Appendix A

Vector Fields

Here we recall some basic notions in differential topology, a full accountof the subject can be found in [30]. We begin with the definition ofTp(M), the tangent space of a smooth manifold M at a point p. Foran open U ⊂ Rn the tangent space at x ∈ U is defined to be TxU =x × Rn. Let (Ui, φi)i be an atlas for M where each φi : U →φ(U) ⊂ M is a homeomorphism. Then TpM is defined to be the set ofequivalence classes [p, v, i] where v ∈ Tφ−1

i (p)Ui and [p, v, i] = [p, v′, j] if

vi = dφ−1j (p)(φ

−1i φj)(vj). Then TpM can be made into a linear vector

space by defining

(1) [v, i] + [u, i] = [v + u, i],

(2) k[v, i] = [kv, i] for k ∈ R.

The tangent space TM is the union⋃x∈M TxM and can be made into a

manifold. One can give an atlas for TM by declaring (TUi, dφi) a chartwhere

dφi(x, v) = [φ(x), v, i]

and the change of coordinates are (φ−1i φj, d(φ−1

i φj)). A vectorfield is a smooth map X : M → TM such that X(p) ∈ TpM or, asis customary, we say that X is a smooth section of the vector bundleπ : TM → M where π([p, v, i]) = p. Notice that the space of vectorfields χ(M) on M is a module over C∞(M) where the scalar product is

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398 Appendix A: Vector Fields

defined by pointwise multiplication i.e.

(f ·X)(p) = f(p)X(p),

and using (2) above. One can consider the derivative of a smooth mapf : M → N at a point p, which is a linear map

dpf : TpM → Tf(p)N

defined using charts (Ui, φi) and (Vj , ψj) for a neighborhood of p andf(p) respectively,

dpf([p, v, i]) = [f(p), dφ−1(p)(ψ−1j f φj)(v), j].

The vector fields on M act on C∞(M) as first-order differentialoperators by

(Xf)(p) = dpf(X(p))

for f ∈ C∞(M) and p ∈M . It is a direct check that

(1) X(fg) = fXg + gXf for all f, g ∈ C∞(M),

(2) X(f + λg) = Xf + λXg for all λ ∈ R,

which says that a vector field on M defines a first-order differentialoperator on C∞(M). In fact one can prove that any first-order differ-ential operator is given by a vector field. Given an atlas (Ui, φi) onM , then a vector field X on φi(Ui) has the local expression X(p) =∑n

i=1 ζi(p)∂/∂xi where ∂/∂xi(p)ni=1 is thought of as a basis for thetangent space TpM induced by the trivialization TUi = Ui×Rn, and ζiare smooth functions defined on φi(Ui). We have

(Xf)(p) =

n∑

i=1

ζi(p)∂f

∂xi(p),

which gives a local expression for the operator defined by X. One cancompose two such operators (vector fields) but of course the result isnot a first-order differential operator. We now define the bracket of twovector fields X and Y as

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Appendix A: Vector Fields 399

[X,Y ] = XY − Y X,

the commutator of differential operators X, Y . Since [·, ·] is clearly skewsymmetric and the Jacobi identity, [[X,Y ], Z]+[[Y,Z], X]+[[Z,X], Y ] =0 is a formal verification, we see that the set of all vector fields is aninfinite dimensional real Lie algebra if the bracket of two vector fields isalso a vector field. The miracle is:

Proposition A.1. [X,Y ] is a vector field.

For purposes of comparison we give two proofs for this, one of whichis classical and one modern in spirit. Typically, the classical one is moreelaborate. It is full of sturm und drang. But, in recompense, it givesmore insight. The modern one is quick and machine-like and has littleinsight. It is merely the verification of a previously established criterion.

Proof. Modern Proof. We use the isomorphism established betweentangent vectors and vector fields. Clearly, [X,Y ] is a linear operatoron functions. We therefore need only verify [X,Y ](fg) = f [X,Y ](g) +[X,Y ](f)g. Indeed,

[X,Y ](fg) = (XY − Y X)(fg) = X(Y (fg)) − Y (X(fg))

= X(Y (f)g) +X(fY (g)) − Y (X(f)g) − Y (fX(g))

= XY (f)g + Y f(Xg) + fY (Xg) +X(fY )g

− Y X(f)g −Xf(Y g) − fX(Y g) − Y (fX)g

= XY (f)g +X(fY )g − Y X(f)g − Y (fX)g

= [X,Y ](f)g + f [X,Y ](g).

