On the Mathieu conjecture for Sp(N) and G2

Zwart, Kevin

doi:10.1063/5.0206983

As a direct continuation of Zwart [J. Math. Phys. 64(10), 101701 (2023)], which is built on the work of Müger and Tuset [Indagationes Math. 35(1), 114 (2024)], we reduce the Mathieu conjecture, formulated by Mathieu [Algèbra Non Commutative, Groupes Quantiques et Invariants, edited by Alex, J. and Cauchon, G. (Société Mathématique de France, Reims, 1997), Vol. 2, pp. 263–279], for Sp(N) and G₂ to a conjecture involving functions over $R^{n} \times {(S^{1})}^{m}$ with $n, m \in N_{0}$ ⁠. The proofs rely on Euler-style parametrizations of these groups, a specific version of the KAK decomposition, which we discuss and prove.

I. INTRODUCTION

In 1997, Olivier Mathieu conjectured the following statement.

Conjecture 1.1

(Ref. 9). Let G be a compact connected Lie group. If f, h are complex valued finite-type functions such that ∫_Gf^Pdg = 0 for all $P \in N$ , then ∫_Gf^Ph dg = 0 for all large enough P.

Just one year after the publication of Mathieu’s paper, Duistermaat and van der Kallen⁵ proved Mathieu’s conjecture for all abelian connected compact groups. Although the conjecture is still an open problem for any non-abelian group, some attempts were made. Dings and Koelink⁴ approached the conjecture for SU(2) by expressing the finite-type functions in terms of explicit matrix coefficients. Influenced heavily by this, Müger and Tuset¹⁰ reduced the Mathieu conjecture for SU(2) to a conjecture about certain Laurent polynomials. In 2023, this author proved,¹⁴ inspired by papers such as Refs. 2 and 12, that a similar reduction can be done for SU(N) and SO(N) to a conjecture regarding Laurent polynomials which also allow for complex square roots.

This paper is a direct continuation of Ref. 14 where we apply the same arguments to the infinite family of groups Sp(N) with $N \in N$ and the exceptional group G₂. As in Ref. 14, this can be achieved by using a generalization of the Euler decomposition which we will call the Euler angles. This decomposition is a more intricate version of the KAK decomposition, a well-known decomposition for connected Lie groups. The decomposition will be proven and applied to the groups Sp(N) and G₂.

After having found the Euler angles, the finite-type functions of Sp(N) and G₂ are considered, and we will reduce the Mathieu conjecture to a similar conjecture to that of Müger and Tuset in Ref. 10 and Zwart¹⁴ with a different weight function. To be more precise, any finite type function of Sp(N) or G₂ reduces to a function $f : C^{n} \times R^{k} \to C$ which can be written as $f (z, x) = \sum_{\vec{m}} c_{\vec{m}} (x) z^{\vec{m}}$ where $\vec{m} = (m_{1}, \dots, m_{k})$ is a multi-index where $m_{i} \in ⋃_{j = 1}^{l} \frac{1}{j} Z$ for each i, for some $l \in N$ ⁠, and $c_{\vec{m}} (x)$ is a polynomial in x₁, …, x_k and $\sqrt{1 - x_{1}^{2}}, \dots, \sqrt{1 - x_{k}^{2}}$ ⁠. Assuming these functions satisfy certain conjectures, the Mathieu conjecture is proven for Sp(N) and G₂. In Sec. II we will focus on the group Sp(N), while in Sec. III the group G₂ will be considered. The final part of the paper is dedicated to proving the generalized Euler angles we use throughout this paper, with the corresponding explicit description of the Haar measure in this parametrization.

II. THE CASE OF Sp(N)

A. The Euler angles decomposition of Sp(N)

In this paper we will discuss Mathieu’s conjecture on Sp(N) and G₂ with $N \in N$ ⁠. We start by recalling Mathieu’s conjecture. To do so, we first introduce the notion of a finite-type function.

Definition 2.1.

Let G be a compact Lie group. A function

f : G \to C

is called a finite-type function if it can be written as a finite sum of matrix components of irreducible continuous representations, i.e.,

f (x) = \sum_{j = 1}^{N} Tr [a_{j} π_{j} (x)],

where (π_j, V_j) is an irreducible continuous representation of G, and a_j ∈ End(V_j).

Conjecture 2.2

(The Mathieu Conjecture⁹). Let G be a compact connected Lie group. If f, h are finite-type functions such that ∫_Gf^Pdg = 0 for all $P \in N$ , then ∫_Gf^Ph dg = 0 for all large enough P.

This section is dedicated to the Lie group Sp(N), while Sec. III is dedicated to G₂. This document is a direct continuation of Ref. 14, and we assume the reader is familiar with the techniques used there, for we will apply them throughout the paper.

In a similar fashion as in Ref. 14, we wish to apply some form of KAK decomposition to our groups in such a way that we can reduce the Mathieu conjecture. As already hinted in Ref. 14, the generalized Euler angles are an example of a more general decomposition, which is stated below in Theorem 2.3. This theorem is key to the rest of the paper, and will be applied to both Sp(N) and G₂. We will prove the theorem in Appendix A to keep this section focussed on Sp(N).

Theorem 2.3

(The Euler Angles Theorem). Let G be a simply connected compact Lie group with finite center, and let

g

be its Lie algebra. Let

θ : g \to g

be an involutive automorphism. Let

k, p

be the +1 and −1 eigenspace of θ, respectively, in such a way that

g = k \oplus p

. Fix a maximal abelian subalgebra

a \subset p

and fix a set of (positive) roots

Δ_{p}

with respect to

a

. Then the mapping

(K / M) \times \exp (A) \times K \to G, (k M, \exp (H), l) \mapsto \exp ({Ad}_{G} (k) H) l

(2.1)

is surjective and a diffeomorphism up to a measure zero set if we replace

A

by

int (A)

⁠. Here K ⊆ G is the connected Lie group with Lie algebra

k

, and

M = Z_{K} (a)

. Here

A

is the closure of the connected component of the following set

a_{+} - {H \in a | α (H) \in π i Z f o r s o m e α \in Δ_{p}}

in such a way that

0 \in A

. Here

a_{+}

is the closed positive Weyl chamber in

a

. In addition, the Haar measure decomposes in this parameterization as follows

\int_{G} h (g) d g = C \int_{K / M} \int_{A} \int_{K} h (k_{1} \exp (H) k_{2}) | J (\exp (H)) | d k_{2} d H d g_{K / M}

for any measurable function

h : G \to C

, where C > 0 is a constant (independent of h), k₂ ∈ K with corresponding Haar measure dk₂, k₁ ∈ k an arbitrary representative of k₁M ∈ K/M with corresponding unique K-invariant measure dg_K/M on K/M, and dH is the measure on

a

. Here

J : A \to C

is given by

J (\exp (H)) ≔ \prod_{α \in Δ_{p}^{+}} \sin (α (i H))

(2.2)

where

Δ_{p}^{+}

is the set of positive roots on

a

⁠.

The rest of the section will be dedicated to applying Theorem 2.3 to the group Sp(N). To do that, we first recall the definition of the compact symplectic group Sp(N).

Definition 2.4.

Let

N \in N

⁠. We define the compact symplectic group as

S p (N) ≔ {A \in U (2 N) | J A J^{T} = A}

where

J = (\begin{matrix} 0 & 1_{N} \\ - 1_{N} & 0 \end{matrix})

and 1_N is the N × N identity matrix.

It is clear that the Lie algebra of Sp(N) is given by

s p (N) = \{(\begin{matrix} U & A \\ - A^{†} & U^{*} \end{matrix})| U \in u (N) and A \in g l (N, C) is a symmetric matrix\} .

(2.3)

Here, we denoted A^† as the adjoint of A, and U* the complex conjugate of U. In addition, since

s p (N) \subset u (2 N)

⁠, the Killing form is just given by the bilinear form ⟨X, Y⟩ = 4 Tr(XY). We note that the map

θ ≔ Ad (\begin{matrix} i 1_{N} & 0 \\ 0 & - i 1_{N} \end{matrix})

has the property

θ \in I n t (s p (N))

and can be seen as a Cartan involution. It is easily seen that with this Cartan involution, we get the following decomposition:

k ≔ \{(\begin{matrix} U & 0 \\ 0 & U^{*} \end{matrix})| U \in u (N)\} ≃ u (N),

(2.4)

p ≔ \{(\begin{matrix} 0 & A \\ - A^{†} & 0 \end{matrix})| A \in g l (N, C) such that A is symmetric\} .

(2.5)

which gives

K ≔ \{(\begin{matrix} U & 0 \\ 0 & U^{*} \end{matrix})| U \in U (N)\} ≃ U (N) .

One can take the following set as maximal abelian subalgebra:

a ≔ \{(\begin{matrix} 0_{N} & \begin{matrix} a_{1} \\ a_{2} \\ ⋱ \\ a_{N} \end{matrix} \\ \begin{matrix} - a_{1} \\ - a_{2} \\ ⋱ \\ - a_{N} \end{matrix} & 0_{N} \end{matrix})| a_{j} \in R for all j\}

where 0_N is the N × N-matrix consisting of only zeros. With that, we can calculate

M = Z_{K} (a)

⁠. Since M ⊆ K, and it must commute with all options of a_j, we see that the only options for x ∈ M is to be of the form

x = diag (e^{i ϕ_{1}}, \dots, e^{i ϕ_{N}}, e^{- i ϕ_{1}}, \dots, e^{- i ϕ_{N}})

for some

ϕ_{j} \in R

⁠. In specific, we see that

Ad (x) (\begin{matrix} 0_{N} & \begin{matrix} a_{1} \\ ⋱ \\ a_{N} \end{matrix} \\ \begin{matrix} - a_{1} \\ ⋱ \\ - a_{N} \end{matrix} & 0_{N} \end{matrix}) = (\begin{matrix} 0_{N} & \begin{matrix} e^{2 i ϕ_{1}} a_{1} \\ ⋱ \\ e^{2 i ϕ_{N}} a_{N} \end{matrix} \\ \begin{matrix} - e^{- 2 i ϕ_{1}} a_{1} \\ ⋱ \\ - e^{- 2 i ϕ_{N}} a_{N} \end{matrix} & 0_{N} \end{matrix}) .

Hence ϕ_j = 0 or π for all j, giving

\begin{array}{l} M & = ⟨ 1_{N}, diag (- 1, 1, \dots, 1, - 1, 1, \dots, 1), diag (1, - 1, \dots, 1, 1, - 1, \dots, 1), \dots, diag (1, 1, \dots, - 1, 1, 1, \dots, - 1) ⟩ \\ ≃ Z_{2}^{N} \end{array}

Therefore, applying Theorem 2.3, we find that

\begin{array}{l} S p (N) & = (K / M) A K \\ ≃ (U (N) / Z_{2}^{N}) \exp (A) U (N) . \end{array}

For the rest of this paper we will work with the latter diffeomorphism, since on U(N) some results regarding the Euler angle decomposition are already known.¹⁴ Let us consider the parametrization of U(N) first.

We remind ourselves of the Euler angle decomposition of SU(N) and on U(N). To describe this decomposition, we require an orthogonal basis for

u (N)

⁠. Let j = 1, 2, …, N − 1 and k = 1, 2, …, 2j and define the matrices

λ_{0}, λ_{j} \in s u (N)

in the following way (in most physics papers the matrices

{i λ_{j}}_{j}

are called Gell-Mann matrices, see e.g. Refs. 2, 12, and 13).

\begin{array}{l} {[λ_{0}]}_{i j} & ≔ i δ_{i, 1} δ_{j, 1} \\ {[λ_{j^{2} - 1 + k}]}_{μ, ν} & ≔ i (δ_{⌈ \frac{k}{2} ⌉, μ} δ_{j + 1, ν} + δ_{j + 1, μ} δ_{⌈ \frac{k}{2} ⌉, ν}) if k is odd, \\ {[λ_{j^{2} - 1 + k}]}_{μ, ν} & ≔ δ_{\frac{k}{2}, μ} δ_{j + 1, ν} - δ_{j + 1, μ} δ_{\frac{k}{2}, ν} if k is even, \\ [λ_{{(j + 1)}^{2} - 1}] & ≔ (\begin{matrix} i 1_{j} & 0 \\ 0 & - i j \\ 0 \\ ⋱ \\ 0 \end{matrix}) . \end{array}

This is an orthonormal basis for $u (N)$ as can easily be checked. With that, we get the following lemma.

Lemma 2.5

[Generalized Euler Angles SU(N)¹⁴]. Let N ≥ 1. Define inductively the mapping

F_{S U (N)} : ([0, π] \times {[0, 2 π]}^{N - 2}) \times ([0, π] \times {[0, 2 π]}^{N - 3}) \times \dots \times ([0, π] \times [0, 2 π]) \times [0, π] \times {[0, \frac{π}{2}]}^{\frac{N (N - 1)}{2}} \times [0, 2 π] \times \dots \times [0, \frac{2 π}{N - 1}] \to S U (N)

by F_SU(1) ≡ 1 and

\begin{aligned} F_{S U (N)} (ϕ_{1}, \dots ϕ_{\frac{N (N - 1)}{2}}, ψ_{1}, \dots, ψ_{\frac{N (N - 1)}{2}}, ω_{1}, \dots, ω_{N - 1}) \\ ≔ (\prod_{2 \leq k \leq N} A (k) (ϕ_{k - 1}, ψ_{k - 1})) \cdot (\begin{matrix} F_{S U (N - 1)} (ϕ_{N}, \dots, ϕ_{\frac{N (N - 1)}{2}}, ψ_{N}, \dots, ψ_{\frac{N (N - 1)}{2}}, ω_{1}, \dots, ω_{N - 2}) & 0 \\ 0 & 1 \end{matrix}) e^{λ_{N^{2} - 1} ω_{N - 1}}, \end{aligned}

(2.6)

where

A (k) (x, y) ≔ e^{λ_{3} x} e^{λ_{{(k - 1)}^{2} + 1} y}

, and

ψ_{j} \in [0, \frac{π}{2}], ω_{j} \in [0, \frac{2 π}{j}]

for all j. Here we denoted the product as

\prod_{2 \leq k \leq N} A (k) (ϕ_{k - 1}, ψ_{k - 1}) ≔ A (2) (ϕ_{1}, ψ_{1}) \dots A (N) (ϕ_{N - 1}, ψ_{N - 1}) .

This mapping is surjective. Moreover it is a diffeomorphism on the interior of the hypercube onto its image which is SU(N) up to a measure zero set.

Corollary 2.6

[Generalized Euler Angles U(N)]. Let

N \in N

. Then the mapping

F_{U (N)} : ([0, π] \times {[0, 2 π]}^{N - 2}) \times ([0, π] \times {[0, 2 π]}^{N - 3}) \times \dots \times ([0, π] \times [0, 2 π]) \times [0, π] \times {[0, \frac{π}{2}]}^{\frac{N (N - 1)}{2}} \times [0, 2 π] \times \dots \times [0, \frac{2 π}{N - 1}] \times [0, 2 π] \to U (N)

defined by

F_{U (N)} (ϕ_{1}, \dots, ω_{N - 1}, ξ) ≔ F_{S U (N)} (ϕ_{1}, \dots, ω_{N - 1}) e^{ξ λ_{0}}

is a surjective map onto U(N), and is a diffeomorphism on the interior of the hypercube onto its image which is U(N) up to a measure zero set.

Proof.

This can be seen by the fact that SU(N) × U(1) ≃ U(N) as manifolds by sending (A, z) ↦ A · diag(z, 1, …, 1) with inverse $A \mapsto (A \cdot diag (\det {(A)}^{- 1}, 1, \dots, 1), \det (A))$ ⁠. Using that x ↦ e^ix with x ∈ [0, 2π] is a diffeomorphism on the interior of the hypercube, and $diag (z, 1, \dots, 1) = e^{i x λ_{0}}$ in this parametrization, gives the corollary.□

Now that we know how to parametrize U(N), we go to

U (N) / Z_{2}^{N}

⁠. To do so, note that the N rightmost matrices in F_U(N)(ϕ₁, …, ξ) are diagonal matrices. Since the symmetric space

U (N) / Z_{2}^{N}

is found by considering the coset elements

g Z_{2}^{N}

which is a finite set, and these sets are found by multiplying diagonal matrices on the right, we can argue that the ω₁, …, ω_N−1, ξ parameters will have a shorter range which will fully describe

U (N) / Z_{2}^{N}

⁠. In other words we can see

U (N) / Z_{2}^{N}

as a submanifold of U(N). To be more precise, we note that

{e^{ξ λ_{0}}|}_{ξ = π} = (\begin{matrix} - 1 \\ 1 \\ ⋱ \\ 1 \end{matrix}) \in Z_{2}^{N}, {e^{ω_{1} λ_{3}} e^{ξ λ_{0}}|}_{ω_{1} = π, ξ = π} = (\begin{matrix} 1 \\ - 1 \\ \dots \\ 1 \end{matrix}) \in Z_{2}^{N}

and so on. We conclude that we can find a parametrization of all elements in

Z_{2}^{N}

this way, hence giving the following parametrization.

