On Discrete Subgroups of Lie Groups

Lectures On Discrete Subgroups Of Lie Groups

By G.D. Mostow

Tata Institute Of Fundamental Research, Bombay 1969 Lectures On Discrete Subgroups Of Lie Groups

By G.D. Mostow

Notes by Gopal Prasad

No part of this book may be reproduced in any form by print, microﬁlm or any other means without written permission form the Tata Institute of Fundamental Research, Colaba, Bombay 5.

Tata Institute of Fundamental Research, Bombay 1969 Introduction

These lectures are devoted to the proof of two theorems (Theorem 8.1, the ﬁrst main theorem and Theorem 9.3). Taken together these theorems provide evidence for the following conjecture:

Let Y and Y′ be complete locally symmetric Riemannian spaces of non-positive curvature having finite volume and having no direct factors of dimensions 1 or 2. If Y and Y′ are homeomorphic, then Y and Y′ are isometric upto a constant factor (i.e., after changing the metric on Y by a constant). The proof of the first main theorem is largely algebraic in nature, relying on a detailed study of the restricted root system of an algebraic group defined over the field R of real numbers. The proof of our second main theorem is largely analytic in nature, relying on the theory of quasi- conformal mappings in n-dimensions. The second main theorem verifies the conjecture above in case Y and Y′ have constant negative curvature under a rather weak supplementary hypothesis. The central idea in our method is to study the induced homeomorphism ϕ of X, the simply covering space of Y and in particular to inves- tigate the action of ϕ at infinity. More precisely our method hinges on the question: Does ϕ induce a smooth mapping ϕ of the (unique) compact orbit X in a Frustenberg-Stake compactification◦ of the symmetric Riemannian◦ space X? There are good reasons to conjecture that not only is ϕ smooth, but 1 ◦ that ϕ G ϕ− = G where G denotes the group of transformations of X induced◦ ◦ by◦G, provided◦ of course◦ that X has no one or two dimensional◦

iii iv factors. The boundary behaviour of ϕ thus merits further investigation. It is a pleasure to acknowledge my gratitude to Mr. Gopal Prasad who wrote up this account of my lectures.

G.D. Mostow Contents

Introduction iii

0 Preliminaries 1

1 Complexiﬁcation of a real Linear Lie Group 9

2 Intrinsic characterization of K and E 17 ∗ 3 R-regular elements 29

4 Discrete Subgroups 37

5 Some Ergodic Properties of Discrete Subgroups 41

6 Real Forms of Semi-simple Algebraic Groups 45

7 Automorphisms of Φ∗ 53

8 The First Main Theorem 55

9 The Main Conjectures and the Main Theorem 61

10 Quasi-conformal Mappings 65

Bibliography 79

Chapter 0 Preliminaries

We start with two deﬁnitions. 1 Symmetric spaces. A Riemannian manifold X is said to be symmetric if x X, there = ∀= ∈ is an isometry σx such that σx(x) x and t Tx σ◦x(t) t, where ∀ ∈ ﬀ − Tx is the tangent space at x and σ◦x denotes the di erential of σx. Locally symmetric spaces. A Riemannian manifold X is locally symmetric if x X, is a neigh- ∀ ∈ borhood Nx which is a symmetric space under induced structure. Remark. Simply connected covering of a locally symmetric complete Riemannian space is a symmetric space (see Theorem 5.6 and Cor 5.7 pp.187-188 [8]).

Now we give an example which suggests that the last condition (that is there are no direct factors of dimensions 1 or 2 in the statement of the conjecture given in the introduction is in a sense necessary).

0.1 Example. Y, Y′ Compact Riemann surfaces of the same genus > 1 and which are not conformally equivalent. By uniformization theory, the simply connected covering space of such Riemann surfaces is analytically equivalent to the interior X of the unit disc in the complex plane. Then Y = Γ X, Y = Γ X where Γ, Γ are fundamental groups \ ′ ′\ ′ of Y, Y′ respectively. The elements of Γ, Γ′ operate analytically on X.

1 2 0. Preliminaries

Letting G denote the group of conformal mappings of X into itself, we have Γ, Γ′ G. It is well known that G is also the group of isometries ⊂ 2 = dz2 2 of X with respect to the hyperbolic metric ds 1 z2 and with respect to this metric X is symmetric Riemannian space of− negative curvature. Hence Y, Y′ are locally symmetric spaces of negative curvature.

If Y, Y′ were isometric then they would be conformally equivalent, which would be a contradiction. We list some facts about linear algebraic groups, these are standard and the proofs are readily available in literature. Perhaps the use of algebraic groups is not indispensable, however we hope that this will simplify the treatment. Let K be an algebraically closed field. For our purpose, we need only consider the case K = C, the field of complex numbers, Definitions. Algebraic set: A subset A of Kn is said to be algebraic if it is the set of zeros of a set of polynomials in K[X1,..., Xn]. n If A is a subset of K , then I(A) will denote the ideal of K[X1, X2,..., Xn] consisting of the polynomials which vanish at every point of A. Zariski topology on Kn: The closed sets are algebraic sets. Field of definition of a set: Let k be a subfield of K and A a subset n of K . If I(A) is generated over K by polynomials in k[X1,... Xn] then A is said to be defined over k or k is a field of definition of A. A subgroup of the group GL(n, K) of non-singular n n matrices × over K is algebraic if it is the intersection with GL(n, K) of a Zariski 3 closed subset of the set of all n n matrices M(n, K). × An algebraic group G is a k-group if G is defined over k, where k is a subfield of K. Terminology. If k = R or C, we shall refer to the usual euclidean space topology as the R-topology for GR or for GC. For a k-group G we write,

Gk = G GL(n, k). ∩ 0. Preliminaries 3

0.2 Theorem. G be an algebraic group then the Zariski connected component of identity is a Zariski-closed, normal subgroup of G of ﬁnite index. ([5] Th 2, Chap. II, pp.86-88).

0.3 Theorem (Rosenlicht). If k is a infinite perfect field, G a connected k-group then Gk is Zariski dense in G ([18] pp.25-50). 0.4 Proposition. If k is a perfect field, any x GL(n, k) can be written ∈ uniquely in the form [Jordan normal form] x = s u where s is semi- simple and u is unipotent; s, u commute. (use Th. 7, pp.71-72 [72])

0.5 Theorem. If k is a field of characteristic zero and G an algebraic k-group then there is a decomposition G = M.U (semi-direct product) where U is a normal unipotent k-subgroup, M is a reductive k-subgroup. Moreover any reductive k-subgroup of G is conjugate to a subgroup of M by an element in Uk. (Th.7.1, pp.217-218, [15]). 0.6 Proposition. If U is a unipotent algebraic subgroup of an algebraic group defined over a field k of characteristic zero, then

1. U is connected ([2] §8, p.46). 4

2. U is hypercentral [Engel-Kolchin] (see LA 5.7 [22]).

3. UR is connected in the R-topology if k R. ⊆ 0.7 Proposition. An abelian reductive group over algebraically closed ﬁeld is diagonalizable.

Deﬁnition. A connected abelian reductive group is called a torus.

0.8 Theorem. Let G be an algebraic k-group. Then

1. The maximal tori are conjugate by an element of G.

2. Every reductive element of Gk lies in a k-torus. 3. A maximal k-torus is a maximal torus.

4. Any maximal torus is a maximal abelian subgroup if G is connected and reductive. 4 0. Preliminaries

Deﬁnitions. A reductive element x GL(n, k) is called k split (or k- ∈ − reductive) if y GL(n, k) such that yxy 1 is diagonal, this is equivalent ∈ − to saying that all the eigen-values of x are in k.

A torus T is called k split if y GL(n, k) with yTy 1 diagonal, − ∈ − equivalently if each element of Tk is k split. − Let G be a reductive group, G˚ its Lie algebra and T be a maximal torus. Consider the adjoint representation

G Aut G˚ → x Adx → ˚ 5 then G˚ = G α Hom(T, C ) α ∈ ∗ where G˚α = y y G˚ Adx(y) = α(x) y x T Hom(T, C ) being | ∈ ∀ ∈ ∗ abelian we will use additive notation.

0.9 Theorem. Let¶ φ = α α Hom(T, C), G˚α , 0,α© , 0 then is called { | ∈ } the set of roots of G on T and we have

1. α φ α φ ∈ ⇒− ∈ 2. α Φ dim εα = 1. ∈ ⇒ 3. G˚α, G˚β = G˚α+ if α,β,α φ |∈ G˚α, G˚β = 0 if α + β < φ î ó 4. There exists a linearly independent set φ such that the roots △ ∈ areî eitheró non-negative integral linear combination or a non positive integral linear combination of elements in . Such a subset is called △ a fundamental system of roots on T.

Remark. A fundamental system of roots can be obtained as follows. Take any linear ordering of Hom(T, C∗) compatible with addition. Let be the set α α φ,α not a sum of two positive elements in φ . △ { ∈ } Notations. LetG be a group and A a subset of G, then Z(A) will denote the centralizer and Norm (A) the normalizer of A. 0. Preliminaries 5

If A and B are two subsets of G

1 A[B] = aba− A, b B . { ∈ ∈ } Deﬁnition. Let T be a maximal torus a of a connected reductive group 6 = Norm(T) G. Z(T) operates trivially on φ. The group W Z(T) is called the Weyl group of G.

0.10 Theorem. The Weyl group operates simply transitively on the set of fundamental systems of roots.

Deﬁnitions. A reductive element x G is k regular if y k - reduc- ∈ − ∀ ∈ tive, dim Z(x) dim Z(y). ≤ A reductive element is called singular if it is not R-regular. Let V be a K-subspace of Km and let k be a subﬁeld of K, then V is a k subspace if V = K(V km)i.e., V km generates the space over K. − ∩ ∩ Let G be a connected reductive k-group and kT a maximal k-split torus. Consider the adjoint representation of kT on G. Then G˚ = G˚α Hom(kT, C∗). α Each G˚α is a k-subspace. The following analogue of the Theorem 0.9 is true.

0.11 Theorem. Let kφ = α G˚α , 0,α , 0 . Then { } 1. αǫkφ αǫkφ ⇒− 2. G˚α, G˚β = G˚α+β if α,β,α + βǫkφ

G˚α, G˚α = 0 if α + βǫkφ î ó 3. There exists a linearly independent subset k kφ such that the roots △⊂ îare eitheró non-negative integral linear combination or non-positive 7 integral linear combination of elements from k k is called a fun- △ △ damental system of restricted roots.