Classical Proof. To see this we need only see that in local coor-dinates so defined [X,Y ] is a smooth first order differential opera-tor. Let f be a smooth function on M and U a neighborhood of p,X(p) =

∑i ηi(p)∂/∂xi and Y (p) =

∑i ζi(p)∂/∂xi. Then

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400 Appendix A: Vector Fields

([X,Y ]f)(p) = (∑

i

ηi∂

∂xi

∑

j

ζj∂f

∂xj−

∑

j

ζj∂

∂xj

∑

i

ηi∂f

∂xi)(p)

=∑

i,j

ηi(p)∂ζj∂xi

(p)∂f

∂xj(p) −

∑

i,j

ζj(p)∂ηi∂xj

(p)∂f

∂xi(p)

+∑

i,j

ηi(p)ζj(p)∂2f

∂xi∂xj(p) −

∑

i,j

ζi(p)ηj(p)∂2f

∂xj∂xi(p).

Since f is smooth, the mixed second partials are equal and so thesecond order terms cancel leaving a first-order operator,

∑

i,j

[ηi(p)ζj∂xi

(p)∂f

∂xj(p) − ζj(p)

ηi∂xj

(p)∂f

∂xi(p)].

A curve in M is a smooth map x : R → M , the tangent vector atevery point on the curve is the vector

x′(t) = dtx(1)

where 1 is thought of as a generator for TtR ' R. Given a vector fieldand a point p, if we can find a smooth curve through p whose tangentvector at every point coincides with the vector field, we call the curvean integral curve. This amounts to solving a differential equation withan initial condition. If we can only find a local curve then we have alocal solution to our differential equation with initial condition.

We now give the form of the fundamental theorem of ordinary dif-ferential equations which will be of use to us. A proof of this can befound in [37] or [69].

Theorem A.2. Let U ⊆M and V ⊆ Rm be neighborhoods of 0 and y0

respectively and v(x, y) be a vector field in M which depends smoothlyon (x, y). For each fixed y ∈ V consider the initial value problem f

′(t) =

v(f(t), y), f(0) = 0, where f : R → M . Then there is an ε > 0 and a

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Appendix A: Vector Fields 401

neighborhood V ′ of y such that there is a unique solution for t ∈ (−ε, ε)and y′ ∈ V ′ to the initial value problem. It depends smoothly on t andy ∈ V ′.

Corollary A.3. Let v(x) be a smooth vector field defined in a neighbor-hood U of 0 in M . Consider the initial value problem f

′(t) = v(f(t)),

f(0) = 0. Then there is an ε > 0 and a unique smooth solution fort ∈ (−ε, ε) to the initial value problem.

We recall that a 1-parameter group of diffeomorphisms is a map,φ : R × M → M , where we write φt(p) instead of φ(t, p), such thatφt is a diffeomorphism for each t ∈ R and t 7→ φt is a homomorphismfrom R → Diff(M) and φ0 = I. A similar definition holds for local1-parameter groups of diffeomorphisms. Namely, there is an interval Iabout 0 in R such that for all p ∈ M , φt(φs(p)) = φt+s(p) whenevers, t and s + t ∈ I. Now a local 1-parameter group of diffeomorphismsgives rise to a vector field on M as follows. For each point p0 ∈ Mconsider the smooth curve φt(p0) through p0. Taking its tangent vectorat each point gives a vector field on M . Conversely, given a vector fieldon M and a point p0, there is always a local 1-parameter group of localdiffeomorphisms φt which is the integral curve to this vector field andfor any smooth function f , limt→0(f φt − f) = Xf .

Proof. Let U, x1, . . . , xn be local coordinates around p0 and assume forsimplicity that for i=1, . . . , n, xi(p0)= 0. Let X=

∑i ηi(x1, . . . , xn)

∂∂xi

in U . Consider the following system of ODE, where i = 1, . . . , n andf1(t), . . . , fn(t) are the unknown functions,

df i

dt= ηi(f

1(t), . . . , fn(t)).

