Spectra and symmetric eigentensors of the Lichnerowicz Laplacian on
  $S^n$
arXiv:0704.1363v1  [math.DG]  11 Apr 2007
7 Spectra and symmetric eigentensors of the
Lichnerowicz Laplacian on Sn
M. Boucetta
Faculté des Sciences et Techniques
BP 549 Marrakech
Morocco
Email: boucetta@fstg-marrakech.ac.ma ∗
Abstract. We compute the eigenvalues with multiplicities of the Lichnerow-
icz Laplacian acting on the space of symmetric covariant tensor fields on the
Euclidian sphere Sn. The spaces of symmetric eigentensors are explicitly
given.
Mathematical Subject Classification (2000):53B21, 53B50, 58C40
Key words: Lichnerowicz Laplacian
1 Introduction
Let (M, g) be a Riemannian n-manifold. For any p ∈ IN, we shall denote
by Γ(⊗pT ∗M), Ωp(M) and SpM the space of covariant p-tensor fields on M ,
the space of differential p-forms on M and the space of symmetric covariant
p-tensor fields on M , respectively. Note that Γ(⊗0T ∗M) = Ω0(M) = S0M =
C∞(M, IR), Ω(M) =
Ωp(M) and S(M) =
Sp(M).
Let D be the Levi-Civita connection associated to g; its curvature tensor
field R is given by
R(X, Y )Z = D[X,Y ]Z − (DXDY Z −DYDXZ) ,
∗Recherche menée dans le cadre du Programme Thématique d’Appui à la Recherche
Scientifique PROTARS III.
http://arxiv.org/abs/0704.1363v1
and the Ricci endomorphism field r : TM −→ TM is given by
g(r(X), Y ) =
g(R(X,Ei)Y,Ei),
where (E1, . . . , En) is any local orthonormal frame.
For any p ∈ IN, the connectionD induces a differential operatorD : Γ(⊗pT ∗M) −→
Γ(⊗p+1T ∗M) given by
DT (X, Y1, . . . , Yp) = DXT (Y1, . . . , Yp) = X.T (Y1, . . . , Yp)−
T (Y1, . . . , DXYj , . . . , Yp).
Its formal adjoint D∗ : Γ(⊗p+1T ∗M) −→ Γ(⊗pT ∗M) is given by
D∗T (Y1, . . . , Yp) = −
DEiT (Ei, Y1, . . . , Yp),
where (E1, . . . , En) is any local orthonormal frame.
Recall that, for any differential p-form α, we have
dα(X1, . . . , Xp+1) =
(−1)j+1DXjα(X1, . . . , X̂j, . . . , Xp+1). (1)
We denote by δ the restriction of D∗ to Ω(M) ⊕ S(M) and we define δ∗ :
Sp(M) −→ Sp+1(M) by
δ∗T (X1, . . . , Xp+1) =
DXjT (X1, . . . , X̂j, . . . , Xp+1).
Recall that the operator trace Tr : Sp(M) −→ Sp−2(M) is given by
TrT (X1, . . . , Xp−2) =
T (Ej, Ej , X1, . . . , Xp−2),
where (E1, . . . , En) is any local orthonormal frame.
The Lichnerowicz Laplacian is the second order differential operator
∆M : Γ(⊗
pT ∗M) −→ Γ(⊗pT ∗M)
given by
∆M(T ) = D
∗D(T ) +R(T ),
where R(T ) is the curvature operator given by
R(T )(Y1, . . . , Yp) =
T (Y1, . . . , r(Yj), . . . , Yp)
{T (Y1, . . . , El, . . . , R(Yi, El)Yj, . . . , Yp) + T (Y1, . . . , R(Yj, El)Yi, . . . , El, . . . , Yp)} ,
where (E1, . . . , En) is any local orthonormal frame and, in
T (Y1, . . . , El, . . . , R(Yi, El)Yj, . . . , Yp),
El takes the place of Yi and R(Yi, El)Yj takes the place of Yj.
This differential operator, introduced by Lichnerowicz in [15] pp. 26, is self-
adjoint, elliptic and respects the symmetries of tensor fields. In particular,
∆M leaves invariant S(M) and the restriction of ∆M to Ω(M) coincides with
the Hodge-de Rham Laplacian, i.e., for any differential p-form α,
∆Mα = (dδ + δd)(α). (2)
We have shown in [6] that, for any symmetric covariant tensor field T ,
∆M (T ) = (δ ◦ δ
∗ − δ∗δ)(T ) + 2R(T ). (3)
Note that if T ∈ S(M) and gl denotes the symmetric product of l copies of
the Riemannian metric g, we have
(Tr ◦∆M)T = (∆M ◦ Tr)T, (4)
∆M(T ⊙ g
l) = (∆MT )⊙ g
l, (5)
where ⊙ is the symmetric product.
