Pierre Degond
email: pierre.degond@math.univ-toulouse.fr

Radial Laplacian on rotation groups

Keywords: Representations, characters, eigenfunctions AMS Subject classification: 22E30, 22E46, 22E70, 81R05

The Laplacian on the rotation group is invariant by conjugation. Hence, it maps class functions to class functions. A maximal torus consists of block diagonal matrices whose blocks are planar rotations. Class functions are determined by their values of this maximal torus. Hence, the Laplacian induces a second order operator on the maximal torus called the radial Laplacian. In this paper, we derive the expression of the radial Laplacian. Then, we use it to find the eigenvalues of the Laplacian, using that characters are class functions whose expressions are given by the Weyl character formula. Although this material is familiar to Lie-group experts, we gather it here in a synthetic and accessible way which may be useful to non experts who need to work with these concepts.

Introduction

Rotation groups are of fundamental physical importance because they encode symmetries of the underpinning space. Recently, they have emerged as important players in the description of large systems of collectively moving self-propelled particles. Such systems are frequently encountered in life science, such as swarming bacteria, migrating cells or schooling fish. While motion of physical particles is determined by their momenta which span a flat space, collectively moving particles offer a greater variety of situations. It often happens that particle dynamics is described in terms of elements of a Lie group or of a homogeneous space (i.e. a quotient of a Lie group by one of its subgroups).

One of the earliest examples of such models is the Vicsek model [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF], where the particles are moving with constant speed. Hence, their dynamics is determined by the direction of their velocity which belongs to the homogeneous space S n-1 ∼ = SO n R/SO n-1 R. More recently, a model in which particle dynamics is determined by an element of the rotation group SO n R has been proposed [START_REF] Degond | A new flocking model through body attitude coordination[END_REF]. Several variants of this model have been explored [START_REF] Degond | Body-attitude coordination in arbitrary dimension[END_REF][START_REF] Degond | Phase transitions and macroscopic limits in a BGK model of body-attitude coordination[END_REF][START_REF] Degond | Alignment of selfpropelled rigid bodies: from particle systems to macroscopic equations[END_REF][START_REF] Degond | Quaternions in collective dynamics[END_REF]. One may find other collective dynamics models involving the rotation group in [START_REF] Fetecau | Emergent behaviors of rotation matrix flocks[END_REF], or involving other groups or homogeneous spaces, such as S 1 × R in [START_REF] Degond | A macroscopic model for a system of swarming agents using curvature control[END_REF], S n × S 1 in [START_REF] Degond | Topological states and continuum model for swarmalators without force reciprocity[END_REF] or the unitary group SU n in [START_REF] Golse | A mean-field limit of the Lohe matrix model and emergent dynamics[END_REF][START_REF] Ha | Emergent dynamics of a generalized Lohe model on some class of Lie groups[END_REF]. More generally, the interplay between collective dynamics and geometry is an active subject, see e.g. [START_REF] Ahn | Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds[END_REF][START_REF] Fetecau | Self-organization on Riemannian manifolds[END_REF].

Here, we focus on the rotation group in relation to the model exposed in [START_REF] Degond | A new flocking model through body attitude coordination[END_REF]. In this paper, a particle system for rigid bodies interacting through body attitude alignment was proposed and a kinetic model was derived when the number of particles is large. Finally, a hydrodynamic model was deduced from the kinetic model assuming that the alignment interaction is strong. At the kinetic level, the interaction operator is of Fokker-Planck type and thus, involves the Laplacian on SO n R. The derivation of the hydrodynamic model made in [START_REF] Degond | A new flocking model through body attitude coordination[END_REF] strongly used the peculiar structure of SO 3 R (Rodrigues formula or the quaternion representation like in [START_REF] Degond | Quaternions in collective dynamics[END_REF]) and thus, was restricted to the case of dimension n = 3. A further step towards an arbitrary dimension n was made in [START_REF] Degond | Body-attitude coordination in arbitrary dimension[END_REF], but in this case, the interaction operator was replaced by a relaxation operator of BGK type, thus avoiding to deal with the Laplacian on SO n R. The goal of this paper is to collect the appropriate mathematical framework in view of extending the result of [START_REF] Degond | A new flocking model through body attitude coordination[END_REF] to arbitrary dimensions. Here we focus on the rotation group. Its use in the model of [START_REF] Degond | A new flocking model through body attitude coordination[END_REF] will be the subject of future work.

Although the material presented here is well-known to Lie-group experts, we were not able to find it anywhere exposed in an accessible fashion to non experts. In particular, we could not find anywhere the expression of the radial Laplacian as exposed in Section 3 below. The present work owes a lot to the presentation made in [START_REF] Faraut | Analysis on Lie Groups, an introduction[END_REF] but [START_REF] Faraut | Analysis on Lie Groups, an introduction[END_REF] restricts itself to the unitary group SU n so we had to reprocess all the details for the rotation groups. By applying the radial Laplacian to the characters of the irreducible representations (which are known thanks to the Weyl character formula), one determines all the eigenvalues of the Laplacian, as explained in Section 4. Ref. [START_REF] Fulton | Representation theory: a first course[END_REF] gives a comprehensive introduction to Lie group and Lie algebra representations. Another point of view, focused on compact Lie group, can be found in [START_REF] Simon | Representations of finite and compact groups[END_REF]. Finally, the role of the radial Laplacian in the spectral theory of the Laplacian is treated in [START_REF] Gurarie | Symmetries and Laplacians: introduction to harmonic analysis, group representations and applications[END_REF][START_REF] Helgason | Groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions[END_REF], however in an abstract way which makes it hardly tractable in practice. Of course, Lie group theory is a very mature subject and there are numerous books and treatises on it. The above is just a small selection. Let us also mention the classical books [START_REF] Chevalley | Theory of Lie groups[END_REF][START_REF] Weyl | The classical groups: their invariants and representations[END_REF]. Other useful references are (without being exhaustive) [START_REF] Adams | Lectures on Lie groups[END_REF][START_REF] Bröcker | Representations of compact Lie groups[END_REF][START_REF] Bump | Lie groups[END_REF][START_REF] Duistermaat | Lie groups[END_REF][START_REF] Hall | Lie groups, Lie algebras, and representations[END_REF][START_REF] Humphreys | Introduction to Lie algebras and representation theory[END_REF][START_REF] Knapp | Lie groups beyond an introduction[END_REF][START_REF] Sepanski | Compact lie groups[END_REF].

The organization of this paper is as follows. Section 2 introduces the basic notations and facts about the rotations groups, the Laplacian operator acting on them and the radial Laplacian. Section 3 is devoted to the computation of the radial Laplacian while Section 4 develops the spectral theory of the Laplacian and radial Laplacian. Finally, a conclusion is drawn in Section 5. Two appendices provide proofs of auxilliary results.

Laplacian on rotation groups

The following presentation is succinct. We refer to e.g. [START_REF] Faraut | Analysis on Lie Groups, an introduction[END_REF] for detail. We denote by SO n R, the group of rotations of R n , i.e. the subset of the space M n R of n × n real matrices defined by

SO n R = {A ∈ M n R | AA T = A T A = I, det A = 1},
where A T denotes the transpose of A, I is the identity matrix of R n and det A is the determinant of A. The space M n R ≈ R 2n is a vector space which can be endowed with a Euclidean structure by means of the following inner product

A • B = 1 2 Tr{A T B}, ∀A, B ∈ M n R, (2.1)
where TrA denotes the trace of the matrix A. We note that (2.1) is the usual Frobenius inner product multiplied by the factor 1 2 . This turns out to be the convenient definition of a matrix inner product when dealing with rotations groups. The set SO n R is at the same time a group for matrix multiplication and an embedded manifold into M n R where matrix multiplication is C ∞ . It is therefore a Lie group. Its tangent space at I is the Lie algebra so n R of antisymmetric matrices endowed with the Lie bracket [P, Q] = P Q -QP , for all P, Q ∈ so n R. Its tangent space at any A ∈ SO n R is given by T A SO n R = A so n R = so n R A. From now on, we will abbreviate T A SO n R into simply T A . As a Lie group, SO n R, can be endowed with a unique (up to a multiplicative factor) Haar measure, i.e. a measure which is invariant by left multiplication. Being compact, SO n R is a modular group. Thus, the Haar measure is also right invariant, as well as invariant by group inversion and hence, by transposition. Also, the Haar measure can be normalized, and the resulting normalized Haar measure is unique.