By the fundamental theorem of ODE, there exists a unique set of func-tions f 1(t, x1, . . . , xn), . . . , f

n(t, x1, . . . , xn), defined for |(x1, . . . , xn)| <δ and |t| < ε such that for all i, f i(0, x1, . . . , xn) = xi. Let x =(x1, . . . , xn) and φt(x) = (f 1(t, x), . . . , fn(t, x)). Clearly, φ0 = I onthis neighborhood. If |x| < δ and |t|, |s| and |t + s| are all less thanε, then x and φs(x), (where s is considered fixed), are both in this

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402 Appendix A: Vector Fields

neighborhood. Hence the n-tuple of functions, gi(t, x) = f i(t + s, x)are also in the neighborhood and satisfy the same ODE, but with ini-tial conditions, gi(0, x) = f i(s, x). By the uniqueness it follows thatgi(t) = f i(t, φs(x)). Hence φtφs = φt + s on this neighborhood. Thuswe have a local 1-parameter group of local diffeomorphisms which is theintegral curve to our original vector field.

Let φ be a diffeomorphism of M and dφ its differential. For a vectorfield X on M , φ∗X will denote the vector field induced by the actionof Diff(M) on X (M) mentioned above. If the 1-parameter group gen-erated by X is φt, then the smooth vector field φ∗X also generates a1-parameter group. It is

φ φt φ−1.

Proof. Now φ φt (φ)−1 is clearly a 1-parameter group of diffeomor-phisms, so let Y be its vector field. We must show Y = φ∗X. Let p ∈Mand q = φ−1(p). Since φt induces X, the vector Xq ∈ Tq is tangent tothe curve φt(q) at t = 0. Therefore (φ∗X)p = φ∗(Xq) ∈ Tp is tangent toφ φt(q) = φ φt φ−1(p).

Corollary A.4. Let φ be a diffeomorphism of M . A vector field is φfixed (i.e. φ∗X = X) if and only if φ commutes with all φt in Diff(M)as t varies.

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Appendix B

The KroneckerApproximation Theorem

Let n be a fixed integer n ≥ 1 and consider n-tuples α1, . . . , αn,where αi ∈ R. We shall say that α1, . . . , αn is generic if whenever∑n

i=1 kiαi ∈ Z for ki ∈ Z, then all ki = 0.Here is an example of a generic set. Let θ be a transcendental real

number and consider the powers, αi = θi. Then for any positive integern, θ1, . . . , θn is generic. For if k1θ

1 + . . . knθn = k, where the ki and

k are integers then since Z ⊆ Q, this is polynomial relation of degreebetween 1 and n which θ satisfies. This is a contradiction.

Proposition B.1. The set α1, . . . , αn is generic if and only if1, α1, . . . , αn is linearly independent over Q.

Proof. Suppose 1, α1, . . . , αn is linearly independent over Q. Let∑ni=1 kiαi = k, where k ∈ Z. We may assume k 6= 0. For if k = 0

then since the subset α1, . . . , αn is linearly independent over Q andki ∈ Q for each i we get ki = 0. On the other hand if k 6= 0 we divideand get

∑ni=1

ki

k αi = 1. So 1 is a Q-linear combination of αi’s. Thiscontradicts our hypothesis regarding linear independence.

Conversely suppose α1, . . . , αn is generic and q + q1α1 + . . . +qnαn = 0, where q and all the qi ∈ Q. If q = 0, then clearing denomi-nators gives a relation k1α1 + . . . + knαn = 0, where ki ∈ Z. Since the

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404 Appendix B: The Kronecker Approximation Theorem

αi are generic and 0 ∈ Z we get each ki = 0. Hence also each qi = 0.Thus 1, α1, . . . , αn is linearly independent over Q. On the other handif q 6= 0, by dividing by q we get, 1 + s1α1 + . . . + snαn = 0, wheresi ∈ Q. Again clear denominators and get k + k1α1 + . . . + knαn = 0,where k and ki ∈ Z. Since k1α1 + . . .+ knαn = −k and the original αiis generic, each ki = 0. Therefore each si is also 0 and thus 1 = 0, acontradiction.

Here is another way to “find” generic sets. We consider R to be avector space over Q. Let B be a basis for this vector space. Then anyfinite subset of this basis gives a generic set after removing 1.

Before proving Kronecker’s approximation theorem we define thecharacter group G of a locally compact abelian group G. Here

G = Hom(G,T)

consists of continous homomorphisms and is equipped with the compact-open topology and pointwise multiplication. G is a locally compactabelian topological group.