Let G be a connected reductive k-group, let kT be a maximal k-split torus in G and let T be a maximal k-torus containing kT. Let be △ a fundamental system of roots on T and k a fundamental system of △ 6 0. Preliminaries

restricted roots on kT. We call and k Coherent if the elements in k △ △ △ are restriction of roots in . If one introduces ordering of the sets φ and △ kφ via lexicographic ordering with respect to and k respectively, the △ △ resulting orders are Coherent the sense: If α φ and α kT > 0 then ∈ α> 0. The existence of Coherent and k can be seen as follows. Let △ △ X = Hom(T, C∗), the group of rational characters of T. Then X is a free abelian group. Ann kT, the subgroup of characters which are trivial on kT is a direct summand of X since kT is connected. Therefore, one can choose a basis χ1,...χr for X such that χ1,...,χs is a base for Ann kT. Now introduce lexicographic ordering on X with respect to this base. The resulting order on φ clearly has the property: If α and β have the same restrictions to kT and if α > 0, then β > 0. Consequently, there is induced an order on kφ compatible with addition. The corresponding fundamental systems and k are Coherent. △ △ Notations. Let k △′ ⊂ △ ′ = Z linear span of ′ {△ } − △ α ′ = ker α kT △ ⊂ α ′ 8 ∈△ Choose an ordering such that k consists of positive roots. △ Put N˚ ( ′) = G˚α △ α>0 α< {△′} N( ) be the complex analytic subgroup of G with Lie algebra △′ N˚ ( ′). Since x G˚α is nilpotent, N( ′) is a unipotent group. △ ∀ ∈ α>0 △ Let

G( ′) = Z(⊥ ′) △ △ P( ′) = Norm (N( ′)) △ △ N = N(φ) P = P(φ) φ = empty set. = = = Mk′ norm (kT) Mk Z(kT) G(φ) W = k Mk′/Mk. 0. Preliminaries 7

The group kW is called Little Weyl Group or relative Weyl group and this operates transitively on the set of fundamental systems of restricted roots.

0.12 Lemma. P( ′) = G( ′) N( ′). △ △ △ G( ) is a maximal reductive subgroup of P( ) and N( ) is a maximal △′ △′ △′ normal unipotent subgroup (called unipotent radical of P( )). △′ 0.13 Theorem (Bruhat’s decomposition). Let G be a connected reduc- 9 tive k-group. Then

1. Gk = Nk M Nk k′

2. The natural map M /M Pk Gk/Pk is bijection. ′ → \ 3. Any unipotent k-subgroup of G is conjugate to a subgroup of N by an element in Gk.

4. Any k-subgroup containing P equals P( ) for some (P is △′ △′ ⊂ △ minimal parabolic k-subgroup). (see [4] or [21]).

Remark. Z(T) is a connected subgroup. More generally if S is any torus in G then Z(S ) is connected.

Now we consider for a moment the special case that k is algebraically closed. In this case G(φ) = Z(T) = T. Since T is a maximal abelian subgroup and P = T N is solvable. Clearly P is connected. It follows at once from assertion 4 of the previous theorem that the connected component of the identity in P( ) contains P and therefore it is seen to △′ coincide with P( ). In particular, every subgroup of G containing T N △′ is connected, and T N is a maximal connected solvable subgroup.

Deﬁnition. A maximal connected solvable subgroup of an algebraic group is called a Borel subgroup. A subgroup containing a Borel subgroup is called Parabolic. 8 0. Preliminaries

0.14 Theorem. The Borel subgroups of an algebraic group are conju- 10 gate under an inner automorphism.

0.15 Theorem. If G is a connected reductive k-group and k is a fun- △ damental system of restricted roots on a maximal k-split torus kT, then P( ) is parabolic for any k . △′ △′ ⊂ △ Proof. Let T be a maximal k-torus containing kT. Since all fundamental systems are conjugate under the Weyl group, it is possible to find a + fundamental system on T which is coherent with k . Let φ denote △ △ the set of positive roots on T defined by a lexicographic ordering with respect to . Then for any ′ k , the Lie algebra of P( ′) contains △ + △ ⊂ △ △ G˚α for all α φ . It follows directly that P( ) is parabolic. ∈ △′ Remark. A subgroup Q is parabolic iff G/Q is a complete variety; equivalently, in case k = C if G/Q is compact in the R-topology.

From the conjugacy of Borel subgroups and the theorem above, it is seen that any parabolic k-subgroup is conjugate to P( ) for some △′ ′ k . Also any parabolic subgroup containing the maximal k-split △ ⊂ △ 1 torus kT is ω[P( )]ω with ω N(kT). △′ − ∈ Since a reductive element of a connected reductive group is k-regular iﬀ it lies in a single maximal torus, we see

0.16 Proposition. A reductive element of a connected reductive k-group G is k regular iff it lies in at least one and at most finitely many parabolic k-subgroups of G, not equal to G. Chapter 1 Complexification of a real Linear Lie Group

Let G be a Lie subgroup of the Lie group of all automorphisms of a 11 real vector space V. Let VC denote the complexification of V (i.e., VC = V C) we identify the elements of G˙, the lie algebra of G, with ⊗ endomorphisms of V. We let G˙C denote the complification of the Lie algebra G˙ and let G◦C denote the analytic group of automorphisms of VC that is determined by G˙C. We identify the endomorphisms of V with their unique endomorphism extension to VC, so that we have G˙ G˙C ⊂ and G GC. Where G is connected component of identify in G. ◦ ⊂ ◦ ◦ Definitions. By the complexification of a real linear Lie group G is meant G◦C. G, it will be denoted by GC. By a f.c.c. group we mean a topological group with finitely many connected components. Suppose that G is a semisimple f.c.c. Lie subgroup of GL(n, R). ∗ Then G˙ RC = G˙C is semisimple. Hence G˙C = [G˙C, G˙C] is an algebraic ∗⊗ Lie algebra [Th. 15, pp. 177-179 [5]]. Since a Zariski-connected subgroup of GL(n, C) is topologically connected, it follows that the complex analytic analytic semisimple group G◦Cis algebraic, and therefore G G◦C ∗ is algebraic. Thus we have 1.1 Theorem. The Zariski closure in GL(n, C) of the semisimple f.c.c. Lie subgroup of GL(n, R) is its complexification.

9 10 1. Complexiﬁcation of a real Linear Lie Group

Deﬁnition. A subset S of GL(n, R) is said to be selfadjoint if tS = S where tS = g tg S ,(tg transpose of g) . { ∈ } R 12 1.2 Theorem. Let G be a semisimple f.c.c. Lie subgroup of GL(n, ) ∗ 1 then x GLn, R) such that xG x− is self adjoint. (for a proof see ∃ ∈ ∗ [12]). Notations. S (n) will denote the set of all real n n symmetric matrices × and P(n) the set of real positive deﬁnite symmetric matrices.

t 1 t 1 t 1 For any g GL(n, R) g = (g g) 2 (g g)− 2 g with (g g) 2 P(n) and t 1 ∈ ∈ (g g) 2 g 0(n, R). − ∈ 1.3 Theorem. Let G be a self adjoint Lie subgroup of GL(N, R). IfG is ∗ ∗ of ﬁnite index in FR, F an algebraic R group (equivalently G◦ = (FR)◦). Then ∗ 1. G = G P(n) G 0(n, R) ∗ { ∗ ∩ }{ ∗ ∩ } 2. G 0(n, R) is a maximal compact subgroup of G ∗ ∩ ∗ 3. G P(n) = exp(G˙ S (n)) (see [12]). ∗ ∩ ∗ ∩ 1.4 Lemma. Let G be a real analytic self adjoint subgroup of GL(n, R), G its Zariski closure∗ in GL(n, C). Let A be a maximal connected abelian subgroup in G P(n). Let T be the Zariski closure of A in GL(n, C), ∗ ∩ then T is a maximal R-split torus in G and A = (TR)◦. Proof. A being a commutative group of R-diagonalizable matrices, is R-diagonalizable. Therefore T, its Zariski closure is R-diagonalizable and hence an abelian subgroup of G.

Since A is self adjoint, the centralizer Z(A) of A in G and therefore also G Z(A) are self adjoint. ∗ ∩ 13 By the previous theorem

G Z(A) = G Z(A) P(n) G Z(A) 0(n, R) ∗ ∩ { ∗ ∩ ∩ } ◦{ ∗ ∩ ∩ } By maximality of A

G Z(A) P(A) = A. ∗ ∩ ∩ 1. Complexiﬁcation of a real Linear Lie Group 11

Hence Z(A) G = A G Z(A) 0(n, R) ∩ ∗ { ∗ ∩ ∩ Since T Z(A), we have ⊂

(TR)◦ = A (TR)◦ 0(n, R) ◦{ ∩ } Also since (TR) is diagonalizable over R, (TR) 0(n, R) is ﬁnite ◦ ◦ ∩ and as (TR)◦ is connected, this consists of identity matrix alone. Thus (TR)◦ = A. 1.5 Lemma. Let G be a semi-simple self adjoint analytic subgroup ∗ of GL(n, R) and let G be its Zariski-closure. Let KR = G 0(n, R), ∩ E = G P(n) and A as above, then ∗ ∩ KR[A] = E.

Proof. Evidently KR[A] E. We will prove the other inclusion. ⊂ First we show that if e, p P(n) and epe 1 P(n) then epe 1 = p. ∈ − ∈ − By the theorem 1.3 we have 14

Z(p) = Z(p) P(n) Z(p) 0(n, R) { ∩ }{ ∩ } and Z(p) P(n) = exp Z(p) S (n) ∩ { ∩ } where Z(p) is centralizer of p. Since

1 1 t 1 1 epe− P(n) epe− = (epe− ) = e− p e ∈ so e2 p = pe2 i.e., e2 Z(p) ∈ Since

e2 = Exp(X) for some X Z˚(p) S (n) ∈ ∩ 1 e = Exp X therefore e Z(p) P(n). 2 ∈ ∩ 1 ∴ ep = pe i.e. epe− = p, as asserted.