The Lichnerowicz Laplacian acting on symmetric covariant tensor fields is of
fundamental importance in mathematical physics (see for instance [9], [20]
and [22]). Note also that the Lichenrowicz Laplacian acting on symmetric
covariant 2-tensor fields appears in many problems in Riemannian geometry
(see [3], [5], [19]...).
On a Riemannian compact manifold, the Lichnerowicz Laplacian ∆M has
discrete eigenvalues with finite multiplicities. For a given Riemannian com-
pact manifold, it may be an interesting problem to determine explicitly the
eigenvalues and the eigentensors of ∆M on M .
Let us enumerate the cases where the spectra of ∆M was computed:
1. ∆M acting on C
∞(M, |C): M is either flat toris or Klein bottles [4], M
is a Hopf manifolds [1];
2. ∆M acting on Ω(M): M = S
n or P n( |C) [10] and [11], M = |CaP 2
or G2/SO(4) [16] and [18], M = SO(n + 1)/SO(2) × SO(n) or M =
Sp(n+ 1)/Sp(1)× Sp(n) [21];
3. ∆M acting on S
2(M) and M is the complex projective space P 2( |C)
[22];
4. ∆M acting on S
2(M) and M is either Sn or P n( |C) [6] and [7];
5. Brian and Richard Millman give in [2] a theoretical method for com-
puting the spectra of Lichnerowicz Laplacian acting on Ω(G) where G
is a compact semisimple Lie group endowed with the Killing form;
6. Some partial results where given in [12]-[14].
In this paper, we compute the eigenvalues and we determine the spaces of
eigentensors of ∆M acting on S(M) in the case where M is the Euclidian
sphere Sn.
Let us describe our method briefly. We consider the (n+ 1)-Euclidian space
IRn+1 with its canonical coordinates (x1, . . . , xn+1). For any k, p ∈ IN, we
denote by SpHδk the space of symmetric covariant p-tensor fields T on IR
satisfying:
1. T =
1≤i1≤...≤ip≤n+1
Ti1,...,ipdxi1⊙. . .⊙dxip where Ti1,...,ip are homogeneous
polynomials of degree k;
2. δ(T ) = ∆IRn+1(T ) = 0.
The n-dimensional sphere Sn is the space of unitary vectors in IRn+1 and the
Euclidian metric on IRn+1 induces a Riemannian metric on Sn. We denote
by i : Sn →֒ IRn+1 the canonical inclusion.
For any tensor field T ∈ Γ(⊗pT ∗IRn+1), we compute i∗(∆IRn+1T )−∆Sn(i
and get a formula (see Theorem 2.1). Inspired by this formula and having
in mind the fact that i∗ :
SpHδk −→ S
pSn is injective and its image is
dense in SpSn (see [10]), we give, for any k, a direct sum decomposition of
SpHδk composed by eigenspaces of ∆Sn. Thus we obtain the eigenvalues and
the spaces of eigentensors with its multiplicities of ∆Sn acting on S(S
n) (see
Section 4).
Note that the eigenvalues and the eigenspaces of ∆Sn acting on Ω(S
n) was
computed in [10] by using the representation theory. In [11], I. Iwasaki and K.
Katase recover the result by a method using the restriction of harmonic tensor
fields and a result in [8]. The formula obtained in Theorem 2.1 combined
with the methods developed in [10] and [11] permit to present those results
in a more precise form (see Section 3).
2 A relation between ∆IRn+1 and ∆Sn
We consider the Euclidian space IRn+1 endowed with its canonical coordinates
(x1, . . . , xn+1) and its canonical Euclidian flat Riemannian metric < , >. We
denote by D be the Levi-Civita covariant derivative associated to < , >. We
consider the radial vector field given by
−→r =
For any p-tensor field T ∈ Γ(⊗pT ∗IRn+1) and for any 1 ≤ i < j ≤ p, we
denote by i−→r ,jT the (p− 1)-tensor field given by
i−→r ,jT (X1, . . . , Xp−1) = T (X1, . . . , Xj−1,
−→r ,Xj, . . . , Xp−1),
and by Tri,jT the (p− 2)-tensor field given by
Tri,jT (X1, . . . , Xp−2) =
T (X1, . . . , Xi−1, El, Xi, . . . , Xj−2, El, Xj−1, . . . , Xp−2),
where (E1, . . . , En+1) is any orthonormal basis of IR
n+1. Note that Tri,jT = 0
if T is a differential form and Tri,jT = TrT if T is symmetric.