Proposition B.2. Let G and H be locally compact abelian groups (writ-ten additively) and β : G×H → T be a nondegenerate, jointly continu-ous bilinear function. Consider the induced map ωG : G → H given byωG(g)(h) = β(g, h). Then ωG is a continuous injective homomorphismwith dense range. Similarly, ωH : H → G given by ωH(h)(g) = β(g, h)is also a continuous injective homomorphism with dense range.

Proof. By symmetry we need only consider the case of ωG. ClearlyωG : G → H is a continuous homomorphism. If ωG(g) = 0 then forall h ∈ H, β(g, h) = 0. Hence g = 0 so ωG is injective. To prove that

ωG(G) is a dense subgroup of H we show that its annihilator inH is

trivial. Identifying H with its second dualH, its annihilator consists

of all h ∈ H so that β(g, h) = 0 for all g ∈ G. By nondegeneracy (thistime on the other side) the annihilator of ωG(G) is trivial. Hence ωG(G)is dense in H (see [70]).

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Appendix B: The Kronecker Approximation Theorem 405

We now come to the Kronecker theorem itself. What it saysis that one can simultaneously approximate (x1, . . . , xn)mod(1) byk(α1, . . . , αn). If we denote by π : R → T the canonical projec-tion with Kerπ = Z, the Kronecker theorem says that any point,(π(x1), . . . , π(xn)) on the n-torus, Tn, can be approximated to any re-quired degree of accuracy by integer multiples of (π(α1), . . . , π(αn)).

Of course a fortiori any point on the torus can be approximatedto any degree of accuracy by real multiples of (π(α1), . . . , π(αn)). Theimage under π of such a line (namely the real multiples of (α1, . . . , αn))is called the winding line on the torus. So winding lines and generic setsalways exist.

Theorem B.3. Let α1, . . . , αn be a generic set, x1, . . . , xn ∈ R andε > 0. Then there exists a k ∈ Z and ki ∈ Z such that |kαi−xi−ki| < ε.

Proof. Consider the bilinear form β : Z × Zn → T given byβ(k, (k1, . . . kn)) = π(k

∑ni=1 kiαi). Then β is additive in each vari-

able separately and of course is jointly continuous since here the groupsare discrete. The statement is equivalent to saying that image of themap ωG : Z→ Zn ' Tn is dense.

We prove that β is nondegenerate. That is if β(k, (k1, . . . , kn)) =0 for all k then (k1, . . . , kn) = 0 and if β(k, (k1, . . . , kn)) = 0 for all(k1, . . . , kn) then k = 0.

If β(k, (k1, . . . , kn)) = 0 for all k, then (k1, . . . , kn) = 0. The hypoth-esis here means just that π(k

∑ni=1 kiαi) = 0, or k

∑ni=1 kiαi is an inte-

ger. Choose any k 6= 0. Then∑n

i=1 kkiαi is an integer, because of ourhypothesis regarding the α’s we conclude all kki = 0 therefore ki = 0.On the other hand, suppose β(k, (k1, . . . , kn)) = 0 for all (k1, . . . , kn),then we show k = 0. Hence we have k

∑ni=1 kiαi is an integer for all

choices of (k1, . . . , kn). Arguing as before suppose k 6= 0. Choose ki notall zero. This gives kk0 = 0 as the αi is a generic set therefore k = 0.

Hence by Proposition B.2 we get an injective homomorphism ωG :Z→ Zn = Tn with dense range. Thus the cyclic subgroup ω(Z) in densein Tn.

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406 Appendix B: The Kronecker Approximation Theorem

Exercise B.4. (1) Show that in R2 a line is winding if and only if ithas irrational slope.

(2) Find the generic sets when n = 1. What does this say about densesubgroups of T?

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Appendix C

Properly DiscontinuousActions

Let Γ × X → X be a (continuous) group action of a locally compactgroup Γ on a locally compact spaceX. We shall say the action is properlydiscontinuous if given a compact set C of X there is a finite subset FCof Γ so that C ∩ (

⋃γ∈C\FC

γC) is empty. In particular, for each pointx ∈ X, the orbit, Γx, has no accumulation point. In particular, Γ mustbe discrete. Also clearly the isotopy group Γx of each point x ∈ X isfinite.

We now look at the converse in the case of an isometric action.

Proposition C.1. Let (X, d) be a metric space on which Γ acts isomet-rically. Suppose each orbit, Γx, has no accumulation points and eachisotopy group Γx is finite. Then Γ acts properly discontinuously.