Now if p E p = Exp X for some X G˙ S (n). ∈ ∈ ∩ 12 1. Complexiﬁcation of a real Linear Lie Group

The Zariski closure of one parameter group Exp tX is a torus which is contained in a maximal R-split torus (say S ). 1 By conjugacy of maximal R-split tori, x G with x(TR) x = ∃ ∈ ◦ − (S R)◦ where T is the Zariski closure of A in G. By the previous lemma 1 (TR) = A and hence p xAx . ◦ ∈ − As G = E KR (see Th 1.3) ◦ we have x = ek with e E, k KR. ∈ ∈ 1 xax− = p for some a A. ∈ Thus ekak 1e 1 = p but kak 1 P(n) hence kak 1 = p. − − − ∈ − 15 Remark. If B is a maximal connected abelian subgroup in K = (KR)◦ then an argument similar to the one used in the above proof∗ yields: K [B] = K . ∗ ∗ Weyl chambers. The connected components of A ker α, where φ − α φ is a restricted root system on T, are called the Weyl chambers∈ associated with G and A. If ∗ is a fundamental system of restricted roots, then A = a a △ △ { ∈ A,α(a) > 1 α is a Weyl chamber. Observe that (Norm T)R oper- ∀ ∈ △} ates on A, for (Norm T)R operates on TR and hence on (TR)◦ = A. If 0 , Xα G˙α then h T ∈ ∀ ∈ 1 Ad h(Xα) = hXαh− = α(h)Xα t 1 1 t t (hXαh− ) = (h− ) Xαh = α(h) Xα t 1 1t i.e. h Xαh− = (α(h))− Xα

t this proves that Xα G˙ α. t ∈ − t Let hα = [Xα, Xα] then hα, Xα, Xα is a base for 3 dimensional split Lie algebra over R. By taking a suitable multiple of Xα, we can assume that t t [hα, Xα] = 2Xα, [hα, Xα] = 2 Xα − t then Exp π/2(Xα Xα) (Norm T). − ∈ 1. Complexiﬁcation of a real Linear Lie Group 13

t 16 Since Xα Xα is skew symmetric it actually belongs to (Norm T) − ∩ K . ∗ t Ad Exp π/2(Xα Xα) is reflection in the Wall corresponding to α , − of the Weyl chamber. This shows that Ad[(Norm T)R K ] contains the reflections in all ∩ ∗ the Walls of the Weyl chambers. 1.6 Theorem. E = K [A¯ ]. ∗ △ Proof. K [A¯ ] = K [(Norm A K )[A¯ ]] ∗ △ ∗ ∩ ∗ △ Since Ad(Norm A K ) contains the reflections in all the walls of ∩ ∗ Weyl chambers (Norm A K )[A¯ ] = A. ∩ ∗ △ ∴ K [A¯ ] = K [A]. ∗ △ ∗ Let X E˙ and let Y be an R-regular element in A, then since K is ∈ ∗ compact, k K such that ∃ ∈ ∗ d(X, k[Y]) = d(X, K [Y]) ∗ where d(X, Y) = Tr (X Y)2 − then d(k[X], Y) = d(X, k[Y]) d(k[X], l[Y]), l K ≤ ∀ ∈ ∗ therefore Z K˙ the real valued function ∀ ∈ ∗

fZ : t d(k[x], Exp tZ[Y]) → = Tr [h[X] Exp(tZ)Y Exp( tZ)]2 − − is minimum at t = 0.

∂ f ∴ 2 = 0. ∂t t=0 which gives 17

Tr (h[X] Y)[Y, Z] = 0 but since Tr Y[Y, Z] = 0 − we have

= Tr Z[k[X], Y] = Tr k[X][Y, Z] = 0, Z K ∀ ∈ ∗ 14 1. Complexiﬁcation of a real Linear Lie Group

hence [k[X], Y] = 0.

R-regularity of Y implies

Z(Y) G˙ S (n) = A˙ ∩ ∩ ∴ k[X] A˙ ∈ this proves that

K [A] E. The other inclusion is obvious. ∗ ⊃ Deﬁnition. An algebraic k-group is said to be k-compact if it contains no k-split connected solvable subgroup, that is a connected group that can be put in triangular form over k.

Remark. If G is a reductive algebraic k-group then the following three conditions are equivalent.

18 1. G is k-compact

2. Gk has no unipotent elements

3. the elements of Gk are reductive.

Exercise. Prove the above equivalences. [Hint (1) (2) (3) is obvious prove (3) (1) by showing: not ⇒ ⇒ ⇒ (1) not (3).] ⇒ The following digression is included just for fun, we need it only in the case k = R.

1.7 Theorem. Let k be a loc. compact ﬁeld of characteristic 0. Then G is k-compact iﬀ it is compact in the k-topology.

Proof. ( ) Let V be the underlying vector space [i.e. G is a subgroup ⇒ of Aut v] 1. Complexiﬁcation of a real Linear Lie Group 15

Let Ed be the set of d dimensional subspaces of V. Then there d is a canonical imbedding Ed ֒ P( (V)) which makes Ed a closed → ∧ n d subvariety of the projective variety P( (V)). The product Ed is a ∧ d=1 n closed subvariety of ( d(V)) (which, by Segre imbedding, itself is d=1 ∧ a closed subvariety of a projective space P of sufficiently large dimen- n sion). Hence Ed is a compact set. d=1 n The set W = (ω1,...ωn) (ω1,...,ωn) Ed ω1 ω2,...αωn is ⊂ d=1 n a closed subvariety (it is called the Flag manifold) of Ed. 1 G operates on W. For a k-rational point ω Wk. Let Tω be the ∈ stabalizer of ω in G then G/Tω = G.ω. Since Tω is k-triangularizable 19 (hence solvable) and as G is k-compact (Tω) = e . Hence Tω is finite ◦ { } [In an algebraic group the connected component of identity is of finite index see Th. 0.2]. Therefore G. = G/T has dimension equal to that of G. 1 Let D Tgω since Tgω = gTωg− . D is a finite (therefore discrete) g ∩Gk normal subgroup∈ and so it is central. Since Tω is finite we can choose g1 gr such that r = D Tgiω = i1 i let ui = gi ω and W be the Zariski closure of G ui in the projective variety. r r i G acts on G.ui w . = ⊂ = i 1 i 1 Since D acts trivially we get a faithful action of G′ = G/D on r (G.ui). = i 1 16 1. Complexification of a real Linear Lie Group

Let v = (u1 ur) and let V be the Zariski closure of the G orbit ′ G′.v of v. Then since V is a irreducible closed set and G′v is open (by Chevalley’s theorem in algebraic geometry) dim(V G v) dim V = − ′ ≤ dim (G orbit of a k-rational element). Therefore V G v has no k- ′ − ′ rational points. = = So Vk (G′v)k Gk′ v ∴ Vk is compact in k-topology. 20 Since the differential of the map G G .v is surjective by the k′ → k′ implicit function theorem for loc. compact fields, this map is compact. Gk But Gk G is open (again by implicit function th.) and so = → k′ D Image of Gk in Gk′ is open (and therefore a closed subgroup). This Gk proves that D is compact, but since D is finite Gk is compact in k- topology. The converse is also true. For if Gk is compact in k-topology G cannot have a unipotent subgroup. (Any unipotent group is isomorphic as an algebraic variety to Kr and its set of k-rational point is kr which is not compact). This proves that any element of Gk is reductive and this by the preceding remark implies that Gk is k-compact. Chapter 2 Intrinsic characterization of K and E ∗

K is a maximal compact subgroup of G , equivalently the complexifi- 21 cation∗ K of K is a maximal R-compact subgroup∗ of G. ∗ E = Exp(G˙ S (n)) ∗ ∩ G˙ S (n) = log E. ∗ ∩ log E is the orthogonal complement to K˙ in G˙ with respect to the Killing form (see [13]). ∗ ∗ 2.1 Theorem. The maximal compact subgroups in a f.c.c. Lie group are conjugate by inner automorphism ([13] or Chapter XV [9]). For gGL(n, R) we have a linear automorphism of S (n)s gstg → which leaves P(n) stable. This operation of GL(n, R) on S (n) is called the canonical action. Now let G be an analytic semi-simple group with finite center and let ρ be a finite∗ dimensional representation of G with finite kernel. By Theorem 1.2 we can assume, after conjugation that ρ(G ) is self adjoint. We set ∗ 1 K = ρ− (ρ(G ) 0(n, R)) ∗ ∗ ∩ K is then a maximal compact subgroup of G , let ϕ : G P(n) denote ∗ ∗ → the map g ρ(g)tρ(g) → 17 18 2. Intrinsic characterization of K and E ∗

22 then t ϕ(g1g2) = ρ(g1)ϕ(g2) ρ(g1). Thus under ϕ, left translation by g corresponds to the canonical action by ρ(g) on P(n). In addition

ϕ(gk) = ϕ(g) for k K ∈ ∗ and ϕ(g1) = ϕ(g2) iﬀ g1K = g2K ∗ ∗ therefore ϕ induces an injection

ϕ : X = G /K P(n). ∗ ∗ → Let [S ] denote the projective space of lines in S (n) and let

Π : S (n) 0 [S ] − → be the natural projection and let ψ = π ϕ and be the composite ◦ G G /K [S ] ∗ → ∗ ∗ →

then ψ is injective because if p1, p2 ϕ(X) with πp1 = πp2, then since ∈ p1, p2 are positive deﬁnite matrices, c > 0 such that ∃

p1 = cp2 n so p1 = c p2 where p = det p. | | | | | | But since p1, p2 ϕ(G ) p1 = p2 =+1. ∈ ∗ | | | | [for G being semi-simple, the commutator [G , G ] = G and so there ∗ ∗ ∗ ∗ 23 does not exist a non-trivial homomorphism of G into an abelian group. ∗ Thus g ρ(g) is a trivial homomorphism of G into R∗]. → | | ∗ This implies that c =+1 i.e., p1 = p2. The map ψ is a G-map that is ψ(gx) = gψ(x) for all g G, x Xs thus ψ(X) is stable under G. ∈ ∈ Definition. If ρ is irreducible over R then ψ(X) is called the stable compactification of X. This of course depends on ρ. Remark. The above compactification was arrived in a measure theoretic way by Frusentenberg [7]. 2. Intrinsic characterization of K and E 19 ∗

We shall now show that X has the structure of a symmetric Rie- mannian space and shall obtain a decomposition for ψ(X) in terms of symmetric Riemannian spaces. On P(n) we introduce a inﬁnitesimal metric

2 1 2 ds = Tr (p− p˙) where p(t) is a diﬀerentiable curve in P(n) andp ˙(t) = dp . dt t It is easy to check that this metric is invariant under the action of GL(n, R) on P(n) and also under the map p p 1. This implies that → − P(n) is a symmetric Riemannian space. (see [14]). Let G be a semi-simple analytic subgroup of GL(n, R), then by The- orem 1.3.∗ G = (G P(n) (G 0(n))). ∗ ∗ ∩ ∗ ∩ Let A be a maximal connected abelian subgroup of P(n) G . ∩ ∗ Since any abelian subgroup of P(n) can be (simultaneously) diago- 24 nalized, we can assume that A D(n) the set of real diagonal matrices. ⊂ Let T be the Zariski closure of A in GL(n, C) then by Lemma 1.4, T is a maximal R-split tours in the Zariski closure G of G in GL(n, C) and ∗

A = (TR)◦.

