The Lichnerowicz Laplacian on compact symmetric spaces

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Differential Geometry

The Lichnerowicz Laplacian on compact symmetric spaces

Mohamed Boucetta

Faculté des sciences et techniques Gueliz, BP 549, 40000 Marrakech, Morocco Received 20 January 2009; accepted after revision 24 June 2009

Available online 23 July 2009 Presented by Étienne Ghys

Abstract

We show that, on a compact symmetric space, the Lichnerowicz Laplacian acting on the space of covariant tensor fields coincides with the Casimir operator and we deduce that, on a compact semisimple Lie group, the Lichnerowicz Laplacian is the mean of the left invariant Casimir operator and the right invariant Casimir operator.To cite this article: M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Résumé

Le laplacien de Lichnerowicz sur les espaces symétriques compacts.On montre que, dans un espace symétrique compact, le laplacien de Lichnerowicz agissant sur l’espace des tenseurs covariants coincide avec l’opérateur de Casimir et on déduit que, dans un groupe de Lie compact semisimple, le laplacien de Lichnerowicz est la moyenne de l’opérateur de Casimir invariant à gauche et celui invariant à droite.Pour citer cet article : M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Version française abrégée

Le laplacien de Lichnerowicz, introduit dans [3], est un opérateur agissant sur les tenseurs covariants d’une variété riemannienne, il est auto-adjoint, elliptique et respecte la symétrie des tenseurs. Sa restriction aux formes différen- tielles coincide avec le laplacien de Hodge–de Rham. D’un autre côté, siGest une algèbre de Lie réelle semisimple compacte, l’élément de Casimir deG est un élémentΩ du centre de l’algèbre enveloppante deG. Il est donné par Ω= −_n

i=1e_i.e_i où(e₁, . . . , e_n)est une base orthonormée de l’opposée de la forme de Killing. SiG agit sur une variétéM, grâce à la dérivée de Lie des champs fondamentaux de cette action,Ωinduit un opérateur différentielΩ_M surM. Dans cette Note, on s’intéresse aux deux situations suivantes :

(i) SoitGun groupe de Lie compact et semisimple. L’élément de CasimirΩ de l’algèbre de Lie deGinduit surG deux opérateursΩ⁺etΩ⁻associés, respectivement, à l’action à gauche et à droite deGsur lui même.

(ii) Une paire symétrique est un couple(G, K)oùGest un groupe de Lie compact et semisimple etKun sous-groupe fermé deGtels qu’il existe une involutionσ deGvérifiantG^σ₀ ⊂K⊂G^σ, oùG^σ= {g∈G, σ (g)=g}etG^σ₀

E-mail address:[email protected].

doi:10.1016/j.crma.2009.06.019

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est la composante neutre deG^σ. L’action deGsurG/Kassocie à l’élément de Casimir de l’algèbre de Lie deG un opérateur notéΩ_(G,K).

Le but de cette Note est de démontrer les deux résultats suivants :

Théorème 0.1. Soit(G, K)une paire symétrique. Alors_L=Ω_(G,K),où_Lest le laplacien de Lichnerowicz sur G/Kassocié à la métrique riemannienne naturelle surG/Kdéduite de la forme de Killing deG.

Théorème 0.2. SoitGun groupe de Lie semisimple compact. AlorsL=¹₂(Ω⁺+Ω⁻),oùLest le laplacien de Lichnerowicz associée à la métrique riemannienne bi-invariante induite par l’opposée de la forme de Killing.

Le Théorème 0.1 généralise un résultat de [2] et le Théorème 0.2 généralise un résultat énoncé sans preuve dans [1].

1. Introduction and main results

It is well known that the Hodge–de Rham Laplacian on compact symmetric spaces coincides with the Casimir operator (see [2]). Also it is well known that, on a compact semisimple Lie group, the Hodge–de Rham Laplacian is the mean of the left invariant Casimir operator and the right invariant Casimir operator (see [1]). In this paper, we generalize these results to the Lichnerowicz Laplacian. Let us recall the definitions of the Lichnerowicz Laplacian and the Casimir operator and state our main results.

Let(M, g)be a Riemanniann-manifold. For anyp∈N, we shall denote byΓ (_p

T^∗M)the space of covariant p-tensor fields on M. The curvature tensor R and the Ricci endomorphism field r:T M →T M associated to the Levi-Civita connectionDare given by

R(X, Y )Z=D_[_X,Y_]−(D_XD_Y −D_YD_X) and g

r(X), Y

= n

i=1

g

R(X, E_i)Y, E_i , where(E₁, . . . , E_n)is a local orthonormal frame.

The connectionDinduces a differential operatorD:Γ (_p

T^∗M)→Γ (_p₊₁

T^∗M)given by DT (X, Y₁, . . . , Y_p)=X.T (Y₁, . . . , Y_p)−

p

j=1

T (Y₁, . . . , D_XY_j, . . . , Y_p).