Proof. If not, there is some compact set C ⊆ X so that C ∩ γ · C isnon-empty for infinitely many γ ∈ Γ. Thus there is a sequence γi ofdistinct elements of Γ with γi(ci) ∈ C, where ci ∈ C. By compactnessthere is a convergent subsequence which we relabel γi(ci) → c ∈ C.Again passing to a subsequence, using compactness of C and relabelingwe find ci → c′, c′ ∈ C. Now

d(γic, c′) ≤ d(γic, γici) + d(γici, c

′),

407

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408 Appendix C: Properly Discontinuous Actions

since Γ acts isometrically d(γic, γici) = d(c, ci) which therefore tends tozero. Also d(γici, c

′) tends to zero. Hence γic→ c′. Since Γc is finite, foreach i there are only finitely many j with γic = γjc. Hence by choosinga subsequence there is a sequence γic→ c′ where the terms are distinct.This contradicts the second condition and proves the result.

However, being properly discontinuous is stronger than being dis-crete. For example consider the action of Z on Tn where n ≥ 2. Thisaction is one in which a discrete group acts by isometries on a (com-pact) metric space. If we have an irrational flow , then every orbit isdense by Kronecker’s approximation theorem. Therefore this action isnot properly discontinuous. Now consider a rational flow. Since it is anaction on a metric space by isometries we only have to check the orbitsare discrete and the isotropy groups are finite. In this case both theseconditions are satisfied so the action is properly discontinuous,

We require the following lemma. Here the group, Homeo(X), thehomeomorphisms of X takes the topology of uniform convergence oncompacta which we call the compact open topology.

Lemma C.2. Let Γ × X → X be a continuous group action where(X, d) is a compact metric space and the countable discrete group, Γ,acts isometrically. Then the image of Γ ∈ Homeo(X) is also discrete.

Proof. Denote the map γ 7→ Φ(γ) by Φ, where Φ(γ)(x) = γ · x, x ∈ X.Then for each γ ∈ Γ, Φ(γ) is a homeomorphism, in fact an isometry,of X. Notice that Φ(γ)(X) = X. For if it were smaller, then applyingΦ(γ−1) would yield a contradiction. Also Φ is evidently a continuoushomomorphism Γ → Homeo(X). To complete the proof we need toshow this map is open. Since Γ is countable discrete the open mappingtheorem will do this if we know the image is locally compact. Now inthe compact open topology a neighborhood of I in the image is givenby N(C, ε), together with the inverses, where C is compact and ε > 0.However, since X is compact we can always take a smaller neighborhoodN0 = N(X, ε) of I. These are the homeomorphisms (actually isometries)h such that d(h(x), x) < ε for all x ∈ X. The condition of beingan isometry automatically shows any such N , in fact all of Isom(X),

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Appendix C: Properly Discontinuous Actions 409

is equicontinuous. Evidently N0 is pointwise bounded. Hence by theAscoli theorem N0 has compact closure so Φ(Γ) is locally compact. Theopen mapping theorem says Φ is open and therefore Φ(Γ) is discrete.

Let G be a connected semisimple Lie group of non-compact type,X = G/K the associated symmetric space. Then G is the connectedcomponent of the isometry group of X. Let Γ be a torsion free discretecocompact subgroup of G. Then Γ, the fundamental group of S = X/Γacts on S and S is a smooth connected manifold locally isometric withX so S is also metric and Γ acts by isometries. The cocompactness ofΓ implies S is compact.

Proposition C.3. The action of Γ on a compact locally symmetricspace S is properly discontinuous.

Proof. If not, there is a point s ∈ S and an infinite number of distinctγi so that Φ(γi)(s) converges to something in S. By Lemma C.2 Φ(Γ)is a discrete subgroup of Homeo(S). Now the set Φ(Γ1) of the γi isequicontinuous since all of Isom(S) acts equicontinuously. Let t ∈ S befixed. Then Γ1(t) ⊆ N(γis, d(s, t)) which is compact since S is. HenceΓ1 is uniformly bounded. Since it is also equicontinuous Γ1 has compactclosure. On the other hand Φ(Γ) is discrete. Therefore Γ1 is finite, acontradiction. This means Γ acts properly discontinuously.

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410 Appendix C: Properly Discontinuous Actions

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Appendix D

The Analyticity of SmoothLie Groups

Here we sketch the proof of the analyticity of a connected smooth Liegroup G. In the complex case this is just a fact of complex analysis sohere we focus on the real case, although the proof that follows worksequally well in the case of complex Lie groups.

