \le \nu \le 4 \} \cup \{\frac{1}{2}(\epsilon _1 {\pm } \epsilon _2 {\pm } \epsilon _3 {\pm } \epsilon _4) : {\pm } \hbox{ independent} \}$ and positive basis $B = \{ \alpha _1 = \frac{1}{2}(\epsilon _1 -\epsilon _2 - \epsilon _3 - \epsilon _4), \alpha _2 = \epsilon _4, , \alpha _3 = \epsilon _3 - \epsilon _4, \alpha _4 = \epsilon _2 - \epsilon _3 \}$. The co-root lattice, central lattice, center and sum of the positive roots are given by:

• $\Lambda _{R\,\check{}} = \langle \frac{1}{2} \sum _{\mu =1}^{4} {\pm } e_\mu , {\pm } 2 e_\mu , {\pm } e_{\mu } {\pm } e_{\nu } : 1 \le \mu \ne \nu \le 4 \rangle$,

• $\Lambda _Z = \Lambda _{R\,\check{}}$,

• $Z = \Lambda _Z / \Lambda _{R\,\check{}} \simeq 1$,

• $2\rho \equiv \sum _{\alpha \in R^+} \alpha = 15 \epsilon _1 + 5 \epsilon _2 + 3 \epsilon _3 + \epsilon _4$.

The corresponding simply connected compact Lie group is also denoted by $F_4$ and it is the unique group of this type. The integral lattice of this group with respect to any bi-invariant metric is given by

$$\begin{aligned} \Lambda _I(F_4) = \Lambda _{R\,\check{}}. \end{aligned}$$

(A.6)

1.2.7 A.2.7. Type $G_2$

As a vector space, we have the two-dimensional subspace $V = \{ v \in {\mathbb{R}}^3 : \epsilon _1(v) + \epsilon _2(v) + \epsilon _3(v) = 0 \}$ of ${\mathbb{R}}^3$. The corresponding root system is given by $R=\{(\epsilon _\mu - \epsilon _\nu ) \upharpoonright V : 1 \le \mu \ne \nu \le 3\} \cup \{ {\pm } 3\epsilon _\mu \upharpoonright V: 1 \le \mu \le 3\}$, which contains the roots of $A_2$. And, observing that for any permutation $\sigma$ we have $3\epsilon _{\sigma (1)} \upharpoonright V = 2 \epsilon _{\sigma (1)} - \epsilon _{\sigma (2)} -\epsilon _{\sigma (3)}$, it follows that the associated co-root system is given by $R\,\check{} = \{ e_\mu {\pm } e_\nu : 1 \le \mu <\nu \le 3 \} \cup \{ {\pm } \frac{1}{3}(2e_1-e_2 -e_3), {\pm } \frac{1}{3}(-e_1 +2e_2-e_3), {\pm } \frac{1}{3} (-e_1-e_2 + 2e_3) \}$. For a Weyl chamber, we choose the component ${\mathcal {C}}$ of $V-\cup \ker (\alpha )$ containing the regular vector $(3, 2, -5)$, which gives the positive roots $R^{+} = \{ (\epsilon _1 - \epsilon _2) \upharpoonright V, (\epsilon _1 - \epsilon _3) \upharpoonright V, (\epsilon _2 - \epsilon _3) \upharpoonright V, 3\epsilon _1 \upharpoonright V, 3\epsilon _2 \upharpoonright V, - 3\epsilon _3 \upharpoonright V \}$ and the corresponding positive basis $B=\{(\epsilon _1 - \epsilon _2) \upharpoonright V, 3\epsilon _2 \upharpoonright V \}$. The co-root lattice, central lattice, center and sum of the positive roots are given by:

• $\Lambda _{R\,\check{}} = \langle \frac{1}{3}(2e_1-e_2 -e_3), \frac{1}{3}(-e_1 +2e_2-e_3), \frac{1}{3}(-e_1-e_2 + 2e_3), e_\mu {\pm } e_\nu : 1 \le \mu < \nu \le 3 \rangle,$

• $\Lambda _Z = \Lambda _{R\,\check{}}$,

• $Z = \Lambda _Z / \Lambda _{R\,\check{}} \simeq 1$,

• $2\rho \equiv \sum _{\alpha \in R^+} \alpha = (5\epsilon _1 +3\epsilon _2 - 5\epsilon _3) \upharpoonright V = 5(\epsilon _1 -\epsilon _3) + 3\epsilon _2 \upharpoonright V$.

