In this section we will x notation and give the denitions, which are needed to study the compact hyperbolic odd dimensional manifolds.
Let G = Spin(d,1) and K = Spin(d) or G = SO0(d,1) and K = SO(d), for d = 2n+ 1, n ∈ N. Then, K is a maximal compact subgroup of G. Let g,k be the Lie algebras of Gand K, respectively. We denote by Θthe Cartan involution of G and θ the dierential of Θ at eG = e, the identity element of G. It holds θ2 = Idg. Hence, there exist subspaces p and k of g, such that p is the eigenspace for the (−1)-eigenvalue and k is the eigenspace for the (+1)-eigenvalue of θ. The Cartan decomposition of g is given by
g=k⊕p. (1.1)
We have
[k,k]⊆k, [k,p]⊆p, [p,p]⊆k.
Let a be a Cartan subalgebra of p, i.e. a maximal abelian subalgebra of p. We consider the subgroup A of G with Lie algebra a. Let M be the centralizer of A inK. Then, ThenM = Spin(d−1)orSO(d−1). Letm be its Lie algebra. Letb be a Cartan subalgebra of m and h a Cartan subalgebra of g.
We consider the complexications
gC:=g⊕ig hC:=h⊕ih mC :=m⊕im.
19
20 CHAPTER 1. PRELIMINARIES We want to use the theorem of the highest weight for the groups K and M. We recall here some basic facts from the representation theory of compact reductive Lie groups.
Let (Φ, V) be a nite dimensional complex representation of a compact linear connected reductive group K, with Lie algebra gK. By [Kna86, Proposition 1.6], we can regard Φ as unitary. Let φ be the dierential of Φ at e. Then, φ(Y) is skew symmetric for every Y ∈ gK. Let j be a Cartan subalgebra of gK. Let (Hi), i = 1, . . . , N be a basis for j. The matrices φ(Hi) are diagonalizable with imaginary eigenvalues, and sinceHi commute with each other, so doφ(Hi). Hence, there exists a simultaneous eigenspace decomposition of V under φ(Hi), which can be extended to an eigenspace decomposition under φ(jC), where jC is the complexied algebra ofj (with real eigenvalues).
Denition 1.1. A weight λ(H) ∈ (jC)∗ of the representation φ is a linear func- tional on jC such that
φ(H)v =λ(H)v, wherev ∈V with v 6= 0.
A weight space Vλ is a subspace of V, which is an eigenspace for the eigenvalue λ(H), i.e.
Vλ :={v ∈V :φ(H)v =λ(H)v,∀H ∈jC}.
The weight space decomposition is a decomposition of V, given by the nite sum over the weights
V =M
λ
Vλ.
Let (gK)C be the complexication of gK. We dene the trace form on (gK)C× (gK)C by B0(X1, X2) := Tr(X1X2). It is a complex-valued symmetric bilinear form. We dene an inner product hã,ãi on(ijC)∗ by
hλ1, λ2i:=B0(Hλ1, Hλ2) =λ1(Hλ2) = λ2(Hλ1). (1.2) Denition 1.2. We call root a non zero weight α for the representationφ = ad : (gK)C→gl(gK). The corresponding root space decomposition is given by
(gK)C= (gK)0⊕ X
α∈∆((gK)C,jC)
(gK)α,
where∆((gK)C,jC)denotes the set of roots for the adjoint representation of(gK)C. Denition 1.3. Let (H1, . . . , Hk) be a basis for ij. We say that α ∈ (jC)∗ is positive if it is real on ij and α = α(Hi) > 0, where i = 1, . . . , r, and α(Hj) = 0 for all j = 1, . . . , r−1.
1.1. COMPACT HYPERBOLIC ODD DIMENSIONAL MANIFOLDS 21 We write α >0 for a positive root α, and α1 > α2, if α1−α2 >0. We denote the set of the positive roots by ∆+((gK)C,jC).