If left and right translations are analytic, to prove the claim it issufficient to prove that multiplication and inversion are analytic in aneighborhood of 1 in G. For suppose we were at a neighborhood of(p, q). Let x1 = p−1x and y1 = q−1y. Then xy−1 = px1y

−11 q−1 =

LpRq−1x1y−11 . If the function (x, y) 7→ xy−1 is analytic at the origin

and, as above, left and right translations are analytic on G, then as acomposition of analytic functions (x, y) 7→ xy−1 is analytic at (p, q).

Now we prove the analyticity in a neighborhood of 1. Since this isa local question and any Lie group is locally isomorphic to a linear Liegroup, as mentioned in Section 1.7, we may assume G is linear. Let Ube a canonical neighborhood of 1 in G. We identify U with an open ballB about 0 in g using Exp which is analytic. Since Exp(x) 7→ Exp(−x)is evidently analytic, i.e. x 7→ −x being linear, it is sufficient to provemultiplication is analytic on U . We can consider each u = (u1, . . . , un) ∈B, where n = dimG. Let z = xy, where x and y ∈ U . Then for eachi, zi = fi(x1, . . . , xn, y1, . . . , yn), fi ∈ C∞(U × U). Now ∂fi

∂yj= δij at

411

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412 Appendix D: The Analyticity of Smooth Lie Groups

(x, y) = (1, 1). However, at y = 1 with x varying ∂fi

∂yjis a function of x:

vij(x) = vij(x1, . . . , xn). If b = (b1, . . . , bn) ∈ B, the 1-parameter groupExp(tb) satisfies the system of differential equations,

dxidt

= Σni=1bivij(x1(t), . . . , xn(t)), xi(0) = 0.

Since Exp(tb) is the unique solution, this system of equations is nothingmore than the matrix differential equation dx

dt = bExp(tb), x(0) = I.Thus the matrix, (vij(x)) = Exp(x) and since x 7→ Exp(x) is analyticso are the vij .

Now the product functions zi = fi(x, y) satisfy a system of partialdifferential equations:

Σjvij(z)∂zj∂xk

(x) = vik(x), i, k = 1, . . . , n,

called the fundamental differential equations of the group, G which de-termine the z’s if the v’s are known and certain integrability conditionsare satisfied. These link the v’s and their derivatives to the structureconstants of g. Since these conditions are necessary and sufficient andG is a smooth Lie group, the vij certainly satisfy these integrabilityconditions. The only question remaining is whether the zi are analytic.But since we know the v’s are analytic, so are the z’s. This follows fromthe Frobenius theorem (see [66] Theorem 211.9).

Finally we prove that left and right translations are analytic. Mul-tiplication is analytic in a neighborhood U of 1 in G. Hence so is lefttranslation Lg on U when g ∈ U . Therefore, because of the way we putthe manifold structure on G, such Lg’s are analytic on all of G. Nowlet g ∈ G be arbitrary. Then g = g1 . . . gn, where each gi ∈ U . HenceLg = Lg1 . . . Lgn , a composition of analytic functions and therefore eachLg is analytic. Similarly each Rg is analytic.