Corollary 2.7

[Parametrization

U (N) / Z_{2}^{N}

]. Let

N \in N

. Then the mapping

{\tilde{F}}_{U (N)} : ([0, π] \times {[0, 2 π]}^{N - 2}) \times ([0, π] \times {[0, 2 π]}^{N - 3}) \times \dots \times ([0, π] \times [0, 2 π]) \times [0, π] \times {[0, \frac{π}{2}]}^{\frac{N (N - 1)}{2}} \times [0, π] \times \dots \times [0, \frac{π}{N - 1}] \times [0, π] \to U (N)

defined by

{\tilde{F}}_{U (N)} (ϕ_{1}, \dots, ω_{N - 1}, ξ) ≔ F_{S U (N)} (ϕ_{1}, \dots, ω_{N - 1}) e^{ξ λ_{0}}

is a surjective map onto

U (N) / Z_{2}^{N}

, and is a diffeomorphism on the interior of the hypercube onto its image which is

U (N) / Z_{2}^{N}

up to a measure zero set.

With this corollary, we get the following lemma which parametrizes Sp(N). We will call this the Euler angles parametrization or the Euler angles decomposition of Sp(N).

Lemma 2.8

[Euler Angles Parametrization of Sp(N)]. Let

N \in N

and define G = Sp(N), fix a maximal abelian subalgebra

a \subseteq p

and consider K ≃ U(N) as before. Then

G ≃ (K / M) A K

diffeomorphic up to a measure zero set, where

A = \exp (A) \subseteq \exp (a)

and

M = Z_{K} (a)

. Here

A = \{(y_{1}, \dots, y_{N}) \in R^{N} |0 \leq y_{j} \leq \frac{π}{2} a n d y_{i} \leq y_{i + 1} f o r a l l i = 1, \dots, N - 1\} .

In addition, let F_U(N) and

{\tilde{F}}_{U (N)}

be as in Corollaries 2.6 and 2.7 respectively. Then we define the mapping

F_{S p (N)} : ([0, π] \times {[0, 2 π]}^{N - 2}) \times ([0, π] \times {[0, 2 π]}^{N - 3}) \times \dots \times ([0, π] \times [0, 2 π]) \times [0, π] \times {[0, \frac{π}{2}]}^{\frac{N (N - 1)}{2}} \times [0, π] \times \dots \times [0, \frac{π}{N - 1}] \times [0, π] \times A \times ([0, π] \times {[0, 2 π]}^{N - 2}) \times ([0, π] \times {[0, 2 π]}^{N - 3}) \times \dots \times ([0, π] \times [0, 2 π]) \times [0, π] \times {[0, \frac{π}{2}]}^{\frac{N (N - 1)}{2}} \times [0, 2 π] \times \dots \times [0, \frac{2 π}{N - 1}] \times [0, 2 π] \to S p (N)

by

\begin{array}{l} F_{S p (N)} ({\tilde{ϕ}}_{1}, \dots, {\tilde{ω}}_{N - 1}, \tilde{ξ}, y_{1}, \dots, y_{N}, ϕ_{1}, \dots ω_{N - 1}, ξ) \\ ≔ (\begin{matrix} {\tilde{F}}_{U (N)} ({\tilde{ϕ}}_{1}, \dots, {\tilde{ω}}_{N - 1}, \tilde{ξ}) & 0 \\ 0 & {\tilde{F}}_{U (N)} {({\tilde{ϕ}}_{1}, \dots, {\tilde{ω}}_{N - 1}, \tilde{ξ})}^{*} \end{matrix}) \exp (\begin{matrix} 0_{N} & \begin{matrix} y_{1} \\ ⋱ \\ y_{n} \end{matrix} \\ \begin{matrix} - y_{1} \\ ⋱ \\ - y_{n} \end{matrix} & 0_{N} \end{matrix}) \cdot \\ (\begin{matrix} F_{U (N)} (ϕ_{1}, \dots, ω_{N - 1}, ξ) & 0 \\ 0 & F_{U (N)} {(ϕ_{1}, \dots, ω_{N - 1}, ξ)}^{*} \end{matrix}) \end{array}

This mapping is surjective onto Sp(N) and is a diffeomorphism on the interior of the hypercube onto its image which is Sp(N) up to a measure zero set.

Finally, the Haar measure dg is in this parametrization given by:

\begin{array}{l} d g & = d g_{K / M} ({\tilde{ϕ}}_{1}, \dots, \tilde{ξ}) d g_{K} (ϕ_{1}, \dots, ξ) J (y_{1}, \dots, y_{N}) d y_{1} \dots d y_{N} \\ = C d g_{K} ({\tilde{ϕ}}_{1}, \dots, \tilde{ξ}) d g_{K} (ϕ_{1}, \dots, ξ) J (y_{1}, \dots, y_{N}) d y_{1} \dots d y_{N} \end{array}

where C > 0 is some constant, dg_K(ϕ₁, …, ξ) is the Haar measure on K ≃ U(N) integrating over the variables ϕ₁, …, ξ, and

J_{S p (N)} (y_{1}, \dots, y_{N}) ≔ \prod_{j = 1}^{N} \sin (2 y_{j}) \prod_{i > k} \sin (y_{i} - y_{k}) \sin (y_{i} + y_{k}) .

Here the integration over the y_j variables goes over exactly the area

A

⁠.

Remark 2.9.

Note that we chose the variables that go over the symmetric space K/M to have a ∼ above their letters for bookkeeping, while the variables going over elements in K or A are without a ∼. This notation is kept throughout the paper.

Proof.

The first equation is proven by applying Theorem 2.3 to our case of Sp(N). Secondly, by applying Corollaries 2.6 and 2.7 to the first equation, we find that

F_{S p (N)} (\tilde{ϕ}, \dots, ξ)

is a surjective map onto Sp(N) and is a diffeomorphism up to a measure zero set. The only thing to prove is the area

A

⁠. To do so, we recall the reduced root system of

s p (N)

with respect to

a

⁠. Let us define the basis of

a

as the set

{e_{j}}_{j = 1, \dots, N}

where

{[e_{j}]}_{k, l} = δ_{k, j} δ_{l, N + j} - δ_{l, j} δ_{k, N + j} .

Define the linear functional

α_{j} \in a^{*}

as

α_{j} (e_{k}) = δ_{j, k}

Then

s p {(N)}_{C}

has a root decomposition, with roots given by

Δ_{p} = {\pm 2 α_{j}, α_{j} \pm α_{k}, - α_{j} \pm α_{k} | j, k = 1, \dots, N, j > k} .

We choose the set of positive roots to be

Δ_{p}^{+} ≔ {2 α_{j}, α_{j} \pm α_{k} | j > k} .

Then it is immediate that

\begin{array}{l} J_{S p (N)} (\exp (H)) & = \prod_{j = 1}^{N} \sin (2 α_{j} (H)) \prod_{j > k} \sin ((α_{j} - α_{k}) (H)) \sin ((α_{j} + α_{k}) (H)) \\ = \prod_{j = 1}^{N} \sin (2 y_{j}) \prod_{j > k} \sin (y_{j} - y_{k}) \sin (y_{j} + y_{k}) \end{array}

where we wrote

H = (\begin{matrix} 0_{N} & \begin{matrix} y_{1} \\ ⋱ \\ y_{n} \end{matrix} \\ \begin{matrix} - y_{1} \\ ⋱ \\ - y_{n} \end{matrix} & 0_{N} \end{matrix}) .

Thus

A

can be described as

A = \{(y_{1}, \dots, y_{N}) \in R^{N}| 0 \leq 2 y_{j} \leq π and 0 \leq y_{j} \pm y_{k} \leq π for all j > k\} .

Note that if

0 \leq y_{j} \leq \frac{π}{2}

then y_j ± y_k ≤ π automatically, so we can drop most inequalities, and only 0 ≤ y_j − y_k adds new information. In other words, we can describe

A

as

\begin{array}{l} A & = \{(y_{1}, \dots, y_{N}) \in R^{N} |0 \leq 2 y_{j} \leq π and y_{k} \leq y_{j} for all j > k\} \\ = \{(y_{1}, \dots, y_{N}) \in R^{N} |0 \leq 2 y_{j} \leq π and y_{i} \leq y_{i + 1} for all i = 1, \dots N - 1\} . \end{array}

which yields the theorem. We finally note that the measure on K/M can be seen to be the Haar measure on K times some positive constant. This can be seen by noting that up to a measure zero set, we can embed K/M into K. This allows us to note that the left-invariant measure on K/M can then be seen as the measure on K as well up to a C^∞-function. But since both dg_K/M and dg_K are left-invariant, this a C^∞ function should be constant.□

B. The Mathieu conjecture on Sp(N)

Now that we have a suitable parametrization for Sp(N), we go to the Mathieu conjecture. We work in the same way as in Ref. 14. By Procesi (Ref. 11, Theorem 8.2.3), all finite-type functions of compact groups

G \subset GL (N, C)

are generated by the matrix entries of G and the inverse of the determinant. With our parametrization of Sp(N) in Theorem 2.8, we can thus describe all finite-type functions almost everywhere. Together with the fact that the determinant of all matrices in Sp(N) is one, we see that any finite type function on Sp(N) can be written as a sum of products of the matrix entries of Sp(N), which is by Theorem 2.8 is described by the matrix entries of K/M,

\exp A

and K. Since the matrix entries of K (and of K/M) are by Corollary 2.6 given by the matrix entries of SU(N) times the determinant of U(N), which again can be interpreted as finite-type functions on SU(N) times the determinant of U(N), we get that a finite type function f on Sp(N) are can be written as

f (g) = \sum_{i, j} c_{i j} f_{i j}^{S U (N)} ({\tilde{ϕ}}_{1}, \dots, {\tilde{ω}}_{N - 1}) e^{i m_{i j} \tilde{ξ}} g_{i j} (y_{1}, \dots, y_{N}) h_{i j}^{S U (N)} (ϕ_{1} \dots, ξ) e^{i n_{i j} ξ}

(2.7)

for almost all g ∈ G, where we identified

g = F_{S p (N)} ({\tilde{ϕ}}_{1}, \dots, ξ)

such that the parameters

{\tilde{ϕ}}_{1}, \dots, ξ

are well-defined,

f_{i j}^{S U (N)}

and

h_{i j}^{S U (N)}

are finite-type functions on SU(N),

m_{i j}, n_{i j} \in Z

and

g_{i j} (y_{1}, \dots, y_{N}) ≔ \sin^{p_{i j}^{1}} (y_{1}) \cos^{q_{i j}^{1}} (y_{1}) \dots \sin^{p_{i j}^{N}} (y_{N}) \cos^{q_{i j}^{N}} (y_{N})

where

p_{i j}^{n} \in N_{0}

for all n = 1, 2, …, N and

q_{i j}^{m} \in {0, 1}

for all m = 1, 2, …, N. We sum over two separate indices to make sure we have all the possible terms and allow for different powers of each term.

For more details regarding the finite-type functions on SU(N), we refer to Ref. 14. In a similar way as in Ref. 14, we have the following proposition.

Proposition 2.10.

Let f be a finite-type function on Sp(N) as in Eq. (2.7). Then for any

P \in N

we have

\begin{array}{l} \int_{S p (N)} f^{P} (g) d g = C^{'} & \int_{{[0, 1]}^{N (N - 1)}} \int_{{(S^{*})}^{N (N + 1)}} \int_{0}^{1} \int_{0}^{ξ_{N}} \dots \int_{0}^{ξ_{2}} \hat{f} {(x_{1}, \dots, x_{N (N - 1)}, z_{1}, \dots, z_{N (N + 1)}, ξ_{1}, \dots, ξ_{N})}^{P} \\ {\tilde{J}}_{S p (N)} (x_{1}, \dots, x_{N (N - 1)}, ξ_{1}, \dots, ξ_{N}) d ξ_{1} \dots, d ξ_{N} \frac{d z_{1}}{z_{1}} \dots \frac{d z_{N (N + 1)}}{z_{N (N + 1)}} d x_{1} \dots d x_{N (N - 1)} \end{array}

where

\begin{array}{l} \hat{f} (x_{1}, \dots, ξ_{N}) ≔ \sum_{i, j} & c_{i j} {\bar{f}}_{i j}^{S U (N)} (x_{1}, \dots, z_{\frac{N (N + 1)}{2} - 1}) z_{\frac{N (N + 1)}{2}}^{\frac{m_{i j}}{2}} {\tilde{g}}_{i j} (ξ_{1}, \dots, ξ_{N}) \cdot \\ \tilde{h_{i j}^{S U (N)}} (x_{\frac{N (N - 1)}{2} + 1}, \dots, z_{N (N + 1) - 1}) z_{N (N + 1)}^{n_{i j}}, \end{array}

and

\begin{array}{l} {\tilde{J}}_{S p (N)} (x_{1}, \dots, x_{N (N - 1)}, ξ_{1}, \dots, ξ_{N}) ≔ & J_{S U (N)} (x_{1}, \dots x_{\frac{N (N - 1)}{2}}) (\prod_{j = 1}^{N} ξ_{j}) \cdot \\ (\prod_{j > k} (ξ_{j}^{2} (1 - ξ_{k}^{2}) - (1 - ξ_{j}^{2}) ξ_{k})) \cdot \\ J_{S U (N)} (x_{\frac{N (N - 1)}{2} + 1}, \dots, x_{N (N - 1)}) . \end{array}

Here C′ > 0 is some constant and

\tilde{h_{i j}^{S U (N)}}

and J_SU(N) are as in Ref. 14, Lemma 2.7. The function J_Sp(N) is defined as in Theorem 2.8, the set S* ≔ S¹\{1} and the function

{\bar{f}}_{i j}^{S U (N)}

is defined as

{\bar{f}}_{i j}^{S U (N)} (x_{1}, \dots, z_{\frac{N (N + 1)}{2} - 1}) = \tilde{f_{i j}^{S U (N)}} (x_{1}, \dots, x_{\frac{N (N - 1)}{2}}, z_{1}, \dots, z_{\frac{N (N - 1)}{2}}, z_{\frac{N (N - 1)}{2} + 1}^{1 / 2}, \dots, z_{\frac{N (N + 1)}{2} - 1}^{1 / 2}),

and

{\tilde{g}}_{i j} (ξ_{1}, \dots, ξ_{N}) = ξ_{1}^{p_{i j}^{1}} {(1 - ξ_{1}^{2})}^{\frac{q_{i j}^{1}}{2}} \dots ξ_{N}^{p_{i j}^{N}} {(1 - ξ_{N}^{2})}^{\frac{q_{i j}^{N}}{2}} .

Proof.