For any permutation σ of {1, . . . , p}, we denote by T σ the p-tensor field
T σ(X1, . . . , Xp) = T (Xσ(1), . . . , Xσ(p)).
For 1 ≤ i < j ≤ p, the transposition of (i, j) is the permutation σi,j of
{1, . . . , p} such that σi,j(i) = j, σi,j(j) = i and σi,j(k) = k for k 6= i, j. Let
T denote the set of the transpositions of {1, . . . , p}.
The sphere i : Sn →֒ IRn+1 is endowed with the Euclidian metric.
Theorem 2.1 Let T be a covariant p-tensor field on IRn+1. Then,
i∗(∆IRn+1T ) = ∆Sni
∗T + i∗
p(1− p)T + (2p− n + 1)L−→r T − L−→r ◦ L−→r T
T σ +O(T )
where O(T ) is given by
O(T )(X1, . . . , Xp) = 2
< Xi, Xj > Tri,j(X1, . . . , X̂i, . . . , X̂j, . . . , Xp)
DXj(i−→r ,jT )(X1, . . . , X̂j, . . . , Xp),
where X̂ designs that X is deleted.
Proof. The proof is a massive computation in a local orthonormal frame
using the properties of the Riemannian embedding of the sphere in the Eu-
clidian space.
We choose a local orthonormal frame of IRn+1 of the form (E1, . . . , En, N)
such that Ei is tangent to S
n for 1 ≤ i ≤ n and N = 1
−→r where r =
x21 + . . .+ x
For any vector field X on IRn+1, we have
DXN =
(X− < X,N > N) , (6)
DNX = [N,X ] +
(X− < X,N > N). (7)
Let ∇ be the Levi-Civita connexion of the Riemannian metric on Sn. We
have, for any vector fields X, Y tangent to Sn,
DXY = ∇XY− < X, Y > N. (8)
Let T be a covariant p-tensor field on IRn+1 and (X1, . . . , Xp) a family of
vector fields on IRn+1 which are tangent to Sn. A direct calculation using
the definition of the Lichnerowicz Laplacian gives
∆IRn+1(T )(X1, . . . , Xp) = D
∗D(T )(X1, . . . , Xp)
−EiEi.T (X1, . . . , Xp) + 2
Ei.T (X1, . . . , DEiXj, . . . , Xp)
+DEiEi.T (X1, . . . , Xp)−
T (X1, . . . , DDEiEiXj , . . . , Xp)
T (X1, . . . , DEiDEiXj, . . . , Xp) −2
T (X1, . . . , DEiXl, . . . , DEiXj, . . . , Xp)
−N.N.T (X1, . . . , Xp) + 2
N.T (X1, . . . , DNXj , . . . , Xp)
+DNN.T (X1, . . . , Xp)−
T (X1, . . . , DDNNXj , . . . , Xp)
T (X1, . . . , DNDNXj, . . . , Xp)− 2
T (X1, . . . , DNXl, . . . , DNXj , . . . , Xp).
(6)-(8) make it obvious that
DDEiEiXj = ∇∇EiEiXj− < ∇EiEi, Xj > N − [N,Xj ] (9)
(Xj− < Xj, N > N),
DEiDEiXj = ∇Ei∇EiXj − (< Ei,∇EiXj > +Ei. < Ei, Xj >)N
< Ei, Xj > Ei, (10)
DNDNX = [N, [N,X ]] +
[N,X ] + (
)(X− < X,N > N)
N. < X,N > N. (11)
By (8)-(10), we get easily, in restriction to Sn,
Ei.T (X1, . . . , DEiXj , . . . , Xp) +DEiEi.T (X1, . . . , Xp)
T (X1, . . . , DDEiEiXj, . . . , Xp)−
T (X1, . . . , DEiDEiXj , . . . , Xp)
Ei.T (X1, . . . ,∇EiXj , . . . , Xp) +∇EiEi.T (X1, . . . , Xp)
T (X1, . . . ,∇∇EiEiXj, . . . , Xp)−
T (X1, . . . ,∇Ei∇EiXj , . . . , Xp)
Xj .T (X1, . . . ,
N , . . . , Xp) + p(n + 1)T (X1, . . . , Xp)− nLNT (X1, . . . , Xp).