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Index

∗, 269

1-parameter subgroup, 32

A∗, xv

At, xv

B(G), 373

Bτ , 343

Bθ, 343

C(G)G, 246

DX(Y ), 279

G-equivalent, 17

G#, 389

G0, xv

GR, 44

H, 264

K, 268

L1(G), 224

L2(G), 224

Mn(C), 3

Mn(R), 3

Mn(k), xv

NG, 218

P , 264, 268

R(G), 236

R(ρ), 238

Tk, 228

Vλ, 313

X(T ), 175, 177

ZG, 218

AdG, xv, 55

Ad, xv

AdG(H), xv

Aut(G), 16

Aut(g), 29, 321

Der(g), 59

Exp, 33

Homeo(X), 408

=(H), 134

=, xv

Ind(H ↑ G, σ), 250

O(n,C), 3

O(n,R), 3

R-points, 44

Rad(G), 390

<, xv

SL(n,C), 3

SL(n,R), 3

SO(n,C), 3

SO(n,R), 3

SO(p, q), 4

Sp(n,C), 4

Sp(n,R), 4

SU(n,C), 4

Spec(T ), xv

U(n,C), 4

421

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422 Index

ad g, xv, 29, 59

ad, xv

ad-nilpotent, 193

H, 264

P, 264

R(G), 225

X (G), 244

χ(M), 398

χρ, 243

exp, 36

gl(V ), 26

gl(n, k), 26

gλ,X , 315

gλ, 315

k, 268

o(n, k), 28

p, 267

gk, 135

gk, 134

nk(V ), 135

s(V ), 135

z(g), 130

sl(n, k), 127

so(n, k), 28

u(n), 28

zg(X), 139

Inn(g), 321

T, 2

Z(p), 10

sp(n,R), 45

rad(g), 136

reg(ρ, g, V ), 317

ρ(X), 29

ρX , 29

ρg, 7

ax+ b-Lie algebra, 27, 137

ax+ b-group, 96l.s.k, xvl.s.C(Ω), 235

p-adic integers, 101-parameter group of diffeomor-

phisms, 4012-step nilpotent, 134, 311, 371

n(V ), 135

min(ρ, g, V ), 317nil(g), 135σ-compact group, 18

action, 15

simply transitive, 15transitive, 15

adjoint, 27adjoint algebra, 29adjoint group, xv

adjoint representation, 29, 55Ado’s theorem, 75, 186affine group, 9

algebra of invariants, 122algebraic group, 43algebraic hull, 389

amenable, 394analytically dense, 392

approximate identity, 235arithmetic, 393automorphism, 29

automorphism group, 7

Baire’s category theorem, 19Baker-Campbell-Hausdorff for-

mula, 40, 74

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Index 423

BCH formula, 74

binomial theorem, 150

block triangular form, 157

Bochner linearization theorem,119

Borel Density Theorem, 377

bounded part, 373

canonical coordinates of the 2ndkind, 82

Cartan criteria, 162

Cartan decomposition, 267, 270

Cartan involution, 343

Cartan relations, 28, 273

Cartan subalgebra, 316

Cartan’s fixed point theorem,273, 293

Cartan’s solvability criterion,162

Cartan, Elie, 261

Cartier, 115

Casimir element, 173

Casimir index, 173

Casimir operator, 173

Cayley-Hamilton theorem, 196

center, 127, 152

central extension, 136

central function, 246

central groups, 223

central ideal, 130

centralizer, 139

character, 243

character group, 404

Chevalley’s Theorem, 125

class function, 246

cocompact, 106

cofinite volume, 106

commutative operators, 154

compact real form, 337

complete reducibility, 173

completely reducible, 186

complexification, 147

conjugation, 338

conjugation relative to the realform, 338

covering map, 10

covering space, 11

density theorem, 379

derivation, 59

inner, 59

derived series, 135

derived subalgebra, 130

diagonalizable, 158

direct sum, 129

distribution, 45

integrable, 45

involutive, 45

smooth, 45

Engel’s theorem, 152

equivariantly equivalent, 17

Erlanger Program, 262

essentially algebraic group, 44

essentially algebraic subgroup,293

exponential, 213

exponential map, 36

exponential submanifolds, 267

external direct sum, 129

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424 Index

faithful representation, 185fiber preserving map, 11field extension, 147finite generation of an algebra,

123first isomorphism theorem, 52first-order differential operator,

398flag manifolds, 23flag of ideals, 156Fourier transform, 240Frobenius reciprocity theorem,

254Frobenius theorem, 46fundamental differential equa-

tions of the group, 412fundamental domain, 374, 375fundamental theorem of invari-

ant theory, 124Furstenberg, 379

general linear group, 3Grassmann Space, 22group

affine, 9complex Lie, 6real Lie, 6topological, 1transformation, 15

group action, 15group homomorphism, 4

Haar measure, 89Hadamard manifold, 273, 290Heisenberg Lie algebra, 132Hilbert basis theorem, 123