We note that

\begin{array}{l} \int_{G} f {(g)}^{P} d g & = \int_{G} {[\sum_{i, j} c_{i j} f_{i j}^{S U (N)} (\tilde{ϕ_{1}}, \dots, {\tilde{ω}}_{N - 1}) e^{i m_{i j} \tilde{ξ}} g_{i j} (y_{1}, \dots, y_{N}) h_{i j}^{S U (N)} (ϕ_{1} \dots, ω_{N - 1}) e^{i n_{i j} ξ}]}^{P} d g \\ = \sum_{\sum β_{i j} = P} \int_{G} \prod_{i, j} c_{i j}^{β_{i j}} {(f_{i j}^{S U (N)} (\tilde{ϕ_{1}}, \dots, {\tilde{ω}}_{N - 1}))}^{β_{i j}} e^{i m_{i j} β_{i j} \tilde{ξ}} {(g_{i j} (y_{1}, \dots, y_{N}))}^{β_{i j}} \cdot \\ {(h_{i j}^{S U (N)} (ϕ_{1} \dots, ω_{N - 1}))}^{β_{i j}} e^{i n_{i j} β_{i j} ξ} d g \\ = \sum_{\sum β_{i j} = P} (\prod_{i, j} c_{i j}^{β_{i j}}) [C \int_{K / M} \prod_{i, j} {(f_{i j}^{S U (N)} ({\tilde{ϕ}}_{1}, \dots, {\tilde{ω}}_{N - 1}))}^{β_{i j}} e^{i m_{i j} β_{i j} \tilde{ξ}} d g_{K}] \cdot \\ [\int_{A} \prod_{i, j} {(g_{i j} (y_{1}, \dots, y_{N}))}^{β_{i j}} J_{S p (N)} (y_{1}, \dots, y_{N}) d y_{1} \dots d y_{N}] \cdot \\ [\int_{K} \prod_{i, j} {(h_{i j}^{S U (N)} (ϕ_{1} \dots, ω_{N - 1}))}^{β_{i j}} e^{i n_{i j} β_{i j} ξ} d g_{K}] . \end{array}

To complete the lemma, we consider each integral separately. Using Ref. 14, Lemma 2.7, we immediately can claim that the last integral equals to

\begin{array}{l} \int_{K} \prod_{i, j} & {(h_{i j}^{S U (N)} (ϕ_{1} \dots, ω_{N - 1}))}^{β_{i j}} e^{i n_{i j} β_{i j} ξ} d g_{K} = \frac{1}{2 (N - 1) i^{\frac{N (N + 1)}{2}}} \int_{{[0, 1]}^{\frac{N (N - 1)}{2}}} \int_{{(S^{*})}^{\frac{N (N + 1)}{2}}} \\ \prod_{i, j} {(\tilde{h_{i j}^{S U (N)}} (x_{1}, \dots, z_{\frac{N (N + 1)}{2} - 1}))}^{β_{i j}} z_{\frac{N (N + 1)}{2}}^{n_{i j} β_{i j}} \cdot J_{S U (N)} (x_{1}, \dots, x_{\frac{N (N - 1)}{2}}) \frac{d z_{1}}{z_{1}} \dots \frac{d z_{\frac{N (N + 1)}{2}}}{z_{\frac{N (N + 1)}{2}}} d x_{1} \dots d x_{\frac{N (N - 1)}{2}}, \end{array}

where

\tilde{h_{i j}^{S U (N)}}

is defined as in Ref. 14, Lemma 2.7, where we recall S* = S¹\{1}. The same arguments can be applied to the first integral, only note that the

{\tilde{ω}}_{j}

go over the range

[0, \frac{π}{j}]

instead of

[0, \frac{2 π}{j}]

which gives a factor of 1/2 per parameter. Therefore, we get that the first integral can be written as

\begin{array}{l} \int_{K} \prod_{i, j} & {(f_{i j}^{S U (N)} ({\tilde{ϕ}}_{1} \dots, \tilde{ξ}))}^{β_{i j}} e^{i m_{i j} β_{i j} {\tilde{ω}}_{N - 1}} d g_{K} = \frac{1}{2^{N} (N - 1) i^{\frac{N (N + 1)}{2}}} \\ \int_{{[0, 1]}^{\frac{N (N - 1)}{2}}} \int_{{(S^{*})}^{\frac{N (N + 1)}{2}}} \prod_{i, j} {({\bar{f}}_{i j}^{S U (N)} (x_{1}, \dots, z_{\frac{N (N + 1)}{2} - 1}))}^{β_{i j}} z_{\frac{N (N + 1)}{2}}^{\frac{m_{i j}}{2} β_{i j}} \cdot J_{S U (N)} (x_{1}, \dots, x_{\frac{N (N - 1)}{2}}) \cdot \\ \frac{d z_{1}}{z_{1}} \dots \frac{d z_{\frac{N (N + 1)}{2}}}{z_{\frac{N (N + 1)}{2}}} d x_{1} \dots d x_{\frac{N (N - 1)}{2}}, \end{array}

where

{\bar{f}}_{i j}^{S U (N)} (x_{1}, \dots, z_{\frac{N (N + 1)}{2} - 1}) = \tilde{f_{i j}^{S U (N)}} (x_{1}, \dots, x_{\frac{N (N - 1)}{2}}, z_{1}, \dots, z_{\frac{N (N - 1)}{2}}, z_{\frac{N (N - 1)}{2} + 1}^{1 / 2}, \dots, z_{\frac{N (N + 1)}{2} - 1}^{1 / 2}) .

Finally note that

\begin{array}{l} \int_{A} \prod_{i, j} {(g_{i j} (y_{1}, \dots, y_{N}))}^{β_{i j}} & J_{S p (N)} (y_{1}, \dots, y_{N}) d y_{1} \dots d y_{N} \\ = \int_{0}^{π / 2} \int_{0}^{y_{N - 1}} \dots \int_{0}^{y_{2}} & \sin^{\sum_{i j} β_{i j} p_{i j}^{1}} (y_{1}) \cos^{\sum_{i j} β_{i j} q_{i j}^{1}} (y_{1}) \dots \sin^{\sum_{i j} β_{i j} p_{i j}^{N}} (y_{N}) \cos^{\sum_{i j} β_{i j} q_{i j}^{N}} (y_{N}) \cdot \\ J_{S p (N)} (y_{1}, \dots, y_{N}) d y_{1} \dots d y_{N} \end{array}

Note that

\begin{array}{l} J (y_{1}, \dots, y_{N}) & = [\prod_{j = 1}^{N} \sin (2 y_{j})] \cdot [\prod_{j > k} \sin (y_{j} - y_{k}) \sin (y_{j} + y_{k})] \\ = [\prod_{j = 1}^{N} 2 \sin (y_{j}) \cos (y_{j})] \cdot [\prod_{j > k} (\sin^{2} (y_{j}) \cos^{2} (y_{k}) - \cos^{2} (y_{j}) \sin^{2} (y_{k}))] . \end{array}

So if we subsitute ξ_j = sin(y_j), we see that

\begin{array}{l} \int_{A} & \prod_{i, j} {(g_{i j} (y_{1}, \dots, y_{N}))}^{β_{i j}} J_{S p (N)} (y_{1}, \dots, y_{N}) d y_{1} \dots d y_{N} \\ = \int_{0}^{1} \int_{0}^{ξ_{N - 1}} \dots \int_{0}^{ξ_{2}} ξ_{1}^{\sum_{i j} β_{i j} p_{i j}^{1}} {(1 - ξ_{1}^{2})}^{\frac{\sum_{i j} β_{i j} q_{i j}^{1}}{2}} \dots ξ_{N}^{\sum_{i j} β_{i j} p_{i j}^{N}} {(1 - ξ_{N}^{2})}^{\frac{\sum_{i j} β_{i j} q_{i j}^{N}}{2}} \cdot \\ [\prod_{j = 1}^{N} 2 ξ_{j}] \cdot [\prod_{j > k} (ξ_{j}^{2} (1 - ξ_{k}^{2}) - (1 - ξ_{j}^{2}) ξ_{k})] d ξ_{1} \dots d ξ_{N} \end{array}

Putting everything together gives:

\begin{array}{l} \int_{S p (N)} f^{P} (g) d g & = \sum_{\sum β_{i j} = P} (\prod_{i, j} c_{i j}^{β_{i j}}) [C \frac{1}{2^{N} (N - 1) i^{\frac{N (N + 1)}{2}}} \int_{{[0, 1]}^{\frac{N (N - 1)}{2}}} \int_{{(S^{*})}^{\frac{N (N + 1)}{2}}} \\ \prod_{i, j} {({\bar{f}}_{i j}^{S U (N)} (x_{1}, \dots, z_{\frac{N (N + 1)}{2} - 1}))}^{β_{i j}} z_{\frac{N (N + 1)}{2}}^{\frac{m_{i j}}{2} β_{i j}} \cdot J_{S U (N)} (x_{1}, \dots, x_{\frac{N (N - 1)}{2}}) \cdot \\ \frac{d z_{1}}{z_{1}} \dots \frac{d z_{\frac{N (N + 1)}{2}}}{z_{\frac{N (N + 1)}{2}}} d x_{1} \dots d x_{\frac{N (N - 1)}{2}}] \cdot \\ [\int_{0}^{1} \int_{0}^{ξ_{N}} \dots \int_{0}^{ξ_{2}} ξ_{1}^{\sum_{i j} β_{i j} p_{i j}^{1}} {(1 - ξ_{1}^{2})}^{\frac{\sum_{i j} β_{i j} q_{i j}^{1}}{2}} \dots ξ_{N}^{\sum_{i j} β_{i j} p_{i j}^{N}} {(1 - ξ_{N}^{2})}^{\frac{\sum_{i j} β_{i j} q_{i j}^{N}}{2}} \cdot \\ (\prod_{j = 1}^{N} 2 ξ_{j}) \cdot (\prod_{j > k} (ξ_{j}^{2} (1 - ξ_{k}^{2}) - (1 - ξ_{j}^{2}) ξ_{k})) d ξ_{1} \dots d ξ_{N}] \cdot \\ [\frac{1}{2 (N - 1) i^{\frac{N (N + 1)}{2}}} \int_{{[0, 1]}^{\frac{N (N - 1)}{2}}} \int_{{(S^{*})}^{\frac{N (N + 1)}{2}}} \prod_{i, j} {({\tilde{h}}_{i j}^{S U (N)} (x_{1}, \dots, z_{\frac{N (N + 1)}{2} - 1}))}^{β_{i j}} z_{\frac{N (N + 1)}{2}}^{n_{i j} β_{i j}} \cdot \\ J_{S U (N)} (x_{1}, \dots, x_{\frac{N (N - 1)}{2}}) \frac{d z_{1}}{z_{1}} \dots \frac{d z_{\frac{N (N + 1)}{2}}}{z_{\frac{N (N + 1)}{2}}} d x_{1} \dots d x_{\frac{N (N - 1)}{2}}] \\ = \frac{C}{2 {(N - 1)}^{2} i^{N (N + 1)}} \int_{{[0, 1]}^{N (N - 1)}} \int_{{(S^{*})}^{N (N + 1)}} \int_{0}^{1} \int_{0}^{ξ_{N}} \dots \int_{0}^{ξ_{2}} \\ [\sum_{i, j} c_{i j} {\bar{f}}_{i j}^{S U (N)} (x_{1}, \dots, z_{\frac{N (N + 1)}{2} - 1}) z_{\frac{N (N + 1)}{2}}^{\frac{m_{i j}}{2}} ξ_{1}^{k_{i j}} {(1 - ξ_{1}^{2})}^{\frac{l_{i j}^{1}}{2}} \dots ξ_{N}^{k_{i j}^{N}} {(1 - ξ_{N}^{2})}^{\frac{l_{i j}}{2}} \cdot \\ {{\tilde{h}}_{i j}^{S U (N)} (x_{\frac{N (N - 1)}{2} + 1}, \dots, z_{N (N + 1) - 1}) z_{N (N + 1)}^{n_{i j}}]}^{P} J_{S U (N)} (x_{1}, \dots x_{\frac{N (N - 1)}{2}}) \cdot \\ (\prod_{j = 1}^{N} ξ_{j}) \cdot (\prod_{j > k} (ξ_{j}^{2} (1 - ξ_{k}^{2}) - (1 - ξ_{j}^{2}) ξ_{k})) J_{S U (N)} (x_{\frac{N (N - 1)}{2} + 1}, \dots, x_{N (N - 1)}) \\ d ξ_{1} \dots d ξ_{N} \frac{d z_{1}}{z_{1}} \dots \frac{d z_{N (N + 1)}}{z_{N (N - 1)}} d x_{1} \dots d x_{N (N - 1)} . \end{array}

Defining

\begin{array}{l} {\tilde{J}}_{S p (N)} & (x_{1}, \dots, x_{N (N - 1)}, ξ_{1}, \dots, ξ_{N}) ≔ J_{S U (N)} (x_{1}, \dots x_{\frac{N (N - 1)}{2}}) (\prod_{j = 1}^{N} ξ_{j}) \cdot \\ (\prod_{j > k} (ξ_{j}^{2} (1 - ξ_{k}^{2}) - (1 - ξ_{j}^{2}) ξ_{k})) J_{S U (N)} (x_{\frac{N (N - 1)}{2} + 1}, \dots, x_{N (N - 1)}) . \end{array}

and defining

C^{'} = \frac{C}{{(- 1)}^{\frac{N (N + 1)}{2}} \cdot 2 {(N - 1)}^{2}}

gives the desired result.□

We note that Proposition 2.10 gives a sum of products of (possible roots of) polynomials, in a similar way as in Ref. 14. Let us give a name for these kind of polynomial.

Definition 2.11.

Let

k, l, N \in N

and

f : {[0, 1]}^{k} \times {(S^{*})}^{l} \to C

⁠. We say f is a

\frac{1}{N}

-admissible function if f can be written as

f (x_{1}, \dots, x_{k}, z_{1}, \dots, z_{l}) = \sum_{\vec{m}} c_{\vec{m}} (x) z^{\vec{m}},

where

\vec{m} = (m_{1}, \dots, m_{l})

is a multi-index where

m_{i} \in ⋃_{j = 1}^{N} \frac{1}{j} Z

⁠, and

c_{\vec{m}} (x) \in C [x_{1}, {(1 - x_{1}^{2})}^{1 / 2}, \dots, x_{k}, {(1 - x_{k}^{2})}^{1 / 2}]

is a complex polynomial in x_i and

\sqrt{1 - x_{i}^{2}}

⁠. We define the collection of

\vec{m}

for which

c_{\vec{m}} \neq 0

the spectrum of f, and it will be denoted by Sp(f).

Remark 2.12.

Note that in Ref. 14, we called these functions SU(N)-admissible functions.

It is clear from Proposition 2.10 that $\hat{f}$ is a $\frac{1}{N}$ -admissible function. Motivated by Refs. 10 and 14, we make the following conjecture.

Conjecture 2.13.

Let

f : {[0, 1]}^{N^{2}} \times {(S^{*})}^{N (N + 1)} \to C

be a

\frac{1}{N}

-admissible function. If

\int_{{[0, 1]}^{N (N - 1)}} \int_{{(S^{*})}^{N (N + 1)}} \int_{0}^{1} \int_{0}^{ξ_{N}} \dots \int_{0}^{ξ_{2}} f^{P} {\tilde{J}}_{S p (N)} d ξ_{1} \dots d ξ_{N} \frac{d z_{1}}{z_{1}} \dots \frac{d z_{N (N + 1)}}{z_{N (N + 1)}} d x_{1} \dots d x_{N (N - 1)} = 0

for all

P \in N

, then

\vec{0}

does not lie in the convex hull of Sp(f).

Theorem 2.14.

Assume Conjecture 2.13 is true. Then the Mathieu conjecture is true for Sp(N).

Proof.

Let f, h be finite-type functions such that ∫_Gf^P(g)dg = 0 for all

P \in N

⁠. We are interested in ∫_Gf^P(g)h(g)dg. Since f, h are finite-type functions, by the previous discussion we can conclude f and h have the form of Eq. . By linearity of the integral we can assume h to be of the form

h (g) = c \cdot h_{1}^{S U (N)} ({\tilde{ϕ}}_{1}, \dots, {\tilde{ω}}_{N - 1}) e^{i m \tilde{ξ}} h_{2} (y_{1}, \dots, y_{n}) h_{3}^{S U (N)} (ϕ_{1}, \dots, ω_{N - 1}) e^{i n ξ}

where

c \in C

⁠, the functions h₁, h₃ are finite-type functions on SU(N),

m, n \in Z

and

h_{2} (y_{1}, \dots, y_{N}) = \cos^{P_{1}} (y_{1}) \sin^{Q_{1}} (y_{1}) \dots \cos^{P_{N}} (y_{N}) \sin^{Q_{N}} (y_{N})

where

K_{j} \in N_{0}

and L_j ∈ {0, 1}. Then the integral is of the form

\begin{align} \int_{G} f^{P} (g) h (g) d g = & \sum_{\sum_{i, j} β_{i j} = P} (c \cdot (\prod_{i, j} c_{i j}^{β_{i j}}) \\ [\int_{K / M} (f_{i j}^{S U (N)} {({\tilde{ϕ}}_{1}, \dots, {\tilde{ω}}_{N - 1})}^{β_{i j}} e^{i m_{i j} β_{i j} \tilde{ξ}} h_{1}^{S U (N)} ({\tilde{ϕ}}_{1}, \dots, {\tilde{ω}}_{N - 1}) e^{i m \tilde{ξ}} d g_{K / M}] \cdot \\ [\int_{A} \prod_{i, j} {(g_{i j} (y_{1}, \dots, y_{N}))}^{β_{i j}} h_{2} (y_{1}, \dots, y_{N}) J_{S p (N)} (y_{1}, \dots, y_{N}) d y_{1} \dots d y_{N}] \cdot \\ [\int_{K} \prod_{i, j} {(h_{i j}^{S U (N)} (ϕ_{1}, \dots, ω_{N - 1}))}^{β_{i j}} e^{i n_{i j} ξ} h_{3}^{S U (N)} (ϕ_{1}, \dots, ω_{N - 1}) e^{i n ξ} d g_{K}]) . \end{align}

(2.8)

Assume to the contrary that ∫_Gf(g)^Ph(g)dg ≠ 0 for infinitely many P. That means that there exists a set

{β_{i j}}_{i, j}

such that the integrals in Eq. are non-zero. In particular, it means that

\int_{K / M} \prod_{i, j} {(f_{i j}^{S U (N)} ({\tilde{ϕ}}_{1}, \dots, {\tilde{ω}}_{N - 1}))}^{β_{i j}} e^{i m_{i j} β_{i j} \tilde{ξ}} h_{1}^{S U (N)} ({\tilde{ϕ}}_{1}, \dots, {\tilde{ω}}_{N - 1}) e^{i m \tilde{ξ}} d g_{K / M} \neq 0

(2.9)

\int_{A} \prod_{i, j} {(g_{i j} (y_{1}, \dots, y_{N}))}^{β_{i j}} h_{2} (y_{1}, \dots, y_{N}) J (y_{1}, \dots, y_{N}) d y_{1} \dots d y_{N} \neq 0

(2.10)

\int_{K} \prod_{i, j} {(h_{i j}^{S U (N)} (ϕ_{1}, \dots, ω_{N - 1}))}^{β_{i j}} e^{i n_{i j} ξ} h_{3}^{S U (N)} (ϕ_{1}, \dots, ω_{N - 1}) e^{i n ξ} d g_{K} \neq 0

(2.11)

Our goal now is to make extensive use of the Haar measure properties to conclude that

\vec{0} \in S p (f)

⁠. Note that any element g ∈ Sp(N) can be written as

g = x \exp (H) k

as in Theorem 2.3, where x ∈ K is a representative of xM, e^H ∈ A and k ∈ K, and the Haar measure can be written as

d g_{S p (N)} = d g_{K / M} J (\exp (H)) d H d g_{K} .