The corresponding simply connected compact Lie group is also denoted by $G_2$ and it is the unique group of this type. The integral lattice of this group with respect to any bi-invariant metric is given by

$$\begin{aligned} \Lambda _I(G_2) = \Lambda _{R\,\check{}}. \end{aligned}$$

(A.7)

1.2.8 A.2.8. Type $E_8$

As a vector space, we have $V ={\mathbb{R}}^8$. The corresponding root system is $R = \{ {\pm } \epsilon _\mu {\pm } \epsilon _\nu : 1 \le \mu< \nu \le 8 \} \cup \{ \frac{1}{2} \sum _{\mu =1}^{8} {\pm } \epsilon _\mu : \hbox{ there are an even number of minus signs} \} = \{ {\pm } \epsilon _\mu {\pm } \epsilon _\nu : 1 \le \mu < \nu \le 8 \} \cup \{ \frac{1}{2} \sum _{\mu =1}^{8} (-1)^{k_{\mu }} \epsilon _\mu : k_\mu = 0,1 \hbox{ and } \sum k_\mu \equiv 0 \mod 2 \}$, it is the union of the 112 roots of $D_8$ with 128 additional roots. The corresponding co-root system is $R\,\check{} = \{ e_\mu {\pm } e_\nu : 1 \le \mu \ne \nu \le 8 \} \cup \{ \frac{1}{2} \sum _{\mu =1}^{8} {\pm } e_\mu : \hbox{ there are an even number of minus signs}\}.$ For a Weyl chamber, we choose the component ${\mathcal {C}}$ of $V - \cup \ker (\alpha )$ containing the vector (0, 1, 2, 3, 4, 5, 6, 23). Then the positive roots are $R^{+} = R_{1}^{+} \cup R_2^+ \cup R_3^+$, where

• $R_1^+ = \{ \frac{1}{2}( \epsilon _8 + \epsilon _7 + \sum _{\mu =1}^{6} (-1)^{k_\mu } \epsilon _\mu : \sum k_\mu \equiv 0 \mod 2 \}$,

• $R_2^+ = \{ \frac{1}{2}( \epsilon _8 - \epsilon _7 + \sum _{\mu =1}^{6} (-1)^{k_\mu } \epsilon _\mu : \sum k_\mu \equiv 1 \mod 2 \}$,

• $R_3^+ = \{ \epsilon _\mu {\pm } \epsilon _\nu : 1 \le \mu < \nu \le 8 \}$.

The sets $R_1$ and $R_2$ each contain 32 elements, while $R_3$ contains 56. As a positive basis for this root system, we have the set B consisting of the elements $\alpha _1 \equiv \frac{1}{2} (\epsilon _1 + \epsilon _8 - \sum _{\mu =2}^{7} \epsilon _\mu )$, $\alpha _2 \equiv e_1 + e_2$, $\alpha _{j+1} \equiv \epsilon _j -\epsilon _{j-1}$ for $j = 2, \ldots , 7$. Now, since the roots of $D_8$ are contained in the roots of $E_8$ we realize that the central lattice of $E_8$ is contained in the central lattice of $D_8$. The co-root lattice, central lattice, center and sum of the positive roots are then given by:

• $\Lambda _{R\,\check{}} = \Lambda _Z = \langle v_1, \ldots , v_7, v_8 \rangle$, where $v_1 \equiv \alpha _{1}^* = \frac{1}{2}( e_1 + e_8 -\sum _{j=2}^{7} e_j)$, $v_2 \equiv \alpha _{2}^*= e_1 + e_2$ and $v_{j+1} \equiv \alpha _{j+1}^* = v_j - v_{j-1}$ for $j = 2, \ldots , 7$,

• $Z = \Lambda _Z / \Lambda _{R\,\check{}} \simeq 1$, and

• $2\rho \equiv \sum _{\alpha \in R^+} \alpha = 32 \epsilon _8 +\sum _{j=1}^{7} 2 (8- j) \epsilon _j$.