Hilbert’s 14th problem, 121, 124

Hilbert’s fifth problem, 7Hilbert-Schmidt inner product,

278homogeneous space, 296

hyperboloid, 303

ideal, 127characteristic, 131nilpotent, 134

solvable, 135identity component, 7index of nilpotence, 134

index of solvability, 135induced representations, 250inner derivations, 59

integral curve, 400integral distribution, 45integral manifold, 45

intertwining operator, 140, 224invariant

vector, 151invariant form, 145, 203invariant measure, 89, 102

invariant set, 17invariant vector, 151involution, 343

irrational flow, 408isometric, 407Iwasawa decomposition, 343

J-M condition, 193

Jacobi identity, 25Jordan decomposition, 158

Kazdan, 366

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Index 425

kernel function, 229

Killing form, 146

Klein’s Erlanger Program, 16

Klein, Felix, 16

Kronecker’s approximation the-orem, 404

lattice, 106

log, 372

Lebesgue measure, 95

left invariant, 31

left invariant subspace, 239

left translation, 16

Levi decomposition, 185

Levi’s splitting theorem, 180

Lie algebra, 25, 31

ax+ b, 27, 137

Heisenberg, 198

abelian, 26

affine, 137

compact type, 203

complete, 139

Heisenberg, 132

linear, 26

nilpotent, 133

reductive, 187

semisimple, 138, 163

simple, 138

solvable, 135

Lie algebra representation, 29

equivalent, 140

Lie bracket, 25

Lie group

compact, 202

exponential, 213, 304

Lie homomorphism, 6, 28

Lie subgroup, 6Lie’s theorem, 153, 156light cone, 300

linear actions, 16linearly reductive, 189log lattice, 372

Lorentz group, 274Lorentz model, 277, 303

Malcev uniqueness theorem, 183

Margulis, 84, 366Margulis Lemma, 84, 85maximal abelian subalgebra,

294

maximal compact subgroup, 270maximal torus, 207minimally almost periodic

groups, 386

modular function, 100monodromy principle, 50Morozov’s lemma, 193

nilradical, 135

niltriangular, 135norm, 228normalizer, 138

operator

Casimir, 173compact, 228nilpotent, 150

self adjoint, 229semisimple, 157

skew symmetric, 27symmetric, 27

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426 Index

operators

finite rank, 228orbit, 17orbit map, 17

orthogonal group, 3

Plancherel theorem, 240polar decomposition, 264polar decomposition theorem,

266

Pontrjagin, 16properly discontinuous, 407

quasi-generator, 208

radical, 136, 390rank, 211, 294, 317

rational form, 148real form, 328, 337

real points, 44regular elements, 317regular measure, 89

representation, 7, 29adjoint, 29, 55admissible, 382

completely reducible, 141,224

equivalent, 140, 224faithful, 7, 29

irreducible, 140, 224Lie algebra, 29reducible, 140

strongly admissible, 382unitary, 224

representation space, 7representative functions, 236

restricted root space, 348

root, 153, 315

root space, 315

root string, 326

root vector, 153, 315

Schur orthogonality relations,226

Schur’s lemma, 141

self-adjoint subgroup, 269

semi direct sum, 131

semi-invariant, 153

semidirect product, 9, 97

semidirect products, 9

semisimple Lie algebra, 163

semisimple operator, 157

Siegel generalized upper halfspace, 263

simply connected, 50

small subgroups, 48

spherical harmonics, 256

stabilizer, 18

Stiefel manifolds, 23

structure constants, 26

subalgebra, 26

symmetric space, 288

symplectic form, 4

symplectic group, 4

system of differential equations,412

theorem

Bochner, 119

Cartan’s fixed point, 273

Cayley-Hamilton, 196

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Index 427

Chevalley normalization,332

Engel, 152first isomorphism, 5, 52, 128Frobenius, 46Harish Chandra, 365Jacobson-Morozov, 192Kronecker’s approximation,

404Lagrange Interpolation, 160Levi’s splitting, 180Lie’s, 153, 156Mahler’s compactness crite-

rion, 363Malcev uniqueness, 183Mostow’s rigidity, 366Mostow-Tamagawa, 365open mapping, 20Peter-Weyl, 237second isomorphism, 52,

130Serre isomorphism, 332spectral, 230third isomorphism, 53, 130Weyl’s finiteness, 118

totally geodesic, 288transformation group, 15

triangular form, 154two-fold transitively, 297two-fold transitivity, 296two-point homogeneous space,

297

uniform lattice, 106uniform subgroup, 106unimodular groups, 96unipotent, 371unitary group, 3universal cover, 11

vector field, 397

weight, 153, 313weight space, 313weight vector, 153weight vectors, 313Weyl group, 218Weyl’s finiteness theorem, 118Weyl’s theorem, 173, 175, 341Whitehead’s lemma, 179

Zariski dense, 377Zassenhaus, 84

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