We denote byD^∗:Γ (_p₊₁

T^∗M)→Γ (_p

T^∗M)its formal adjoint given by D^∗T (Y₁, . . . , Y_p)= −

n

j=1

DT (E_i, E_i, Y₁, . . . , Y_p).

The Lichnerowicz Laplacian is the differential operatorL:Γ (p

T^∗M)→Γ (p

T^∗M)given by L(T )=D^∗D(T )+R(T ),

whereR(T )is the curvature operator given by R(T )(Y₁, . . . , Y_p)=

p

j=1

T

Y₁, . . . , r(Y_j), . . . , Y_p

− n

l=1

i<j

T

Y₁, . . . , E_l, . . . , R(Y_i, E_l)Y_j, . . . , Y_p +T

Y₁, . . . , R(Y_j, E_l)Y_i, . . . , E_l, . . . , Y_p ,

where, inT (Y₁, . . . , E_l, . . . , R(Y_i, E_l)Y_j, . . . , Y_p),E_ltakes the place ofY_i andR(Y_i, E_l)Y_jtakes the place ofY_j. This operator, introduced by Lichnerowicz in [3, p. 26], is self-adjoint, elliptic and respects the symmetries of tensor fields. The restriction of _Lto the space of differential forms Ω^∗(M) coincides with the Hodge–de Rham Laplacian.

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LetGbe real compact semisimple Lie algebra. We denote by−,the Killing form ofG. Recall that the Casimir element ofGis the elementΩbelonging to the center of the enveloping algebraU(G)ofGgiven byΩ= −_n

i=1e_i.e_i, where(e₁, . . . , e_n)is any orthonormal basis with respect to,. IfG acts on a differentiable manifoldM, i.e., there exists a Lie algebras homomorphismρ:G→X(M),Ω gives rise to a differential operatorΩ_ρ onMgiven by, for anyT ∈Γ (_∗

T^∗M),Ω_ρ(T )= −_n

i=1Lρ(ei)◦Lρ(ei)T, whereLis the Lie derivative. In this Note, we will study the two following cases:

(i) LetGbe a compact semisimple Lie group of dimensionnand letG=T_eGbe its Lie algebra. For anyu∈Gwe denote byu⁺andu⁻respectively the left and the right invariant vector field associated tou. The Casimir element ΩofGinduces two differential operators onGgiven by

Ω⁺= − n

i=1

L_e⁻

i ◦L_e⁻

i and Ω⁻= −

n

i=1

L_e⁺

i ◦L_e⁺

i , where(e₁, . . . , e_n)is any orthonormal basis with respect to,.

(ii) A compact symmetric pair is a couple(G, K)whereGis a compact semisimple Lie group of dimensionnand Kis a closed subgroup ofGsuch that there exists an involutive automorphismσsuch thatG^σ₀⊂K⊂G^σ where G^σ= {a∈G: σ (a)=a}andG^σ₀ is the connected component of the identity inG^σ. Let(G,K)be the Lie algebras of(G, K). We haveG=K⊕K^⊥whereK^⊥is the,-orthogonal toKand

K,K^⊥

⊂K^⊥ and K^⊥,K^⊥

⊂K. (1)

We denote byπ:G→G/K the natural projection, we puto=π(e) and we consider the natural action ofG onG/K. For any vectoru∈G, we denote byu˜ the corresponding fundamental vector field onG/K given by

˜

u(π(a))=_dt^d_|_t₌₀exp(t u).π(a). The Casimir element Ω of G and the action of G on the quotient G/K give rise to a differential operatorΩ_(G,K) onG/Kgiven by Ω_(G,K)= −_n

i=1Le_˜i ◦Le_˜i, where(e₁, . . . , e_n)is any orthonormal basis of,. On the other hand, the restriction of, toK^⊥ gives rise to an invariant Riemannian metricg_BonG/Kand, for anyu∈G,u˜is a Killing vector field. We can state now our main results.

Theorem 1.1.Let(G, K)be a compact symmetric pair. Then _L=Ω_(G,K),

where_Lis the Lichnerowicz Laplacian onG/Kassociated to the Riemannian metricg_B.

A particular case of a compact symmetric pair is given by the Cartesian productG×Gof a compact semisimple Lie groupGand its diagonal(G)= {(a, a)∈G×G:a∈G}. By applying Theorem 1.1 to this pair, we will deduce the following result:

Theorem 1.2.LetGbe a compact semisimple Lie group. Then _L=1

2

Ω⁺+Ω⁻ ,

whereLis the Lichnerowicz Laplacian onGassociated to the biinvariant Riemannian metric onGinduced by the negative of the Killing form.