As an embedded manifold in M n R, SO n R is also endowed with a Riemannian innerproduct and metric (for Riemannian geometry, we refer to [START_REF] Carmo | Riemannian geometry[END_REF][START_REF] Gallot | Riemannian geometry[END_REF]). For two elements AP , and AQ in T A , with P and Q in so n R, the Riemannian inner-product is given by (AP, AQ) = AP • AQ = P • Q. On oriented manifolds, the Riemannian metric gives rise to a Riemannian volume form ω and thus to a positive measure written dω by abuse of notation. As any Lie group is orientable [START_REF] Warner | Foundations of differentiable manifolds and Lie groups[END_REF]Section 4.3,Example (a)], SO n R is thus endowed with a Riemannian volume form ω. On general Lie groups, the Riemannian measure dω and the Haar measure may not be multiple of each other. However, they are indeed multiple of each other on SO n R as stated in the following Lemma, whose proof is shown in Appendix A: Lemma 2.1. Let dω and dµ be the Riemannian measure and the Haar measure on SO n R respectively. Then, there exists a constant C n > 0 such that dω = C n dµ.

On a Riemannian manifold, we can define the gradient and divergence operators. The gradient of a smooth map

SO n R → R, A → f (A) at A is an element ∇f (A) of T A defined by ∇f (A) • X = df A (X), ∀X ∈ T A , (2.2)
where df A is the differential of f at A. We recall that df A is a linear map T A → R and df A (X) is the image of the tangent vector X by this map. A smooth tangent vector field

ϕ is a smooth map SO n R → M n R, A → ϕ(A) such that ϕ(A) ∈ T A , for all A ∈ SO n R.
For instance, the gradient of a smooth map SO n R → R is a smooth vector field. The divergence ∇ • ϕ of the tangent vector field ϕ is defined by duality as follows:

SOnR ∇ • ϕ ψ dω = - SOnR ϕ • ∇ψ dω,
where ψ is an arbitrary smooth map from SO n R to R. Here, we note the use of the Riemannian measure in the integrals. However, because of Lemma 2.1, the Riemannian measure dω can be replaced by the Haar measure dµ.

The Riemannian Laplacian ∆ M is defined as the composition of the gradient and the divergence, i.e.

∆ M f = ∇ • (∇f ).
If ϕ is a tangent vector field, and f is a real function on SO n R, it is customary in differential geometry [START_REF] Warner | Foundations of differentiable manifolds and Lie groups[END_REF]Section 1.22,formula (6)] to define the map ϕ(f

), SO n R → R, by ϕ(f )(A) =: df A ϕ(A) = ∇f (A) • ϕ(A). (2.3)
In other words, ϕ(f )(A) is the directional derivative of f in the direction of ϕ at A. For i, j in {1, . . . , n}, let E ij = e i ⊗e j ∈ M n R where e i is the i-th canonical basis vector of R n and ⊗ is the tensor product. In other words, (E ij ) kℓ = δ ij δ kℓ , where δ is the Kronecker delta. Then, √ 2E ij (i,j)∈{1,...,n} 2 is an orthonormal basis of M n R for the inner product (2.1). Denote by P T A the orthogonal projection of M n R on T A . We can easily check that

P T A M = A A T M -M T A 2 , ∀M ∈ M n R, ∀A ∈ SO n R. (2.4)
Then, by [25, Theorem 3.1.4], we have

∆ M f (A) = 2 (i,j)∈{1,...,n} 2 (P T A E ij ) 2 (f )(A), ∀A ∈ SO n R, (2.5)
where the square means that the operator P T A E ij (see (2.3) for its definition) is acted twice on f . The right-hand side of (2.5) can be shown to be independent of the choice of the orthonormal basis of M n R. As is clear from above, this definition of the Laplacian does not depend on the Lie-group structure of SO n R. On Lie groups, there is an alternate concept of the Laplacian (see [START_REF] Faraut | Analysis on Lie Groups, an introduction[END_REF]Section 8.2]). We first remind that the matrix exponential maps so n R into SO n R (and actually onto, but this will not be used). For any X ∈ so n R, ρ(X) denotes the linear map:

C ∞ (SO n R) → C ∞ (SO n R), f → ρ(X)f such that ρ(X)f (A) = d dt f (Ae tX ) | t=0 .
It is readily seen that

(ρ(X)f )(A) = df A (AX) = ∇f (A) • AX = AX(f ). (2.6)
In particular, ρ is linear with respect to X. The map ρ:

so n R → L C ∞ (SO n R) , X → ρ(X)
is actually a Lie algebra representation, i.e. on top of linearity, it satisfies

ρ([X, Y ])f = [ρ(X), ρ(Y )]f, ∀X, Y ∈ so n R, ∀f ∈ C ∞ (SO n R),
where L C ∞ (SO n R) is the space of linear maps from C ∞ (SO n R) into itself. Let us now introduce F ij = E ij -E ji = e i ∧ e j , where the wedge product of two vectors a and b is their anti-symmetrized tensor product:

a ∧ b = a ⊗ b -b ⊗ a.
Then, (F ij ) 1≤i<j≤n is an orthonormal basis of so n R for the inner product (2.1). The Lie group Laplacian ∆ G is defined for a function f ∈ C ∞ (SO n R) as follows:

(∆ G f )(A) = 1≤i<j≤n d 2 dt 2 f (Ae tF ij ) t=0 = 1≤i<j≤n ρ(F ij ) 2 f (A). (2.7)
It can be shown that the orthonormal basis (F ij ) 1≤i<j≤n can be replaced by any other orthonormal basis of so n R without changing the operator ∆ G . Denote by "Ad" and "ad" the adjoint representations of SO n R and so n R, respectively. Ad maps SO n R into the group Aut(so n R) of automorphisms of so n R while ad maps linearly so n R into the space L(so n R) of endomorphisms of so n R. They are defined for

A ∈ SO n R and X, Y ∈ so n R by Ad(A)(Y ) = AY A -1 , ad(X)(Y ) = [X, Y ] = XY -Y X.
Ad is a Lie-group representation of SO n R while ad is a Lie-algebra representation of so n R, i.e. they satisfy Ad(A)Ad(B) = Ad(AB), [ad(X), ad(Y )] = ad([X, Y ]), for all A, B ∈ SO n R and all X, Y ∈ so n R. The latter formula is a re-expression of the Jacobi identity of the Lie bracket using its skew-commutativity. Now, for A ∈ SO n R, we have A -1 = A T , and thus the inner product on so n R satisfies Ad

(A)(X) • Ad(A)(Y ) = X • Y, ad(Z)(X) • Y + X • ad(Z)(Y ) = 0.
(2.8)

An inner product satisfying (2.8) is said to be invariant. Furthermore, denote by L and R the left and right regular representations of SO n R on C ∞ (SO n R) defined for any

A, B ∈ SO n R and any f ∈ C ∞ (SO n R) by (L(B)f )(A) = f (B -1 A) and (R(B)f )(A) = f (AB). We have L(B)(∆ G f ) = ∆ G (L(B)f ),
f (gAg -1 ) = f (A), ∀A, g ∈ SO n R.
A class function thus only depends on the conjugation classes. On SO n R conjugation classes are well-known thanks to classical results on matrix reduction. Let θ ∈ R. We define the planar rotation matrix R θ by

R θ = cos θ -sin θ sin θ cos θ .
For (θ 1 , . . . , θ p ) ∈ R p , A θ 1 ,...,θp denotes the following matrices defined by blocks:

-in the case n = 2p, p ≥ 2,

A θ 1 ,...,θp =       R θ 1 0 R θ 2 . . . 0 R θ p       ∈ SO 2p R, (2.9)
-in the case

n = 2p + 1, p ≥ 1, A θ 1 ,...,θp =          R θ 1 0 0 R θ 2 . . . . . . . . . 0 R θ p 0 0 . . . . . . 0 1          ∈ SO 2p+1 R.
(2.10)

We write Θ = (θ 1 , θ 2 , . . . , θ p ) ∈ R p . We define the subset T of SO 2p R or SO 2p+1 R by