On other hand, also by using (8), we have
T (X1, . . . , DEiXl, . . . , DEiXj, . . . , Xp) =
T (X1, . . . , DEiXl, . . . ,∇EiXj, . . . , Xp)−
T (X1, . . . , DXjXl, . . . ,
N , . . . , Xp) =
T (X1, . . . ,∇EiXl, . . . ,∇EiXj, . . . , Xp)−
T (X1, . . . ,
N , . . . ,∇XlXj, . . . , Xp)
T (X1, . . . , DXjXl, . . . ,
N , . . . , Xp) =
T (X1, . . . ,∇EiXl, . . . ,∇EiXj , . . . , Xp)
T (X1, . . . , DXjXl, . . . ,
N , . . . , Xp)−
T (X1, . . . ,
N , . . . , DXlXj , . . . , Xp)
< Xl, Xj > T (X1, . . . ,
N , . . . ,
N , . . . , Xp).
So we get, in restriction to Sn, since DNN = 0
∆IRn+1(X1, . . . , Xp)−∇
∗∇T (X1, . . . , Xp) =
p(n + 1)T (X1, . . . , Xp)− nLNT (X1, . . . , Xp)− 2
DXj(iN,jT )(X1, . . . , X̂j, . . . , Xp)
< Xl, Xj > T (X1, . . . ,
N , . . . ,
N , . . . , Xp)−N.N.T (X1, . . . , Xp)
N.T (X1, . . . , DNXj, . . . , Xp)−
T (X1, . . . , DNDNXj , . . . , Xp)
T (X1, . . . , DNXi, . . . , DNXj, . . . , Xp).
Remark that, in restriction to Sn, the following equality holds
DXj (iN,jT )(X1, . . . , X̂j, . . . , Xp) =
DXj (i−→r ,jT )(X1, . . . , X̂j, . . . , Xp).
Now by using (7) and (11) and by taking the restriction to Sn, we have
N.T (X1, . . . , DNXj, . . . , Xp) =
N.T (X1, . . . , [N,Xj], . . . , Xp) + 2
)T (X1, . . . , Xj, . . . , Xp)
N.T (X1, . . . , Xj, . . . , Xp)− 2
N(< Xj , N >)T (X1, . . . ,
N , . . . , Xp) =
N.T (X1, . . . , [N,Xj], . . . , Xp)− 2pT (X1, . . . , Xp) + 2pN.T (X1, . . . , Xj, . . . , Xp)
N(< Xj , N >)T (X1, . . . ,
N , . . . , Xp).
T (X1, . . . , DNDNXj, . . . , Xp) =
T (X1, . . . , [N, [N,Xj], . . . , Xp)− 2
N(< Xj, N >)T (X1, . . . ,
N , . . . , Xp).
T (X1, . . . , DNXi, . . . , DNXj, . . . , Xp) =
T (X1, . . . , [N,Xi], . . . , [N,Xj ], . . . , Xp) +
p(p− 1)
T (X1, . . . , Xp)
T (X1, . . . , Xi, . . . , [N,Xj], . . . , Xp) +
T (X1, . . . , [N,Xi], . . . , Xj , . . . , Xp).
So we get, in restriction to Sn
−N.N.T (X1, . . . , Xp) + 2
N.T (X1, . . . , DNXj , . . . , Xp)
T (X1, . . . , DNDNXj, . . . , Xp)− 2
T (X1, . . . , DNXi, . . . , DNXj, . . . , Xp) =
−LN ◦ LNT (X1, . . . , Xp) + 2pLNT (X1, . . . , Xp)− p(1 + p)T (X1, . . . , Xp).
The curvature of Sn is given by
R(X, Y )Z =< X, Y > Z− < Y,Z > X and r(X) = (n− 1)X.
Hence, a direct computation gives that the curvature operator is given by
R(T )(X1, . . . , Xp) = p(n− 1)T (X1, . . . , Xp) + 2
T σ(X1, . . . , Xp)
< Xi, Xj > T (X1, . . . , El, . . . , El, . . . , Xp).
Finally, we get
i∗(∆IRn+1T ) = ∆Sni
∗T + i∗ (p(1− p)T + (2p− n)LNT − LN ◦ LNT
T σ +O(T )
One can conclude the proof by remarking that
i∗(LNT ) = i
∗(L−→r T ) and i
∗(LN ◦LNT ) = −i
∗(L−→r T )+ i
∗(L−→r ◦L−→r T ).
Q.E.D.
Corollary 2.1 Let α be a differential p-form on IRn+1. Then
i∗(∆IRn+1α) = ∆Sni
∗α + i∗
(2p− n+ 1)L−→r α− L−→r ◦ L−→r α− 2di−→r α
Corollary 2.2 Let T be a symmetric p-tensor field on IRn+1. Then
i∗(∆IRn+1T ) = ∆Sni
∗T + i∗
2p(1− p)T + (2p− n + 1)L−→r T − L−→r ◦ L−→r T
− 2δ∗(i−→r T ) + 2Tr(T )⊙ <,>
where ⊙ is the symmetric product.