Let be a fundamental system of restricted roots on T. There is △ a natural faithful representation of GL(n, C) and therefore of G on Cn. In this section the complex vector space Cn considered as a G-module under this representation will be denoted by V. From the representation theory of semi-simple Lie algebras we have

V = V ⊕ where ′s are “weights” (more precisely, restricted weights) on T. The highest weight will be denoted by . Also we know that any other ◦ weight is of the form = nαα, where each nα is a non-negative ◦ − integer. 20 2. Intrinsic characterization of K and E ∗

For h A we have clearly ∈ △ . .. 0 ψ(h) = π ((n))2 . Ü 0 ..ê

25 After a conjugation we can assume that the ﬁrst diagonal entry is ( (h))2. ◦ So . 1 .. 0 ψ(h) = π (( )(h))2 − ◦ . Ü 0 ..ê

Let hn be a sequence in A such that the sequence ψ(hn) is con- { } △ vergent in the projective space [S (n)]. If necessary by passing to a sub- sequence, we can assume that α lim α(hn) exists in R and n ∀ ∈ △ →∞ ∪ {∞} is equal to ℓα. For a weight = nαα, if we deﬁne Supp = ◦ − α hα , 0 , then clearly the diagonal entry in lim ψ(hn), corresponding { } n ﬀ →∞ = to the weight is zero i Supp contains some α with Iα . ∞ Notations. For a non-empty subset of , we write △′ △

V( ′) = V △ Supp ⊂△′

p = the projection of V on V( ′) with kernal V △′ △ Supp 1 ′ = △ π ′ π(p ′ hp ′ ) for h S (n) and let ψ ′ be the composite G △ △ P △ ∈ △ ∗ → G /K P(n) △′ [S (n)]. (K = G 0(n, R)). ∗ ∗ → −−−→ ∗ ∗ ∩

26 Since V( ′) is stable under A, we note that p h = hp = p hp . △ △′ △′ △′ △′ The preceding remarks establish

2.2 Lemma. ψ(A ) = ψ (A ) △′ △ △ △′⊂△ 2. Intrinsic characterization of K and E 21 ∗

Also, if K = G 0(n, R) ∗ ∗ ∩ we have by theorem 1.6

E = K [A¯ ] ∗ △ G = E K ∗ ∗ = K [A¯ ] K ∗ △ ∗ ∴ ψ(X) = ψ(G ) = ψ(K [A¯ ] K ) = ψ(K [A¯ ]) ∗ ∗ △ ∗ ∗ △ = ψ(K [A ]) = πψK [A ] = π(K [ϕA ]) ∗ △ ∗ △ ∗ △ = K (πϕA ) = K πϕ(A ) ∗ △ ∗ △

For h, h T˙ < h, h >= Tr(hh ) is a inner product on T˙. This inner ′ ∈ ′ ′ product induces an inner product on Hom(T˙, C) and hence its restriction on Hom(T, C ) ֒ Hom(T˙, C). This restriction will again be denoted ∗ → by <,>.

2.3 Lemma. If G˙αV = 0, then the following two conditions are equivalent.

1. G˙ αV = 0. − 2. <,α>= 0.

t Proof. We can choose Xα G˙α such that Tr (Xα Xα) = 1 set 27 ∈ t [Xα, Xα] = h′ , then for h T˙ we have α ∈ = = t < h, hα′ > Tr hhα′ Tr h[Xα, Xα] tr t = Tr [h, Xα] Xα = α(h) Tr Xα Xα = α(h) ∴ = hα′ hα where hα is the dual of α in the inner product. Therefore for any weight , <,α>= (hα).

By considering the representation of 3-dimensional simple Lie al- t gebra generated by Xα, hα, Xα on V+nα the result follows immedi- { } n Z ately. (see [10] or pp. IV-3 to IV-6 of∈ [23]). 22 2. Intrinsic characterization of K and E ∗

Deﬁnition. Let Eρ = α α < α, >= 0 { ∈ △ ◦ A subset ′ of is said to be ρ connected if ′ is connected △ △ − △ ∪{ ◦} in the sense of Dynkin’s diagram of lies in Eρ. △′ For we set = α α Eρα β for β . The △′ ⊂ △ △′ △′ ∪{ ∈ ∀ ∈ △′} following is an easy consequence of the previous lemma. 2.4 Lemma. A subset of is ρ-connected iﬀ there is a weight with △′ △ support = . △′ Proof. By induction on s = the cardinality of . If s = 1 the result △′ follows at once from lemma 2.3. If s > 1 then contains a ρ-connected △′ subset ′′ of cardinal s 1, and hence there is a weight ′′ = n1α1 △ − △ ◦ − − ns 1αs 1, where ′′ = α1,...,αs 1. − − △ −

28 Let αs ′ ′′. Then < ,αs >=< ,αs > nk <αk,αs > 0 ∈ △ − △ ◦ − ≥ and is not zero since αs is ρ-connected. Hence αs is a weight △′′ ∪{ } − of support . △′ 2.5 Corollary. V( ) = V (largest ρ-conn. subset in ) and △′ △′ ψ(A ) = ψ (A ) △ △′ △ ′ △ρ⊂△conn. △′− 2.6 Lemma. π(E) = G π(1). ∗ ′ ′ △ ρ△⊂△conn. △′ − Proof.

π(E) = ψ(E) = πK [A ] ∗ △ = ψK [A ] = K ψ(A ) ∗ △ ∗ △ = K ψ (A¯ ) = K ψ (A¯ ) ∗ △′ △ ∗ △′ △ ′ ′ △ρ⊂△conn ρ△⊂△conn. △′ − △′ − = K (A¯ ψ (1)) = (K A¯ )ψ (1) ∗ △ △′ ∗ △ △′ ρ conn △′ − ρ△⊂△conn. △′ − 2. Intrinsic characterization of K and E 23 ∗

= G π (1). ∗ △′ ′ ρ△⊂△conn. △′ − Since is finite there are only finitely many subsets . So this △ △′ ⊂ △ lemma in particular shows that ψ(X) = π(E) consists of a finite number 29 of G orbits. ∗

2.7 Lemma. (i) For h and v : V( ′), hv = (h)v ∈⊥△′ △ ◦

(ii) G˙αV( ) = 0 if α> 0 and αψ △′ {△}

(iii) G˙αV( ) = 0 if α △′ ∈{△′ − △} Proof. Parts (i) and (ii) are immediate. (iii) follows from Lemma 2.3 and (ii) of this lemma.

For each restricted root α, set Gα the group generated by Exp X, X { ∈ G˙α . For a subset of let G ( ) be the group generated by Gα, } △′ △ ′ △′ α and let K( ) be the subgroup generated by Exp(X tX)X ∈ {△′} △′ − ∈ G˙α, α and maximum R-compact subgroup of Z(T). G ( ) is ∈ {△′} ′ △′ semisimple. We write

G ( ′) = G( ′) G ; K ( ′) = K( ′) G ∗ △ △ ∩ ∗ ∗ △ △ ∩ ∗ G′ ( ′) = G′( ′) G ; K = K ( ) = K( ) G ∗ △ △ ∩ ∗ ∗ ∗ △ △ ∩ ∗ and K′( ′) = K G′( ′). ∗ △ ∗ ∩ △

It is easy to see that G( ′) = G′( ′) Z(T); G′ ( ′) = (G′( ′)R)◦ △ △ ∗ △ △ but G ( ′) need not be connected. Also K ( ′) and K′( ′) are maximal ∗ △ ∗ △ ∗ △ compact subgroups of G ( ′) and G′ ( ′) respectively. ∗ △ ∗ △ Remarks.

1 (i) Since g Gα implies Z gZ Gα Z Z(T) we have Gα Z(T) = ∈ − ∈ ∀ ∈ Z(T) Gα. (ii) Since ( ) , roots in ( ) = roots in . △ − △′ ⊥ △′ {△′}−{△′} {△′ − △′} 24 2. Intrinsic characterization of K and E ∗

(iii) The Lie algebras of G( ) and P( ) are respectively 30 △′ △′

Z(T) + G˙α and Z(T) + Gα + Gα

α ′ α>0 α<0 ∈{△ } α ∈{△′}

(iv) P( ) is connected and for we have P( )P( ). Now we △′ △′ ⊃ △′′ △′ △′′ prove following results, which allow us to determine the G orbits in ψ(X). ∗

2.8 Lemma.

(i) The stabalizer of V( ) is P( ) △′ △′ (ii) The stabalizer of P( ′) π (1) is P( ′). △ △′ △ (iii) The stabalizer of the point π (1) in G is △′ ∗

G ( ′ ′) K ( ′)N ( ′) (⊥ ′ A). ∗ △ − △ ∗ △ ∗ △ △ ∩ Proof.