Since dg_K is just the Haar measure on K, for any smooth function

f : G \to C

we have

\begin{array}{l} \int_{G} f (g) d g & = \int_{G} f (x \exp (H) k) | J (\exp (H)) | d g_{K / M} d H d g_{K} \\ = \int_{G} f (x \exp (H) k h) | J (\exp (H)) | d g_{K / M} d H d g_{K} \\ = \int_{G} f (x \exp (H) h k) | J (\exp (H)) | d g_{K / M} d H d g_{K} \end{array}

for any h ∈ K. Choosing k ∈ K we see that kh, hk ∈ K, and therefore by previous argument we see that only Eq. would change. Also note that dg_K/M is just the Haar measure on K as well up to a factor. In the same way we thus see that

\begin{array}{l} \int_{G} f (g) d g & = \int_{G} f (x h \exp (H) k) | J (\exp (H)) | d g_{K / M} d H d g_{K} \\ = \int_{G} f (h x \exp (H) k) | J (\exp (H)) | d g_{K / M} d H d g_{K} \end{array}

where we used the left- and right-invariance of dg_K/M. Because of the diffeomorphism up to a measure zero in Theorem 2.3, we have that only Eq. would change. Being able to change Eqs. and independently, we can conclude that Eqs. – are all independent equations.

But note that the Eqs. and are statements of finite-type functions on U(N). Expanding the finite-type functions on SU(N), in the same line as in Ref. 14, gives

\begin{array}{l} f_{i j}^{S U (N)} (ϕ_{1}, \dots, ω_{N - 1}) ≔ & c_{i j} e^{i k_{i j}^{1} ϕ_{1}} \sin^{a_{i j}^{1}} (ψ_{1}) \cos^{b_{i j}^{1}} (ψ_{1}) \dots e^{i k_{i j}^{N - 1} ϕ_{N - 1}} \sin^{a_{i j}^{N - 1}} (ψ_{N - 1}) \cdot \\ \cos^{b_{i j}^{N - 1}} (ψ_{N - 1}) f_{i j}^{S U (N - 1)} (ϕ_{N}, \dots, ω_{N - 2}) e^{i l_{i j}^{N - 1} ω_{N - 1}} \end{array}

and

\begin{array}{l} h_{1}^{S U (N)} (ϕ_{1}, \dots, ω_{N - 1}) & = c_{1} e^{i K_{1} ϕ_{1}} \sin^{A_{1}} (ψ_{1}) \cos^{B_{1}} (ψ_{1}) \dots e^{i K_{N - 1} ϕ_{N - 1}} \sin^{A_{N - 1}} (ψ_{N - 1}) \cdot \\ \cos^{B_{N - 1}} (ψ_{N - 1}) (h_{S U (N - 1)}) (ϕ_{N}, \dots, ω_{N}) e^{i L_{N - 1} ω_{N - 1}} \end{array}

where

f_{i j}^{S U (N - 1)}

and h_SU(N−1) are defined inductively, and the indices satisfy

\begin{array}{l} k_{i j}^{1}, \dots, k_{i j}^{N - 1}, k_{i j}^{N}, \dots, k_{i j}^{N (N - 1) / 2}, l_{i j}^{1}, \dots, l_{i j}^{N - 1}, K_{1}, \dots, K_{N (N - 1) / 2}, L_{1}, \dots, L_{N - 1} \in Z, \\ a_{i j}^{1}, \dots a_{i j}^{N (N - 1) / 2}, A_{1}, \dots, A_{N (N - 1)} \in N_{0}, b_{i j}^{1}, \dots b_{i j}^{N (N - 1) / 2}, B_{1}, \dots, B_{N (N - 1) / 2} \in {0, 1} . \end{array}

Applying the same steps as in the proof of Ref. 14, Theorem 2.11, we find that

\sum_{i, j} β_{i j} k_{i j}^{m} + K_{m} = 0, \sum_{i, j} β_{i j} l_{i j}^{n} + L_{n} = 0

for all

m = 1, \dots, \frac{N (N - 1)}{2}

and n = 1, …, N. For more details we refer to Ref. 14. Using the same arguments for

{\tilde{f}}_{i j}^{S U (N)}

⁠, assigning a tilde on the indices of

{\tilde{f}}_{i j}^{S U (N)}

⁠, we find that

\sum_{i, j} β_{i j} {\tilde{k}}_{i j}^{m} + {\tilde{K}}_{m} = 0, \sum_{i, j} β_{i j} {\tilde{l}}_{i j}^{n} + {\tilde{L}}_{n} = 0

for all

m = 1, \dots, \frac{N (N - 1)}{2}

and n = 1, …, N. Finally, we note that

\int f {(x e^{H} k)}^{P} | J (\exp (H)) | d k_{M} d H d k = \int f (x e^{H} k \exp (\begin{matrix} t λ_{0} & 0 \\ 0 & 0 \end{matrix})) | J (\exp (H)) | d k_{M} d H d k,

which must be equivalent to the map

\tilde{ξ} \mapsto \tilde{ξ} + t

⁠. This mapping can only be invariant for all

t \in R

if

\sum_{i j} n_{i j} β_{i j} + n = 0 .

In the same way we get

\int f {(x e^{H} k)}^{P} | J (\exp (H)) | d k_{M} d H d k = \int f (x \exp (\begin{matrix} t λ_{0} & 0 \\ 0 & 0 \end{matrix}) e^{H} k) | J (\exp (H)) | d k_{M} d H d k,

which gives in the same way

\sum_{i, j} m_{i j} β_{i j} + m = 0 .

In other words, adding it all together, we get that

\sum_{i j} (\sum_{m} k_{i j}^{m} + {\tilde{k}}_{i j}^{m} + n_{i j} + m_{i j}) β_{i j} = - (\sum_{m} (K_{m} + {\tilde{K}}_{m}) + n + m) .

Multiplying both sides with

\frac{1}{P}

and letting P go to zero, gives a sequence that converges to 0. Since the convex hull is a closed set, 0 ∈ Conv(Sp(f)). But this is a contradiction with Conjecture 2.13. So ∫_Sp(N)f(g)^Ph(g)dg ≠ 0 for only finitely many P, so ∫_Sp(N)f(g)^Ph(g)dg = 0 for P large enough.□

III. THE CASE OF G₂

A. The Euler angle decomposition of G₂

Next, we shift our focus to G₂. The structure of this section will be the same as before: we first find an Euler angles decomposition and then translate the Mathieu conjecture to a conjecture on ${[0, 1]}^{m} \times {(S^{1})}^{n}$ for some $n, m \in N$ ⁠. However, G₂ is not by definition defined as a closed subgroup of $GL (n, C)$ for some n. Although that is not a necessity to apply Theorem 2.3, we will do so to make much of the calculations actually computable. For that, we make extensive use of Ref. 3 which gives an explicit embedding of G₂ into $GL (7, R)$ and a way to do the Euler angles decomposition.

The group G₂ can be seen as the set of automorphisms on the octonions.¹ Let $O$ be the set of octonions. We can see $O$ as a 8-dimensional vectorspace over $R$ ⁠, the spanning vectors being e₀, …, e₇ where e₀ corresponds to the real unit, and e_j to imaginary units for all j = 1, …, 7. There exists a natural multiplication on $O$ which we will not need for our purpose. For more information on the octonions we refer to Ref. 1. However, it is known¹ that G₂ can be seen as the automorphism group of $O$ ⁠. Since A is invertible and is linear, it must leave e₀ fixed and can only permute e₁, …, e₇. Therefore one can consider G₂ as a closed subset of $GL (7, R)$ ⁠.

Since the Lie algebra of automorphisms on an algebra A is the set of derivations on A, the Lie algebra $g_{2} \subset g l (7, R)$ should therefore be the set of derivations on $O$ ⁠. One can compute these, and the generators are given in Appendix B.

We note that G₂ is a simply connected compact simple Lie group, hence we can apply Theorem 2.3. In the same way as in Sp(N), we first find a Cartan involution. We choose the analytic involutive automorphism given by

θ = Ad (\begin{matrix} 1_{3} \\ - 1_{4} \end{matrix}) .

It is clear that

θ \in I n t (g_{2})

and one can easily see that only matrices of the form

(\begin{matrix} A & 0 \\ 0 & B \end{matrix})

⁠, where

A \in g l (3, R)

and

B \in g l (4, R)

⁠, are invariant under θ. The Lie algebra

k

is then given by

k ≔ {s p a n}_{R} (λ_{1}, λ_{2}, λ_{3}, λ_{8}, λ_{9}, λ_{10}) = k_{1} \oplus k_{2}

where

k_{1} ≔ {s p a n}_{R} (λ_{1}, λ_{2}, λ_{3})

and

k_{2} ≔ {s p a n}_{R} (λ_{8}, λ_{9}, λ_{10})

⁠, both isomorphic to the simple Lie algebra

s u (2)

and

[k_{1}, k_{2}] = 0

⁠. Note that the matrices

{λ_{j}}_{j}

are defined in Appendix B. This defines our K to be

K ≔ ⟨ e^{k} ⟩ .

Since

[k_{1}, k_{2}] = 0

one would expect K to be isomorphic to SU(2) × SU(2). Let us define the latter set by

\hat{K} ≔ 〈 e^{k_{1}} 〉 \times 〈 e^{k_{2}} 〉 ≃ S U (2) \times S U (2) .

However, we note that

K \neq \hat{K}

since the group homomorphism

f : \hat{K} \to GL (7, R), f (x, y) = x y

has ker f = {(1₇, 1₇), (diag(1₃, −1₄), diag(1₃, −1₄))}. So we see that

K ≃ (S U (2) \times S U (2)) / Z_{2} ≃ S O (4)

where

Z_{2} ≃ \ker f

and

\hat{K}

is the double cover of K.

So to apply Theorem 2.3, let us first describe the Euler angle decomposition of K. The Euler angle decomposition on $\hat{K}$ can be described as follows.

Lemma 3.1.

Let

\hat{K}

be as before, then up to a measure zero set there is a diffeomorphism such that

\hat{K} ≃ K_{1} A_{1} K_{1}^{'} \times K_{2} A_{2} K_{2}^{'}

as manifolds. Here

\begin{array}{l} K_{1} & ≔ {e^{x λ_{3}} | x \in [0, π]}, K_{2} ≔ {e^{x λ_{8}} | x \in [0, π]} \\ A_{1} & ≔ \{e^{x λ_{2}} |x \in [0, \frac{π}{2}]\}, A_{2} ≔ \{e^{x λ_{9}} |x \in [0, \frac{π}{2}]\} \\ K_{1}^{'} & ≔ {e^{x λ_{3}} | x \in [0, 2 π]}, K_{2}^{'} ≔ {e^{x λ_{8}} | x \in [0, 2 π]} . \end{array}

Proof.

The fact that

C

can be seen as

R^{2}

⁠, where multiplication with i can seen as applying the matrix

(\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix})

we see that SU(2) can be readily been embedded into SO(4), and we know that SU(2) can be described by the Euler angle decomposition as in, for example Refs. 4, 10, and 14, by

S U (2) ≃ K_{1} A K_{2}

where

K_{1} = \{\exp (\begin{matrix} i x & 0 \\ 0 & - i x \end{matrix})| x \in [0, π)\}, A = \{\exp (\begin{matrix} 0 & x \\ - x & 0 \end{matrix})| x \in [0, π)\}

and

K_{2} = \{\exp (\begin{matrix} i x & 0 \\ 0 & - i x \end{matrix})| x \in [0, 2 π)\} .

Applying this to every component gives the result.□

This way, we see that if

k \in \hat{K}

then

k = (e^{ϕ_{1} λ_{3}} e^{ψ_{1} λ_{2}} e^{ω_{1} λ_{3}}, e^{ϕ_{2} λ_{8}} e^{ψ_{2} λ_{9}} e^{ω_{2} λ_{8}})

where ϕ₁, ϕ₂ ∈ [0, π], ψ₁, ψ₂ ∈ [0, π/2] and ω₁, ω₂ ∈ [0, 2π]. Note that

e^{π λ_{3}} = diag (1_{3}, - 1_{4})

⁠, and since

e^{k_{1}} e^{k_{2}} = e^{k_{2}} e^{k_{1}}

⁠, we can pull the mod

Z_{2}

operation into the first component, i.e., we get a unique description of g ∈ K up to a measure zero set if we restrict the range of ω₁ to ω₁ ∈ [0, π]. So we see that the following lemma holds.

Lemma 3.2

(Euler angle decomposition of K). Let K be as before. Define the mapping

F_{K} : [0, π] \times [0, \frac{π}{2}] \times {[0, π]}^{2} \times [0, \frac{π}{2}] \times [0, 2 π] \to K

by

F_{K} (ϕ_{1}, ψ_{1}, ω_{1}, ϕ_{2}, ψ_{2}, ω_{2}) ≔ e^{ϕ_{1} λ_{3}} e^{ψ_{1} λ_{2}} e^{ω_{1} λ_{3}} e^{ϕ_{2} λ_{8}} e^{ψ_{2} λ_{9}} e^{ω_{2} λ_{8}} .

Then this mapping is a smooth surjective map, and is a diffeomorphism up to a measure zero set.

With the Euler angle decomposition of K found, we note that the pair (G₂, K) is the Riemannian symmetric pair associated with

(g_{2}, θ)

⁠. To get the rest of the data we need for Theorem 2.3, we need to find a maximal abelian subalgebra

a \subseteq p

⁠, where

p ≔ {s p a n}_{R} (λ_{4}, λ_{5}, λ_{6}, λ_{7}, λ_{11}, λ_{12}, λ_{13}, λ_{14}) .

We choose the maximal abelian subalgebra to be

a ≔ {s p a n}_{R} (λ_{5}, λ_{11}) .

To apply Theorem 2.3, we need to calculate M. By definition,

x \in M = Z_{K} (a)

if and only if x ∈ K and

Ad (x) H = H

for all

H \in a

⁠. A direct computation shows that, if we define

σ ≔ (\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 \end{matrix}), η ≔ (\begin{matrix} - 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 \end{matrix}),

then M = {1, σ, η, ση}. With that, we get the following lemma.

Lemma 3.3

(Euler Angle decomposition of G₂). Consider the compact simple Lie group G₂. Let

K = ⟨ e^{k} ⟩

be as before. Then

G_{2} ≃ (K / M) A K

diffeomorphic up to a measure zero set, where

A = \exp A

and M = {1, σ, η, ση} as before. In addition, define the mapping

F_{G_{2}} : [0, π] \times [0, \frac{π}{4}] \times [0, \frac{π}{2}] \times [0, π] \times [0, \frac{π}{2}] \times [0, 2 π] \times A \times [0, π] \times [0, \frac{π}{2}] \times {[0, π]}^{2} \times [0, \frac{π}{2}] \times [0, 2 π] \to G_{2}

by

\begin{array}{l} F_{G_{2}} & ({\tilde{ϕ}}_{1}, {\tilde{ψ}}_{1}, {\tilde{ω}}_{1}, {\tilde{ϕ}}_{2}, {\tilde{ψ}}_{2}, {\tilde{ω}}_{2}, y_{1}, y_{2}, ϕ_{1}, ψ_{1}, ω_{1}, ϕ_{2}, ψ_{2}, ω_{2}) \\ = e^{{\tilde{ϕ}}_{1} λ_{3}} e^{{\tilde{ψ}}_{1} λ_{2}} e^{{\tilde{ω}}_{1} λ_{3}} e^{{\tilde{ϕ}}_{2} λ_{8}} e^{{\tilde{ψ}}_{2} λ_{9}} e^{{\tilde{ω}}_{2} λ_{8}} e^{y_{1} λ_{5}} e^{y_{2} λ_{11}} e^{ϕ_{1} λ_{3}} e^{ψ_{1} λ_{2}} e^{ω_{1} λ_{3}} e^{ϕ_{2} λ_{8}} e^{ψ_{2} λ_{9}} e^{ω_{2} λ_{8}} \end{array}

where we repeat

{\tilde{ϕ}}_{1}, {\tilde{ϕ}}_{2}, ϕ_{1}, ω_{1}, ϕ_{2} \in [0, π]

,

{\tilde{ψ}}_{1} \in [0, \frac{π}{4}]

,

{\tilde{ω}}_{1}, {\tilde{ψ}}_{2}, ψ_{1}, ψ_{2} \in [0, \frac{π}{2}]

and

{\tilde{ω}}_{2}, ω_{2} \in [0, 2 π]

and

y_{1}, y_{2} \in A

for clarity. Here

A = \{(y_{1}, y_{2}) \in R^{2} |0 \leq y_{i} \leq \frac{π}{2} f o r i = 1, 2 a n d s u c h t h a t y_{2} \leq \frac{1}{3} y_{1}\} .