The corresponding simply connected compact Lie group is also denoted by $E_8$ and it is the unique group of this type. The integral lattice of this group with respect to any bi-invariant metric is given by

$$\begin{aligned} \Lambda _I(E_8) = \Lambda _{R\,\check{}}. \end{aligned}$$

(A.8)

1.2.9 A.2.9. Type $E_7$

We will describe this root system in terms of $E_8$. Letting $v_1, \ldots , v_8$ and $\alpha _1 , \ldots , \alpha _8$ be as in A.2.8, the vector space V will be the 7-dimensional subspace of ${\mathbb{R}}^8$ spanned by $v_1, \ldots , v_7$. In other words, V is spanned by $e_1, \ldots , e_6,$ and $e_7 - e_8$, so it consists of the vectors in ${\mathbb{R}}^8$ where the $e_7$ and $e_8$ coordinate are opposite. The corresponding root system R is the restriction of the roots in $E_8$ to V and it has positive basis $B = \{ \alpha _1, \ldots , \alpha _7\}$. Specifically, we have $R = \{ {\pm } (\epsilon _\mu {\pm } \epsilon _\nu ) \upharpoonright V : 1 \le \mu < \nu \le 6 \} \cup \{{\pm } (\epsilon _7 - \epsilon _8) \upharpoonright V\} \cup \{ {\pm } \frac{1}{2}((\epsilon _7 - \epsilon _8) \upharpoonright V + \sum _{j=1}^{6} (-1)^{k_j} \epsilon _j) : \sum k_j \equiv 1 \mod 2 \}$. The co-root system is $R\,\check{} = \{ {\pm } (e_\mu {\pm } e_\nu ) : 1 \le \mu < \nu \le 6 \} \cup \{ {\pm } (e_7 - e_8) \} \cup \{ {\pm } \frac{1}{2} (e_7 - e_8 + \sum _{j=1}^{6} (-1)^{k_j} e_j ): \sum k_j \equiv 1 \mod 2 \}$. For a Weyl chamber, we choose the component ${\mathcal {C}}$ of $V - \cup _{\alpha \in R} \ker (\alpha )$ containing the regular vector $6e_1 + 5e_2 + 4e_3 + 3 e_4 + 2e_5 + 1e_6 + 11(e_7 - e_8)$. Then, the positive roots are given by the set $R^{+} = R_{1}^{+} \cup R_2^+ \cup R_3^+$, where

• $R_1^+ = \{ (\epsilon _7 - \epsilon _8) \upharpoonright V \}$,

• $R_2^+ = \{ (\epsilon _\mu {\pm } \epsilon _\nu ) \upharpoonright V: 1 \le \mu < \nu \le 6 \} \}$,

• $R_3^+ = \{ \frac{1}{2}( \epsilon _7 - \epsilon _8 + \sum _{\mu =1}^{6} (-1)^{k_\mu } \epsilon _\mu : \sum k_\mu \equiv 1 \mod 2\}$.

The set $R_1^+$ has one element, the set $R_2^+$ contains 30 elements and $R_3^+$ contains 32 elements. As a positive basis for this root system, we have $B = \{\alpha _1, \ldots , \alpha _7\}$. The co-root lattice, central lattice, center and sum of the positive roots are then given by:

• $\Lambda _{R\,\check{}} = \langle v_1, \ldots , v_6, v_7 \rangle$,

• $\Lambda _Z = \langle v_1, \ldots , v_6, v_7, F \rangle$, where

  $$\begin{aligned} F = \frac{1}{2}(e_7 -e_8) + e_1 + e_2 + e_3 = -v_1 +\frac{1}{2} v_2 - v_3 - v_4 - \frac{3}{2} v_5 - v_6, \end{aligned}$$

• $Z = \Lambda _Z / \Lambda _{R\,\check{}} = \langle \overline{F} \rangle \simeq {\mathbb{Z}}_2$,

• $2 \rho \equiv \sum _{\alpha \in R^+} \alpha = \sum _{j=1}^{6} 2(6-j) \epsilon _j + 17(\epsilon _7 - \epsilon _8) \upharpoonright V$.