(i) It is clear that the stabalizer of V( ) contains P( ) hence is a △′ △′ parabolic group and therefore it is connected. V( ) is stable under △′ a connected subgroup H iﬀ it is stable under H˙ . From this it can be easily proved that the stabalizer is P( ). △′ (ii) Let S be the Stabalizer of P( ′)π (1) and S the Stabalizer of △ △′ △′ π (1) △′

Clearly S P( ′). If x stabalizer P( ′)π ′ (1) x.π ′ (1) for some ⊃ △ 1 △ △ △1 p P( ′), this implies that p− x.π (1) = π (1) i.e. p− x S . ∈ △ △′ △′ ∈ △′ Hence S = P( ′) (S S ). △ △′ ∩ We ﬁrst prove that S P( ′). △′ ⊂ △ 31 t = R If g S ′ then gp ′ g cp ′ for some c ∈ △ △ △ ∈ t 1 i.e. gp = cp ( g)− △′ △′ 2. Intrinsic characterization of K and E 25 ∗

t 1 So gV = gp V = cp ( g)− V V △′ △′ △′ △′ △′ ⊂ △′

∴ g the stabalizer of V( ′) = P( ′). △ △ ∴ S P( ′) △′ ⊂ △ ∴ S P( ′).P( ′) = P( ′). ⊂ △ △ △ From parts (ii) and (iii) of Lemma 2.7 it follows almost immediately that N( ′) S ′ and Gα S ′ and Gα S ′ , α ′ ′ (& So △ ⊂ △ ⊂ △ = ⊂ △= ∀ ∈ {△ − △ } G′( ′ ′) S ) i.e., N( ′). π (1) π (1) G′( ′ ′)π (1). Also △ −△ ⊂ △′ △ △′ △′ △ −△ △′ from the remark (ii) after §2.7, we get Gα Z(G ( )) α . ⊂ ′ △′ ∀ ∈{△′ − △′} Now we prove that G ( ) S . ′ △′ − △′ ⊂ For α ′ ′ ∈{△ − △ } Gα P( ′)π (1)Gα G( ′)N( ′).π (1) △ △′ △ △ △′ = GαG( ′)π (1) △ △′ = GαG′( ′)Z(T)π (1) △ △′ = GαG′( ′)GαZ(T)π (1) △ △′ = G′( ′)Z(T) Gαπ (1) = G′( ′)Z(T)π (1) △ △′ △ △′ P( ′)π (1). ⊂ △ △′

This proves that α Gα S and therefore G ( ) S . 32 ∀ ∈{△′−△′} ⊂ ′ △′−△′ ⊂ From the Lie algebra considerations it is easy to see that the group given by G ( ) and P( ) is P( ). ′ △′ − △′ △′ △′ ∴ P( ) S . This proves that S = P( ). △′ ⊂ △′ (iii) In (ii) we proved S P( ′). △′ ⊂ △ As P( ′) = N( ′) G( ′) △ △ △ and S N( ′) N( ′) △′ ⊃ △ ⊃ △ we have S = N( ′) (S G( ′)) △′ △ △′ ∩ △ since G( ′) = G′( ′ ′) G( ′) and since △ △ − △ △ G′( ′ ′) S △− △ ⊂ △′ G( ′) S = G′( ′ ′) G( ′) S △ ∩ △′ △ − △ { △ ∩ △′ } 26 2. Intrinsic characterization of K and E ∗

clearly S K( ′)R and S T = ⊥ ′ △′ ⊃ △ △′ ∩ △

(G( ′)S )R = (S )R (G( ′))R = (S )R(K( ′)RAK( ′)R) △ △′ △′ ∩ △ △′ △ △ = K( ′)R (⊥ ′ A) △ △ ∩ ∴ (S ) G = N ( ′) (S G( ′)) △′ ∩ ∗ ∗ △ △′ ∩ △ = N ( ′) G′ ( ′ ′) K ( ′) ⊥ ′ A . ∗ △ ∗ △ − △ ∗ △ △ ∩ 33

2.9 Lemma.

(1) G = K P ( ′) ′ ∗ ∗ ∗ △ ∀△ ⊂ △

(2) dimension of K ( ′) N( ′) is independent of ′. ∗ △ △ △

Proof. (i) Since K P ( ′) G and since for ′ ′′P ( ′) ∗ ∗ △ ⊂ ∗ △ ⊂ △ ∗ △ ⊂ P ( ′′) it is suﬃcient to prove that ∗ △ G = K P (φ). ∗ ∗ ∗ As a vector space K (˙ ) + P (˙φ) = G˙ ∗ △ ∗ ∗ By implicit function theorem K P (φ) is open in G . Also it is closed, since K is compact. Connectedness∗ ∗ of G implies∗ the result. ∗ ∗

i (ii) For a positive root α, let X , be a basis of G˙α then the set { α} Xi tXi Xi { α − α}∪ { α} α ′ α>0 α>∈{△0 } α<′ {△ }

is a basis for the Lie algebra of K ( ′) N( ′). This shows that ∗ △ △ dim(K ( ′)N( ′)) = dim G˙α . ∗ △ △ α> 0 2. Intrinsic characterization of K and E 27 ∗

34 Deﬁnition. If A is a right H-set and B a left H-set, where H is a group, then A H B denotes the set (A B)/R where R is the equivalence relation × × (ah, h 1b) (a, b), h H. − ∼ ∀ ∈ By the previous lemma

G = K P ( ′) ′ ∗ ∗ ∗ △ ∀△ ⊂ △ So the G orbit of ∗ π (1) = G π (1) △′ ∗ △′ = K P ( ′)π (1) ∗ ∗ △ △′ = K (P ( ′)π (1)) ∗ ∗ △ △′ P ( ′) ∗ △ .K (by part (iii) of Lemma 2.8). ≈ K ( ′)N ( ′)(⊥ ′ A) ∗ ∗ △ ∗ △ △ ∩ G ( ) = ∗ △′ K . K ( ′) × ∗ ∗ △ Since P ( ′) = G′ ( ′) N ( ′) (Z(T) G ). If we put ∗ △ ∗ △ ∗ △ ∩ ∗ G′ ( ′) ∗ △ = X( ′) K ( ′) △ ∗ △

we have the G orbit of π ′ (1) X( ′) K ∗ △ ≈ △ × K (∗′) This is compact iff X( ) is a single∗ △ point set, equivalently iff △′ G′ ( ′) = K′( ′) i.e., iff ′ = φ. ∗ △ ∗ △ △ = K 35 Then the orbit is the compact set X K (∗φ) π ′ (1). Also from part ◦ ∗ △ (iii) of Lemma 2.8 it is clear that dim S dim S if ′ ′′. △′ ≥ △′′ △ ⊂ △ So we have proved.

Theorem (Satake). ψ(X) consists of a ﬁnite number of G orbits. Among ∗ these there is a unique compact orbit X , also characterized as the orbit of minimum dimension. ◦

Chapter 3 R-regular elements

When k = R and G is a semi-simple algebraic R-group we can give 36 another description of reductive R-regular elements. Let G be a semi-simple R-group without loss of generality we can (and we will) assume that G is self adjoint (cf. [13]). Let T be a maximal R-split torus in G. Let A = (TR)◦. We can assume that A P(n) [see Lemma 1.4]. Let be a funda- ⊂ △ mental system of restricted roots on T, let

At = x x A α(x) > t α { ∈ 1 ∀ ∈ △} then A = A . △ Z(T)R = Z(A)R = A.(Z(A) 0(n, R)) ∩ we put Z(A) 0(n, R) = L. ∩ Then L is the unique maximum compact-subgroup of Z(A)R. The only R-regular elements in A are those in (Norm A)[A ]. More gener- △ ally the R-regular elements in Z(T)R are of the form m.a with m L and ∈ a (Norm A)[A ]. For given such an element it lies in P( ′) for any ∈ △ △ . Moreover if P is parabolic and m.a P then Z(m.a) P △′ ⊂ △ ∈ ⊂

∴ T P&P = P( ′) for some ′ ⊂ △ △ ⊂ △ This implies that m.a is R-regular. Since all the max R-split tori are conjugate by an element from GR 37 it follows that the set of reductive R-regular elements in G is GR[L.A ]. △ 29 30 3. R-regular elements

3.1 Lemma (Polar decomposition). If x is a reductive element of GL (n, R) then x can be written uniquely in the form x = p.k with p, k ∈ GL(n, R), the eigenvalues of p are positive, the eigenvalues of k are of absolute value 1 and pk = kp.

Proof. Let V be the underlying complex vector space.

V = Vλ, whee λ varies over the eigenvalues of x. ⊕ λ Let

p : v λ V for v Vλ → | | ∈ λ and k : v v. for v Vλ. → λ ∈ | | Then p, k satisfy the requirements of the lemma.

Deﬁnition. p is called the polar part of x.

The polar decomposition provides the following characterization of R-regular elements.

Proposition. A reductive element is R regular iﬀ its polar part is R- − regular.

The rest of this section will be devoted to the proof of the

3.2 Theorem. Let G be a semi-simple R group and y be an R-regular − reductive element in GR. Then there is an algebraic subset S y, not containing 1, such that for all large n, xyn is R-regular, provided x ∈ GR S y. − 38 We introduce the following new notations:

+ N = The unipotent analytic subgroup with Lie algebra G˙α α> 0 N− = The unipotent analytic subgroup with Lie algebra G˙α α< 0 LC = The Zariski closure of L= maximal R-compact subgroup of Z(T). + F = NR NR − 3. R-regular elements 31 we have Bruhat’s decomposition + G = N−(Norm T)N + GR = NR−NR. we need following Lemmas.

3.3 Lemma. Let V be a ﬁnite dimensional vector space and let vi V ∈ and di GL(V)i = 1, 2,... ∈ Assume

(i) lim vi = v = lim divi and i i →∞ →∞ 1 (ii) (di 1) are bounded uniformly in i then v = 0. − − Proof. Set wi = (di 1)vi then wi 0 as i − → →∞ 1 ∴ vi = (di 1)− wi 0 i.e.,v = 0. − →

3.4 Lemma. Let K be a compact subset of GR, let W and UA be neigh- bourhoods of 1 in GR and A respectively. Let t > 1, then there is a nbd. 39 U of 1 in G such that

t (kW)[LaUA] k[La].U k K, a A . ⊃ ∀ ∈ ∈ Proof. Since the rank of the map (g, b) g(b)of(W F) LA1 into → ∩ × GR at (1, b) equals the dimension of [G˙R, L˙ + A˙] + L˙ + A˙ = G˙ the map is open in a nbd. of (1, b). By taking a open subset of UA we can assume 1 t that a′ UA t− <α(a′) < t α and UA is compact. Then a A ∀ ∈1 ∀ ∈ △ ∀ ∈ aUA A . If necessary, by passing to a open subset, we can assume ⊂ that the above map has maximal rank on (W F) LA1, W is compact ∩ × and W Norm A Z(A) . Then the set kW[LaUA] is a nbd. of identity. ∩ ⊂ ◦ It remains only to show that

1 (k[ma])− (kW)[LaUA] is a nbd. of identity. a At,k K ∈mL∈ ∈ 32 3. R-regular elements

Since for any nbd. U of 1, k Kk[U], for K compact, is a nbd. of 1, ∩ ∈ it is suﬃcient to show that

1 (*) (ma)− W[LaUA] is a nbd. of 1. a At,m L ∈ ∈ Let

π : GR GR/A → set W = π(W)

deﬁne fm,a : W UA L G × × 1 → fm,a :(WA, a′, m′) (ma)− (w(m′aa′)) ‹ → 40 then (*) is equivalent to ‹

(**) Image fm,a is a nbd. of 1. a At,m L ∈ ∈ It is easy to see that the condition (**) fails iﬀ there is a sequence of t points xi W UA L and a sequence (mi, ai) L A such that xi ∈ × × ∈ × → boundary of = W UA L in GR/A A L and lim fm ,a (xi) = 1. Hence × × × × i i i ﬃ →∞ = to prove (**) it su ces to show that if lim fmi,ai (wiA, ai′, mi′) 1 with i →∞= = ai′ UA, mi′ L and if lim(wi, ai′, mi, mi′) (w, a′, m, m′) then w 1 and ∈ ∈ ‹ a′ = 1. For then it will follow that

(wiA, a′, m′) (A, 1, m) i i → which is not a boundary point of W UA L. × × The previous statement is equivalent to

1 If (miai) (wi[m aia ]) 1 and if − i′ i′ →‹ (***) (wi, ai′, mi, m1′ ) (w, a′, m, m′) (W F) UA L L  → ∈ ∩ × × × then w = 1 and a′ = 1.  We prove (***) From the uniqueness of Bruhat’s decomposition it follows that NR− + + LANR, being the image of NR LA NR under a homeomorphism, is − × × open (invariance of domain). 3. R-regular elements 33

41 Let b A be close enough to the identity 1, so that ∈ + w[b] NR−LANR w W. ∈ ∀ ∈ Then

+ wi[b] = piciqi pi NR−, ci LA, qi NR ∈ ∈ ∈ + and w[b] = pcq p NR− c LA, q NR. ∈ ∈ ∈ = 1 = 1 Set bi miai then bi− vi(wi[(mi′aiai′)− ]) = 1 where vi bi− (wi[mi′aiai′])

∴ 1 = 1 1 1 (bi− wi)[b] vi(wi[(mi′aiai′)− ])wibwi− (wi[mi′aiai′])vi− . = viwi[b].