In Section 2, we give a complete proof of Theorems 1.1 and 1.2.

2. Proofs of Theorems 1.1 and 1.2

The proof of Theorem 1.1 requires some preparation. The proof of the following lemma is straightforward:

Lemma 2.1.Let(G,,)be a real Lie algebra endowed with a biinvariant scalar product, i.e., for anyu, v, w∈G, [u, v], w + v,[u, w] =0.Then:

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(i) for any bilinear formωonG, for anyu, v∈Gand for any orthonormal basis(e₁, . . . , e_n), we have n

i=1

ω

[u, e_i],[v, e_i]

= n

i=1

ω [v, e_i], u , e_i

= n

i=1

ω

e_i, [u, e_i], v

;

(ii) for any subalgebra K satisfying (1), for any orthonormal basis (e₁, . . . , e_r) of K, any orthonormal basis (f₁, . . . , f_s)ofK^⊥and for anyu∈K^⊥, we have_r

i=1[e_i,[e_i, u]] =_s

i=1[f_i,[f_i, u]].

Let us give now some properties of the Riemannian manifold(G/K, g_B)where(G, K)is a compact symmetric pair. We denote byDthe Levi-Civita connection associated tog_B.

Note first that, for anyu, v∈K^⊥and for anyz∈K, gB

[˜z,u˜],v˜

(o)+gB

[˜z,v˜],u˜

(o)=0. (2)

On the other hand, from (1), we get that, for anyu, v∈K^⊥,

[ ˜u,v˜](o)=0. (3)

Moreover, since for anyu, v, w∈G,u,˜ v,˜ w˜ are Killing vector fields we have 2gB(D_u_˜v,˜ w)˜ =g_B

[ ˜u,v˜],w˜ +g_B

[ ˜u,w˜],v˜ +g_B

[˜v,w˜],u˜

. (4)

We deduce from (3) and (4) that, for anyu, v∈K^⊥,

(D_u_˜v)(o)˜ =0. (5)

The following lemma summarizes some well-known results on curvature of symmetric spaces:

Lemma 2.2.Let(G/K, g_B)be the Riemannian manifold associated with a compact symmetric pair andDits Levi- Civita connection. Then:

(i) for anyu, v, w∈K^⊥,(D_u_˜D_v_˜w)(o)˜ = [˜v,[ ˜u,w˜]](o);

(ii) for anyu, v, w∈K^⊥, the tensor curvatureRand the Ricci endomorphism fieldrofg_B are given by R(u,˜ v)˜ w(o)˜ = [ ˜u,v˜],w˜

(o), r(u)(o)˜ = − s

j=1

[ ˜u,f˜_j],f˜_j (o),

where(f1, . . . , fs)is any orthonormal basisK^⊥.

Proof of Theorem 1.1. We will show that for any covariant tensor fieldT,L(T )−ΩG,K(T )vanishes atoand we deduce the vanishing ofL(T )−Ω(G,K)(T )onG/Ksince bothLandΩ(G,K)are invariant. Choose an orthonormal basis(e1, . . . , er)ofKand an orthonormal basis(f1, . . . , fs)ofK^⊥. LetT be a covariantp-tensor field. We have

Ω(G,K)(T )= − r

i=1

Le_˜i◦Le_˜i(T )− s

j=1

L_f_˜_j◦L_f_˜_j(T ).

By expandingL_e_˜_i◦L_e_˜_i(T )and by using the fact thate˜i(o)=0, we get for anyu1, . . . , up∈K^⊥ Le_˜_i◦Le_˜_i(T )(u˜₁, . . . ,u˜_p)(o)= −

j=1

T

˜

u₁, . . . , [˜e_i,u˜_j],e˜_i

, . . . ,u˜_p (o)

+2

l<j

T

˜

u1, . . . ,[˜e_i,u˜_l], . . . ,[˜e_i,u˜_j], . . . ,u˜_p

(o). (6)

On the other hand, by expandingL_f_˜_j ◦L_f_˜_j(T ), we get

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L_f_˜_j ◦L_f_˜_j(T )(u˜₁, . . . ,u˜_p)= ˜f_j.f˜_j.T (u˜₁, . . . ,u˜_p)−2 p

i=1

f˜_j.T

˜

u₁, . . . ,[ ˜f_j,u˜_i], . . . ,u˜_p

− p

i=1

T

˜

u1, . . . , [ ˜f_j,u˜_i],f˜_j

, . . . ,u˜_p +2

l<i

T

˜

u₁, . . . ,[ ˜f_j,u˜_l], . . . ,[ ˜f_j,u˜_i], . . . ,u˜_p .

