The First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces

0.15 MB·English

by  Jean-Louis Milhorat

THE FIRST EIGENVALUE OF THE DIRAC OPERATOR ON
COMPACT SPIN SYMMETRIC SPACES
5
0
0
2 JEAN-LOUISMILHORAT

n
a Abstract. We give a formula for the first eigenvalue of the Dirac operator
J actingonspinorfieldsofaspincompactirreduciblesymmetricspaceG/K.

4
2

]
G
D 1. Introduction
. It is well-known that symmetric spaces provide examples where detailed infor-
h
t mation on the spectrum of Laplace or Dirac operators can be obtained. Indeed,
a
for those manifolds, the computation of the spectrum can be (theoretically) done
m
using group theoretical methods. However the explicit computation is far from
[
being simple in general and only few examples are known. On the other hand,

1 many results require some information about the first (nonzero) eigenvalue, so it
v seems interesting to get this eigenvalue without computing all the spectrum. In
0
that direction, the aim of this paper is to prove the following formula for the first
1
eigenvalue of the Dirac operator:
4
1
Theorem1.1. LetG/K beacompact,simply-connected, n-dimensionalirreducible
0
5 symmetricspacewithGcompactandsimply-connected, endowedwiththemetricin-
0 ducedbytheKilling form ofGsign-changed. AssumethatGandK have samerank
/ andthatG/K hasaspinstructure. Letβ ,k =1,...,p,betheK-dominantweights
h k
t occurring in the decomposition into irreducible components of the spin representa-
a
tion under the action of K. Then the square of the first eigenvalue of the Dirac
m
operator is
:
v
i (1) 2 min kβkk2+n/8,
X 1≤k≤p
r
a where k·k is the norm associated to the scalar product <, > induced by the Killing
form of G sign-changed.
Remark1.2. TheproofusesalemmaofR.Parthasarathyin[Par71], whichallows
to express (1) in the following way. Let T be a fixed common maximal torus of G
and K. Let Φ be the set of non-zero roots of G with respect to T. Let δ , (resp.
G
δ ) be the half-sum of the positive roots of G, (resp. K), with respect to a fixed
K
lexicographic ordering in Φ. Then the square of the first eigenvalue of the Dirac
operator is given by
(2) 2kδ k2+2kδ k2−4 max <w·δ ,δ >+n/8,
G K G K
w∈W
where W is a certain (well-defined) subset of the Weyl group of G.
1
2 JEAN-LOUISMILHORAT
2. The Dirac Operator on a Spin Compact Symmetric Space
We first review some results about the Dirac operator on a spin symmetric
space,cf.forinstance[CFG89]or[B¨ar91]. Adetailedsurveyonthesubjectmaybe
found, among other topics, in the reference [BHMM]. Let G/K be a spin compact
symmetric space. We assume that G/K is simply connected, so G may be chosen
to be compact and simply connected and K is the connected subgroup formed by
the fixed elements of an involution σ of G, cf. [Hel78]. This involution induces the
Cartan decomposition of the Lie algebra G of G into
G=K⊕P,
where K is the Lie algebraof K and P is the vector space {X ∈G; σ ·X =−X}.
∗
ThisspacePiscanonicallyidentifiedwiththetangentspacetoG/K atthepointo,
o being the class of the neutral element of G. We also assume that the symmetric
space G/K is irreducible, so all the G-invariant scalar products on P, hence all
the G-invariant Riemannian metrics on G/K are proportional. We consider the
metricinducedbytheKillingformofGsign-changed. Withthismetric,G/K isan
EinsteinspacewithscalarcurvatureScal=n/2. Thespinconditionimpliesthatthe
homomorphism α : K → SO(P) ≃ SO , k 7→ Ad (k) lifts to a homomorphism
n G |P
α:K →Spinn,cf.[CG88]. Letρ:Spinn →HomC(Σ,Σ)bethespinrepresentation.
The composition ρ◦α defines a “spin” representation of K which is denoted ρ .
e K
The spinor bundle is then isomorphic to the vector bundle
e
Σ:=G× Σ.
ρK
Spinor fields on G/K are then viewed as K-equivariant functions G → Σ, i.e.
functions:
Ψ:G→Σ s.t. ∀g ∈G, ∀k ∈K , Ψ(gk)=ρ (k−1)·Ψ(g).