T = {A Θ | Θ ∈ R p }. T is an abelian subgroup of SO 2p R or SO 2p+1 R. Consider the map ψ: Θ → A Θ . Since we have A Θ A Θ ′ = A Θ+Θ ′ , A 0 = I, ψ is a surjective group morphism R p → T whose kernel is (2πZ) p . Thus, it induces a group isomorphism ψ, T =: (R/(2πZ)) p → T.
We note that T ∼ = (S 1 ) p is a p-dimensional torus. It can be shown that T is maximal among abelian subgoups of SO n R. For that reason, it is called a maximal torus. Then, by matrix reduction theory, any element of

SO 2p R or SO 2p+1 R is conjugate to an element of T. Let f be a C ∞ class function on SO 2p R or SO 2p+1 R. Defining φ f (Θ) = f (A Θ ), we have f (A) = φ f (Θ), ∀A such that A ∼ A Θ , (2.11)
and φ f ∈ C ∞ (T ). Thus, the knowledge of a class function only requires the knowledge of a function of Θ ∈ T . This involves a dramatic reduction of complexity. However, elements of T which are conjugate to a given A are not unique. The list of such elements differs in the even and odd dimensional cases: -In the case of SO 2p R, (θ 1 , . . . , θ p ) can undergo any permutations of {1, . . . , p}. Permutation of θ i and θ j involves conjugation by the permutation matrix which exchanges e 2i-1 and e 2j-1 on the one hand and e 2i and e 2j on the other hand. Since the corresponding permutation is the product of two transpositions, it has signature +1 and so, the associated permutation matrix is an element of SO 2p R. But one can also exchange e 2i-1 and e 2i on the one hand, and e 2j-1 and e 2j on the other hand. Again the corresponding permutation matrix is an element of SO 2p R and conjugating A Θ by it amounts to making the change (. . . , θ i , . . . , θ j , . . .) → (. . . , -θ i , . . . , -θ j , . . .).

Note that we couldn't just conjugate with the permutation matrix associated with the single transposition which exchanges e 2i-1 and e 2i because the associated matrix is of determinant -1 and does not belong to SO 2p R. Thus, only changes of an even number of signs in the θ i 's is allowed. We denote by G the group of bijections T → T generated by the transformations (. . . , θ i , . . . , θ j , . . .) → (. . . , θ j , . . . , θ i , . . .) and (. . . , θ i , . . . , θ j , . . .) → (. . . , -θ i , . . . , -θ j , . . .) where (i, j) ranges over all pairs of integers in {1, . . . , p} such that i ̸ = j.

-In the case of SO 2p+1 R, (θ 1 , . . . , θ p ) can again undergo any permutations of {1, . . . , p} for the same reason as in the even-dimensional case. But one can also conjugate by a matrix which exchanges e 2i-1 and e 2i and bears a -1 in the last diagonal element (which was not possible in the odd-dimensional case). This matrix has determinant unity and is thus an element of SO 2p+1 R. Conjugating A Θ by it amounts to making the change (. . . , θ i , . . .) → (. . . , -θ i , . . .). Thus, changes of an arbitrary number of signs of the θ i 's are allowed in this case. In this case, G is generated by the transformations (. . . , θ i , . . . , θ j , . . .) → (. . . , θ j , . . . , θ i , . . .) where (i, j) ranges over all pairs of integers in {1, . . . , p} such that i ̸ = j and by (. . . , θ i , . . .) → (. . . , -θ i , . . .), where i ranges over all integers in {1, . . . , p}.

In both the even and odd dimensional cases, all conjugations mapping two elements of T are generated by the conjugations described above. To see that we introduce the set of all the elements of SO n R which conjugate an element of T into an element of T. This set is a group called the normalizer of T and is defined by

N (T) = g ∈ SO n R | gTg -1 = T .
Let us introduce the group homomorphism Φ: N (T) → Aut(T), g → Φ g (where Aut(T) denotes the group of group automorphisms of T), such that Φ g (h) = ghg -1 , ∀h ∈ T. This homomorphism has kernel equal to T (that T ⊂ ker Φ is obvious. The converse is due to the fact that T is a maximal abelian subgroup of SO n R). Thus, Φ induces a group isomorphism Φ: N (T)/T → im(Φ) ⊂ Aut(T). Now, define the isomorphism Ψ:

Aut(T) → Aut(T ), φ → ψ-1 • φ • ψ, ∀φ ∈ Aut(T)
, where ψ: T → T was defined above. Then, Ψ • Φ is an isomorphism between N (T)/T and the group of transformations of T that are induced by conjugations. It turns out that the group N (T)/T is well-known to be isomorphic to the Weyl group W [START_REF] Fulton | Representation theory: a first course[END_REF]Claim 26.15]. W is a finite subsgroup of the orthogonal group of the space R p . If R p is endowed with the canonical basis (e i ) p i=1 , then W is generated by the orthogonal symmetries in the hyperplanes {e i ± e j } ⊥ , with i, j ∈ {1, . . . , p}, i ̸ = j in the case n = 2p and by the orthogonal symmetries in the hyperplanes {e i ± e j } ⊥ and {e i } ⊥ in the case n = 2p + 1 [18, p. 271]. But it is readily checked that the group G of transformations of T defined above is also generated by the same transformations. Hence, G = W showing the claim.

It results that if f ∈ C ∞ is a class function, the function φ f ∈ C ∞ (T ) associated with it through (2.11) must be invariant by the action of W. Hence, φ f is a function defined on the quotient T ′ =: T /W. For simplicity, we will drop the prime and from now on, T will actually mean T ′ . Now, because ∆ is left and right invariant, it is invariant by conjugation (indeed, conjugation by g is the composition

L(g -1 )R(g -1 )). It results that if f is a class function, ∆f is a class function. For any φ ∈ C ∞ (T ), there exists a unique function f ∈ C ∞ (SO n R) such that φ = φ f . Indeed, f is given by f (A) = φ(Θ), for any Θ such that A ∼ A Θ . Then, we can define a linear operator L: C ∞ (T ) → C ∞ (T ) by Lφ f = φ ∆f , ∀f ∈ C ∞ (T).
The operator L is called the radial Laplacian. The goal of this paper is to find an explicit expression of L and to state some of its properties.

3 Radial Laplacian

Expression

The expression of the radial Laplacian is given by the following: Theorem 3.1. The radial Laplacian on the rotation groups is given by the following expressions: (i) Case of SO 2p R for p ≥ 2:

L = p j=1 ∂ 2 ∂θ 2 j + 1≤j<k≤p 2 cos θ k -cos θ j sin θ j ∂ ∂θ j -sin θ k ∂ ∂θ k ) ; (3.1) (ii) Case of SO 2p+1 R for p ≥ 1: L = p j=1 ∂ 2 ∂θ 2 j + sin θ j 1 -cos θ j ∂ ∂θ j + 1≤j<k≤p 2 cos θ k -cos θ j sin θ j ∂ ∂θ j -sin θ k ∂ ∂θ k ) . (3.2)
We note that in the case p = 1, the second sum simply disappears in (3.2). We leave aside the case SO 2 R which is trivial, the group being abelian.

To prove this theorem, we will first review the cases of SO 3 R and SO 4 R and then, generalize the analysis to the general cases SO 2p R and SO 2p+1 R. We will adopt the method presented in [START_REF] Faraut | Analysis on Lie Groups, an introduction[END_REF] (cf Proof of Proposition 8.3.3 and Section 10.2). This method relies on the following lemma:

Lemma 3.2. If f is a class function on SO n R, for all X ∈ so n R and all A ∈ SO n R, we have ρ Ad(A -1 )X -X 2 f (A) -ρ [Ad(A -1 )X, X] f (A) = 0. (3.3)
Proof. If f is a class function, we have f (A) = f (e tX Ae -tX ), for all A ∈ SO n R and all X ∈ so n R. This formula can be recast into f (A) = f (Ae tAd(A -1 )X e -tX ). Taking the second derivative of this formula with respect to t at t = 0 and using that

d 2 dt 2 f (Ae tX e tY ) t=0 = ρ(X + Y ) 2 f (A) + ρ [X, Y ] f (A), ∀X, Y ∈ so n R, (3.4)
we get (3.3). Eq. (3.4) is proved in [START_REF] Faraut | Analysis on Lie Groups, an introduction[END_REF]Section 8.2 (d)].

We will use the following formula, whose proof is straightforwad:

[F ij , F kℓ ] = δ jk F iℓ + δ iℓ F jk -δ ik F jℓ -δ jℓ F ik . (3.5)
We will also use the notation {A, B} for the anti-commutator of two operators A, B, i.e.

{A, B} = AB + BA.