Now by expandingLf_jT (u˜₁, . . . ,[ ˜f_j,u˜_i], . . . ,u˜_p)and by using (3), we get L_f_˜_j ◦L_f_˜_j(T )(u˜1, . . . ,u˜p)(o)= ˜fj.f˜j.T (u˜1, . . . ,u˜p)(o)+

p

i=1

T

˜

u1, . . . , [ ˜fj,u˜i],f˜j

, . . . ,u˜p

(o). (7)

On the other hand

D^∗D(T )(u˜₁, . . . ,u˜_p)(o)

= s

i=1

− ˜f_i.f˜_i.T (u˜₁, . . . ,u˜_p)(o)+2 p

j=1

f˜_i.T (u˜₁, . . . , D_f_˜

iu˜_j, . . . ,u˜_p)(o)

+D_f_˜

if˜_i.T (u˜₁, . . . ,u˜_p)(o)− p

j=1

T (u˜₁, . . . , D_D

fi˜f˜_iu˜_j, . . . ,u˜_p)(o)

− p

j=1

T (u˜1, . . . , D_f_˜

iD_f_˜

iu˜j, . . . ,u˜p)(o)−2

l<j

T (u˜1, . . . , D_f_˜

iu˜l, . . . , D_f_˜

iu˜j, . . . ,u˜p)(o)

. By using (5), the relationf_i.T (u˜₁, . . . , D_f_˜

iu˜_j, . . . ,u˜_p)(o)=T (u˜₁, . . . , D_f_˜

iD_f_˜

iu˜_j, . . . ,u˜_p)(o)and Lemma 2.1, we get

D^∗D(T )(u˜1, . . . ,u˜p)(o)= s

i=1

− ˜fi.f˜i.T (u˜1, . . . ,u˜p)(o)+ p

j=1

T

˜

u1, . . . , f˜i,[ ˜fi,u˜j] , . . . ,u˜p

(o)

(7)= − s

i=1

L_f_˜_i◦L_f_˜_i(T )(u˜₁, . . . ,u˜_p)(o). (8) By using Lemma 2.2, we get that the curvature operator is given by

R(T )(u˜₁, . . . ,u˜_p)(o)

= − s

i=1

p

j=1

T

˜

u1, . . . , [ ˜uj,f˜i],f˜i

, . . . ,u˜p

(o)

− s

l=1

i<j

T

˜

u1, . . . ,f˜l, . . . , [ ˜ui,f˜l],u˜j

, . . . ,u˜p

(o)+T

˜

u1, . . . , [ ˜uj,f˜l],u˜i

, . . . ,f˜l, . . . ,u˜p

(o) .

Now, by using Lemma 2.1, the fact thate˜_i(o)=0, (3) and (6), one can see easily that R(T )(u˜₁, . . . ,u˜_p)(o)= −

r

i=1

Le_˜_i◦Le_˜_i(T )(u˜₁, . . . ,u˜_p)(o). (9) From (8) and (9) we deduce that_LandΩ_(G,K)coincides atoand by invariance on allG/Kwhich completes the proof. 2

Proof of Theorem 1.2. We define the mapφ:(G×G)/(G)→Gbyφ([g, h])=gh⁻¹.This map is an isometry between the symmetric space(G×G)/(G)andGendowed with the biinvariant metric induced by the negative of

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Killing form. Moreover,φis equivariant with respect to the natural action ofG×Gon(G×G)/(G)and the action ofG×GonGgiven byρ(g, h)a=gah⁻¹. According to Theorem 1.1, the Lichnerowicz Laplacian ofGcoincides with the Casimir operator associated to the actionρ. Let compute this operator. Note that the fundamental vector field onGassociated to(u, v)∈G×Gis the vector fieldu⁺−u⁻and for any orthonormal basis(e₁, . . . , e_n)ofG,

1

2(e₁, e₁), . . . ,1

2(e_n, e_n),1

2(e₁,−e₁), . . . ,1

2(e_n,−e_n)

is an orthonormal basis ofG×G. Hence Ω_ρ= −1

4 n

i=1

L_e⁺

i−e_i⁻◦L_e⁺

i−e⁻_i −1 4

n

i=1

L_e⁺

i+e_i⁻◦L_e⁺

i+e⁻_i =1 2

Ω⁺+Ω⁻

which completes the proof. 2 References

[1] B.L. Beers, R.S. Millman, The spectra of the Laplace–Beltrami operator on compact, semisimple Lie groups, Amer. J. Math. 99 (4) (1975) 801–807.

[2] A. Ikeda, Y. Taniguchi, Spectra and eigenforms of the Laplacian onSⁿandPⁿ(C), Osaka J. Math. 15 (3) (1978) 515–546.

[3] A. Lichnerowicz, Propagateurs et commutateurs en relativité générale, Inst. Hautes Etude Sci. Publ. Math. 10 (1961) 5–56.