K
Let L2 (G,Σ) be the Hilbert space of L2 K-equivariant functions G → Σ. The
K
Dirac operator D extends to a self-adjoint operator on L2 (G,Σ). Since it is an
K
elliptic operator, it has a (real) discrete spectrum. Now if the spinor field Ψ is an
eigenvector of D for the eigenvalue λ, then the spinor field σ∗·Ψ is an eigenvector
for the eigenvalue −λ, hence the spectrum of the Dirac operatoris symmetric with
respect to the origin. Thus the spectrum of D may be deduced from the spectrum
ofits squareD2. Bythe Peter-Weyltheorem,the naturalunitary representationof
G on the Hilbert space L2 (G,Σ) decomposes into the Hilbert sum
K
⊕ V ⊗Hom (V ,Σ),
γ K γ
γ∈G
b
where G is the set of equivalence classes of irreducible unitary complex represen-
tationsbof G, (ρ ,V ) represents an element γ ∈G and Hom (V ,Σ) is the vector
γ γ K γ
space of K-equivariant homomorphisms Vγ →Σ,bi.e.
Hom (V ,Σ)={A∈Hom(V ,Σ)s.t.∀k ∈K,A◦ρ (k)=ρ (k)◦A}.
K γ γ γ K
The injection V ⊗Hom (V ,Σ)֒→L2 (G,Σ) is given by
γ K γ K
v⊗A7→ g 7→(A◦ρ (g−1))·v .
(cid:16) γ (cid:17)
NotethatV ⊗Hom (V ,Σ)consistsofC∞spinorfieldstowhichtheDiracoperator
γ K γ
can be applied. The restriction of D2 to the space V ⊗Hom (V ,Σ) is given by
γ K γ
THE FIRST EIGENVALUE OF THE DIRAC OPERATOR... 3
the Parthasaratyformula, [Par71]:
Scal
(3) D2(v⊗A)=v⊗(A◦C )+ v⊗A,
γ
8
where C is the Casimir operator of the representation (ρ ,V ). Now since the
γ γ γ
representation is irreducible, the Casimir operator is a scalar multiple of identity,
C =c id,wheretheeigenvaluec onlydependsofγ ∈G. HenceifHom (V ,Σ)6=
γ γ γ K γ
{0}, cγ +n/16 belongs to the spectrum of D2. Let ρKb= ⊕ρK,k be the decompo-
sition of the spin representation K → Σ into irreducible components. Denote by
m(ρ ,ρ ) the multiplicity of the irreducible K-representation ρ in the rep-
γ|K K,k K,k
resentation ρ restricted to K. Then
γ
dim Hom (V ,Σ)= m(ρ ,ρ ).
K γ X γ|K K,k
k
So the spectrum of the square of the Dirac operator is
(4) Spec(D2)={c +n/16; γ ∈Gs.t.∃k s.t.m(ρ ,ρ )6=0}.
γ γ|K K,k
b
3. Proof of the result
We assume that G and K have same rank. Let T be a fixed common maximal
torus. Let Φ be the set of non-zero roots of the group G with respect to T. Ac-
cording to a classical terminology, a root θ is called compact if the corresponding
rootspace is containedin KC (that is, θ is a rootof K with respect to T)and non-
compactifthe rootspaceis containedinPC. LetΦ+G be the setofpositiverootsof
G, Φ+ be the set of positive roots of K, andΦ+ be the set of positive noncompact
K n
roots with respect to a fixed lexicographic ordering in Φ. The half-sums of the
positive roots of G and K are respectively denoted δ and δ and the half-sum
G K
of noncompact positive roots is denoted by δ . The Weyl group of G is denoted
n
W . The space of weights is endowed with the W -invariant scalar product <, >
G G
induced by the Killing form of G sign-changed.
Let
(5) W :={w∈W ; w·Φ+ ⊃Φ+}.
G G K
ByaresultofR.Parthasaraty,cf.lemma2.2in[Par71],thespinrepresentationρ
K
of K decomposes into the irreducible sum
(6) ρ = ρ ,
K M K,w
w∈W
where ρ has for dominant weight
K,w
(7) β :=w·δ −δ .
w G K
Now define w0 ∈W such that
(8) kβ k2 = min kβ k2,
w0 w
w∈W
and
(9)
if there exists a w1 6=w0 ∈W such that kβw1k2 = min kβwk2, then βw1 ≺βw0,
w∈W
where ≺ is the usual ordering on weights.