Since vi 1 and wi w we have → → 1 = lim(bi− wi)[b] w[b] i →∞ 1 = i.e., lim bi− wi[b] lim wi[b] i i →∞ →∞ 1 1 1 but b− [wi[b]] = b− [pi] ci b− [qi] i i i so

1 1 1 lim b− [wi[b]] = lim b− [pi] ci b− [qi] i i i = lim pi ci qi

∴ 1 = = lim bi− [pi] p lim pi 1 = = and lim bi− [qi] q lim qi.

+ Since N , N− are nilpotent. By induction on the lengths of the de- 42 + scending central series of N−, N and using the previous lemma we get

lim pi = p = 1 = q = lim qi. ∴ w[b] = c LA ∈ 34 3. R-regular elements

since A is connected any nbd. of 1 in A generates A we have

w[A] LA ⊂ ∴ w[A] = A w W Norm A Z(A) ∈ ∩ ⊂ ∴ w W LA F. ∈ ∩ ∩ But from Bruhat’s decomposition W LA F = 1 . ∩ ∩ { } ∴ w = 1. From (***) 1 b− (wi[m′aia′]) 1. i i i → Since wi 1 we have → 1 bi− m1′ aiai′ 1 1 → ∴ mi− mi′ ai′ 1 L → ∈ ∴ a′ 1 since L A = 1 . i → ∩ { } i.e., a′ = 1.

43 This proves the Lemma.

3.5 Lemma. Let C be a compact subset of NR− and let t > 1. Then there exists a compact subset K NR such that ⊂ − Cb K[b] b LAt. ⊂ ∈

Proof. (By induction on the length of the derived series of NR−). Set N = NR− Ni+1 = [Ni, Ni] ◦ Suppose N is abelian. ◦ Then

1 1 ub = v[b] iﬀ ub = vbv− b− b 1 1 i.e., u = vb v− b− 3. R-regular elements 35

= v ad b v (written in additive from) − = (1 adb)v. − 44 Iff v = (1 adb) 1v where adb(x) = bxb 1. − − − Since (1 adb) 1 is uniformly bounded for − − t 1 b LA , (1 adb)− C ∈ b LAt − ∈ is a subset of a compact set K. This proves the statement for the case when N is abelian. ◦ In general, given a fixed b LAt, by applying the above argument ∈ to N /N1, we can find elements v N and n N1 such that nub = v[b]; ◦ ∈ ◦ ∈ moreover since N = NR−, as u varies over compact set C, n and v vary over compact sets.◦ 1 ub = n v[b] = v[n1b] where n1 N1 and varies over a compact set − ∈ K1. By induction n1b = v1[b] and v1 varies over a compact set as n1 t varies over compact set K1 and b over LA . We have ub = v[n1b] = v[v1[b]] = vv1[b]

as both v, v1 vary over compact sets v v1 varies over a compact set proving the Lemma. Now we prove Theorem 3.2. t We can assume (see pp. 36-37) that y LA , t > 1. Let S y = + + ∈ G N Z(A)N . Then since S y Z(A)N is union of N orbits of lower − − − dimensions in GZ(A)N, S y is Zariski closed in G. Let + x GR S y = NR−Z(Z)RNR ∈ − then 45

+ + + x = u−bu with b LA, u− NR− &u NR n + n ∈ n n ∈+ n ∈ xy = u−bu y = (u−by )(y− u y ) n n + n = v[by ](y− u y ) for some v K (by Lemma 3.5). ∈ 36 3. R-regular elements

Since y is R-regular reductive element, given a nbd. U of 1, n (U) n + n ∃ ◦ such that (y− u y ) U for n > n (U). Hence by Lemma 3.4, n (y, x) ∈ ◦ ∃ ◦ such that xyn G[LAt] if n > n (y, x). ∈ ◦ 3.6 Lemma. S y contains no conjugacy class of G.

Proof. Suppose E S y with G[E] = E. Since S y is Zariski closed we ⊂ + can also assume that E = E . Let 0 = G E, then for g N ∗ − ∈ + g[N−Z(A)N ] = g[G S y] g[G E] G E = 0 + + + − ⊂ + − ⊂ − ∴ N [N−Z(A)N ] = N N−Z(A)N 0. ⊂

But

+ + + + + + N N−Z(A)N = N N−Z(A)Z(A)N = N Z(A)N−Z(A)N + + + 1 = N Z(A)N N−Z(A)N = JJ− 0 ⊂ where J = N+Z(A)N+

Since J is Zariski open in G, for any g G, gJ J, being intersection ∈ ∩ of two Zariski open (hence dense) sets, is nonempty. Therefore

1 g JJ− 0. ∈ ⊂ ∴ 0 = G ∴ E = G = 0 = φ.

46 This proves the assertion. Chapter 4 Discrete Subgroups

In this and the following sections we will use the following notations. 47 G will denote a semi-simple (complex analytic) algebraic R group. − GR = G GL(n, R) and G = G◦R. For any subset S of G, S ∗ and S¯ are ∩ ∗ respectively the Zariski closure and the closure in R-topology of S in G. We state the following useful Theorem, for a proof the reader is referred to [3] or [16]. 4.1 Theorem. Let G be a connected algebraic R group with no R- − compact factors and let Γ be a R-closed subgroup of GR. IfGR/Γ has an GR-invariant ﬁnite measure, then Γ is Zariski dense in G.

Here after we assume that Γ is a closed subgroup of GR such that GR/Γ has an GR invariant finite measure and G has no R-compact factors. Now we prove a few “density” results. 4.2 Lemma. If Γ is the set of reductive R-regular elements in Γ then ◦ Γ∗ = G. ◦ Proof. We first show that Γ is non-empty. ◦ Fix an element x of LAt, t > 1, let U be a symmetric nbd. of 1 in G. Then since the sett xnUΓ. **** have same non zero measure and since the total measure is finite, at least two of them intersect 48 let xmUΓ xkUΓ , φ for k > m ∩ k m then Γ Ux − U , φ ∩ 37 38 4. Discrete Subgroups

i.e. for some n 1 ≥ Γ UxnU , φ. ∩ but UxnU U[xn] U2. ⊂ If U is suﬃciently small by Lemma 3.4 U[xn] U2 G[LAt]. This ⊂ implies that Γ G[LAt] , φ ∩ ∴ Γ Γ G[LAt] is non-empty. ◦ ⊃ ∩ Let γ Γ , if γ Γ S y then by Theorem 3.2, ◦ ∈ ◦ ∈ − ◦ γγn G[LAt] for all n > n (γ). ◦ ∈ ◦ Set

B = γn; n > n (γ) ◦ ◦ ◦ then B B B hence B∗ B∗ = B∗. ◦ ◦ ⊂ ◦ ◦ ◦ ◦ Since the ideal of polynomials vanishing on B is stable under trans- ◦ 1 lation by x B∗ and therefore under translation by x− for x B∗ (see ∈ ◦ ∈ ◦ Lemma 1 on p. 80 [50]), we have

1 (B∗)− B∗ ◦ ⊂ ◦ 49 Therefore

1 (B∗)− (B∗) B∗. ◦ ◦ ⊂ ◦ ∴ 1 B∗. ∈ ◦ Also since

γB Γ ◦ ∈ ◦ γB∗ Γ∗ ◦ ⊂ ◦ 1 ∴ γ = γ Γ∗. ∈ ◦

This proves that Γ S y Γ∗. But since S y is a Zariski closed − ◦ ⊂ ◦ ◦ proper subset of G, Γ S y is Zariski dense and therefore Γ∗ = G. − ◦ ◦ 4. Discrete Subgroups 39

4.3 Lemma. Let γ1 Γ, set ∈ n Γ1 = γ γ Γ,γ,γ Γ for n > n (γ) ∈ ∈ ◦ ◦ Γ = n Γ and 2 γ γ 1 n n (γ) ∈ ≥ ◦ then Γ∗ =Γ∗ = G. ◦ ◦ Proof. Since the Zariski closure of xn, n n for x G is a group (see n{ ≥ ◦} ∈ the proof of previous lemma) x : x , n n∗ . { ≥ ◦} This shows that Γ1 Γ∗. ⊂ 2 Hence it is suﬃcient to prove that Γ1 is Zariski dense in G. Given y Γ , since by Lemma 3.6, S y does not contain any conjugacy classes, 50 ∈ ◦ γ such that γ[γ1] < S y. ∃ ∴ Ty = γ,γ γ1 S y is a proper algebraic subset of G. | |∈ For any γ Γ Ty, γ[y1] < S y so ∈ − n γ[γ1]y Γ for all n > m (γ) ◦ ◦ 1 ∈n 1 n 1 γγ1γ− y γ ∴ γ1γ− y γ γ− Γ γ =Γ ◦ ◦ ◦ 1 ∈n ∈ i.e. γ1(γ− [y]) Γ for n > m (γ)) ◦ ◦ 1 ∈ i.e. γ− [y] Γ1 if γ < Ty ∈1 ∴ γ < T − implies γ[y] γ1 y ∈ 1 ∴ Γ∗ (Γ T − )∗[y]. 1 ⊃ − y Bur Γ T 1 is dense in G, so − y−

y G∗[y] Γ∗ ∈ ⊂ 1 ∴ Γ Γ 1∗. ◦ ⊂ Γ =Γ = By the previous lemma we have 1∗ ∗ G. Following is a reﬁnement of the above result◦

4.4 Lemma. Let S be a proper algebraic subset of G, let n be a positive 2 n integer and γ1 Γ. Then γ Γ S such that γ ,γ ,...,γ and 2 ∈ n Γ ∃ ◦ ⊂ ◦ − ◦ ◦ ◦ γ1γ ,γ1γ ,...,γ1γ4 S. ◦ ◦ ∈ ◦ − 40 4. Discrete Subgroups

m m Proof. m, S m = x x G,γ1 x S x x G, x S is a proper 51 ∀ { ∈ ∈ }∪{ ∈ ∈ } algebraic subset of G. Hence S 1 S 2 ... S n is a proper algebraic subset. Since Γ2 is ∪ ∪ ∪ Zariski dense, we can ﬁnd a γ in Γ2 S 1 S 2 S n. Obviously such ◦ − ∪ ∪ a γ satisﬁes the requirements of the lemma. ◦ 4.5 Lemma. Let G be a semi-simple R-group and let RT be a maximal R split torus. Let T be a maximal R torus containing RT. Set A = − − (RT)◦R, H = (TR)◦. Then

(G [Γ] H)∗ = T. ∗ ∩ Proof. Z(A)R = Z(RT)R = L.A.