3.2 Proof of Theorem 3.1: case of SO 3 R This is the case SO 2p+1 R with p = 1. Thus, the matrix (2.10) has a single block R θ . So, the vector Θ has a single coordinate, θ and we use the notation A θ for A Θ . Let f be a C ∞ class function on SO 3 R and φ f = φ f (θ) the corresponding function of θ. Define

G + = 1 √ 2 (F 13 + F 23 ) = 1 √ 2 (e 1 ⊗ e 3 + e 2 ⊗ e 3 -e 3 ⊗ e 1 -e 3 ⊗ e 2 ), G -= 1 √ 2 (F 13 -F 23 ) = 1 √ 2 (e 1 ⊗ e 3 -e 2 ⊗ e 3 -e 3 ⊗ e 1 + e 3 ⊗ e 2 ).
Then, (F 12 , G + , G -) is still an orthonormal basis of so 3 R. We choose this basis to express the Laplacian of a C ∞ function f on SO 3 R:

∆f (A) = ρ(F 12 ) 2 f (A) + ρ(G + ) 2 f (A) + ρ(G -) 2 f (A). (3.6)
We now use (3.3) with A = A θ and X = G + and X = G -successively. We have

Ad(A -θ )G + = cos θ G + + sin θ G -, Ad(A -θ )G -= -sin θ G + + cos θ G -.
Then, using the linearity of the map X → (ρ(X)f )(A) for any A ∈ SO 2 R, we get:

ρ Ad(A -θ )G + -G + 2 f (A θ ) = (cos θ -1) 2 ρ(G + ) 2 + sin 2 θ ρ(G -) 2 +(cos θ -1) sin θ ρ(G + ), ρ(G -) f (A θ ), ρ Ad(A -θ )G --G -2 f (A θ ) = (cos θ -1) 2 ρ(G -) 2 + sin 2 θ ρ(G + ) 2 -(cos θ -1) sin θ ρ(G + ), ρ(G -) f (A θ ).
Furthermore, using (3.5), we get

[Ad(A -θ )G + , G + ] = [Ad(A -θ )G -, G -] = sin θ [G -, G + ] = sin θ [F 13 , F 23 ] = -sin θ F 12 .
Thus, adding the equations (3.3) corresponding to G + and G -together leads to

2(1 -cos θ) ρ(G + ) 2 + ρ(G -) 2 f (A θ ) = -2 sin θ ρ(F 12 )f (A θ ). (3.7)
Thus, with (3.6), we get

(∆f )(A θ ) = ρ(F 12 ) 2 - sin θ 1 -cos θ ρ(F 12 ) f (A θ ). (3.8)
Now, we remark that e tF 12 = A -t . Thus, by the definition of ρ:

ρ(F 12 )f (A θ ) = d dt f (A θ e tF 12 ) t=0 = d dt f (A θ-t ) t=0 (3.9)
= d dt φ f (θ -t) t=0 = - d dθ φ f (θ). (3.10)
Inserting this expresion into (3.8), we get the expression of the radial Laplacian on SO 3 R:

L = d 2 dθ 2 + sin θ 1 -cos θ d dθ ,
which corresponds to (3.2) in the case p = 1.

Proof of Theorem 3.1: case of SO 4 R

This is the case SO 2p R with p = 2. Thus, the matrix (2.9) has a two blocks R θ 1 and

R θ 2 . So, Θ = (θ 1 , θ 2 ). Let f be a C ∞ class function on SO 4 R and φ f = φ f (θ 1 , θ 2 ) the corresponding function of (θ 1 , θ 2 ).
We now define

H + = 1 √ 2 (F 13 + F 24 ), H -= 1 √ 2 (F 13 -F 24 ), K + = 1 √ 2 (F 14 + F 23 ), K -= 1 √ 2 (F 14 -F 23 )
Then, (F 12 , F 34 , H + , H -, K + , K -) is still an orthonormal basis of so 4 R. We choose this basis to express the Laplacian of a C ∞ function f on SO 4 R:

∆f (A) = ρ(F 12 ) 2 + ρ(F 34 ) 2 + ρ(H + ) 2 + ρ(H -) 2 + ρ(K + ) 2 + ρ(K -) 2 f (A). (3.11)
We now use (3.3) with A = A θ 1 ,θ 2 and X = H + , H -, K + , K -successively. We have

Ad(A -θ 1 ,-θ 2 )H + = (c 1 c 2 + s 1 s 2 )H + -(c 1 s 2 -s 1 c 2 )K -, Ad(A -θ 1 ,-θ 2 )H -= (c 1 c 2 -s 1 s 2 )H --(c 1 s 2 + s 1 c 2 )K + , Ad(A -θ 1 ,-θ 2 )K + = (c 1 s 2 + s 1 c 2 )H -+ (c 1 c 2 -s 1 s 2 )K + , Ad(A -θ 1 ,-θ 2 )K -= (c 1 s 2 -s 1 c 2 )H + + (c 1 c 2 + s 1 s 2 )K -,
with c i = cos θ i , s i = sin θ i for i = 1, 2. Then, we get:

ρ Ad(A -θ 1 ,-θ 2 )H + -H + 2 f (A θ 1 ,θ 2 ) = (c 1 c 2 + s 1 s 2 -1) 2 ρ(H + ) 2 +(c 1 s 2 -s 1 c 2 ) 2 ρ(K -) 2 -(c 1 c 2 + s 1 s 2 -1) (c 1 s 2 -s 1 c 2 ) ρ(H + ), ρ(K -) f (A θ 1 ,θ 2 ), ρ Ad(A -θ 1 ,-θ 2 )H --H -2 f (A θ 1 ,θ 2 ) = (c 1 c 2 -s 1 s 2 -1) 2 ρ(H -) 2 +(c 1 s 2 + s 1 c 2 ) 2 ρ(K + ) 2 -(c 1 c 2 -s 1 s 2 -1) (c 1 s 2 + s 1 c 2 ) ρ(H -), ρ(K + ) f (A θ 1 ,θ 2 ), ρ Ad(A -θ 1 ,-θ 2 )K + -K + 2 f (A θ 1 ,θ 2 ) = (c 1 s 2 + s 1 c 2 ) 2 ρ(H -) 2 +(c 1 c 2 -s 1 s 2 -1) 2 ρ(K + ) 2 +(c 1 s 2 + s 1 c 2 ) (c 1 c 2 -s 1 s 2 -1) ρ(H -), ρ(K + ) f (A θ 1 ,θ 2 ), ρ Ad(A -θ 1 ,-θ 2 )K --K -2 f (A θ 1 ,θ 2 ) = (c 1 s 2 -s 1 c 2 ) 2 ρ(H + ) 2 +(c 1 c 2 + s 1 s 2 -1) 2 ρ(K -) 2 +(c 1 s 2 -s 1 c 2 ) (c 1 c 2 + s 1 s 2 -1) ρ(H + ), ρ(K -) f (A θ 1 ,θ 2 ).
Furthermore, using (3.5), we get

[Ad(A -θ 1 ,-θ 2 )H + , H + ] = [Ad(A -θ 1 ,-θ 2 )K -, K -] = -(c 1 s 2 -s 1 c 2 )[K -, H + ] = (c 1 s 2 -s 1 c 2 )(F 12 -F 34 ), [Ad(A -θ 1 ,-θ 2 )H -, H -] = [Ad(A -θ 1 ,-θ 2 )K + , K + ] = -(c 1 s 2 + s 1 c 2 )[K + , H -] = -(c 1 s 2 + s 1 c 2 )(F 12 + F 34 ),
Thus, adding the equations (3.3) corresponding to H + and K -(resp. H -and K + ) together leads to

ρ(H + ) 2 + ρ(K -) 2 f (A θ 1 ,θ 2 ) = sin(θ 2 -θ 1 ) 1 -cos(θ 2 -θ 1 ) ρ(F 12 ) -ρ(F 34 ) f (A θ 1 ,θ 2 ), ρ(H -) 2 + ρ(K + ) 2 f (A θ 1 ,θ 2 ) = - sin(θ 1 + θ 2 ) 1 -cos(θ 1 + θ 2 ) ρ(F 12 ) + ρ(F 34 ) f (A θ 1 ,θ 2 ),
Thus, with (3.11), we get