4 JEAN-LOUISMILHORAT
Lemma 3.1. The weight
βG :=w−1·β =δ −w−1·δ ,
w0 0 w0 G 0 K
is G-dominant.
Proof. Let ΠG = {θ1,...,θr} ⊂ Φ+G be the set of simple roots. It is sufficient to
prove that 2<βwG0,θi> is a non-negative integer for any simple root θ . Since T is
<θi,θi> i
a maximal common torus of G and K, β , which is an integral weight for K is
w0
alsoanintegralweightforG. Nowsincethe WeylgroupW permutestheweights,
G
βG = w−1·β is also a integral weight for G, hence 2<βwG0,θi> is an integer for
w0 0 w0 <θi,θi>
any simple root θ . So we only have to prove that this integer is non-negative.
i
Let θ be a simple root. Since 2<δG,θi> = 1, (see for instance § 10.2 in [Hum72])
i <θi,θi>
and since the scalar product <·,·> is W -invariant, one gets
G
(10) 2<βwG0,θi > =1−2<δK,w0·θi >.
<θ ,θ > <θ ,θ >
i i i i
Suppose first that w0·θi ∈ΦK. If w0·θi is positive then w0·θi is necessarilya K-
simple root. Indeed let Π ={θ′,...,θ′}⊂Φ+ be the set of K-simple roots. One
K 1 l K
hasw0·θi =Plj=1bijθj′,wherethebij arenon-negativeintegers. Butsincew0 ∈W,
there are l positive roots α1,...,αl in Φ+G such that w0·αj = θj′, j = 1,...,l. So
l r
θ = b α . Now each α is a sum of simple roots a θ , where the
i Pj=1 ij j j Pk=1 jk k
a are non-negative integers. So θ = b a θ . By the linear independence
jk i Pj,k ij jk k
of simple roots, one gets b a = 0 if k 6= i, and b a = 1. Hence
Pj ij jk Pj ij ji
there exists a j0 such that bij0 = aj0i = 1, the other coefficients being zero. So
w0 ·θi = θj′0 is a K-simple root. Now since 2<δ<Kθ,iw,θ0i·>θi> = 2<<wδ0K·θ,iw,w0·0θ·iθ>i> = 1,
one gets 2<βwG0,θi> = 0, hence 2<βwG0,θi> ≥ 0. Now, the same conclusion holds if
<θi,θi> <θi,θi>
w0 ·θi is a negative root of K, since 2<δ<Kθ,iw,θ0i·>θi> = −2<δK<,θ−i,wθi0>·θi> = −1, hence
2<βwG0,θi> =2.
<θi,θi>
Suppose now that w0·θi ∈/ ΦK, that is w0·θi is a noncompact root. This implies
that w0σi, where σi is the reflectionacrossthe hyperplane θi⊥, is an element of W.
Let α1,...,αm be the positive roots in Φ+G such that w0·αj = α′j, where the α′j,
j =1,...,marethepositiverootsofK. Sinceσ permutesthepositiverootsother
i
than θ , (cf. for instance Lemma B, § 10.2 in [Hum72]), and since θ can not be
i i
one of the roots α1,...,αm (otherwise w0·θi ∈ Φ+K), each root σi·αj is positive.
So w0σi ∈W since w0σi·(σi·αj)=α′j, j =1,...,m.
We now claim that 2<βwG0,θi> < 0, which is equivalent to 2<δK,w0·θi> > 1, is
<θi,θi> <θi,θi>
impossible.
Suppose that
<δK,w0·θi >
(11) 2 >1.
<θ ,θ >
i i
Since δ can be expressed as δ = l c θ′, where the c are nonnegative, there
exists aKK-simple root θj′ suchKthatP<i=θ1j′,wi 0i·θi > 0, andisince 2<<θj′θ,′w,θ0·′θ>i> is an
j j
THE FIRST EIGENVALUE OF THE DIRAC OPERATOR... 5
integer, this implies that
<θj′,w0·θi >
(12) 2 ≥1.
<θ′,θ′ >
j j
So θj′ −w0 ·θi is a root (cf. for instance § 9.4 in [Hum72]). Moreover, from the
bracketrelation[K,P]⊂P,itisanoncompactroot. Now±(θj′−w0·θi)isapositive
noncompact root, so by the description of the weights of the spin representation
ρ , (they are of the form: δ −(a sum of distinct positive noncompact roots), cf.
K n
§2 in [Par71]),
(w0·δG−δK)±(θj′ −w0·θi) is a weight of ρK.