Since

H = (H L) A ∩ L◦[H] = L◦[H L] = L◦.A ∩ and G [G [Γ] H] = G [Γ G [H]] ∗ ∗ ∩ ∗ ∩ ∗ = G [Γ G [L◦.A]] ∗ ∩ ∗ G [Γ ] Γ . ⊃ ∗ ◦ ⊃ ◦

By taking Zariski closure we get, since G∗ = G ∗

G =Γ∗ = (G [G [Γ] H])∗ = G[(g [Γ] H∗)]. ◦ ∗ ∗ ∩ ∗ ∩ Therefore

dim G = dim G[(G [Γ] H)∗] = dim G/Z(x) + dim(G [Γ] H)∗ ∗ ∩ ∗ ∩ for some x H. ∈

52 Since dim Z(x) dim T, we ﬁnd dim(G (Γ) H)∗ = dim T and thus ≥ ∗ ∩ (G [Γ] H)∗ = T. ∗ ∩ Chapter 5 Some Ergodic Properties of Discrete Subgroups

5.1 Lemma (Mautner). Given a group B A, where B is an additive group 53 of reals or complex numbers and A is an infinite cyclic subgroup of the multiplicative group of complex numbers a with a < 1 and assume that | | 0 is group operation is a b a 1 = a.b ◦ ◦ − ordinary multiplication in C Let V be a Hilbert space and let ρ be a unitary representation of B A on V, then any element v V whose line is fixed under A is fixed under ∈ B.

Proof. Since ρ is unitary

ρ(a)v = αv with a = 1 for b B | | ∈ <ρ(b)v, v > =<ρ(a)ρ(b)v,ρ(a)v > 1 =<ρ(a)ρ(b)ρ(a− )ρ(a)v,ρ(a)v > 1 1 =<ρ(a b a− )αv,αv >=<ρ(a b a− )v, v > ◦ ◦ ◦ ◦ So for n positive ∀ n n <ρ(b)v, v > =<ρ(a b a− )v, v > ◦ ◦ =<ρ(an.b)v, v >

41 42 5. Some Ergodic Properties of Discrete Subgroups

as n →∞ <ρ(b)v, v > =< v, v > ∴ ρ(b)v = v.(use Schwarz’s inequality).

54 This proves the assertion.

5.2 Lemma. Let G be an analytic semis-simple group having no compact factors. Let ρ be a unitary representation of G on a Hilbert space V, let x be a reductive R-regular element in G, if for some element v V, ∈ ρ(x)v = αv then ρ(G)v = v. Proof. Take the decomposition of G with respect to x. Let A be the group generated by x and B a root space. The previous Lemma applies.

Remark. The above result holds for any x not contained in a compact subgroup (see [11]). 5.3 Theorem. Let x be a reductive R regular element of G. Then x − operates ergodically on G Γ, i.e. any measurable subset of G Γ stable under left translation by x∗ is either of measure zero or its complement∗ has measure zero. Proof. Let V = L 2(G /Γ). ∗ Since the measure on G /Γ is G -invariant, the canonical action of G on V is unitary. ∗ ∗ ∗ Let Z G /Γ with xZ Z and let v be the characteristic function of ⊂ ∗ ⊂ Z. Then since measure of x 1Z Z is zero − − x.v = v.

55 Therefore by the previous lemma

G v = v. ∗ ∴ v = 1 almost every where or v = 0 almost every where.

This implies that either Z or G/Γ Z has zero measure. − 5. Some Ergodic Properties of Discrete Subgroups 43

Remark. Let M be a separable topological measure space [i.e. the open sets are measurable and have positive measure] and let f : M M be + → a measurable transformation. Let A = f n, n = 1, 2,... . Then if f is + { } ergodic, for almost all p M, A p is dense in M. ∈

[Proof : Let Ui be a denumerable base of open sets. Let Wi = p p + { } { | ∈ M, A p Ui = φ then Wi is measurable. Also p Wi f p Wi ∩ } ∈ ⇒ ∈ therefore fWi Wi. Since f is ergodic and Ui M Wi, Wi is of ⊂ ⊂ − measure zero. ∞ ∴ E = Wi has measure 0 = i1 + p < E implies A p Ui , φ i and this proves that for almost all p M, + ∩ ∀ ∈ A p is dense].

5.4 Theorem. Let G be a semi-simple analytic linear group. Let Γ be a subgroup such that∗ G /Γ has a ﬁnite invariant measure. Let P ∗ be a R-parabolic subgroup of G∗ (= the complexiﬁcation of G ). Set P = P G then ΓP = G . ∗ ∗ ∗ ∩ ∗ ∗ ∗

Proof. Let T be a maximal R split torus in P. Let x TR◦ such that 56 − + ∈ for any restricted root α on T with Gα U the unipotent radical of ⊂ P,α(x) > 1.

+ Let U− be the opposite (i.e. U˙ − = G˙ α where U˙ = G˙α) of + − U and K a maximal compact subgroup of G . Also let W− be a nbd. ∗ ∗ of 1 in U− G whose logarithm is a convex set. Since W−P is a nbd. ∩ ∗ ∗ of 1 in G , and K is compact a nbd. W of 1 in G with W W−P ∗ ∗ ∃ ∗ ⊂ ∗ and K [W] = W. ∗ UxnWΓ is stable under x and contains an non-empty open set, hence by Theorem 5.3 it diﬀers from G Γ in a set of measure zero. Therefore = n n(k) such that ∃ 1 1 n W− k− Γ x WΓ , φ ∩ ∴ Γ kWxnW , φ, n n ∩ n n but x W x W−P = x W− x− P W−P ⊂ ∗ ∗ ⊂ ∗ 44 5. Some Ergodic Properties of Discrete Subgroups

(using the convexity of logarithm of W−)

∴ Γ kWW−P , φ ∩ ∗ ∴ ΓP meets kWW−. ∗ This proves that K ΓP . ∴ K P ΓP G . ∗ ⊂ ∗ ∗ ∗ ⊂ ∗ ⊂ ∗ But we know that K P = G . ∗ ∗ ∗ Therefore G = ΓP . ∗ ∗ Chapter 6 Real Forms of Semi-simple Algebraic Groups

In this and the following sections, G will denote a semi-simple R-groups 57 RT a maximal R-split Torus, T a maximal R torus containing RT; R a − △ fundamental system of restricted roots on RT, a fundamental system of roots on T whose restriction to RT consists of R 0 (such a can + △∪{ } △ always be found). Φ, Φ will denote respectively the set of roots and the set of positive roots and Φ∗ the set of positive roots whose restriction to RT is non-zero. will denote the subset of consisting of those roots △◦ △ which are constant on RT. Given α Φ we deﬁne α Φ by the formula α (¯x) = α(x) x T, ∈ ′ ∈ ′ ∀ ∈ x¯ is complex conjugate of x. Then for any α α′ = α on . ∈ △◦ − △ − △◦ We can deﬁne a permutation σ by

= + nββ α′ σ(α) β nβ non-negative integers. ∈△◦ Satake’s Diagrams of semi-simple R groups. − In Dynkin’s diagram every root in is denoted by a back circle △◦ • and every root of by a white circle . If α then the white △−△◦ ◦ ∈△−△◦ } " circles corresponding to α and σ(x) are joined by a arrow .

Deﬁnition. GR is said to be R-simple if (GR)◦ has no proper normal subgroups of positive dimension.

45 46 6. Real Forms of Semi-simple Algebraic Groups

If GR is R-simple, but G is not simple then G˙ = restriction H˙ = 58 H˙ R C, where H˙ is a simple Lea algebra over C/R ⊗ Thus G˙ = H˙ H˙ , and the diagram of G˙ consists of two copies ⊕ of Dynkin’s diagram of H˙ , with vertices corresponding under complex conjugation joined by arrows

0 0 0 ... 0

Real forms of semi-simple Lie groups have been determined by F- Gautmacher (cf. Matsbornik (47) V. 5 (1939) pp. 217-249). The following is a complete list of C-simple R-groups (cf. [1], [20] & [24]). 1 = ♯ p = ♯R △ △

Group Type of p △ R △ A I SL(l + 1, R) 0-0-0-0-...-0 A1 p = 1 + 00-0-0-0-0-0----00 0 = l 1 A II S U∗(l 1) Ap p −2 A III S U(p, q) 0 0 0 0

(p < q, p + q = l + 1) B p 1/2 p ≤ 0 0 0 0 0 0 0 ... 0 l+1 l+1 0 = l+1 S U 2 , 2 Cp p 2 0 0 0 ... 0 BI S 0(p, 2l + 1 p) 0-0-0-...-0-0-0-...0 0 0 0 0 B p 1 − p ≤ CI S p(n, R) 0-0-0-----0 0 Cl 0-0-0-0----0-0-0-00 0 0 0 0 l 1 C II S p(p, l p) Bp p −2 , l − odd≤ S p(l/2, l/2) 0-0-0-...-0 0 0 0 Cp p = l/2, l even 0 DI S O(p, 2l p) 0-0-0-0...-0-0-...0 0 0 B p l 2 − 0 p ≤ − 0 S O(l.l) 0-0-0-...-0 D 0 1 6. Real Forms of Semi-simple Algebraic Groups 47

0 S O(l 1, l + 1) 0-0-0-0-...-0 Bl 1 − 0 − 0 D III S O (21) 0-0-0-...-0-0-0 B p = l 1 ∗ 0 p −2

S O∗(21) Cp p = l/2

E6 A2

E7 C3

E7 F4

E8 F4

F4 F4 ∨1 G2 G2 59

Deﬁnition. R-rank of an algebraic group is the dimension of a maximal 60 R-split torus.