Then

F_{G_{2}}

is surjective and, up to a measure zero set, a diffeomorphism onto G₂.

Finally, the Haar measure dg decomposes as follows

d g = C J_{G_{2}} (y_{1}, y_{2}) d g_{K / M} d y_{1} d y_{2} d g_{K}

where C > 0 is some constant, dg_K is the Haar measure on K, dg_K/M the unique left-invariant measure on K/M, both given by

d g_{K} = \cos (ψ_{1}) \sin (ψ_{1}) \cos (ψ_{2}) \sin (ψ_{2}) d ϕ_{1} d ψ_{1} d ω_{1} d ϕ_{2} d ψ_{2} d ω_{2},

(3.1)

d g_{K / M} = \cos ({\tilde{ψ}}_{1}) \sin ({\tilde{ψ}}_{1}) \cos ({\tilde{ψ}}_{2}) \sin ({\tilde{ψ}}_{2}) d {\tilde{ϕ}}_{1} d {\tilde{ψ}}_{1} d {\tilde{ω}}_{1} d {\tilde{ϕ}}_{2} d {\tilde{ψ}}_{2} d {\tilde{ω}}_{2}

(3.2)

and

J_{G_{2}} (y_{1}, y_{2}) ≔ \sin (y_{1} - 3 y_{2}) \sin (y_{1} - y_{2}) \sin (y_{1} + y_{2}) \sin (y_{1} + 3 y_{2}) \sin (2 y_{1}) \sin (2 y_{2}) .

Here the integration over the y_j variables goes exactly over the area

A

⁠.

Proof.

The first part of the Theorem is proven by just applying Theorem 2.3 to G₂. To prove the claims for

F_{G_{2}}

⁠, we need to find a parametrization on K, on A and on K/M. The parametrization on K is given by Lemma 3.2, and the parametrization of A is immediate since

a

is abelian. So we only need to find a parametrization of K/M. We recall that M is discrete and finite, so we can describe K/M as a subset of K. So let g ∈ K. Using Lemma 3.2 to get the decomposition

g = e^{ϕ_{1} λ_{3}} e^{ψ_{1} λ_{2}} e^{ω_{1} λ_{3}} e^{ϕ_{2} λ_{8}} e^{ψ_{2} λ_{9}} e^{ω_{2} λ_{8}}

⁠, we see immediately that

g σ = e^{ϕ_{1} λ_{3}} e^{ψ_{1} λ_{2}} e^{(ω_{1} + \frac{π}{2}) λ_{3}} e^{ϕ_{2} λ_{8}} e^{ψ_{2} λ_{9}} e^{(ω_{2} + \frac{π}{2}) λ_{8}} .

In the same way, after some calculations, we see that

g η = e^{ϕ_{1} λ_{3}} e^{(ψ_{1} + \frac{π}{2}) λ_{2}} e^{(- \frac{π}{4} - ω_{1}) λ_{3}} e^{ϕ_{2} λ_{8}} e^{(ψ_{2} + \frac{π}{2}) λ_{9}} e^{(\frac{π}{4} - ω_{2}) λ_{8}} .

Note we need to decompose this more, for in our decomposition

ψ_{j} \in [0, \frac{π}{2}]

and gη is not described by this range. However, note that

e^{ψ_{1} λ_{2}} = e^{\frac{π}{2} λ_{3}} e^{(π - ψ_{1}) λ_{2}} e^{\frac{π}{2} λ_{3}}

and the similarly for

e^{ψ_{2} λ_{9}}

⁠. In other words, we see that

g η = e^{(ϕ_{1} + \frac{π}{2}) λ_{3}} e^{(\frac{π}{2} - ψ_{1}) λ_{2}} e^{(\frac{π}{4} - ω_{1}) λ_{3}} e^{(ϕ_{2} + \frac{π}{2}) λ_{8}} e^{(\frac{π}{2} - ψ_{2}) λ_{9}} e^{(\frac{3 π}{4} - ω_{2}) λ_{8}} .

Therefore, the equivalence relation on K to go to K/M is given by

\begin{array}{l} g = g σ & \Leftrightarrow (ϕ_{1}, ψ_{1}, ω_{1}, ϕ_{2}, ψ_{2}, ω_{2}) \sim (ϕ_{1}, ψ_{1}, ω_{1} + \frac{π}{2}, ϕ_{2}, ψ_{2}, ω_{2} + \frac{π}{2}) \\ g = g η & \Leftrightarrow (ϕ_{1}, ψ_{1}, ω_{1}, ϕ_{2}, ψ_{2}, ω_{2}) \sim (ϕ_{1} + \frac{π}{2}, \frac{π}{2} - ψ_{1}, \frac{π}{4} - ω_{1}, ϕ_{2} + \frac{π}{2}, \frac{π}{2} - ψ_{2}, \frac{π}{4} - ω_{2}) \end{array}

So if we reduce our interval of ω₁ to

ω_{1} \in [0, \frac{π}{2}]

and the interval of ψ₁ to

ψ_{1} \in [0, \frac{π}{4}]

⁠, we can describe kM ∈ K/M explicitly by

k M = e^{ϕ_{1} λ_{3}} e^{ψ_{1} λ_{2}} e^{ω_{1} λ_{3}} e^{ϕ_{2} λ_{8}} e^{ψ_{2} λ_{9}} e^{ω_{2} λ_{8}}

for some unique ϕ₁, ϕ₂, ∈ [0, π],

ψ_{1} \in [0, \frac{π}{4}]

⁠,

ω_{1}, ψ_{2} \in [0, \frac{π}{2}]

and ω₂ ∈ [0, 2π], up to a measure zero set.

Now we are in the position to apply Theorem 2.3, by applying both our Euler angle decompositions of K, as given in Lemma 3.2, and K/M which was given above. This gives that any g ∈ G₂ can be written uniquely, up to a measure zero set, as

\begin{array}{l} g & = k_{1} a k_{2} \\ = e^{{\tilde{ϕ}}_{1} λ_{3}} e^{{\tilde{ψ}}_{1} λ_{2}} e^{{\tilde{ω}}_{1} λ_{3}} e^{{\tilde{ϕ}}_{2} λ_{8}} e^{{\tilde{ψ}}_{2} λ_{9}} e^{{\tilde{ω}}_{2} λ_{8}} e^{y_{1} λ_{5}} e^{y λ_{11}} e^{ϕ_{1} λ_{3}} e^{ψ_{1} λ_{2}} e^{ω_{1} λ_{3}} e^{ϕ_{2} λ_{8}} e^{ψ_{2} λ_{9}} e^{ω_{2} λ_{8}} \end{array}

where

{\tilde{ϕ}}_{1}, {\tilde{ϕ}}_{2}, ϕ_{1}, ω_{1}, ϕ_{2} \in [0, π]

⁠,

{\tilde{ψ}}_{1} \in [0, \frac{π}{4}]

⁠,

{\tilde{ω}}_{1}, {\tilde{ψ}}_{2}, ψ_{1}, ψ_{2} \in [0, \frac{π}{2}]

and

{\tilde{ω}}_{2}, ω_{2} \in [0, 2 π]

and

y_{1}, y_{2} \in A

⁠, where

A

is as in Theorem 2.3.

Note that both K and K/M can be seen as open subsets of SU(2) × SU(2), hence the Haar measure is given by the Haar measure on SU(2) × SU(2) restricted to K and K/M times some constant, i.e., there exist C′, C″ > 0 such that the Haar measure on K and the left-invariant measure on K/M are given by

C^{'} d g_{K} = C^{″} d g_{K / M} = d g_{S U (2) \times S U (2)} = \cos (ψ_{1}) \sin (ψ_{1}) \cos (ψ_{2}) \sin (ψ_{2}) d ϕ_{1} d ϕ_{2} d ψ_{1} d ψ_{2} d ω_{1} d ω_{2}

proving the Haar measure decomposition. To get an explicit form of

A

⁠, and the corresponding Haar measure on

\exp A

⁠, we recall the root decomposition of

g_{2}

with respect to

a

⁠. The details of the root decomposition are in Appendix B. Recall

a = s p a n (λ_{5}, λ_{11})

⁠. So we define the linear functionals

α, β \in a^{*}

as

\begin{array}{l} α (λ_{5}) & ≔ 0, α (λ_{11}) ≔ 2 \\ β (λ_{5}) & ≔ 1, β (λ_{11}) ≔ - 3 . \end{array}

As will be shown in Appendix B, the functionals iα, iβ are the simple roots, and the set of positive roots are given by

Δ_{p}^{+} = {i α, i β, i (α + β), i (2 α + β), i (3 α + β), i (3 α + 2 β)} .

It is then immediate from Eq. that for H = y₁λ₅ + y₂λ₁₁ we get

\begin{align} J_{G_{2}} (\exp (H)) = \sin (2 y_{2}) \sin (y_{1} - 3 y_{2}) \sin (y_{1} + 2 y_{2} - 3 y_{2}) \sin (y_{1} + 4 y_{2} - 3 y_{2}) \sin (y_{1} + 6 y_{2} - 3 y_{2}) \cdot \sin (2 y_{1} + 6 y_{1} - 6 y_{1}) \\ = \sin (2 y_{2}) \sin (2 y_{1}) \sin (y_{1} - 3 y_{2}) \sin (y_{1} - y_{2}) \sin (y_{1} + y_{2}) \sin (y_{1} + 3 y_{2}) \sin (2 y_{2}) . \end{align}

With that, we see that

A = {(y_{1}, y_{2}) \in R^{2} | 0 \leq 2 y_{i} \leq π for i = 1, 2 and such that 0 \leq y_{1} \pm y_{2} \leq π and 0 \leq y_{1} \pm 3 y_{2} \leq π} .

Note that this means that

0 \leq y_{i} \leq \frac{π}{2}

⁠, hence y₁ ± y₂ ≤ π does not give any new information. In the same way we see that 0 ≤ y₁ − 3y₂ gives that

y_{2} \leq \frac{1}{3} y_{1}

which is also satisfied by 0 ≤ y₁ − y₂. Finally if

y_{1} \leq \frac{π}{2}

then

y_{1} + 3 y_{2} \leq \frac{π}{2} + \frac{π}{2}

⁠. In other words, we can equally describe

A

by

A = \{(y_{1}, y_{2}) \in R^{2} |0 \leq y_{i} \leq \frac{π}{2} for i = 1, 2 and such that y_{2} \leq \frac{1}{3} y_{1}\}

(3.3)

which concludes the proof.□

B. The Mathieu conjecture on G₂

In this section, we will follow the same reasoning as in Sec. III B to reduce the Mathieu conjecture to a more abelian version. We note that

G_{2} \subset S L (7, C)

⁠, hence the finite-type functions of G₂ are generated by the matrix entries and the inverse determinant. Since we have, up to a measure zero set, a description of the matrix entries using Lemma 3.3, we see that any finite type function f is given by, using the diffeomorphism

g = F_{G_{2}} ({\tilde{ϕ}}_{1}, \dots, ω_{2})

⁠,

\begin{aligned} f (g) = \sum_{j = 1}^{M} \sum_{i = 1}^{Q} & c_{i j} e^{i k_{i j}^{1} {\tilde{ϕ}}_{1}} \sin^{l_{i j}^{1}} ({\tilde{ψ}}_{1}) \cos^{m_{i j}^{1}} ({\tilde{ψ}}_{1}) e^{i k_{i j}^{2} {\tilde{ω}}_{1}} e^{i k_{i j}^{3} {\tilde{ϕ}}_{2}} \sin^{l_{i j}^{3}} ({\tilde{ψ}}_{2}) \cos^{m_{i j}^{3}} ({\tilde{ψ}}_{2}) \cdot \\ e^{i k_{i j}^{4} {\tilde{ω}}_{2}} \sin^{l_{i j}^{4}} (y_{1}) \cos^{m_{i j}^{4}} (y_{1}) \sin^{l_{i j}^{5}} (y_{2}) \cos^{m_{i j}^{5}} (y_{2}) e^{i k_{i j}^{5} ϕ_{1}} \sin^{l_{i j}^{6}} (ψ_{1}) \cos^{m_{i j}^{6}} (ψ_{1}) \cdot \\ e^{i k_{i j}^{6} ω_{1}} e^{i k_{i j}^{7} ϕ_{2}} \sin^{l_{i j}^{7}} (ψ_{2}) \cos^{m_{i j}^{7}} (ψ_{2}) e^{i k_{i j}^{8} ω_{2}} . \end{aligned}

(3.4)

Here

k_{i j}^{p} \in Z

⁠,

l_{i j}^{p} \in N_{0}

and

m_{i j}^{p} \in {0, 1}

for all i, j, p. We reduced

m_{i j}^{p}

to be in {0, 1} by using sin(x)² + cos(x)² = 1. In a similar way as in Sec. II B, we can rewrite the condition

\int_{G_{2}} f^{P} d g

in the following way.

Proposition 3.4.

Let f be a finite-type function on G₂ as in Eq. (3.4). Then for any

P \in N

we have

\int_{G_{2}} f^{P} (g) d g = C^{'} \int_{{(S^{*})}^{8}} \int_{{[0, 1]}^{5}} \int_{0}^{S (ξ_{1})} {\tilde{f}}_{G_{2}}^{P} {\tilde{J}}_{G_{2}} (x_{1}, \dots, x_{4}, ξ_{1}, ξ_{2}) d ξ_{1} d ξ_{2} d x_{1} \dots d x_{4} \frac{d z_{1}}{z_{1}} \dots \frac{d z_{8}}{z_{8}}

where

C^{'} = \frac{C}{2^{5} \sqrt{2}} > 0

is some constant,

\begin{array}{l} {\tilde{f}}_{G_{2}} (z_{1}, \dots, z_{7}, x_{1}, \dots, x_{4}, ξ_{1}, ξ_{2}) ≔ \sum_{j = 1}^{M} \sum_{i = 1}^{Q} & c_{i j} z_{1}^{\frac{k_{i j}^{1}}{2}} {(\frac{x_{1}}{\sqrt{2}})}^{l_{i j}^{1}} {(1 - \frac{x_{1}^{2}}{2})}^{\frac{m_{i j}^{1}}{2}} z_{2}^{\frac{k_{i j}^{2}}{4}} z_{3}^{\frac{k_{i j}^{3}}{2}} x_{2}^{l_{i j}^{3}} {(1 - x_{2}^{2})}^{\frac{m_{i j}^{3}}{2}} \cdot \\ z_{4}^{k_{i j}^{4}} ξ_{1}^{l_{i j}^{4}} {(1 - ξ_{1}^{2})}^{\frac{m_{i j}^{4}}{2}} ξ_{2}^{l_{i j}^{5}} {(1 - ξ_{2}^{2})}^{\frac{m_{i j}^{5}}{2}} z_{5}^{\frac{k_{i j}^{5}}{2}} x_{3}^{l_{i j}^{6}} {(1 - x_{3}^{2})}^{\frac{m_{i j}^{6}}{2}} \cdot \\ z_{6}^{\frac{k_{i j}^{6}}{2}} z_{7}^{\frac{k_{i j}^{7}}{2}} x_{4}^{l_{i j}^{7}} {(1 - x_{4}^{2})}^{\frac{m_{i j}^{7}}{2}} z_{8}^{k_{i j}^{8}}, \end{array}

and

\begin{aligned} {\tilde{J}}_{G_{2}} (x_{1}, \dots, x_{4}, ξ_{1}, ξ_{2}) ≔ & ξ_{1} ξ_{2} [ξ_{1}^{2} (16 {(1 - ξ_{2}^{2})}^{3} + 9 (1 - ξ_{2}^{2}) - 24 {(1 - ξ_{2}^{2})}^{2}) \\ - (1 - ξ_{1}^{2}) {(3 ξ_{2} - 4 ξ_{2}^{2})}^{2}] [ξ_{1}^{2} (1 - ξ_{2}^{2}) - (1 - ξ_{1}^{2}) ξ_{2}^{2}] x_{1} x_{2} x_{3} x_{4} . \end{aligned}

(3.5)

and finally

S (ξ_{1}) ≔ \frac{1}{2} (\sqrt[3]{\sqrt{ξ_{1}^{2} - 1} - ξ_{1}} + \frac{1}{\sqrt[3]{\sqrt{ξ_{1}^{2} - 1} - ξ_{1}}}) .

Proof.