The associated simply connected compact Lie group will also be denoted by $E_7$ and all other groups of this type are of the form $U = E_7/ \Gamma$, where $\Gamma \le Z(E_7) = \langle \overline{F} \rangle \simeq {\mathbb{Z}}_2$. The integral lattice of U with respect to any bi-invariant metric is

$$\begin{aligned} \Lambda _I(E_7 / \Gamma ) = \left\{ \begin{array}{ll} \Lambda _{R\,\check{}} &{}\quad \hbox{if}\quad \Gamma \ \hbox{is trivial}, \\ \Lambda _Z &{} \quad \hbox{if}\quad \Gamma = \langle \overline{F} \rangle \simeq {\mathbb{Z}}_2. \end{array} \right. \end{aligned}$$

(A.9)

1.2.10 A.2.10. Type $E_6$

We will describe this root system in terms of $E_8$. As a vector space, we take V to be the 6-dimensional subspace of ${\mathbb{R}}^8$ spanned by $v_1, \ldots , v_6$ where the $v_i$’s are as in A.2.8. One can check that in this case V is the orthogonal complement of $e_6+e_8$ in the 7-dimensional vector space associated to $E_7$; therefore, V has a basis given by $e_1 , \ldots , e_5, e_6+e_7-e_8$. The dual space $V^*$ is the linear span of $\alpha _1, \ldots , \alpha _6$ and the root system R consists of the roots of $E_8$ that have nonzero restriction to V. More explicitly, we obtain $R = \{ {\pm } (\epsilon _\mu {\pm } \epsilon _\nu ) \upharpoonright V : 1 \le \mu < \nu \le 5 \} \cup \{ {\pm } \frac{1}{2}(\epsilon _6 +\epsilon _7 - \epsilon _8 + \sum _{j=1}^{5} (-1)^{k_j} \epsilon _j ): \sum _{j=1}^{5} k_j \equiv 1 \mod 2 \}$ and the co-roots are given by $R\, \check{} = \{ {\pm } (e_\mu {\pm } e_\nu ) \upharpoonright V : 1 \le \mu < \nu \le 5 \} \cup \{ {\pm } \frac{1}{2}(e_6 + e_7 - e_8 +\sum _{j=1}^{5} (-1)^{k_j} e_j ) : \sum _{j=1}^{5} k_j \equiv 1 \mod 2 \}$. We choose our Weyl chamber ${\mathcal {C}}$ to be the component containing the regular vector $5e_1 + 4e_2 + 3e_3 + 2e_4 +1e_1 + 6(e_6 + e_7 - e_8)$ with corresponding positive roots $R^+ =R_1^+ \cup R_2^+$, where

• $R_1^+ = \{ \epsilon _i {\pm } \epsilon _j : 1 \le i < j \le 5 \}$,

• $R_2^+ = \{ \frac{1}{2}(\epsilon _6 + \epsilon _7 - \epsilon _8 +\sum _{j=1}^{5} (-1)^{k_j} e_j : \sum _{j=1}^{5} k_j \equiv 1 \mod 2\}$.

One can check that the co-root lattice, central lattice, center and the sum of the positive roots are as follows:

• $\Lambda _{R\,\check{}} = \langle v_1, \ldots , v_5, v_6 \rangle$,

• $\Lambda _{Z} = \langle v_1, \ldots , v_5, v_6, F \rangle$, where

  $$\begin{aligned} F = \frac{2}{3}(e_6 + e_7 - e_8) = \frac{2}{3}\left( -2v_1 -\frac{3}{2}v_2 -\frac{5}{2}v_3 -3v_4 -2v_5 -v_6 + v_7\right) , \end{aligned}$$

• $Z = \Lambda _Z / \Lambda _{R\,\check{}} = \langle \overline{F} \rangle \simeq {\mathbb{Z}}_3$,

• $2 \rho \equiv \sum _{\alpha \in R^+} \alpha = \sum _{j=1}^{5} 2(5-j) \epsilon _j + 8 (\epsilon _6 + \epsilon _7 - \epsilon _8)$.

The associated simply connected compact Lie group will also be denoted by $E_6$ and all other groups of this type are of the form $U = E_6/ \Gamma$, where $\Gamma \le Z(E_6) = \langle \overline{F} \rangle \simeq {\mathbb{Z}}_3$. The integral lattice of U with respect to any bi-invariant metric is

$$\begin{aligned} \Lambda _I(E_6 / \Gamma ) = \left\{ \begin{array}{ll} \Lambda _{R\,\check{}} &{} \quad \hbox{if}\quad \Gamma \ \hbox{is trivial}, \\ \Lambda _Z &{} \quad \hbox{if}\quad \Gamma = \langle \overline{F} \rangle \simeq {\mathbb{Z}}_3. \end{array}\right. \end{aligned}$$

(A.10)