(∆f )(A θ 1 ,θ 2 ) = ρ(F 12 ) 2 + ρ(F 34 ) 2 + sin(θ 2 -θ 1 ) 1 -cos(θ 2 -θ 1 ) ρ(F 12 ) -ρ(F 34 ) - sin(θ 1 + θ 2 ) 1 -cos(θ 1 + θ 2 ) ρ(F 12 ) + ρ(F 34 ) f (A θ 1 ,θ 2 ). (3.12)
Now, similarly to (3.10), we have

ρ(F 12 )f (A θ 1 ,θ 2 ) = - ∂φ f ∂θ 1 (θ 1 , θ 2 ), ρ(F 34 )f (A θ 1 ,θ 2 ) = - ∂φ f ∂θ 2 (θ 1 , θ 2 ).
Inserting it into (3.12) leads to

L = ∂ 2 ∂θ 2 1 + ∂ 2 ∂θ 2 2 + sin(θ 1 -θ 2 ) 1 -cos(θ 1 -θ 2 ) ∂ ∂θ 1 - ∂ ∂θ 2 + sin(θ 1 + θ 2 ) 1 -cos(θ 1 + θ 2 ) ∂ ∂θ 1 + ∂ ∂θ 2 .
This can be recast into

L = ∂ 2 ∂θ 2 1 + ∂ 2 ∂θ 2 2 + 2 cos θ 2 -cos θ 1 sin θ 1 ∂ ∂θ 1 -sin θ 2 ∂ ∂θ 2 ,
which corresponds to (3.1) in the case p = 1.

3.4 Proof of Theorem 3.1: case of SO 2p R, p ≥ 1

For 1 ≤ j < k ≤ p, we define

H + jk = 1 √ 2 (F 2j-1 2k-1 + F 2j 2k ), H - jk = 1 √ 2 (F 2j-1 2k-1 -F 2j 2k ), (3.13)
K + jk = 1 √ 2 (F 2j-1 2k + F 2j 2k-1 ), K - jk = 1 √ 2 (F 2j-1 2k + F 2j 2k-1 ). (3.14)
Then, (F 2j-1 2j ) j=1,...,p , (H + jk , H - jk , K + jk , K - jk ) 1≤j<k≤p is still an orthonormal basis of so 2p R. We choose this basis to express the Laplacian of a C ∞ function f on SO 2p R:

∆f (A) = p j=1 ρ(F j j+1 ) 2 + 1≤j<k≤p ρ(H + jk ) 2 + ρ(H - jk ) 2 + ρ(K + jk ) 2 + ρ(K - jk ) 2 f (A).
(3.15) We note that the computations of Ad(A -Θ )H ± jk and Ad(A -Θ )K ± jk only involve the four 2 × 2 matrix subblocks corresponding to line and column indices belonging to {2j -1, 2j} × {2j -1, 2j}, {2j -1, 2j} × {2k -1, 2k}, {2k -1, 2k} × {2j -1, 2j} and {2k -1, 2k} × {2k -1, 2k}. Thus, the computation reduces to computations on 4 × 4 matrices that are identical to what was done in the case of SO 4 R. Thus, we can directly write that for any 1 ≤ j < k ≤ p, we have

ρ(H + jk ) 2 + ρ(K - jk ) 2 + ρ(H - jk ) 2 + ρ(K + jk ) 2 f (A Θ ) (3.16) = - 2 cos θ k -cos θ j sin θ j ρ(F 2j-1 2j ) -sin θ k ρ(F 2k-1 2k ) f (A Θ ). (3.17)
Now, similarly to (3.10), for j ∈ {1, . . . , p}, we have

ρ(F 2j-1 2j )f (A Θ ) = - ∂φ f ∂θ j (Θ).
Inserting these into (3.15), we get (3.1).

3.5 Proof of Theorem 3.1: case of SO 2p+1 R, p ≥ 1

For 1 ≤ j < k ≤ p, we define H ± jk and K ± jk in a similar way as in (3.13), (3.14). For j ∈ {1, . . . , p}, we define

G + j = 1 √ 2 (F 2j-1 2p+1 + F 2j 2p+1 ), G - j = 1 √ 2 (F 2j-1 2p+1 -F 2j 2p+1 ).
Then, (F 2j-1 2j , G + j , G - j ) j=1,...,p , (H + jk , H - jk , K + jk , K - jk ) 1≤j<k≤p is still an orthonormal basis of so 2p+1 R. We choose this basis to express the Laplacian of a C ∞ function f on SO 2p+1 R:

∆f (A) = p j=1 ρ(F j j+1 ) 2 + ρ(G + j ) 2 + ρ(G - j ) 2 + 1≤j<k≤p ρ(H + jk ) 2 + ρ(H - jk ) 2 + ρ(K + jk ) 2 + ρ(K - jk ) 2 f (A). (3.18)
The computations involving H ± jk and K ± jk are identical to those of the preceding section. Those involving G ± j are identical to those done in Section 3.2 for SO 3 R. Thus, we can collect (3.17) for 1 ≤ j < k ≤ p and (3.7) with G ± replaced by G ± j , θ by θ j and F 12 by F 2j-1 2j , for all j ∈ {1, . . . , p}, and get (3.2).

4 Some properties of the radial Laplacian

Alternate expressions

As a preliminary to this section, we state the Weyl integration formula [29, Theorems IX.9.4 & IX.9.5]. We introduce the following functions:

Π 2p (Θ) = 1≤j<k≤p cos θ j -cos θ k , for p ≥ 2, ( 4
f (A) dA = 1 (2π) p [0,2π] p f (A Θ ) u n (Θ) dΘ, (4.4)
with dΘ = dθ 1 . . . dθ p .

For a smooth function φ: T → R p , ∇ Θ φ denotes the vector field T → R p of coordinates ( ∂φ ∂θ i ) p i=1 . Similarly, for a smooth vector field X :

T → R p of coordinates (X i ) p i=1 , the divergence ∇ Θ • X is the scalar p i=1 ∂X i ∂θ i . The Laplacian ∆ Θ is naturally defined by ∆ Θ = ∇ Θ • ∇ Θ φ.
Finally, we introduce the constants

γ 2p = p j=1 (p -j) 2 , γ 2p+1 = p j=1 (p -j + 1 2 ) 2 . (4.5)
We note that γ 2p (for p ≥ 2) and γ 2p+1 (for p ≥ 1) are strictly positive constants. We have the following (ii) Case n = 2p + 1, p ≥ 1: we have

∇ Θ u 2p+1 (Θ) = k̸ =j 2 cos θ k -cos θ j + 1 1 -cos θ j sin θ j u 2p+1 (Θ) p j=1 . (4.7)
(iii) In both the cases n = 2p (p ≥ 2) and n = 2p + 1 (p ≥ 1), we have

∆ Θ Π n = -γ n Π n . (4.8) Proof. (i) We compute ∂ ∂θ j log u 2p = 2 ∂ ∂θ j log Π 2p = 2 ∂ ∂θ j 1≤k<ℓ≤p log(cos θ k -cos θ ℓ ) = - j<ℓ≤p 2 sin θ j cos θ j -cos θ ℓ + 1≤k<j 2 sin θ j cos θ k -cos θ j = k̸ =j 2 sin θ j cos θ k -cos θ j ,
which leads to (4.6) .

(ii) We remark that, up to an unimportant multiplicative constant, we have

u 2p+1 (Θ) = 1≤k<ℓ≤p cos θ k -cos θ ℓ 2 p k=1 (1 -cos θ k ),
and we proceed like in the previous point.

(iii) case n = 2p: this proof is adapted from that of [START_REF] Faraut | Analysis on Lie Groups, an introduction[END_REF]Lemma 12.5.2]. We have, up to an unimportant multiplicative constant,

Π 2p (Θ) =
k<ℓ (e iθ k + e -iθ k -e iθ ℓ -e -iθ ℓ ). (4.9)

We note that the right-hand side of (4.9) is the Vandermonde determinant of the unknowns (e iθ k + e -iθ k ) p k=1 . We recall that if (X 1 , . . . , X p ) are elements of C, the Vandermonde determinant is

V (X 1 , . . . , X n ) = det (X p-ℓ k ) kℓ = k<ℓ (X k -X ℓ ).
Thus, we have

Π 2p (Θ) = V e iθ 1 + e -iθ 1 , . . . , e iθp + e -iθp = det (e iθ k + e -iθ k ) p-ℓ kℓ . (4.10)
By developing the powers involved in all the terms of the determinant and using elementary manipulations on the rows and columns (see [18, Eq. (24.39)]), one realizes that

Π 2p (Θ) = det (e i(p-ℓ)θ k + e -i(p-ℓ)θ k ) kℓ = σ∈Sp (-1) σ p k=1 e i(p-σ(k))θ k + e -i(p-σ(k))θ k ,
where S p is the symmetric group of p elements and for σ ∈ S p , (-1) σ stands for its signature. Now, taking two derivatives with respect to θ j , we get

∂ 2 Π 2p ∂θ 2 j (Θ) = - σ∈Sp (-1) σ (p -σ(j)) 2 p k=1 e i(p-σ(k))θ k + e -i(p-σ(k))θ k .
Summing over j and realizing that

p j=1 (p -σ(j)) 2 = p j=1 (p -j) 2 = γ 2p ,
is independent of σ, we finally get (4.8) for n = 2p.