The proof goes analoguous to the Proof of Proposition 2.10 or the proof of Ref. 14, Lemma 2.7. By Lemma 3.3 and Eq. we have

\begin{array}{l} \int_{G_{2}} f {(g)}^{P} d g = C \int_{F^{- 1} (G_{2})} & [\sum_{j = 1}^{M} \sum_{i = 1}^{Q} c_{i j} e^{i k_{i j}^{1} {\tilde{ϕ}}_{1}} \sin^{l_{i j}^{1}} ({\tilde{ψ}}_{1}) \cos^{m_{i j}^{1}} ({\tilde{ψ}}_{1}) e^{i k_{i j}^{2} {\tilde{ω}}_{1}} e^{i k_{i j}^{3} {\tilde{ϕ}}_{2}} \sin^{l_{i j}^{3}} ({\tilde{ψ}}_{2}) \cdot \\ \cos^{m_{i j}^{3}} ({\tilde{ψ}}_{2}) e^{i k_{i j}^{4} {\tilde{ω}}_{2}} \sin^{l_{i j}^{4}} (y_{1}) \cos^{m_{i j}^{4}} (y_{1}) \sin^{l_{i j}^{5}} (y_{2}) \cos^{m_{i j}^{5}} (y_{2}) e^{i k_{i j}^{5} ϕ_{1}} \cdot \\ {\sin^{l_{i j}^{6}} (ψ_{1}) \cos^{m_{i j}^{6}} (ψ_{1}) e^{i k_{i j}^{6} ω_{1}} e^{i k_{i j}^{7} ϕ_{2}} \sin^{l_{i j}^{7}} (ψ_{2}) \cos^{m_{i j}^{7}} (ψ_{2}) e^{i k_{i j}^{8} ω_{2}}]}^{P} \cdot \\ 16 J_{G_{2}} (y_{1}, y_{2}) \sin ({\tilde{ψ}}_{1}) \cos ({\tilde{ψ}}_{1}) \sin ({\tilde{ψ}}_{2}) \cos ({\tilde{ψ}}_{2}) \sin (ψ_{1}) \cos (ψ_{1}) \cdot \\ \sin (ψ_{2}) \cos (ψ_{2}) d {\tilde{ϕ}}_{1} \dots d ω_{2} \end{array}

We will follow the steps as in Ref. 14. In short, we use the multinomium theorem to expand the exponent of the sum of matrix components, and then we use the following equations

\int_{0}^{2 π} e^{i \frac{k}{l} x} d x = \frac{1}{i} \int_{S^{*}} z^{\frac{k}{l}} \frac{d z}{z}, \int_{0}^{π / 2} \sin^{k + p} (ϕ) \cos^{l + q} (ϕ) d ϕ = \int_{0}^{1} x^{k + p} {(1 - x^{2})}^{\frac{l + q - 1}{2}} d x

for

p, q, k, l \in N_{0}

with l > 0 to get to integrals over S* and [0, 1]. Then we use the multinomial of Newton again to get the desired result. Note that there are a few integrals of the form

\int_{0}^{π} e^{ikx} d x

⁠, one integral of the form

\int_{0}^{π / 2} e^{ikx} d x

and one of the form

\int_{0}^{π / 4} \cos^{k} (x) \sin^{l} (x) d x

⁠. The first integral can be transformed to an integral over S* by subsituting u = 2x, which gives

\int_{0}^{π} e^{ikx} d x = \frac{1}{2} \int_{0}^{2 π} e^{i \frac{k}{2} u} d u = \frac{1}{2 i} \int_{S^{*}} z^{\frac{k}{2}} \frac{d z}{z} .

In the same way, we see that a subsitution of v = 4x gives

\int_{0}^{π / 2} e^{ikx} d x = \frac{1}{4} \int_{0}^{2 π} e^{i \frac{k}{4} v} d v = \frac{1}{4 i} \int_{S^{*}} z^{\frac{k}{4}} \frac{d z}{z} .

Finally, the last integral can be transformed to

\int_{0}^{π / 4} \cos^{k} (x) \sin^{l} (x) d x = \int_{0}^{\frac{1}{\sqrt{2}}} x^{k} {(1 - x^{2})}^{\frac{l - 1}{2}} d x = \frac{1}{\sqrt{2}} \int_{0}^{1} {(\frac{x}{\sqrt{2}})}^{k} {(1 - \frac{x^{2}}{2})}^{\frac{l - 1}{2}} d x .

This allows us to transform all the integrals, which give

\begin{array}{l} \int_{G_{2}} f {(g)}^{P} d g & = \frac{C}{\sqrt{2} \cdot 2^{7}} \int_{{(S^{*})}^{8}} \int_{{[0, 1]}^{4}} \int_{A} [\sum_{j = 1}^{M} \sum_{i = 1}^{Q} c_{i j} z_{1}^{\frac{k_{i j}^{1}}{2}} {(\frac{x_{1}}{\sqrt{2}})}^{l_{i j}^{1}} {(1 - \frac{x_{1}^{2}}{2})}^{\frac{m_{i j}^{1}}{2}} z_{2}^{\frac{k_{i j}^{2}}{4}} z_{3}^{\frac{k_{i j}^{3}}{2}} x_{2}^{l_{i j}^{3}} \cdot \\ {(1 - x_{2}^{2})}^{\frac{m_{i j}^{3}}{2}} z_{4}^{k_{i j}^{4}} \sin^{l_{i j}^{4}} (y_{1}) \cos^{m_{i j}^{4}} (y_{1}) \sin^{l_{i j}^{5}} (y_{2}) \cos^{m_{i j}^{5}} (y_{2}) z_{5}^{\frac{k_{i j}^{5}}{2}} x_{3}^{l_{i j}^{6}} {(1 - x_{3}^{2})}^{\frac{m_{i j}^{6}}{2}} \cdot \\ {z_{6}^{\frac{k_{i j}^{6}}{2}} z_{7}^{\frac{k_{i j}^{7}}{2}} x_{4}^{l_{i j}^{7}} {(1 - x_{4}^{2})}^{\frac{m_{i j}^{7}}{2}} z_{8}^{k_{i j}^{8}}]}^{P} J_{G_{2}} (y_{1}, y_{2}) x_{1} x_{2} x_{3} x_{4} d y_{1} d y_{2} d x_{1} \dots d x_{4} \frac{d z_{1}}{z} \dots \frac{d z_{8}}{z_{8}} . \end{array}

Finally we would like to change the integrals going over y₁ and y₂. Putting in the area we found for

A

⁠, we get that

\begin{array}{l} \int_{G_{2}} f {(g)}^{P} d g & = \frac{C}{\sqrt{2} \cdot 2^{7}} \int_{{(S^{*})}^{8}} \int_{{[0, 1]}^{4}} \int_{0}^{π / 2} \int_{0}^{y_{1} / 3} [\sum_{j = 1}^{M} \sum_{i = 1}^{Q} c_{i j} z_{1}^{\frac{k_{i j}^{1}}{2}} {(\frac{x_{1}}{\sqrt{2}})}^{l_{i j}^{1}} {(1 - \frac{x_{1}^{2}}{2})}^{\frac{m_{i j}^{1}}{2}} z_{2}^{\frac{k_{i j}^{2}}{4}} z_{3}^{\frac{k_{i j}^{3}}{2}} \cdot \\ x_{2}^{l_{i j}^{3}} {(1 - x_{2}^{2})}^{\frac{m_{i j}^{3}}{2}} z_{4}^{k_{i j}^{4}} \sin^{l_{i j}^{4}} (y_{1}) \cos^{m_{i j}^{4}} (y_{1}) \sin^{l_{i j}^{5}} (y_{2}) \cos^{m_{i j}^{5}} (y_{2}) z_{5}^{\frac{k_{i j}^{5}}{2}} x_{3}^{l_{i j}^{6}} {(1 - x_{3}^{2})}^{\frac{m_{i j}^{6}}{2}} \cdot \\ {z_{6}^{\frac{k_{i j}^{6}}{2}} z_{7}^{\frac{k_{i j}^{7}}{2}} x_{4}^{l_{i j}^{7}} {(1 - x_{4}^{2})}^{\frac{m_{i j}^{7}}{2}} z_{8}^{k_{i j}^{8}}]}^{P} J_{G_{2}} (y_{1}, y_{2}) x_{1} x_{2} x_{3} x_{4} d y_{2} d y_{1} d x_{1} \dots d x_{4} \frac{d z_{1}}{z} \dots \frac{d z_{8}}{z_{8}} \\ = \frac{C}{\sqrt{2} \cdot 2^{7}} \int_{{(S^{*})}^{8}} \int_{{[0, 1]}^{4}} \int_{0}^{π / 2} \int_{0}^{\sin (y_{1} / 3)} [\sum_{j = 1}^{M} \sum_{i = 1}^{Q} c_{i j} z_{1}^{\frac{k_{i j}^{1}}{2}} {(\frac{x_{1}}{\sqrt{2}})}^{l_{i j}^{1}} {(1 - \frac{x_{1}^{2}}{2})}^{\frac{m_{i j}^{1}}{2}} z_{2}^{\frac{k_{i j}^{2}}{4}} z_{3}^{\frac{k_{i j}^{3}}{2}} \cdot \\ x_{2}^{l_{i j}^{3}} {(1 - x_{2}^{2})}^{\frac{m_{i j}^{3}}{2}} z_{4}^{k_{i j}^{4}} \sin^{l_{i j}^{4}} (y_{1}) \cos^{m_{i j}^{4}} (y_{1}) ξ_{2}^{l_{i j}^{5}} {(1 - ξ_{2}^{2})}^{\frac{m_{i j}^{5}}{2}} z_{5}^{\frac{k_{i j}^{5}}{2}} x_{3}^{l_{i j}^{6}} {(1 - x_{3}^{2})}^{\frac{m_{i j}^{6}}{2}} z_{6}^{\frac{k_{i j}^{6}}{2}} z_{7}^{\frac{k_{i j}^{7}}{2}} \cdot \\ {x_{4}^{l_{i j}^{7}} {(1 - x_{4}^{2})}^{\frac{m_{i j}^{7}}{2}} z_{8}^{k_{i j}^{8}}]}^{P} f (y_{1}, ξ_{2}) x_{1} x_{2} x_{3} x_{4} d ξ_{2} d y_{1} d x_{1} \dots d x_{4} \frac{d z_{1}}{z} \dots \frac{d z_{8}}{z_{8}} . \end{array}

where

\begin{array}{l} f (y_{1}, ξ_{2}) & = 2 ξ_{2} \sin (2 y_{1}) [\sin^{2} (y_{1}) (16 {(1 - ξ_{2}^{2})}^{3} + 9 (1 - ξ_{2}^{2}) - 24 {(1 - ξ_{2}^{2})}^{2}) - \cos^{2} (y_{1}) {(3 ξ_{2} - 4 ξ_{2}^{2})}^{2}] \cdot \\ [\sin^{2} (y_{1}) (1 - ξ_{2}^{2}) - ξ_{2}^{2} \cos^{2} (y_{1})] \end{array}

Note now that doing the subsitution ξ₁ = sin(y₁) is possible, with only the boundary of the ξ₂ integral is not easily transformable. However, it follows that

\sin (y_{1} / 3) = \frac{1}{2} (\sqrt[3]{\sqrt{\sin^{2} (y_{1}) - 1} - \sin (y_{1})} + \frac{1}{\sqrt[3]{\sqrt{\sin^{2} (y_{1}) - 1} - \sin (y_{1})}}) .

Let us call the right-handed side S(sin(y₁)), then we see that sin(y₁/3) = S(sin(y₁)) which is some function depending on sin(x). Therefore we can apply the subsitution, and we this get the final form

\begin{array}{l} \int_{G_{2}} f {(g)}^{P} d g & = \frac{C}{\sqrt{2} \cdot 2^{7}} \int_{{(S^{*})}^{8}} \int_{{[0, 1]}^{5}} \int_{0}^{S (ξ_{1})} [\sum_{j = 1}^{M} \sum_{i = 1}^{Q} c_{i j} z_{1}^{\frac{k_{i j}^{1}}{2}} {(\frac{x_{1}}{\sqrt{2}})}^{l_{i j}^{1}} {(1 - \frac{x_{1}^{2}}{2})}^{\frac{m_{i j}^{1}}{2}} z_{2}^{\frac{k_{i j}^{2}}{4}} z_{3}^{\frac{k_{i j}^{3}}{2}} x_{2}^{l_{i j}^{3}} \cdot \\ {(1 - x_{2}^{2})}^{\frac{m_{i j}^{3}}{2}} z_{4}^{k_{i j}^{4}} ξ_{1}^{l_{i j}^{4}} {(1 - ξ_{1}^{2})}^{\frac{m_{i j}^{4}}{2}} ξ_{2}^{l_{i j}^{5}} {(1 - ξ_{2}^{2})}^{\frac{m_{i j}^{5}}{2}} z_{5}^{\frac{k_{i j}^{5}}{2}} x_{3}^{l_{i j}^{6}} {(1 - x_{3}^{2})}^{\frac{m_{i j}^{6}}{2}} z_{6}^{\frac{k_{i j}^{6}}{2}} z_{7}^{\frac{k_{i j}^{7}}{2}} x_{4}^{l_{i j}^{7}} \cdot \\ {{(1 - x_{4}^{2})}^{\frac{m_{i j}^{7}}{2}} z_{8}^{k_{i j}^{8}}]}^{P} {\tilde{J}}_{G_{2}} (ξ_{1}, ξ_{2}) x_{1} x_{2} x_{3} x_{4} d ξ_{2} d ξ_{1} d x_{1} \dots d x_{4} \frac{d z_{1}}{z} \dots \frac{d z_{8}}{z_{8}} \end{array}

where

\begin{array}{l} {\tilde{J}}_{G_{2}} (ξ_{1}, ξ_{2}) & = 4 ξ_{1} ξ_{2} [ξ_{1}^{2} (16 {(1 - ξ_{2}^{2})}^{3} + 9 (1 - ξ_{2}^{2}) - 24 {(1 - ξ_{2}^{2})}^{2}) - (1 - ξ_{1}^{2}) {(3 ξ_{2} - 4 ξ_{2}^{2})}^{2}] \cdot \\ [ξ_{1}^{2} (1 - ξ_{2}^{2}) - (1 - ξ_{1}^{2}) ξ_{2}^{2}] \end{array}

which proves the lemma after noting

\frac{4 C}{\sqrt{2} \cdot 2^{7}} = \frac{C}{\sqrt{2} \cdot 2^{5}} = C^{'}

⁠.□

Remark 3.5.

One might wonder why we chose to write the dependencies of ${\tilde{ϕ}}_{1}, \dots ω_{2}$ in f in the way we did, i.e., why we chose a sum over $e^{i k {\tilde{ϕ}}_{1}}$ instead of $\sin^{l} ({\tilde{ϕ}}_{1}) \cos^{m} ({\tilde{ϕ}}_{1})$ when the integrals that come up in Proposition 3.4 are completely different. For in the end, the choice of $e^{i k {\tilde{ϕ}}_{1}}$ over $\sin^{l} ({\tilde{ϕ}}_{1}) \cos^{m} ({\tilde{ϕ}}_{1})$ should not change the finite-type function. The idea is to write those parameters as e^ikx only if the Jacobian J is independent of x, and keep the rest as sin^l(x)cos^m(x). This way the subsitutions are easily done.

In the same spirit as in Sec. III B, we see that ${\tilde{f}}_{G_{2}}$ is a $\frac{1}{4}$ -admissible function. To solve the Mathieu conjecture on G₂, we assume the following conjecture.

Conjecture 3.6.

Let

f : {[0, 1]}^{6} \times {(S^{*})}^{8} \to C

be a

\frac{1}{4}

-admissible function. If

\int_{{(S^{*})}^{8}} \int_{{[0, 1]}^{5}} \int_{0}^{S (x_{5})} f^{P} {\tilde{J}}_{G_{2}} = 0

for all

P \in N

, then

\vec{0}

does not lie in the convex hull of Sp(f).

Theorem 3.7.

Assume Conjecture 3.6 is true. Then the Mathieu conjecture is true for G₂.

Proof.