Theorem 1.4 (The theorem of highest weight). Let K be a compact linear con- nected reductive group. Apart from equivalence, the irreducible representations Φ of Kstand in one-to-one correspondence with the highest weightsλ∈(jC)∗ (largest weight in the ordering) of Φλ.
Proof. See [Kna86, Theorem 4.28].
We turn now to the case, where G= Spin(d,1),g denotes its Lie algebra, and p and a are as in the beginning of this section. Let(X1, X2) be the inner product on g×g, dened by
(X1, X2) := −Re(B0(X1, θX2)). (1.3) The adjoint operator ad(p) is a symmetric operator on g with respect to the inner product (1.2). Hence, if we restrict ad to a, we get a commuting family of symmetric transformations on g, which can be simultaneously diagonalized. Let
∆(g,a), ∆+(g,a) ⊂ ∆(g,a) be the sets of the restricted, respectively, positive restricted roots of the system (g,a). We dene
n:= M
λ∈∆+(g,a)
gλ.
Proposition 1.5. Let g be the Lie algebra of G. Then, g decomposes as g=k⊕a⊕n,
whereain an abelian subalgebra, nis nilpotent, a⊕nis solvable, and [a⊕n,a⊕n] = n.
Proof. See [Kna86, Proposition 5.10].
Theorem 1.6 (Iwasawa decomposition of the Lie group G). Let A and N be the analytic subgroups of Gwith Lie algebrasaandn. Then,A, N, and AN are simply connected closed subgroups of G, and the multiplication map K ×A×N → G, given by (k, a, n)→kan, is a dieomorphism onto.
Proof. See [Kna86, Theorem 5.12].
Let C(X, Y) := Re(B0(X, Y)) be the real part of the trace form on g×g. We choose a basis (Xi) for g and set cij = C(Xi, Xj). Then, since C(ã,ã) is a non-degenerate form, the matrix C := (cij) is non-singular. We denote the inverse matrix of C by C−1 = (cij) := (cij)−1. We put Xj = P
cijXi, so that Xi =P
cjiXj. Let U(gC)be the universal enveloping algebra of gC.
22 CHAPTER 1. PRELIMINARIES Denition 1.7. We dene the Casimir elementΩ∈U(gC) by
Ω :=X
i,j
XiXj.
Proposition 1.8. The Casimir element Ω is independent of the basis (Xi). Fur- thermore, it satises Ad(g)Ω = Ω for all g ∈G and hence is in the center Z(gC) of U(gC).
Proof. This is proved in [Kna86, Proposition 8.6].
Let(ã,ã)be the inner product ong, dened by (1.3). Let(Xi)be an orthonormal basis of p and (Yj) an orthonormal basis of k, with respect to this inner product.
By Denition 1.7 we have
Ω =X
i
Xi2−X
j
Yj2 ΩK =−X
j
Yj2.
HereΩKdenotes the Casimir element, which corresponds to the restriction(ã,ã)|kìk. It lies in the center Z(k) of the universal enveloping algebra U(k)of k.
If we consider a nite dimensional unitary irreducible representation (τ, Vτ) of K, then, sinceΩ∈Z(k), Schur's Lemma (cf. [Kna86, Proposition 1.5]) assures us that τ(ΩK) acts by a scalar λτ, called the Casimir eigenvalue of τ. Then,
τ(ΩK) =λτIdVτ.
The group SO0(d,1)acts transitively on the hyperbolic space:
Hd={(x1, . . . , xd+1)∈Rd+1 :x21−x22. . .−x2d+1 = 1, x1 >0}.
The stabilizer of the point (1,0, . . . ,0) is SO(d), which is a maximal compact subgroup of SO0(d,1). G and K are the universal covering groups of SO0(d,1) and SO(d), respectively. We set
Xe :=G/K.
LetTeKXe be the tangent space ofXe ateK ∈Xe. There is a canonical isomorphism TeKXe ∼=p.
LetB(X, Y) be the Killing form ong×g dened by B(X, Y) = Tr(ad(X)◦ad(Y)).