3 Eigenvalues and eigenforms of ∆Sn acting
on Ω(Sn)
In this section, we will use corollary 2.1 and the results developed in [10] to
deduce the eigenvalues and the spaces of eigenforms of ∆Sn acting on Ω
∗(Sn).
We recover the results of [10] and [11] in a more precise form.
Let ∧pHk be the space of all coclosed harmonic homogeneous p-forms of
degree k on IRn+1. A differential form α belongs to ∧pHk if δ(α) = 0 and α
can be written
1≤i1<...<ip≤n+1
αi1...ipdxi1 ∧ . . . ∧ dxip ,
where αi1...ip are harmonic polynomial functions on IR
n+1 of degree k. For
any α ∈ ∧pHk, we have
L−→r α = di−→r α + i−→r dα = (k + p)α. (12)
We have (see [10]),
∧pHk −→ Ω
p(Sn)
is injective and its image is dense.
For any α ∈ ∧pHk, we put
ω(α) = α−
di−→r α. (13)
Lemma 3.1 We get a linear map ω : ∧pHk −→ ∧
pHk which is a projector,
i.e., ω ◦ ω = ω. Moreover,
Kerω = d(∧p−1Hk+1), Imω = ∧
pHk ∩Keri−→r ,
and hence
∧pHk = ∧
pHk ∩Keri−→r ⊕ d(∧
p−1Hk+1).
The following lemma is an immediate consequence of Corollary 2.1 and (12).
Lemma 3.2 1. For any α ∈ ∧pHk ∩Keri−→r , we have
∗α = (k + p)(k + n− p− 1)i∗α.
2. For any α ∈ d(∧p−1Hk+1), we have
∗α = (k + p)(k + n− p+ 1)i∗α.
Remark 3.1 We have
(k+ p)(k+n− p− 1) = (k′ + p)(k′ +n− p+1) ⇔ k = k′ +1 and n = 2p.
The following Table gives explicitly the spectra of ∆Sn and the spaces of
eigenforms with its multiplicities . The multiplicity was computed in [11].
Table I
p The eigenvalues The space of eigenforms Multiplicity
p = 0 k(k + n− 1), k ∈ IN ∧0Hk
(n+k−2)!(n+2k−1)
k!(n−1)!
1 ≤ p ≤ n, (k + p)(k + n− p− 1), ω(∧pHk)
(n+k−1)!(n+2k−1)
p!(k−1)!(n−p−1)!(n+k−p−1)(k+p)
n 6= 2p k ∈ IN∗
(k + p)(k + n− p+ 1), d(∧p−1Hk+1)
(n+k)!(n+2k+1)
(p−1)!k!(n−p)!(n+k−p+1)(k+p)
k ∈ IN
1 ≤ p ≤ n, (k + p)(k + p+ 1)
n = 2p k ∈ IN ω(∧pHk+1)⊕ d(∧
p−1Hk+1)
2(2p+k)!(2p+2k+1)
p!(p−1)!k!(k+p+1)(k+p)
4 Eigenvalues and eigentensors of ∆Sn acting
on S(Sn)
This section is devoted to the determination of the eigenvalues and the spaces
of eigentensors of ∆Sn acting on S(S
Let SpPk be the space of T ∈ S
p(IRn+1) of the form
1≤i1≤...≤ip≤n+1
Ti1...ipdxi1 ⊙ . . .⊙ dxip,
where Ti1...ip are homogeneous polynomials of degree k. We put
SpHδk = S
pPk ∩Ker∆IRn+1 ∩Kerδ and S
pHδ0k = S
pHδk ∩KerTr.
In a similar manner as in [10] Lemma 6.4 and Corollary 6.6, we have
SpPk = S
pHδk ⊕ (r
2SpPk−2 + dr
2 ⊙ Sp−1Pk−1), (14)
SpHδk −→ S
is injective and its image is dense in SpSn.
Now, for any k ≥ 0, we proceed to give a direct sum decomposition of SpHδk
consisting of eigenspaces of ∆Sn and, hence, we determine completely the
eigenvalues of ∆Sn acting on S
p(Sn). This will be done in several steps.
At first, we have the following direct sum decomposition:
SpHδk = S
pHδ0k ⊕
Sp−2lHδ0k ⊙ <,>
l, (15)
where <,>l is the symmetric product of l copies of <,>.
The task is now to decompose SpHδ0k as a sum of eigenspaces of ∆Sn and
get, according to (5), all the eigenvalues. This decomposition needs some
preparation.