Let f, h be finite-type functions such that ∫_Gf^P(g)dg = 0 for all

P \in N

⁠. We are interested in ∫_Gf^P(g)h(g)dg. As discussed, f and h have the form of Eq. . By linearity we can again assume h to be of the form

\begin{array}{l} h (g) & = c e^{i K_{1} {\tilde{ϕ}}_{1}} \sin^{L_{1}} ({\tilde{ψ}}_{1}) \cos^{M_{1}} ({\tilde{ψ}}_{1}) e^{i K_{2} {\tilde{ω}}_{1}} e^{i K_{3} {\tilde{ϕ}}_{2}} \sin^{L_{3}} ({\tilde{ψ}}_{2}) \cos^{M_{3}} ({\tilde{ψ}}_{2}) \cdot \\ e^{i K_{4} {\tilde{ω}}_{2}} \sin^{L_{4}} (y_{1}) \cos^{M_{4}} (y_{1}) \sin^{L_{5}} (y_{2}) \cos^{M_{5}} (y_{2}) e^{i K_{5} ϕ_{1}} \sin^{L_{6}} (ψ_{1}) \cos^{M_{6}} (ψ_{1}) \cdot \\ e^{i K_{6} ω_{1}} e^{i K_{7} ϕ_{2}} \sin^{L_{7}} (ψ_{2}) \cos^{M_{7}} (ψ_{2}) e^{i K_{8} ω_{2}} . \end{array}

Using the multinomial theorem, we get that

\int_{G_{2}} f^{P} (g) h (g) d g

can be written as

\begin{aligned} \int_{G_{2}} f^{P} (g) h (g) d g = & \sum_{\sum β_{i j} = P} (\binom{P}{β_{1, 1}, \dots, β_{M, Q}}) \int_{G_{2}} c \cdot (\prod_{j = 1}^{M} \prod_{i = 1}^{Q} c_{i j}^{β_{i j}}) e^{i (\sum_{i, j} β_{i j} k_{i j}^{1} + K_{1}) {\tilde{ϕ}}_{1}} \cdot \\ \sin^{\sum_{i, j} β_{i j} l_{i j}^{1} + L_{1}} ({\tilde{ψ}}_{1}) \cos^{\sum_{i, j} β_{i j} m_{i j}^{1} + M_{1}} ({\tilde{ψ}}_{1}) e^{i (\sum_{i, j} β_{i j} k_{i j}^{2} + K_{2}) {\tilde{ω}}_{1}} e^{i (\sum_{i, j} k_{i j}^{3} + K_{3}) {\tilde{ϕ}}_{2}} \cdot \\ \sin^{\sum_{i, j} β_{i j} l_{i j}^{3} + L_{3}} ({\tilde{ψ}}_{2}) \cos^{\sum_{i, j} β_{i j} m_{i j} + M_{3}} ({\tilde{ψ}}_{2}) e^{i (\sum_{i, j} β_{i j} k_{i j}^{4} + K_{4}) {\tilde{ω}}_{2}} \sin^{\sum_{i, j} β_{i j} l_{i j}^{4} + L_{4}} (y_{1}) \cdot \\ \cos^{\sum_{i, j} β_{i j} m_{i j}^{4} + M_{4}} (y_{1}) \sin^{\sum_{i, j} β_{i j} l_{i j}^{5} + L_{5}} (y_{2}) \cos^{\sum_{i, j} β_{i j} m_{i j}^{5} + M_{5}} (y_{2}) e^{i (\sum_{i, j} β_{i j} k_{i j}^{5} + K_{5}) ϕ_{1}} \cdot \\ \sin^{\sum_{i, j} β_{i j} l_{i j}^{6} + L_{6}} (ψ_{1}) \cos^{\sum_{i, j} β_{i j} m_{i j}^{6} + M_{6}} (ψ_{1}) e^{i (\sum_{i, j} β_{i j} k_{i j}^{6} + K_{6}) ω_{1}} e^{i (\sum_{i, j} β_{i j} k_{i j}^{7} + K_{7}) ϕ_{2}} \\ \sin^{\sum_{i, j} β_{i j} l_{i j}^{7} + L_{7}} (ψ_{2}) \cos^{\sum_{i, j} β_{i j} m_{i j}^{7} + M_{7}} (ψ_{2}) e^{i (\sum_{i, j} β_{i j} k_{i j}^{8} + K_{8}) ω_{2}} J_{G_{2}} (y_{1}, y_{2}) d ϕ_{1} \dots d {\tilde{ω}}_{2} \end{aligned}

(3.6)

Let us assume

\int_{G_{2}} f^{P} (g) h (g) d g \neq 0

for infinitely many P. Then there exists at least one set of non-negative integers

{β_{i j}}_{i, j}

such that ∑_i,jβ_ij = P and such that the integral in Eq. is non-zero. Applying Proposition 3.4 to this integral, we thus get that

\begin{array}{l} 0 \neq \int_{{[0, 1]}^{l}} \int_{{(S^{*})}^{7}} & z_{1}^{\frac{\sum_{i, j} β_{i j} k_{i j}^{1} + K_{1}}{2}} {(\frac{x_{1}}{\sqrt{2}})}^{\sum_{i, j} β_{i j} l_{i j}^{1} + L_{1}} {(1 - \frac{x_{1}^{2}}{2})}^{\frac{\sum_{i, j} β_{i j} m_{i j}^{1} + M_{1}}{2}} z_{2}^{\frac{\sum_{i, j} β_{i j} k_{i j}^{2} + K_{2}}{4}} z_{3}^{\frac{\sum_{i, j} k_{i j}^{3} + K_{3}}{2}} \\ x_{2}^{\sum_{i, j} β_{i j} l_{i j}^{3} + L_{3}} {(1 - x_{2}^{2})}^{\frac{\sum_{i, j} β_{i j} m_{i j} + M_{3}}{2}} z_{4}^{\sum_{i, j} β_{i j} k_{i j}^{4} + K_{4}} ξ_{1}^{\sum_{i, j} β_{i j} l_{i j}^{4} + L_{4}} \cdot \\ {(1 - ξ_{1}^{2})}^{\frac{\sum_{i, j} β_{i j} m_{i j}^{4} + M_{4}}{2}} ξ_{2}^{\sum_{i, j} β_{i j} l_{i j}^{5} + L_{5}} {(1 - ξ_{2}^{2})}^{\frac{\sum_{i, j} β_{i j} m_{i j}^{5} + M_{5}}{2}} z_{5}^{\frac{\sum_{i, j} β_{i j} k_{i j}^{5} + K_{5}}{2}} \cdot \\ x_{3}^{\sum_{i, j} β_{i j} l_{i j}^{6} + L_{6}} {(1 - x_{3}^{2})}^{\frac{\sum_{i, j} β_{i j} m_{i j}^{6} + M_{6}}{2}} z_{6}^{\frac{\sum_{i, j} β_{i j} k_{i j}^{6} + K_{6}}{2}} z_{7}^{\frac{\sum_{i, j} β_{i j} k_{i j}^{7} + K_{7}}{2}} \\ x_{4}^{\sum_{i, j} β_{i j} l_{i j}^{7} + L_{7}} {(1 - x_{4}^{2})}^{\frac{\sum_{i, j} β_{i j} m_{i j}^{7} + M_{7}}{2}} z_{8}^{\sum_{i, j} β_{i j} k_{i j}^{8} + K_{8}} {\tilde{J}}_{G_{2}} (x_{1}, \dots, ξ_{2}) d ξ_{1} \dots \frac{d z_{8}}{z_{8}} . \end{array}

To conclude the proof, we will show that

\sum_{i, j} β_{i j} k_{i j}^{a} + K_{a} = 0

for all a = 1, …, 8 and then make use of Conjecture 3.6. To do this, we will make extensive use of the Haar measure properties, i.e., make use of

\int_{G_{2}} f (g) d g = \int_{G_{2}} f (g k) d g = \int_{G_{2}} f (k g) d g

for any measurable function f and any k ∈ G₂. We will denote this as that the functions g ↦ gk and the function g ↦ kg are invariant mappings. We make extensive use of our parametrization given in Lemma 3.3 to see what the invariances of these functions mean for the possible parameters of Eq. .

We note that sending g ↦ exp(tλ₃)g is an invariant mapping for all

t \in R

⁠. Going over to our parametrization this is equivalent to

F_{G_{2}} ({\tilde{ϕ}}_{1}, {\tilde{ϕ}}_{2}, \dots, ω_{2}) \mapsto F_{G_{2}} ({\tilde{ϕ}}_{1} + t, {\tilde{ϕ}}_{2}, \dots, ω_{2})

by bijectivity of

F_{G_{2}}

up to a measure zero set, which would be equivalent to stating

{\tilde{ϕ}}_{1} \mapsto {\tilde{ϕ}}_{1} + t

is an invariant mapping for the integral given in Eq. . Since t was chosen arbitrarily, this can only be true if

\sum_{i, j} β_{i j} k_{i j}^{1} + K_{1} = 0 .

The same argument holds that g ↦ g exp(tλ₈) is an invariant mapping. By the same arguments this is equivalent to ω₂ ↦ ω₂ + t is an invariant mapping which again can only be true if

\sum_{i, j} β_{i j} k_{i j}^{8} + K_{8} = 0 .

Next, we remember that

d g = J (x, y) d g_{K / M} d y_{1} d y_{2} d k

where dy₁dy₂ is the measure on

A

⁠, dg_K/M is the unique left-invariant measure on K/M and dk is the Haar measure on K. This means that if we parametrize g ∈ G₂ as g = xak where x ∈ K/M,

a = \exp (H) \in \exp (A)

and k ∈ K, then for any measurable function f we have

\begin{array}{l} \int_{G_{2}} f (g) d g = \int_{K / M} \int_{A} (\int_{K} f (x a k) d k) d y_{1} d y_{2} d g_{K / M} & = \int_{K / M} \int_{A} (\int_{K} f (x a h k) d k) d y_{1} d y_{2} d g_{K / M} \\ = \int_{K / M} \int_{A} (\int_{K} f (x a k h) d k) d y_{1} d y_{2} d g_{K / M} . \end{array}

for any h ∈ K. In other words, the mapping g = xak ↦ xahk for any h ∈ K is also an invariant mapping. This way, the mapping g = xak ↦ xa exp(tλ₃)k is equivalent to sending ϕ₁ ↦ ϕ₁ + t which is invariant. This can only be the case in Eq. if

\sum_{i, j} β_{i j} k_{i j}^{5} + K_{5} = 0 .

Next, remember that

K ≃ S U (2) \times S U (2) / Z_{2}

⁠. In other words, it is the product of two commuting groups. So

e^{t λ_{8}} k = e^{ϕ_{1} λ_{3}} e^{ψ_{1} λ_{2}} e^{ω_{1} λ_{3}} e^{t λ_{8}} e^{ϕ_{2} λ_{8}} e^{ψ_{2} λ_{9}} e^{ω_{2} λ_{8}} = e^{ϕ_{1} λ_{3}} e^{ψ_{1} λ_{2}} e^{ω_{1} λ_{3}} e^{(ϕ_{2} + t) λ_{8}} e^{ψ_{2} λ_{9}} e^{ω_{2} λ_{8}}

if

k = e^{ϕ_{1} λ_{3}} e^{ψ_{1} λ_{2}} e^{ω_{1} λ_{3}} e^{ϕ_{2} λ_{8}} e^{ψ_{2} λ_{9}} e^{ω_{2} λ_{8}}

⁠. Thus by previous arguments we see that the mapping g = xak ↦ xa exp(tλ₈)k is equivalent to the map ϕ₂ ↦ ϕ₂ + t being invariant, which can only be the case if

\sum_{i, j} β_{i j} k_{i j}^{7} + K_{7} = 0 .

Finally, we note that the measure dg_K/M = dk as well, hence the same arguments can be reused to argue that the mapping g = xak ↦ xhak for any h ∈ K is also an invariant mapping. Applying that to g = xak ↦ x exp(tλ₃)ak is then equivalent to sending

{\tilde{ω}}_{1} \mapsto {\tilde{ω}}_{1} + t

which can only be the case if

\sum_{i, j} β_{i j} k_{i j}^{2} + K_{2} = 0 .

In the same way, the mapping g = xak ↦ x exp(tλ₈)ak is equivalent to sending

{\tilde{ω}}_{2} \mapsto {\tilde{ω}}_{2} + t

which should be invariant, meaning that

\sum_{i, j} β_{i j} k_{i j}^{4} + K_{4} = 0,

and the map g ↦ g exp(tλ₃) is invariant which is equivalent to sending

{\tilde{ω}}_{1} \mapsto {\tilde{ω}}_{1} + t

which can only be true if

\sum_{i, j} β_{i j} k_{i j}^{6} + K_{6} = 0 .

Finally, using the invariance of the mapping g ↦ exp(tλ₈)g being equivalent to

{\tilde{ϕ}}_{2} \mapsto {\tilde{ϕ}}_{2} + t

being invariant, we can conclude that

\sum_{i, j} β_{i j} k_{i j}^{3} + K_{3} = 0 .

In other words, we have that

\sum_{i, j} β_{i j} k_{i j}^{a} + K_{a} = 0

for all a = 1, …, 8. This means that

- (K_{1}, \dots, K_{8}) = \sum_{i, j} β_{i j} (k_{i j}^{1}, \dots, k_{i j}^{8}) .

Diving both terms by

P \in N

gives

\frac{1}{P} (K_{1}, \dots, K_{8}) = \frac{1}{P} \sum_{i, j} β_{i j} (k_{i j}^{1}, \dots, k_{i j}^{8}) .

Taking the limit P → ∞ gives that the left hand side goes to 0. Hence there exists a limiting sequence in Sp(f) that converges to 0. Since the convex hull is a closed set, it means 0 ∈ Conv(Sp(f)) which is a contradiction with out assumption. So it cannot be that

\int_{G_{2}} f^{P} h \neq 0

for infinitely many P, hence proving the theorem.□

ACKNOWLEDGMENTS

The author would like to thank Michael Müger for the continued help with the project, and the valuable discussions. He also wishes to thank Erik Koelink for the suggestion of looking at these two specific groups, and the valuable support.

AUTHOR DECLARATIONS

Conflict of Interest

The author has no conflicts to disclose.

Author Contributions

Kevin Zwart: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

APPENDIX A: PROOF OF THEOREM 2.3

In this section, we prove Theorem 2.3. For completeness we restate the theorem. The proof is based on Refs. 7 and 8.

Theorem 2.3

(The Euler Angles Theorem). Let G be a simply connected compact Lie group with finite center, and let

g

be its Lie algebra. Let

θ : g \to g

be an involutive automorphism. Let

k, p

be the +1 and −1 eigenspace of θ, respectively, in such a way that

g = k \oplus p

. Fix a maximal abelian subalgebra

a \subset p

and fix a set of (positive) roots

Δ_{p}

with respect to

a

. Then the mapping

(K / M) \times \exp (A) \times K \to G, (k M, \exp (H), l) \mapsto \exp ({Ad}_{G} (k) H) l

(A1)

is surjective and a diffeomorphism up to a measure zero set if we replace

A

by

int (A)

⁠. Here K ⊆ G is the connected Lie group with Lie algebra

k

, and

M = Z_{K} (a)

. Here

A

is the closure of the connected component of the following set

a_{+} - {H \in a | α (H) \in π i Z f o r s o m e α \in Δ_{p}}

in such a way that

0 \in A

. Here

a_{+}

is the closed positive Weyl chamber in

a

. In addition, the Haar measure decomposes in this parameterization as follows

\int_{G} h (g) d g = C \int_{K / M} \int_{A} \int_{K} h (k_{1} \exp (H) k_{2}) | J (\exp (H)) | d k_{2} d H d g_{K / M}

for any measurable function

h : G \to C

, where C > 0 is a constant (independent of h), k₂ ∈ K with corresponding Haar measure dk₂, k₁ ∈ k an arbitrary representative of k₁M ∈ K/M with corresponding unique K-invariant measure dg_K/M on K/M, and dH is the measure on

a

. Here

J : A \to C

is given by

J (\exp (H)) ≔ \prod_{α \in Δ_{p}^{+}} \sin (α (i H))

(A2)

where

Δ_{p}^{+}

is the set of positive roots on

a

⁠.

Proof.

We define the mapping

f : K / M \times \exp (A) \times K \to G, f (k M, \exp (H), l) ≔ \exp (Ad (k) H) l .

We claim that this mapping is a surjection, and up to a measure zero set a diffeomorphism.

First we note that f is well-defined. Indeed, taking any two representatives k, l ∈ kM gives k = lm with m ∈ M. Then it is clear Ad(k)H = Ad(l)Ad(m)H = Ad(l)H. In addition, we note that f(kM, exp(H), l) = k exp(H)k⁻¹l. According to Helgason (Ref. 7, Theorem VII.8.6)

G = K \exp (A) K,

and if

k \exp (H) l \in K \exp (A) K

then H is unique. This show surjectivity of f. Now assume that

u ≔ f (k_{1} M, \exp (H_{1}), l_{1}) = f (k_{2} M, \exp (H_{2}), l_{2}) .

Then H₁ = H₂ ≕ H as mentioned before, and so we are left with

\begin{array}{l} \exp (Ad (k_{1}) H) l_{1} & = \exp (Ad (k_{2}) H) l_{2} \\ k_{1} \exp (H) k_{1}^{- 1} l_{1} & = k_{2} \exp (H) k_{2}^{- 1} l_{2} . \end{array}

We remember that

θ : g \to g

is an automorphism, and G is simply-connected, hence we can lift θ to an automorphism Θ: G → G by Θ(exp(X)) = exp(θX). Then Θ(a) = a⁻¹ for all

a \in \exp p

and Θ(k) = k for all k ∈ K. Therefore we see that

\begin{array}{l} u {(Θ (u))}^{- 1} & = k_{1} \exp (H) k_{1}^{- 1} l_{1} l_{1}^{- 1} k_{1} \exp (H) k_{1}^{- 1} = k_{1} \exp (2 H) k_{1}^{- 1} \\ = k_{2} \exp (2 H) k_{2}^{- 1} . \end{array}

The last equality sign shows that

\begin{array}{l} \exp (2 H) & = k_{2}^{- 1} k_{1} \exp (2 H) {[k_{2}^{- 1} k_{1}]}^{- 1} \\ = \exp (Ad (k) 2 H) \end{array}

where we defined

k ≔ k_{2}^{- 1} k_{1}

⁠.

Let us now assume that $α (H) \notin π i Z$ for all $α \in Δ_{p}$ ⁠, i.e., $H \in i n t (A)$ ⁠. Using a similar argument as in Knapp (Ref. 8, Theorem 7.36) we get that the equality exp(Ad(k)2H) = exp(2H) requires Ad(k)H = H.