Now, (w0 ·δG −δK)+(θj′ −w0 ·θi) can not be a weight of ρK. Otherwise since
σi ·δG = δG −θi, (w0σi ·δG −δK)+θj′ is a weight of ρK. But since w0σi ∈ W,
µ:=w0σi·δG−δK isadominantweightofρK. Soµisadominantweightbutnotthe
highestweightofanirreduciblecomponentofρ . Hencethereexistsanirreducible
K
representation of ρ with dominant weight λ =w·δ −δ , w ∈W, whose set of
K G K
weightsΠcontainsµ. Furthermoreµ≺λ. Nowsinceµ∈Π,kµ+δ k2 ≤kλ+δ k2,
K K
with equality only if µ = λ, (cf. for instance Lemma C, §13.4 in [Hum72]). But
kµ+δ k2 =kδ k2 =kλ+δ k2, so µ=λ, contradicting the fact that µ≺λ.
K G K
Thus only
(13) µ0 :=(w0·δG−δK)−(θj′ −w0·θi),
can be a weight of ρ . Now one has
K
kµ0k2 = kw0·δG−δK +w0·θik2
−2 <w0·δG−δK +w0·θi,θj′ >+kθj′k2.
Since w0·δG−δK is a dominant weight, < w0·δG−δK,θj′ >≥ 0, and from (12),
2 <w0·θi,θj′ >−kθj′k2 ≥0, so
kµ0k2 ≤k(w0·δG−δK)+w0·θik2.
Now
k(w0·δG−δK)+w0·θik2 = kw0·δG−δKk2
+2 <δ −w−1·δ ,θ >+kθ k2.
G 0 K i i
But, as we supposed 2<βwG0,θi> <0, one has 2<δG−w0−1·δK,θi> ≤−1, so
<θi,θi> kθik2
2 <δ −w−1·δ ,θ >+kθ k2 ≤0, hence
G 0 K i i
k(w0·δG−δK)+w0·θik2 ≤kw0·δG−δKk2,
so
kµ0k2 ≤kw0·δG−δKk2.
Now,beingaweightofρK,µ0isconjugateundertheWeylgroupofKtoadominant
weight of ρK, say w1·δG−δK, with w1 ∈W. Note that w1 6=w0, otherwise since
µ0 ≺w1·δG−δK,(cf. LemmaA,§13.2in[Hum72]),thenoncompactrootθj′−w0·θi
should be a linear combination with integral coefficients of compact simple roots.
But, by the bracket relation [K,K]⊂K, that is impossible. Thus, by the definition
of w0, cf. (8), kw0·δG−δKk2 ≤kw1·δG−δKk2 =kµ0k2, so
kµ0k2 =kw1·δG−δKk2 =kw0·δG−δKk2.
But by the condition (9), the last equality is impossible, otherwise since µ0 ≺
w1 ·δG −δK and w1 ·δG −δK ≺ w0 ·δG −δK, the noncompact root θj′ −w0 ·θi
6 JEAN-LOUISMILHORAT
should be a linear combination with integral coefficients of compact simple roots.
Hence 2<<βθwGi0,θ,θi>i> ≥0 also if w0·θi ∈/ ΦK. (cid:3)
Nowlet(ρ0,V0)beanirreduciblerepresentationofGwithdominantweightβwG0.
Thefactthatβw0 =w0·βwG0 isaweightofρ0 isanindicationthatρ0|K maycontain
the irreducible representationρ . This is actually true:
K,w0
Lemma 3.2. With the notations above,
m(ρ0|K,ρK,w0)≥1.
Proof. Let v0 be the maximal vector in V0, (it is unique up to a nonzero scalar
multiple). Let g0 ∈T be a representative of w0. Then g0·v0 is a weight vector for
the weight β , since for any X in the Lie algebra T of T:
w0
X ·(g0·v0) = ddt(cid:16)(exp(tX)g0)·v0(cid:17)|t=0 = ddt(cid:16)(cid:16)g0g0−1exp(tX)g0(cid:17)·v0(cid:17)|t=0
=g0·(cid:16)(cid:16)Ad(g0−1)·X(cid:17)·v0(cid:17) =βwG0(w0−1·X)(g0·v0)
=(w0·βwG0)(X)(g0·v0) =βw0(X)(g0·v0).