Lemma 4.1 Let T ∈ SpPk and h ∈ IN
∗. Then we have the following formu-
1. δ∗(i−→r T )− i−→r δ
∗(T ) = (p− k)T ;
2. δ∗(h)(i−→r T )− i−→r δ
∗(h)(T ) = h(p− k + h− 1)δ∗(h−1)(T );
3. δ∗(i−→r
hT )− i−→r
hδ∗(T ) = h(p− k − h+ 1)i−→r
h−1T,
where i−→r
︷ ︸︸ ︷
i−→r ◦ . . . ◦ i−→r and δ
∗(h) =
︷ ︸︸ ︷
δ∗ ◦ . . . ◦ δ∗ .
Proof. The first formula is easily verified and the others follow by induction
on h. Q.E.D.
Now, we will construct two linear maps pδ∗ : S
pPk −→ S
pPk for k ≤ p, and
p−→r : S
pPk −→ S
pPk for k ≥ p satisfying:
1. pδ∗ ◦ pδ∗ = pδ∗ , Kerpδ∗ = i−→r (S
p+1Pk−1), Impδ∗ = Kerδ
∗ ∩ SpPk;
2. p−→r ◦ p−→r = p−→r , Kerp−→r = δ
∗(Sp−1Pk+1), Imp−→r = Keri−→r ∩ S
The procedure is to put, for T ∈ SpPk,
pδ∗(T ) =
αsi−→r
sδ∗(s)(T ), and p−→r (T ) =
∗(s)(i−→r
sT ),
and find (α0, . . . , αk) and (β0, . . . , βp) such that the required properties are
satisfied.
A straightforward computation using Lemma 4.1 gives
δ∗(pδ∗(T )) =
(αs − (s+ 1)(k − p− s− 2)αs+1)i−→r
sδ∗(s+1)(T ),
i−→r (p−→r (T )) =
(βs − (s+ 1)(p− k − s− 2)βs+1)δ
∗(s)(i−→r
s+1T ).
Hence, we define pδ∗ and p−→r as follows:
pδ∗(T ) =
αsi−→r
sδ∗(s)(T )
α0 = 1 and αs − (s+ 1)(k − p− s− 2)αs+1 = 0 for 1 ≤ s ≤ k − 1;
p−→r (T ) =
∗(s)(i−→r
β0 = 1 and βs − (s+ 1)(p− k − s− 2)βs+1 = 0 for 1 ≤ s ≤ p− 1.
From this definition and by using Lemma 4.1, one can check easily that pδ∗
and p−→r satisfy the required properties.
On other hand, it is easy to check that we have, for any symmetric tensor
field T on IRn+1,
∆IRn+1(i−→r T ) = i−→r ∆IRn+1(T ) + 2δT, (16)
δ(i−→r T ) = i−→r δ(T )− Tr(T ), (17)
Tr(δ∗(T )) = −2δ(T ) + δ∗(Tr(T )), (18)
Tr(i−→r T ) = i−→r Tr(T ). (19)
From these formulas and from (3), one deduce easily that pδ∗(S
pHδ0k ) ⊂
SpHδ0k and p−→r (S
pHδ0k ) ⊂ S
pHδ0k and thus one get the following direct sum
decompositions:
SpHδ0k = S
pHδ0k ∩Kerδ
∗ ⊕ i−→r
Sp+1Hδ0k−1
, if k ≤ p, (20)
SpHδ0k = S
pHδ0k ∩Keri−→r ⊕ δ
Sp−1Hδ0k+1
, if k ≥ p. (21)
These decompositions are far for being sufficient and, in order to obtain
a more sharp direct sum decompositions of SpHδ0k , we need the following
lemma.
Lemma 4.2 1. For k < p, i−→r : S
pPk −→ S
p−1Pk+1 is injective.
2. For k > p, δ∗ : SpPk −→ S
p+1Pk−1 is injective.
3. For k = p, Kerδ∗ = Keri−→r .
Proof.
1. Let T ∈ SpPk such that i−→r T = 0. The second formula in Lemma 4.1
gives, for any h ≥ 1,
i−→r δ
∗(h)(T ) = −h(p− k + h− 1)δ∗(h−1)(T ).
Since δ∗(h)(T ) = 0 for h ≥ k + 1 and h(p − k + h − 1) 6= 0 for any
h ≥ 1, we get form this relation that δ∗(h−1)(T ) = 0 for any h ≥ 1, in
particular for h = 1, we get T = 0.