(iii) case n = 2p + 1: the proof is similar. We note that, up to an unimportant multiplicative constant, we have

Π 2p+1 = k<ℓ (e iθ k + e -iθ k -e iθ ℓ -e -iθ ℓ ) p k=1 (e iθ k /2 -e -iθ k /2 ) (4.11) = det (e i(p-ℓ+ 1 2 )θ k + e -i(p-ℓ+ 1 2 )θ k ) kℓ ,
where the second inequality again requires elementary manipulations of the determinant (see [START_REF] Fulton | Representation theory: a first course[END_REF]Exercise 24.27]. The remainder of the proof is similar.

We now provide alternate expressions of the radial Laplacian in the following (ii) In the case n = 2p + 1, for any φ ∈ C ∞ (T ), we have

Lφ = p j=1 ∂ 2 φ ∂θ 2 j + k̸ =j 2 cos θ k -cos θ j + 1 1 -cos θ j sin θ j ∂φ ∂θ j . ( 4

.13)

(iii) In both cases n = 2p (for p ≥ 2) and n = 2p + 1 (for p ≥ 1), we have

Lφ = 1 u n ∇ Θ • u n ∇ Θ φ , (4.14)
= 1 Π n ∆ Θ + γ n (Π n φ), (4.15)
Proof. (i) and (ii): formulas (4.12) and (4.13) are obtained from (3.1) and (3.2) respectively, by exchanging i and j inside the second terms of the sums over i < j.

(iii) Inserting (4.6) and (4.7) into (4.12) and (4.13) respectively, we can write

Lφ = ∆ Θ φ + 1 u n ∇ Θ u n • ∇ Θ φ = 1 u n ∇ Θ • (u n ∇ Θ φ),
which is (4.14). Equivalently, we can write

Lφ = 1 Π 2 n ∇ Θ • (Π 2 n ∇ Θ φ) = 1 Π n ∇ Θ Π n • ∇ Θ φ + ∇ Θ • (Π n • ∇ Θ φ) = 1 Π n ∇ Θ Π n • ∇ Θ φ + ∇ Θ • ∇ Θ (Π n φ) -φ∇ Θ Π n = 1 Π n ∆ Θ (Π n φ) -φ∆ Θ (Π n ) ,
which, with (4.8), leads to (4.15).

Spectral theory and inversion of the radial Laplacian

In this and the following sections, we denote by G either of the groups SO 2p R or SO 2p+1 R.

We state a few facts concerning the spectral decomposition of the Laplacian -∆ and the radial Laplacian -L.

We first start with the Laplacian -∆. We know [START_REF] Gallot | Riemannian geometry[END_REF] that the spectrum of -∆ on a Riemannian manifold consists of an increasing sequence of discrete eigenvalues λ 1 < λ 2 < . . . λ n . . ., tending to +∞, that the associated eigenspaces are finite dimensional and that their direct sum is dense in L 2 (G). Furthermore, λ 1 = 0 and its eigenspace is onedimensional and consists of the constant functions. However, in the case of the Laplacian on G, we can be more specific about the eigenspaces. We first need to introduce a few concepts from representation theory.

Here, we refer to [START_REF] Faraut | Analysis on Lie Groups, an introduction[END_REF]Ch. 6]. A representation (V, π) of G is a pair with V a vector space over C and π a group morphism G → Aut(V ) where Aut(V ) is the group of linear automorphisms of V . If W is a subspace of V such that π(g)W = W , ∀g ∈ G, then (W, π) where π(g) = π(g)| W is a representation called a subrepresentation of V . An irreducible representation is a representation which has no proper subrepresentation. When G is compact, any irreducible representation is finite dimensional and any finite-dimensional representation can be decomposed in the direct sum of irreducible representations. Two representations (V, π) and (V ′ , π ′ ) are equivalent if and only if there exists a linear isomorphism A: V → V ′ which intertwines the two representations, i.e. Aπ(g) = π ′ (g)A, ∀g ∈ G. We denote by Ĝ the set of equivalence classes of irreducible representations of G. For any λ ∈ Ĝ, let us choose a representative (V λ , π λ ) in λ. We denote by M λ the space of functions f : G → C such that there exists A ∈ Aut(V λ ) and f (g) = Tr{Aπ λ (g)}.

Then, the Peter-Weyl theorem asserts that

L 2 (G) = λ∈ Ĝ M λ , (4.16)
where L 2 (G) is the space of square integrable functions on G. Furthermore [15, Prop. 8.2.1], all functions f ∈ M λ are eigenfunctions of the Laplace operator associated with the same eigenvalue. So, the decomposition (4.16) is the eigenspace decomposition of the Laplace operator on G.

Thanks to this, we can easily solve the following problem:

given h ∈ L 2 (G), find f such that -∆f = h, (4.17)
Obviously, h must be chosen to satisfy the compatibility equation

G h(x) dx = 0, (4.18)
where we now denote the normalized Haar measure on G by dx. A usual method is to use a variational formulation. Supposing f ∈ H 2 (G) (where

H k (G) is defined recursively by H k (G) = {f ∈ H k-1 (G) | ∇f ∈ H k-1 (G)}, and H 0 (G) = L 2 (G)
) and taking a test function v ∈ H 1 (G), multiplying (4.17) by v and using Stokes formula, we get

G ∇f • ∇v dx = G h v dx, ∀v ∈ H 1 (G). (4.19)
Thanks to (4.18), Eq. (4. [START_REF] Gallot | Riemannian geometry[END_REF]) is equivalent to the same formulation but with v taken in

H 1 0 (G) where H 1 0 (G) = {v ∈ H 1 (G) | G v(x) dx = 0}
. Now, since clearly, a solution f to (4.17) is defined up to constants, it is reasonable to search for a unique f in the space H 1 0 (G). Indeed, we have the following Theorem 4.4. Given h ∈ L 2 (G) satisfying the compatibility condition (4.18), the variational formulation

f ∈ H 1 0 (G), (4.20)
G ∇f • ∇v dx = G h v dx, ∀v ∈ H 1 0 (G), (4.21)
has a unique solution, referred to as the variational solution of the problem -∆f = h. Furthermore, the H 1 norm of f is controlled by the L 2 norm of h.

Proof. By the Lax-Milgram theorem, it is enough to show a Poincare inequality. But because there is a spectral gap, i.e. λ 2 > 0, we have

G |∇f | 2 dx ≥ λ 2 G |f | 2 dx, ∀f ∈ H 1 0 (G).
This shows the result.

Remark 4.1. Thanks to the Peter-Weyl theorem and a Plancherel theorem [15, Thm 6.4.2], we could express the Fourier coefficients of f in terms of the Fourier coefficients of h and of the eigenvalues of ∆, where the Fourier coefficient f (λ) is defined as an endomorphism of V λ given by f

(λ) = G f (x) π λ (x -1 ) dx.
Now, we focus on class functions. The character of a representation (V, π) is the map G → C defined by χ π (g) = Tr{π(g)}, ∀g ∈ G. It is a class function. Now, let λ ∈ Ĝ and denote by χ λ the character of any representative (V λ , π λ ) of λ. Then, the subspace of M λ consisting of class functions is one-dimensional and spanned by χ λ . The system {χ λ } λ∈ Ĝ is a Hilbert basis of L 2 (G) class , the subspace of L 2 (G) consisting of class functions [15, Prop. 6.5.3] (the fact that {χ λ } λ∈ Ĝ is an orthonormal system is known as Schur's orthogonality relations). Now, since χ λ is an eigenfunction of ∆ and is a class function, φ λ =: φ χ λ is an eigenfunction of the radial Laplacian L (we recall that for a class function f , φ f is defined by (2.11)).