We claim that

Z_{g} (H) = m \oplus a

where

m = Z_{k} (a)

⁠. For if

X \in Z_{g} (H)

⁠, we see that

X \in g_{C}

can be written as

X = X_{0} + \sum_{α \in Δ_{p}} X_{α}

where

X_{α} \in g_{α}

and

X_{0} \in g_{0} = m_{C} \oplus a_{C}

⁠. Then we note that

ad (H) X = 0 + \sum_{α \in Δ_{p}} α (H) X_{α}

but ad(H)X = 0 by definition of X. So α(H)X_α = 0. However

H \in a_{+}

so X_α = 0. This shows that

X \in m \oplus a

and so

Z_{g} (H) = m \oplus a

⁠. In particular

Z_{p} (H) = a

⁠. In this particular case it means that

Ad (k) a = Z_{p} (Ad (k) H) = Z_{p} (H) = a .

So

Ad (k) a = a

and so

k \in N_{K} (a)

⁠.

Since $H \in a_{+}$ ⁠, using (Ref. 6, Lemma VII.2.2) there exists an s ∈ W(U, K) such that Ad(k)H = s · H, where $W (U, K) ≔ N_{K} (a) / Z_{K} (a)$ is the analytic Weyl group. Since H lies in the positive Weyl chamber, the only reflection satisfying s · H = H is s = 1. So Ad(k) = 1, which means that $k \in Z_{K} (a)$ ⁠.

Going back to the definition of k, we thus see that k₂ = k₁k with $k \in Z_{K} (a) = M$ ⁠. In other words k₂M = k₁M. It is them immediate that l₁ = l₂ and therefore we find that f is a bijection on $i n t (A)$ ⁠. Note that $A$ and $i n t (A)$ differ only by a measure zero set.

Note that f is smooth. This can be seen by first noting that the map $ψ : K \times A \to g$ given by (k, H) ↦ Ad(k)H is smooth. Hence by the universal properties of the quotient, the map $\tilde{ψ} : K / M \times A \to g$ ⁠, given by (kM, H) ↦ Ad(k)H, is also smooth. Therefore we see that the map $\exp ◦ \tilde{ψ}$ is a smooth map, and multiplication is smooth, hence f is smooth.

So

f : K / M \times \exp (i n t (A)) \times K \to G

is bijective and smooth. We only need to show that the inverse is smooth, or that the tangent map is bijective everywhere, i.e., f is regular. We will show the latter. Let π: K → K/M and ρ: G → G/K the canonical maps. We remind ourselves that the map

Φ : K / M \times A \to G / K, (k M, H) \mapsto ρ [\exp ({Ad}_{G} (k) H)]

is a regular map at every point

(k M, H) \in K / M \times i n t (A)

(Ref. 7, Proposition VII.3.2). Now note that on

i n t (A)

the exponential

\exp : i n t (A) \to \exp (i n t (A))

is a diffeomorphism. To prove this, note that since

a

is abelian, the differential of the exponent is just the identity. It is also injective, for if exp(H) = exp(H′) then exp(H − H′) = e. This means Ad(exp(H − H′)) is the identity mapping on

g

which in turn means

α (H) - α (H^{'}) \in 2 π i Z

for all

α \in Δ_{p}

⁠. If there exists an

α \in Δ_{p}

such that α(H) − α(H′) = 2πin with

n \in Z - {0}

then note that either H or H′ is not in

A

since it means |α(H)| or |α(H′)| is bigger or equal than π which is impossible by definition of

i n t (A)

⁠. If α(H) − α(H′) = 0 for all

α \in Δ_{p}

then note that

Δ_{p}

is a root system, this means that

Δ_{p}

span

a^{*}

⁠, hence we must have H − H′ = 0. So H = H′. Hence we conclude that

Φ : K / M \times \exp (i n t (A)) \to G / K, (k M, \exp (H)) \mapsto ρ [\exp ({Ad}_{G} (k) H)]

is a regular map at every point

(k M, \exp (H)) \in K / M \times \exp (i n t (A))

⁠.

Note that

Ad (k) H \in p

for all k ∈ K, and

T_{e K} (G / K) ≃ g / k ≃ p

and thus

T_{g K} (G / K) = (T_{e K} τ (g)) [T_{e K} (G / K)]

where τ(g): xK ↦ gxK is a diffeomorphism for all g ∈ G. This means that

T_{g K} (G / K) ≃ p

for all gK ∈ G/K. Since

T_{e} ρ : g \mapsto g / k ≃ p

by the same inclusion, we can conclude from this that the mapping

\tilde{Φ} : K / M \times \exp (A) \to \exp (p), (k M, \exp (H)) \mapsto \exp ({Ad}_{G} (k) H)

is a regular map. Now we note that

f : K / M \times \exp A \times K \to G, (k M, \exp (H), l) \mapsto \exp ({Ad}_{G} (k) H) l

can be written as

f (k M, \exp (H), l) = μ (\tilde{Φ} (k M, \exp (H)), l)

where μ: G × G → G, (g, h) ↦ gh is the multiplication map. In this way, we see that

T_{k M, \exp (H), l} f (X, Q, Y) = T_{k M, \exp (H)} \tilde{Φ} (X, Q) + Y

which is clearly a bijection, since

g = k \oplus p

⁠. In other words, we have that f is a smooth regular bijection, i.e., a diffeomorphism.

To get the decomposition of the Haar measure, we apply the techniques of Ref. 6, Sec. X.1.5 which gives that

\int_{G} h (g) d g = C \int_{K / M} \int_{A} \int_{K} (h ◦ f) (k_{1} M, \exp (H), k_{2}) |\prod_{α \in Δ_{p^{+}}} \sin (α (i H))| d k_{2} d H d k_{M}

where C > 0 is some constant. Here dk₂ is the Haar measure on K, dH is the Haar measure on

a

and dk_M is the unique left-invariant measure on K/M. We can rewrite this as

\begin{array}{l} \int_{G} h (g) d g & = C \int_{K / M} \int_{A} \int_{K} (h ◦ f) (k_{1} M, \exp (H), k_{2}) |\prod_{α \in Δ_{p^{+}}} \sin (α (i H))| d k_{2} d H d k_{M} \\ = C \int_{K / M} \int_{A} \int_{K} h (\exp (Ad (k_{1}) H) k_{2}) |\prod_{α \in Δ_{p^{+}}} \sin (α (i H))| d k_{2} d H d k_{M} \\ = C \int_{K / M} \int_{A} \int_{K} h (k_{1} \exp (H) k_{1}^{- 1} k_{2}) |\prod_{α \in Δ_{p^{+}}} \sin (α (i H))| d k_{2} d H d k_{M} . \end{array}

Using the fact that dk₂ is a Haar measure, we can write this as

\int_{G} h (g) d g = C \int_{K / M} \int_{A} \int_{K} h (k_{1} \exp (H) k_{2}) |\prod_{α \in Δ_{p^{+}}} \sin (α (i H))| d k_{2} d H d k_{M} .

which proves the theorem.□

APPENDIX B: SOME RESULTS REGARDING G₂

This section is based on Ref. 3. We can see G₂ as the group of automorphisms on the octonions. This already shows that G₂ can be embedded into

GL (7, C)

⁠, but we want to be explicit enough to make sure that the construction done in the main document can be verified explicitly. The basis elements for

g_{2} ≔ T_{e} G_{2}

are given by

\begin{array}{l} λ_{1} & ≔ (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{matrix}), λ_{2} ≔ (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}), λ_{3} ≔ (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}), \\ λ_{4} & ≔ (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}), λ_{5} ≔ (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & 0 & 0 & 0 \end{matrix}), λ_{6} ≔ (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}), \\ λ_{7} & ≔ (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}), λ_{8} ≔ (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 2 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{matrix}), λ_{9} ≔ (\begin{matrix} 0 & - 2 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 \end{matrix}), \\ λ_{10} & ≔ (\begin{matrix} 0 & 0 & - 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}), λ_{11} ≔ (\begin{matrix} 0 & 0 & 0 & - 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}), λ_{12} ≔ (\begin{matrix} 0 & 0 & 0 & 0 & - 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & 0 & 0 & 0 \end{matrix}), \\ λ_{13} & ≔ (\begin{matrix} 0 & 0 & 0 & 0 & 0 & - 2 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}), λ_{14} ≔ (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & - 2 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - 1 & 0 & 0 \\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}) . \end{array}

Next, we briefly discuss the root decomposition of

g_{2}

⁠. It is clear that if we define the involution

θ = Ad (\begin{matrix} 1_{3} \\ - 1_{4} \end{matrix})

that

g_{2} = k \oplus p

where

k ≔ {s p a n}_{R} (λ_{1}, λ_{2}, λ_{3}, λ_{8}, λ_{9}, λ_{10})

and

p ≔ {s p a n}_{R} (λ_{4}, λ_{5}, λ_{6}, λ_{7}, λ_{11}, λ_{12}, λ_{13}, λ_{14}) .

In specific, note that [λ₅, λ₁₁] = 0. Choosing

a ≔ {s p a n}_{R} (λ_{5}, λ_{11})

gives a maximal abelian subalgebra in

p

⁠. One can compute all commutators, and find that in the basis {λ₁, …, λ₄, λ₆, …, λ₁₀, λ₁₂, …, λ₁₄} the operators ad(λ₅) and ad(λ₁₁) look like

\begin{array}{l} ad (λ_{5}) & = (\begin{matrix} 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 \end{matrix}), \\ ad (λ_{11}) & = (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & - 2 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & - 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & - 2 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \end{matrix}) \end{array}

Computing the eigenvalues and eigenvectors of these matrices give the roots of

g_{2}

⁠. Let us define the following linear functionals

α, β \in a^{*}

by

\begin{array}{l} α (λ_{5}) & = 0, α (λ_{11}) = 2 \\ β (λ_{5}) & = 1, β (λ_{11}) = - 3 . \end{array}

We claim iα and iβ are the simple roots for the complex Lie algebra

{(g_{2})}_{C}

⁠. In fact, the root system is given by

Δ_{p} = {i α, i β, \pm i (α + β), \pm i (2 α + β), \pm i (3 α + β), \pm i (3 α + 2 β)},

as was expected. Choosing the positive roots as all the roots having +, we find that α, β are the simple roots. The root spaces are given by

\begin{array}{l} g_{i α} & = C (λ_{3} - 2 i λ_{4} + λ_{8}), g_{- i α} = C (λ_{3} + 2 i λ_{4} + λ_{8}), \\ g_{i β} & = C (i λ_{1} + i λ_{2} - λ_{6} - λ_{7} - i λ_{9} - λ_{10} - λ_{13} + λ_{14}), \\ g_{i β} & = C (i λ_{1} - i λ_{2} - λ_{6} - λ_{7} + i λ_{9} - λ_{10} - λ_{13} + λ_{14}), \\ g_{i (α + β)} & = C (3 i λ_{1} - 3 i λ_{2} - 3 λ_{6} + 3 λ_{7} + i λ_{9} - i λ_{10} + λ_{13} + λ_{14}), \\ g_{- i (α + β)} & = C (- 3 i λ_{1} + 3 i λ_{2} - 3 λ_{6} + 3 λ_{7} - i λ_{9} + i λ_{10} + λ_{13} + λ_{14}), \\ g_{i (2 α + β)} & = C (- 3 i λ_{1} - 3 i λ_{2} + 3 λ_{6} + 3 λ_{7} - i λ_{9} - i λ_{10} - λ_{12} + λ_{14}), \\ g_{- i (2 α + β)} & = C (3 i λ_{1} + 3 i λ_{2} + 3 λ_{6} + 3 λ_{7} + i λ_{9} + i λ_{10} - λ_{12} + λ_{14}), \\ g_{i (3 α + β)} & = C (- i λ_{1} + i λ_{2} + λ_{6} - λ_{7} + i λ_{9} - i λ_{10} + λ_{13} + λ_{14}), \\ g_{- i (3 α + β)} & = C (i λ_{1} - i λ_{2} + λ_{6} - λ_{7} - i λ_{9} + i λ_{10} + λ_{13} + λ_{14}), \\ g_{i (3 α + 2 β)} & = C (- 3 i λ_{3} + i λ_{8} + 2 λ_{12}), g_{- i (3 α + 2 β)} = C (3 i λ_{3} - i λ_{8} + 2 λ_{12}) . \end{array}

REFERENCES

1.

Baez

,

J. C.

, “

The octonions

,”

Bull. Am. Math. Soc.

39

(

2

),

145

–

205

(

2002

).

https://doi.org/10.1090/s0273-0979-01-00934-x

Google Scholar

Crossref

2.

Bertini

,

S.

,

Cacciatori

,

S. L.

, and

Cerchiai

,

B. L.

, “

On the Euler angles for SU(N)

,”

J. Math. Phys.

47

(

4

),

043510

(

2006

).

https://doi.org/10.1063/1.2190898

Google Scholar

Crossref

3.

Cacciatori

,

S. L.

and

Cerchiai

,

B. L.

, “

Exceptional groups, symmetric spaces and applications

,” arXiv:0906.0121 (

2009

).

Google Scholar

4.

Dings

,

T.

and

Koelink

,

E.

, “

On the Mathieu conjecture for SU(2)

,”

Indagationes Math.

26

(

1

),

219

–

224

(

2015

).

https://doi.org/10.1016/j.indag.2014.10.001

Google Scholar

Crossref

5.

Duistermaat

,

J.

and

Van Der Kallen

,

W.

, “

Constant terms in powers of a Laurent polynomial

,”

Indagationes Math.

9

(

2

),

221

–

231

(

1998

).

https://doi.org/10.1016/s0019-3577(98)80020-7

Google Scholar

Crossref

6.

Helgason

,

S.

,

Differential Geometry and Symmetric Spaces

(

Academic Press, Inc.

,

1962

).

Google Scholar

7.

Helgason

,

S.

,

Differential Geometry, Lie Groups and Symmetric Spaces

(

Academic Press, Inc.

,

1978

).

Google Scholar

8.

Knapp

,

A. W.

,

Lie Groups Beyond an Introduction

, 2nd ed. (

Birkhäuser

,

Boston

,

2002

), ISBN: 0-8176-4259-5.

Google Scholar

9.

Mathieu

,

O.

, “

Some conjectures about invariant theory and their applications

,” in

Algèbre Non Commutative, Groupes Quantiques et Invariants

, edited by

Alex

,

J.

and

Cauchon

,

G.

(

Société Mathématique de France

,

Reims

,

1997

), Vol.

2

, pp.

263

–

279

.

Google Scholar

10.

Müger

,

M.

and

Tuset

,

L.

, “

The Mathieu conjecture for SU(2) reduced to an abelian conjecture

,”

Indagationes Math.

35

(

1

),

114

–

118

(

2024

).

https://doi.org/10.1016/j.indag.2023.10.001

Google Scholar

Crossref

11.

Procesi

,

C.

,

Lie Groups: An Approach Through Invariants and Representations

(

Springer

,

New York

,

2007

), ISBN: 978-0-387-28929-8.

Google Scholar

12.

Tilma

,

T.

,

Byrd

,

M.

, and

Sudarshan

,

E.

, “

A parametrization of bipartite systems based on SU(4) Euler angles

,”

J. Phys. A: Math. Gen.

35

(

48

),

10445

–

10465

(

2002

).

https://doi.org/10.1088/0305-4470/35/48/315

Google Scholar

Crossref

13.

Tilma

,

T.

and

Sudarshan

,

E.

, “

Generalized Euler angle parametrization for SU(N)

,”

J. Phys. A: Math. Gen.

35

(

48

),

10467

–

10501

(

2002

).

https://doi.org/10.1088/0305-4470/35/48/316

Google Scholar

Crossref

14.

Zwart

,

K.

, “

On the Mathieu conjecture for SU(N) and SO(N)

,”

J. Math. Phys.

64

(

10

),

101701

(

2023

).

https://doi.org/10.1063/5.0157709

Google Scholar

Crossref

On the Mathieu conjecture for Sp(N) and G₂

I. INTRODUCTION

II. THE CASE OF Sp(N)

A. The Euler angles decomposition of Sp(N)

B. The Mathieu conjecture on Sp(N)

III. THE CASE OF G₂

A. The Euler angle decomposition of G₂

B. The Mathieu conjecture on G₂

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: PROOF OF THEOREM 2.3

APPENDIX B: SOME RESULTS REGARDING G₂

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

On the Mathieu conjecture for Sp(N) and G2

I. INTRODUCTION

II. THE CASE OF Sp(N)

A. The Euler angles decomposition of Sp(N)

B. The Mathieu conjecture on Sp(N)

III. THE CASE OF G2

A. The Euler angle decomposition of G2

B. The Mathieu conjecture on G2

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: PROOF OF THEOREM 2.3

APPENDIX B: SOME RESULTS REGARDING G2

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only

On the Mathieu conjecture for Sp(N) and G₂

III. THE CASE OF G₂

A. The Euler angle decomposition of G₂

B. The Mathieu conjecture on G₂

APPENDIX B: SOME RESULTS REGARDING G₂