2. The same argument as 1. using the third formula in Lemma 4.1.
3. Let T ∈ SpPp such that i−→r T = 0. From Lemma 4.1, we get i−→r δ
∗(T ) =
0. Since δ∗(T ) ∈ Sp+1Pp−1 and from 1. we deduce that δ
∗(T ) = 0 and
hence Keri−→r ⊂ Kerδ
∗. The same argument using Lemma 4.1 and 2.
will give the other inclusion. Q.E.D.
By combining (20) and (21) with Lemma 4.2, we obtain the following lemma.
Lemma 4.3 We have:
1. if k < p
SpHδ0k =
Sp+lHδ0k−l ∩Kerδ
2. if k > p
SpHδ0k =
Sp−lHδ0k+l ∩Keri−→r
3. If k = p, for any 0 ≤ l ≤ p,
Sp+lHδ0p−l ∩Kerδ
= δ∗l
Sp−lHδ0p+l ∩Keri−→r
SpHδ0p =
Sp+lHδ0p−l ∩Kerδ
Sp−lHδ0p+l ∩Keri−→r
Now, we use Corollary 2.2 to show that the decompositions of SpHδ0k given
in Lemma 4.3 are composed by eigenspaces of ∆Sn.
Theorem 4.1 We have:
1. If k ≤ p, for any 0 ≤ q ≤ k and any T ∈ i−→r
(k−q)
Sp+k−qHδ0q ∩Kerδ
∗T = ((k + p)(n + p+ k − 2q − 1) + 2q(q − 1)) i∗T ;
2. If k ≥ p, for any 0 ≤ q ≤ p and for any T ∈ δ∗(p−q)
SqHδ0k+p−q ∩Keri−→r
∗T = ((k + p)(n+ p+ k − 2q − 1) + 2q(q − 1)) i∗T.
Proof.
1. Let T = i−→r
(k−q)(T0) with T0 ∈ S
p+k−qHδ0q ∩ Kerδ
∗. We have from
Corollary 2.2
∗T = i∗
2p(p− 1)T + (n− 2p− 1)L−→r T + L−→r ◦ L−→r T
+2δ∗(i−→r T )− 2Tr(T )⊙ <,>
We have
TrT = 0, L−→r = (k + p)T and L−→r ◦ L−→r T = (k + p)
Moreover, by using Lemma 4.1, we have
2δ∗(i−→r T ) = 2δ
∗(i−→r
(k−q+1)T0)
δ∗(T0)=0
= 2(k − q + 1)(p+ k − q − q − k + q − 1 + 1)i−→r
(k−q)T0
= 2(k − q + 1)(p− q)T.
Hence
∗T = (2p(p−1)+(n−2p−1)(k+p)+(k+p)2+2(p−q)(k−q+1))i∗T.
One can deduce the desired relation by remarking that
2p(p− 1) + 2(p− q)(k − q + 1) = 2(k + p)(p− q) + 2q(q − 1).
2. This follows by the same calculation as 1. Q.E.D.
From the fact that
SpHδk −→ S
is injective and its image is dense in SpSn, from (15), and from Lemma 4.3
and Theorem 4.1, note that we have actually proved that the eigenvalues of
∆Sn acting on S
pSn belongs to
{(k + p− 2l)(n+ p+ k − 2l − 2q − 1) + 2q(q − 1),
k ∈ IN, 0 ≤ l ≤ [
], 0 ≤ q ≤ min(k, p− 2l)
Our next goal is to sharpen this result by computing dimSpHδ0k ∩Kerδ
k ≤ p and dimSpHδ0k ∩Keri−→r if k ≥ p.
Lemma 4.4 We have the following formulas:
1. dimSpHδk = dimS
pPk − dimS
pPk−2 − dimS
p−1Pk−1 + dimS
p−1Pk−3,
2. dimSpHδ0k = dimS
pHδk − dimS
p−2Hδk ,
3. dim(SpHδ0k ∩Kerδ
∗) = dimSpHδ0k − dimS
p+1Hδ0k−1 (k ≤ p),
4. dim(SpHδ0k ∩Keri−→r ) = dimS
pHδ0k − dimS
p−1Hδ0k+1 (k ≥ p).
Note that we use the convention that SpPk = S
pHδk = S
pHδ0k = 0 if k < 0 or
p < 0.
Proof.
1. The formula is a consequence of (14), the relation
(r2SpPk−2) ∩ (dr
2 ⊙ Sp−1Pk−1) = r
2(dr2 ⊙ Sp−1Pk−3)
and the fact that dr2 ⊙ . : SpPk −→ S
p+1Pk+1 is injective.