We introduce L 2 (T ), the set of functions φ:

T → R, such that T |φ(Θ)| 2 u n (Θ) dΘ < ∞ and H k (T ), defined recursively by φ ∈ H k (T ) ⇐⇒ (φ ∈ H k-1 (T ) and ∇ Θ φ ∈ H k-1 (T )), with H 0 (T ) = L 2 (T ).
T ∇ Θ φ • ∇ Θ v u n dΘ = T η v u n dΘ, ∀v ∈ H 1 0 (T ), (4.24)
has a unique solution referred to as the variational solution of the problem -Lφ = η. Furthermore, the H 1 norm of φ is controlled by the L 2 norm of η.

Similarly to the case of the inversion of -∆, we can also recover the Fourier coefficients of φ (in the Hilbert basis {φ λ } λ∈ Ĝ) from those of η and the eigenvalues of the Laplace operator. The next section is devoted to the computation of the eigenvalues of ∆ and L.

Characters and eigenvalues of the Laplacian

As seen in the previous section, characters are at the same time eigenfunctions of the Laplacian and class functions. Expressions of the characters on the maximal torus T are available thanks to the Weyl character formula. Applying the radial Laplacian to them, we will deduce expressions of the eigenvalues of the Laplacian. All this is expressed in the following theorem: Theorem 4.6. (i) Case n = 2p for p ≥ 2. The irreducible representations of SO 2p R are indexed by a p-tuple of integers λ = (λ 1 , . . . , λ p ) ∈ Z p such that

λ 1 ≥ λ 2 ≥ . . . ≥ λ p-1 ≥ |λ p | ≥ 0. ( 4.25)
Let χ λ be its character and define φ λ =: φ χ λ with φ f defined by (2.11). Then φ λ is given by the Weyl character formula [18, Eq. (24.40)]:

φ λ (Θ) = det (e iθ k (λ ℓ +δ ℓ ) + e -iθ k (λ ℓ +δ ℓ ) ) kℓ Π 2p (Θ) + det (e iθ k (λ ℓ +δ ℓ ) -e -iθ k (λ ℓ +δ ℓ ) ) kℓ Π 2p (Θ) , (4.26)
with δ ℓ = p -ℓ, ∀ℓ ∈ {1, . . . , p}. We denote by δ the p-tuple δ = (δ ℓ ) p ℓ=1 . (ii) Case n = 2p + 1 for p ≥ 1. The irreducible representations of SO 2p+1 R are indexed by a p-tuple of integers λ = (λ 1 , . . . , λ p ) ∈ Z p such that

λ 1 ≥ λ 2 ≥ . . . ≥ λ p-1 ≥ λ p ≥ 0. (4.27)
Then, the Weyl character formula [START_REF] Fulton | Representation theory: a first course[END_REF]Eq. (24.28)] is written in this case:

φ λ (Θ) = det (e iθ k (λ ℓ +δ ℓ ) -e -iθ k (λ ℓ +δ ℓ ) ) kℓ Π 2p+1 (Θ) , (4.28)
with δ ℓ = p -ℓ + 1 2 , ∀ℓ ∈ {1, . . . , p}. We denote by δ the p-tuple δ = (δ ℓ ) p ℓ=1 . (iii) In both the cases n = 2p for p ≥ 2 and n = 2p + 1 for p ≥ 1, φ λ and χ λ are eigenfunctions of -L and -∆ respectively, associated with the eigenvalue κ λ given by

κ λ = ∥λ + δ∥ 2 -∥δ∥ 2 , (4.29)
where ∥ • ∥ is the Euclidean norm in R p . Note that ∥δ∥ 2 = γ 2p or γ 2p+1 according to the cases considered (see Eq. (4.5)).

Proof. (i) & (ii) lie at the heart of Lie group representation theory and we refer the reader to e.g. [START_REF] Fulton | Representation theory: a first course[END_REF]. Here, we just make sure that Eqs. (4.26) and (4.28) make sense, i.e. that they define functions φ λ in T , in other words that φ λ are 2π periodic in each variable θ j and invariant by the Weyl group. 2π-periodicity follows immediately from the assumption that λ ∈ Z p . Indeed, in the case n = 2p, changing one of the θ j into θ j +2π leaves both the numerator and the denominator of (4.26) invariant. In the case n = 2p + 1, both change sign so that their quotient (4.28) remains invariant. In fact, it can be shown that λ ∈ Z p is also a necessary condition for φ λ to be 2π-periodic.

Invariance by the Weyl group will be shown by showing the following stronger property, which is an interesting observation in its own right:

Case n = 2p + 1: there exists a symmetric polynomial P (X 1 , . . . , X p ) such that φ λ (Θ) = P (e iθ 1 + e -iθ 1 , . . . , e iθp + e -iθp ), (4.30)

Case n = 2p: there exist two symmetric polynomials P (X 1 , . . . , X p ) and Q(X 1 , . . . , X p ) such that φ λ (Θ) = P (e iθ (2.11) for the description of the Weyl group for even and odd orthogonal groups).

In the case n = 2p + 1, we first focus on the numerator of (4.28). We have

e iθ k (λ ℓ +δ ℓ ) -e -iθ k (λ ℓ +δ ℓ ) = e iθ k (λ ℓ +p-ℓ+ 1 2 ) -e -iθ k (λ ℓ +p-ℓ+ 1 2 ) = (e i θ k 2 -e -i θ k 2 ) 1 + λ ℓ +p-ℓ m=1 (e imθ k + e -imθ k ) .
It is easy to show by induction that there exists a polynomial p m of degree m such that e imθ k + e -imθ k = p m (e iθ k + e -iθ k ).

Thus, there exists a polynomial q m of degree m such that

e iθ k (λ ℓ +δ ℓ ) -e -iθ k (λ ℓ +δ ℓ ) = (e i θ k 2 -e -i θ k 2 ) q λ ℓ +p-ℓ (e iθ k + e -iθ k ). It follows that det (e iθ k (λ ℓ +δ ℓ ) -e -iθ k (λ ℓ +δ ℓ ) ) kℓ = p k=1 (e i θ k 2 -e -i θ k 2 ) det q λ ℓ +p-ℓ (e iθ k + e -iθ k ) kℓ (4.32) = p k=1 (e i θ k 2 -e -i θ k 2 ) R(e iθ 1 + e -iθ 1 , . . . , e iθp + e -iθp ),
where R(X 1 , . . . , X p ) is an alternating polynomial (i.e. it is changed in its opposite when X j and X k are exchanged, for any pair j ̸ = k). Thanks to (4.11), we get φ λ (Θ) = R(e iθ 1 + e -iθ 1 , . . . , e iθp + e -iθp )

k<ℓ (e iθ k + e -iθ k -e iθ ℓ -e -iθ ℓ ) . (4.33)

From (4.32), it is clear that any factor (X j -X k ) for j < k divides R. Since all these factors are mutually prime polynomials, the denominator divides the numerator. Hence, the right-hand side of (4.33) is a polynomial. Since both the numerator and denominator of (4.33) are alternating, their quotient is symmetric, which ends the proof in the case n = 2p + 1.

The case n = 2p is similar. The first term at the right-hand side of (4.26) gives rise to the first term in (4.31). Indeed, the numerator is clearly an alternating polynomial of (e iθ k + e -iθ k ) p k=1 and the denominator too thanks to (4.10). Furthermore, each factor of the denominator divides the numerator and are mutually prime, so the quotient is a symmetric polynomial. The second term at the right-hand side of (4.26) gives rise to the second term in (4.31) thanks to similar computations as those made in the case n = 2p+1.

(iii) The general methodology is outlined in [21, §5.7] but the proof is quite abstract. On the other hand, proving (4.29) is quite easy, using (4.15). Let us consider the case n = 2p for instance. With (4.26), we have

Π 2p φ λ (Θ) = σ∈Sp (-1) σ p j=1 e iℓ σ(j) θ j + e -iℓ σ(j) θ j + σ∈Sp (-1) σ p j=1
e iℓ σ(j) θ j -e -iℓ σ(j) θ j . Now, like in the proof of Lemma 4.2 (iii), we have

-∆ Θ (Π 2p φ λ ) = p k=1 ℓ 2 k Π 2p φ λ .
Then, with (4.15) and (4.8), we get (4.29). The case n = 2p + 1 is similar.