In order to prove the result, we only have to prove that g0·v0 is a maximal vector
(for the action K), hence is killed by root-vectors corresponding to simple roots
of K. So let θ′ be a simple root of K and E′ be a root-vector corresponding to
i i
that simple root. Since w0 ∈ W, there exists a positive root αi ∈ Φ+G such that
w0·αi =θi′. Then Ei :=Ad(g0−1)(Ei′) is a root-vectorcorrespondingto the rootαi
since for any X in T
[X,Ei] =[X,Ad(g0−1)(Ei′)] =Ad(g0−1)·[Ad(g0)(X),Ei′]
=Ad(g0−1)·[w0·X,Ei′] =(cid:16)(w0−1·θi′)(X)(cid:17)Ad(g0−1)·Ei′
=α (X)E .
i i
But since v0 is killed by the action of the root-vectors corresponding to positive
roots in Φ+, one gets
G
Ei′·(g0·v0) = ddt(cid:16)(cid:16)g0g0−1exp(tEi′)g0(cid:17)·v0(cid:17)|t=0
= ddt(cid:16)(cid:16)g0exp(cid:16)tAd(g0−1)·Ei′(cid:17)(cid:17)·v0(cid:17)|t=0
=g0·(cid:16)Ei·v0(cid:17)
=0.
Hence the result. (cid:3)
From the result (4), we may then conclude:
Lemma 3.3.
2kβ k2+n/8,
w0
is an eigenvalue of the square of the Dirac operator.
Proof. By the Freudenthal’s formula, the Casimir eigenvalue c of the representa-
γ0
tion (ρ0,V0) is given by
kβwG0 +δGk2−kδGk2 =3kδGk2+kδKk2−4 <w0·δG,δK > .
On the other hand
kβw0k2 =kδGk2+kδKk2−2 <w0·δG,δK > .
THE FIRST EIGENVALUE OF THE DIRAC OPERATOR... 7
Hence
c =2kβ k2+kδ k2−kδ k2.
γ0 w0 G K
Now, the Casimir operator of K acts on the spin representation ρ as scalar mul-
K
tiplication by kδ k2−kδ k2, (cf. lemma 2.2 in [Par71]). Indeed, each dominant
G K
weight of ρ being of the form w·δ −δ , w ∈W, the eigenvalue of the Casimir
K G K
operator on each irreducible component is given by:
k(w·δ −δ )+δ k2−kδ k2 =kw·δ k2−kδ k2 =kδ k2−kδ k2.
G K K K G K G K
On the other hand, the proof of the formula (3) shows that the Casimir operator
ofK acts on the spin representationρ as scalarmultiplication by Scal =n/16(cf.
K 8
[Sul79]), hence
(14) kδ k2−kδ k2 =n/16.
G K
So
c +n/16=2kβ k2+n/8.
γ0 w0
(cid:3)
In order to conclude, we have to prove that
Lemma 3.4.
2kβ k2+n/8,
w0
is the lowest eigenvalue of the square of the Dirac operator.
Proof. Let γ ∈ G such that there exists w ∈ W such that m(ρ ,ρ ) ≥ 1. Let
γ|K K,w
β be thedominbantweightofρ . First,sincetheWeylgrouppermutestheweights
γ γ
of ρ , w−1·β =δ −w−1·δ is a weight of ρ . Hence
γ w G K γ
kβ +δ k2 ≥kw−1·β +δ k2,
γ G w G
(cf. for instance Lemma C, §13.4 in [Hum72]). So, from the Freudenthal formula,
c =kβ +δ k2−kδ k2 ≥kw−1·β +δ k2−kδ k2.
γ γ G G w G G
But, using (14)
kw−1·β +δ k2−kδ k2 =2kβ k2+kδ k2−kδ k2 =2kβ k2+n/16.
w G G w G K w
Hence by the definition of β ,
w0
c ≥2kβ k2+n/16≥2kβ k2+n/16.
γ w w0
Hence the result. (cid:3)
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[Hel78] S.Helgason,DifferentialGeometry, LieGroups, and Symmetric Spaces,PureandAp-
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8 JEAN-LOUISMILHORAT
[Sul79] S.Sulanke,Die Berechnung desSpektrums desQuadrates des Dirac-Operators auf der
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Laboratoire Jean Leray, UMRCNRS 6629,D´epartementde Math´ematiques,Univer-
sit´ede Nantes, 2,ruede la Houssini`ere, BP92208,F-44322NANTES CEDEX 03
E-mail address: [email protected]