2. The formula is a consequence of (15).
3. The formula is a consequence of (20) and Lemma 4.2.
4. The formula is a consequence of (21) and Lemma 4.2. Q.E.D.
A straightforward calculation using Lemma 4.4 and the formula
dimSpPk =
(n+ p)!
(n+ k)!
gives dimSpHδ0k ∩ Kerδ
∗ if k ≤ p and dimSpHδ0k ∩ Keri−→r if k ≥ p. We
summarize the results on the following Table.
Table II
Space Dimension Conditions on k and p
S0Hδ0k ∩Keri−→r
(n+ k − 2)!(n+ 2k − 1)
k!(n− 1)!
k ≥ 0
SpHδ00 ∩Kerδ
(n+ p− 2)!(n+ 2p− 1)
p!(n− 1)!
p ≥ 0
S1Hδ0k ∩Keri−→r
(n + k − 3)!k(n + 2k − 1)(n+ k − 1)
(n− 2)!(k + 1)!
k ≥ 1
SpHδ01 ∩Kerδ
(n + p− 3)!p(n+ 2p− 1)(n+ p− 1)
(n− 2)!(p+ 1)!
p ≥ 1
SpHδ0k ∩Kerδ
(n + k − 4)!(n+ p− 3)!(n+ p+ k − 2)
k!(p+ 1)!(n− 1)!(n− 2)!
(n− 2)(n+ 2k − 3)(n+ 2p− 1)(p− k + 1) 2 ≤ k ≤ p
SpHδ0k ∩Keri−→r
(n + k − 3)!(n+ p− 4)!(n+ p+ k − 2)
(k + 1)!p!(n− 1)!(n− 2)!
(n− 2)(n+ 2k − 1)(n+ 2p− 3)(k − p+ 1) k ≥ p ≥ 2
Remark 4.1 Note that, for n = 2, we have
dim(SpHδ0k ∩Kerδ
∗) = 0 for 2 ≤ k ≤ p,
dim(SpHδ0k ∩Keri−→r ) = 0 for k ≥ p ≥ 2.
For simplicity we introduce the following notations.
(k, l, q) ∈ IN3, 0 ≤ l ≤ [
], 0 ≤ k ≤ p− 2l, 0 ≤ q ≤ k
(k, l, q) ∈ IN3, 0 ≤ l ≤ [
], k > p− 2l, 0 ≤ q ≤ p− 2l
V kq,l = i−→r
Sp−2l+k−qHδ0q ∩Kerδ
⊙ <,>l for (k, l, q) ∈ S0,
W kq,l = δ
∗(p−2l−q)
SqHδ0p−2l+k−q ∩Keri−→r
⊙ <,>l for (k, l, q) ∈ S1.
Let us summarize all the results above.
Theorem 4.2 1. For n = 2, we have:
(a) The set of the eigenvalues of ∆S2 acting on S
pS2 is
(k + p− 2l)(p+ k − 2l + 1), k ∈ IN, 0 ≤ l ≤ [
(b) The eigenspace associated to the eigenvalue λ(k, l) = (k + p −
2l)(k + p− 2l + 1) is given by
Vλ(k,l) =
min(l,[ k
V k−2a0,l−a ⊕ V
k+1−2a
1,l−a
if 0 ≤ k ≤ p− 2l
min(l,[ k
W k−2a0,l−a ⊕W
k+1−2a
1,l−a
if k > p− 2l;
(c) The multiplicity of λ(k, l) is given by
m(λ(k, l)) = 2(min(l, [
]) + 1)(1 + 2p+ 2k − 4l).
2. For n ≥ 3, we have:
(a) The set of the eigenvalues of ∆Sn acting on S
pSn is
{(k + p− 2l)(n + p+ k − 2l − 2q − 1) + 2q(q − 1),
k ∈ IN, 0 ≤ l ≤ [
], 0 ≤ q ≤ min(k, p− 2l)
(b) The space
SpHδk = (
(k,l,q)∈S0
V kq,l)⊕ (
(k,l,q)∈S1
W kq,l)
is dense in SpSn and, for any (k, q, l) ∈ S0 (resp. (k, q, l) ∈ S1),
V kq,l (resp. W
q,l) is a subspace of the eigenspace associated to the
eigenvalue (k + p− 2l)(n+ p+ k − 2l − 2q − 1) + 2q(q − 1);
(c) The dimensions of V kq,l and W
q,l are given in Table II since
dimV kq,l = dim
Sp−2l+k−qHδ0q ∩Kerδ
for (k, l, q) ∈ S0,
dimW kq,l = dim
SqHδ0p−2l+k−q ∩Keri−→r
for (k, l, q) ∈ S1.
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