We end this section with some terminology. The p-tuple λ lies on the lattice Z p , which is called the weight lattice. Any λ ∈ Z p is called a weight. The weights satisfying the restrictions (4.25) or (4.27) are called dominant weights, and the regions of R p defined by the inequalities (4.25) or (4.27) are called the closed Weyl chambers. All irreducible representations of SO n R induce irreducible representations of the Lie algebra so n R. The converse is not true. The weight lattice of so n R is equal to Z p ∪ ( 1 2 + Z) p . The Weyl chambers of so n R and SO n R on the other hand are the same. All representations of so n R lift to representations of the spin group Spin n R which is a simply connected doublysheeted covering of SO n R and is such that SO n R = Spin n R/{±1}. Half-integer weights correspond to representations of Spin n R that are not trivial on {±1} and thus, cannot induce representations of SO n R (but some authors call these representations doubledvalued representations of SO n R).

Conclusion

In this work, we have reviewed elements of the spectral theory of the Laplacian on rotation groups. One key element is to find the expression of the radial Laplacian, which is the restriction of the Laplacian on the maximal torus. Adequate expressions of this radial Laplacian, together with the use of the Weyl chararcter formula lead to explicit expressions of the eigenvalues of the Laplacian. Although the material presented in these notes is familiar to Lie group experts, it is not easily found in the literature in a concise and accessible way. We hope these notes will prove useful to those working with rotation groups without being experts of Lie group theory.

A Proof of Lemma 2.1

By uniqueness of the Haar measure, to show that there is a positive constant C such that dω = Cdµ, it is enough to show that the Riemannian volume form is left invariant. First, we define an orientation of so n R associated with the lexicographic ordering of the orthonormal basis (F ij ) 1≤i<j≤n . For A ∈ SO n R, the tangent space T A is oriented by the same lexicographic ordering of the basis (AF ij ) i<j , and this gives rise to a global orientation of SO n R. We note that (AF ij ) i<j is an orthonormal basis of T A precisely because A is an orthogonal matrix. Then, we construct the Riemannian volume form of SO n R associated with this orientation, following the construction of [START_REF] Warner | Foundations of differentiable manifolds and Lie groups[END_REF]Section 4.10]. We denote by (ω ij (A)) i<j the dual basis of (AF ij ) i<j . The inner-product on T A generates a pairing between T A and its dual T * A . This pairing is such that the image ω X ∈ T * A of an element X ∈ T A is such that ⟨ω X , Y ⟩ = X • Y for all Y ∈ T A , where the bracket indicates duality between T * A and T A . In turn, this pairing induces an inner product on T * A , such that ω X • ω Y = X • Y . Thus, since (AF ij ) i<j is an orthonormal basis of T A , (ω ij (A)) i<j is an orthonormal basis of T * A , so ω(A) = i<j ω ij (A) is the volume form where i<j stands for the exterior product of the forms ω ij in lexicographic order. Now, we show that this volume form is left invariant. Let σ ∈ SO n R and ℓ σ the morphism of SO n R such that ℓ σ (A) = σA. The form ω is left invariant if and only if δℓ σ (ω) = ω for all σ ∈ SO n R, where δℓ σ (ω) is the pullback of ω by ℓ σ . Let (AX ij ) i<j , with X ij ∈ so n R for all i < j be elements of T A . Then, δℓ σ (ω) is defined by where N = n(n-1)

2

is the dimension of T A , S N is the permutation group of N elements, ε(τ ) is the signature of the permutation τ and where the elements of {1, . . . , N } are labeled by the pairs (i, j) with 1 ≤ i < j ≤ n. Hence, τ (i, j) denotes the image of the element (i, j) of {1, . . . , N } by the permutation τ . Now, if we show that ω ij (σA) σAX = ω ij (A) AX , ∀A, σ ∈ SO n R, ∀X ∈ so n R, ∀(i, j) | 1 ≤ i < j ≤ n, (A.1) then, we obviously deduce that δℓ σ (ω) = ω, which concludes the proof. To prove (A.1), it is enough to prove it for all basis vectors of a basis of so n R. We take the basis (F kℓ ) k<ℓ . We have ω ij (σA) σAF kℓ = δ ij δ kℓ because (ω ij (σA)) i<j is the dual basis of (σAF kℓ ) k<ℓ . But this being true for all σ, it is true for σ = I which gives ω ij (A) AF kℓ = δ ij δ kℓ . Hence, ω ij (σA) σAF kℓ = ω ij (A) AF kℓ , which shows (A.1).

B Proof of Lemma 2.2

From (2.4) and (2.6), we have, for any C ∞ function f on So n R and any A ∈ So n R:

(P T A E ij )(f )(A) = ρ A T E ij -E T ij A 2 f (A).
A straightforward computation shows that

A T E ij -E T ij A = n k=1 A ik F jk ,
where A ik denote the entries of the matrix A. Since ρ(X) is linear with respect to X, we have

ρ(A T E ij -E T ij A) = n k=1
A ik ρ F jk , and ρ(A

T E ij -E T ij A) 2 = k,ℓ A ik A im ρ F jk ρ F jm .
Now, with (2.5), we get

∆ M f = 1 2 i,j,k,ℓ A ik A im ρ F jk ρ F jm = 1 2 j,k ρ F jk 2 = 1≤j<k≤n ρ F jk 2 = ∆ G f,
where, in the second equality, we have used that A is orthogonal, in third equality, that F ji = -F ih and in the fourth one, Eq. (2.7). This ends the proof.

Lemma 4 . 2 .

 42 (i) Case n = 2p, p ≥ 2: we have ∇ Θ u 2p (Θ) = k̸ =j 2 cos θ k -cos θ j sin θ j u 2p (Θ)

Proposition 4 . 3 .

 43 (i) In the case n = 2p, for any φ ∈ C ∞ (T ), we have

  With these notations, (φ λ ) λ∈ Ĝ is a Hilbert basis of L 2 (T ) consisting of eigenfunctions of L. It follows that L has the same eigenvalues as ∆. Consequently, analogously to the case of the Laplacian, we can define a variational solution of the problem -Lφ = η. Let us define H 1 0 (T ) = {v ∈ H 1 (T ) | T v(Θ) u n (Θ) dΘ = 0}. Then, we have Theorem 4.5. Given η ∈ L 2 (T ) satisfying the compatibility condition T η(Θ) u n (Θ) dΘ = 0, (4.22)

  δℓ σ (ω)(A) AX 12 , AX 13 , . . . , AX n-1 n = ω(σA) dℓ σ (A)(AX 12 ), dℓ σ (A)(AX 13 ), . . . , dℓ σ (A)(AX n-1 n ) ,where dℓ σ (A) is the differential of ℓ σ at A. It is readily checked that dℓ σ (A)(AX) = σAX, for any X ∈ so n R. Thus,δℓ σ (ω)(A) AX 12 , AX 13 , . . . , AX n-1 n = ω(σA)(σAX 12 , σAX 13 , . . . , σAX n-1 n ) = τ ∈S N ε(τ ) i<j ω ij (σA) σAX τ (i,j) ,

  We have ∆ M = ∆ G .As a consequence, we can drop the subscript M or G of ∆. Two elements A and B of SO n R are said to be conjugate if and only if there exists g ∈ SO n R such that B = gAg -1 . Conjugation is an equivalence relation and we write A ∼ B whenever A and B are conjugate. A class function is a function on SO n R which is invariant by conjugation, i.e. which satifies:

	which means that the Lie-group Laplacian
	is left-invariant. Because the inner product on so n R is invariant (see (2.8)), we have
	R(B)(∆ G f ) = ∆ G (R(B)f ), which means that the Lie-group Laplacian is also right-
	invariant.
	It turns out that the two concepts of Laplacians coincide on SO n R as stated in the
	following lemma, proved in Appendix B.
	Lemma 2.2.

  Proposition 4.1 (Weyl integration formula). Let n ∈ N, n ≥ 3. Let p ∈ N such that n = 2p or n = 2p + 1. For any integrable class function f on SO n R, we have

	We define
	u 2p =	2 (p-1) 2 p!	Π 2 2p , for p ≥ 2;	u 2p+1 =	2 p(p-1) p!	Π 2 2p+1 , for p ≥ 1.	(4.3)
	Then, we have
		SOnR
									.1)
	Π 2p+1 (Θ) =	1≤j<k≤p	cos θ j -cos θ k	p j=1	sin	θ j 2	, for p ≥ 1.	(4.2)