1.1. COMPACT HYPERBOLIC ODD DIMENSIONAL MANIFOLDS 23 It is a symmetric bilinear from. We consider the inner product hã,ãi0, induced by the Killing form
hY1, Y2i0 := 1
2(d−1)B(Y1, Y2), Y1, Y2 ∈g. (1.4) The restriction of hã,ãi0 top satises
hAd(k)(Y1),Ad(k)(Y2)i0 =hY1, Y2i0, Y1, Y2 ∈p, k ∈K. (1.5) We consider now the left translation Lg(x) = gx, g ∈ G, x ∈ Xe. We dene a riemannian metric on Xe by the inner product
hX1, X2i=hdLg−1(X1), dLg−1(X2)i0, X1, X2 ∈TxX.e (1.6) By (1.5) the right hand side of (1.6) is independent of the chosen representative of the coset gK. Therefore, the riemannian metric is G-invariant. Then,
Xe ∼=Hd.
Let Γ⊂G be a discrete torsion-free cocompact subgroup of G. Then, X := Γ\G/K = Γ\Xe
is a compact hyperbolic manifold of dimension d, with universal covering Xe. We equip X with the riemannian metric, induced by the inner product (1.4). Then, X has constant negative sectional curvature −1.
Let∆+(g,a)be the set of positive roots of the system(g,a). Then, ∆+(g,a) = {α}. Let M0 = NormK(A). We dene the restricted Weyl group (analytically) as the quotient
WA:=M0/M.
Then, WA has order 2.
LetHR ∈asuch thatα(HR) = 1. With respect to the inner product (1.3), HR has norm 1. We dene
A+:={exp(tHR) :t ∈R+}. (1.7) We dene also
ρ:= 1 2
X
α∈∆+(g,a)
dim(gα)α, (1.8)
ρm := 1 2
X
α∈∆+(mC,b)
α. (1.9)
24 CHAPTER 1. PRELIMINARIES The inclusion i: M ,→ K induces the restriction map i∗: R(K) → R(M), where R(K), R(M) are the representation rings over Z of K and M, respectively. Let K,b Mcbe the sets of equivalent classes of irreducible unitary representations of K and M, respectively. By the theorem of the highest weight (Theorem 1.4), the representations τ ∈ K, σb ∈ Mc are parametrized by their highest weights ντ, νσ, respectively. Then,
ντ = (ν1, . . . , νn),
whereν1 ≥. . .≥νn and νi, i= 1, . . . , nare all integers or all half integers (that is νi =qi/2, qi ∈Z) and
νσ = (ν1, . . . , νn−1, νn), (1.10) whereν1 ≥. . .≥νn−1 ≥ |νn|andνi, i= 1, . . . , nare all integers or all half integers.
Lets be the spin representation of K, given by
s: K →End(∆2n)⊕End(∆2n)−pr→End(∆2n)
where ∆2n := C2k such that n = k, and pr denotes the projection onto the rst component (cf. [Fri00, p.14]). We set for abbreviation S = ∆2n. Let (s+, S+), (s−, S−)be the half spin representations ofM, whereS± := ∆±(cf. [Fri00, p.22]).
The highest weight of s is given by νs = (1
2, . . . ,1 2), and the highest weights ofs+, s− are
νs+ = (1
2, . . . ,1
2) (1.11)
νs− = (1
2, . . . ,−1
2), (1.12)
respectively. Letw∈WAbe a non trivial element ofWA, andmw a representative of w inM0. Then the action of WA on Mcis dened by
(wσ)(m) := σ(m−1w mmw), m ∈M, σ∈M .c
Ifνσ = (ν1, . . . , νn−1, νn) is the highest weight ofσ, then the highest weight ofwσ is given by
νwσ = (ν1, . . . , νn−1,−νn).
Specically, for the half spin representationss± we have νws± = (1
2, . . . ,∓1
2). (1.13)
Hence,
ws±=s∓. (1.14)