From the diagrams above, we can excerpt the diagrams of groups of R-rank 1 and we list the dimension of the restricted root spaces. Let = α △ − △◦ { } Associated symmetric space diagram dim G˙2 dim G˙

0 0 1 Hyperbolic ... 0 2l 1  −  ...... 0 2l 2 − Hermitian hyperbolic  ...... 1 2l 2 − Quaternianic hyperbolic ... 3 4l 8 − Cayley hyperbolic 7 8 Now we collect some results whose proof require examining these 61 diagrams. 48 6. Real Forms of Semi-simple Algebraic Groups

T 6.1 Lemma. If N is the set of automorophisms of G stablizing T and R, then any automorphism τ of restricted root system RΦ is induced by an element of N. Proof. The result is true for inner automorphisms (i.e. elements in little 1 Weyl group). For given any element σ N(RT), both σTσ and T are ∈ − contained in Z(RT) and therefore are maximal R-tori of the connected algebraic group Z(RT). By the conjugacy of maximal tori, z Z(RT) ∃ ∈ such that 1 1 zσTσ− z− = T i.e., zσ N(T) ∈ The inner automorphism given by zσ is the desired element of N.

In case τ, is an outer automorphism we can without loss of generality assume that τ(R ) = R . △ △ From the usual root diagrams it is clear that only the restricted root systems of type A, E and D1 admit an outer automorphism. In case A and E the automorphism is an inner automorphism com- posed with the “opposition” map α α; since each of these extend to → − Φ, then so does the outer automorphism. If the restricted diagram is of type Dl then the Satake diagram shows that = R ; that is the group splits over R and hence the conclusion is △ △ hypothesis.

62 6.2 Lemma. If [Z(RT), Z(RT)] = J then J˙ = J˙1 + J˙2 is simple (possibly zero) and J˙1 is sum of compact Lie algebras of rank 1. Proof. We remark ﬁrst that is a fundamental system of roots for J. △◦ Now simply observe that the diagram of satisﬁes the condition re- △◦ quired by the conclusion.

Note. J˙2 is of rank 1 only if the group is S p(1, 2).

6.3 Lemma. Let W∗ be the subgroup of the Weyl group of T, which stabalizer RT. Then W is irreducible on J˙1 T˙ and J˙2 T.˙ ∗ ∩ ∩ Proof. J˙2 T˙ is a cartan subalgebra of the simple Lie algebra J˙2 and, ∩ as is well known, the Weyl group of a simple Lie algebra operates ir- reducibly of its associated Cartan subalgebra. It remains only to prove that W is irreducible on J˙1 T˙. ∗ ∩ 6. Real Forms of Semi-simple Algebraic Groups 49

Inspection of the position of in , as pictured in the diagrams △◦ △ shows that J˙1 is contained in a subalgebra of type A2p 1 with the diagram . − As is known, the Weyl group of A2p 1 SL(2p) is the group of − ≈ 2p permutations of the standard basis vectors e1,..., e2p in C . The roots ;1 of J˙1 become identified with α2i 1 α2i, i = 1,... p where αi △◦ th { − − } denotes the i matrix coefficient. Clearly the stabalizer of J˙1 contains the conjugation by the matrix sending each. e2i 1 e2π(i) 1 and e2i − → − → e)2π(i), for any permutation π of 1,..., p . Since these automorphisms { } of J˙1 induce the full symmetric group on the elements of the set ,1, △◦ we conclude that W is irreducible on J˙1 T˙. ∗ ∩ 6.4 Lemma. Let G1, G2 be two C-simple R-groups. Assume τ : T1 63 Φ Φ ˙ →˙ T2 is an isomorphism sending RT1 RT2 and 1∗ 2∗. Then G1 G2 τ → → ≈ and RT˙1 can be induced by an isomorphism of G˙1 and G˙2. | Proof. Suppose first that p, the R-rank of G1 and G2 is one. Then the τ-corresponding restricted root spaces must have the same dimension. The listed values in our table for dim Gα and dim G2α show that these determine the group of R-rank 1. Thus G˙1 G˙2 in the rank 1 case. ≈ Moreover, the isomorphism θ of G˙1 to G˙2 can be taken so as to map the restricted root spaces of G˙1,α of G˙1 to the restricted root space G˙2,τ(α) of G˙2. It follows at once that θ and τ induce the same map on RT˙ and thus the lemma is proved for p = 1.

Suppose now that ♯R > 1. We need only consider the case that △ the groups are not split over R (i.e., , R ), otherwise the result is a △ △ well-known theorem of Weyl. Assuming therefore that , R we ﬁnd △ △ that the restricted root diagrams are of type A, B, C, F4. In neither of these cases does the Dynkin diagram of R have a branch point. There- △ fore given a Satake diagram of a non R-split group, one can form a △ sequence of the subdiagrams (1) (2) ... (p) = such that △ ⊂ △ ⊂ △ △ (k) (a) ♯R = k △ (k+1) (k) k+1 k+1 (b) = D where ♯RD = 1. △ △ ∩ (c) Dk Dk+1 = Dk+1 (k). ∩ ∩ △ 50 6. Real Forms of Semi-simple Algebraic Groups

64 Given now the two groups G1 and G2 and the map τ, we decompose = (p 1) p the Satake diagram i of Gi as above, getting i i − Di . By △ (p 1) (p △1) △ (p 1)∪ induction there are isomorphisms θ − : 1 − 2 − and θp : p p △ → △ D1 D2 induced by isomorphisms of the corresponding Lie algebras ˙(p →1) p 1 p 1 ˙(p 1) ˙ p Gi − and Fi . Let ϕ denote the restriction of θ−p θ − to G1 − Fi . p (p 1) p p ∩ Set = − D . Then is connected since the root diagram △i, △i ∩ i △i, ◦ ◦ p = p 1 p has no loops. In fact by property (c) i, Di − Di and is in fact △ △ ◦ ∩ a connected component in i, , the diagram of the R-compact part of △ ◦ Z(Ti). An additional inspection of the diagrams shows that no connected component of the diagram of Z(T1) admits an (outer) automorphisms. (p 1) p Hence ϕ is an inner automorphisms of G˙ − F˙ and thus extends to 1 ∩ 1 an automorphisms χ of G˙1. Replacing θp by θp χ, we obtain the derived isomorphisms of G˙1 onto G˙2. The following is an easy consequence of previous lemma. 6.5 Theorem. Let G be a semisimple R-group having no compact factors. Let τ : T T be an isomorphism which stabilizes RT and Φ . → ∗ Then there exists an automorphism θ of G such that θ τ stabilizes T and on RT it is identity. 6.6 Lemma. Let G be a semisimple R-group with no R-compact factors. We also assume that GR is simple of R-rank 1. If τ is an automorphisms 65 of T stabilizing RT and Φ , then it stabilizes J˙ T,˙ J˙1 T˙ and J˙2 T.˙ ∗ ∩ ∩ ∩ Proof. Let 2 B∗ = α α Φ ∈ ∗ Then τ preserves B∗. Let B, B denote the killing forms of G and Z(T) ◦ respectively then B = B + 2B∗. So any two subspaces of T˙ orthogonal ◦ with respect to both B and B are orthogonal with respect to B∗. ◦

Let X, X J˙1 and Y J˙2, then ′ ∈ ∈ B([X, X′], Y) = B(X′, [X, Y]) = 0 − and since [J˙2, J˙2] = J˙2, we have B(J˙1, J˙2) = 0 similarly B (J˙1, J˙2) = 0 − ◦ ∴ B∗(J˙1 T˙, J˙2 T˙) = 0. ∩ ∩ 6. Real Forms of Semi-simple Algebraic Groups 51

Composing τ with an automorphisms of G˙, we can assume by the Theorem 6.5 that τ induces identity on RT. Thus τ stabilizes the set of all roots having a non-trivial restriction on RT and we can assume accordingly that R consists of a single element, and that the set of △ positive roots in RΦ is either α¯ or α,¯ 2¯α . Let S denote the set of { } { } roots restricting toα ¯ τ stabilizes Φ and therefore also the set S S of ∗ − diﬀerences of roots in S . These diﬀerences clearly lie in linear span of , conversely, given any root β we shall show that β occurs in {△◦} ∈ {△◦} S S . We can assume β> 0. − The hypothesis that G contains no R- compact factors is tantamount 66 to the hypothesis that Gα, α S generates G. Hence < β, S >, 0. { ± ∈ } Let α be the least root in S for which < β,α >, 0. Then σβ(α) = + = 2<α,β> α q(α,β)β is a root where q(α,β) −<β,β> is a positive integer. Thus α + q(α,β)β S and β S S . Hence S S = as asserted. ∈ ∈ − { − } {△◦} Therefore τ stabilizes the intersection of the kernels of the linear functions in e.g. τ stabilizes Z(J) T˙. Now J˙ T˙ is the orthogonal △◦ ∩ ∩ complement of Z(J) T˙ with respect to both killing forms B and B and ∩ ◦ therefore with respect to B∗ = B 2B . Since τ stabilizes B∗, it stabilizes − ◦ J˙ T˙. ∩ Having assumed that G has R-rank 1, we see that τ is simple in all cases except G = Cl(l = 3) or G = D3. In the second case J˙ = J˙1, J˙2 = 0 and the Lemma is established. In case G = Cl(l = 3) the diagram is

and the roots in Φ∗ having the same restriction to RT as 2α2 are 2α2 + ...+2αl 1 +αl; α1 +2α2 +... 2αl 1 +α1, 2α1 +2α2 +...+2αl 1 +αl. Since − − − τ permutes this set, it permutes the diﬀerences and therefore τα1 = α1. ± Hence τ stabilizes T˙2 = ker α1 J˙, and therefore stabilizes T˙1 which is ∩ the orthogonal complement of T˙2 in T˙ J˙ with respect to B . The proof ∩ ∗ of the Lemma is now complete.

Chapter 7 Φ Automorphisms of ∗

7.1 Lemma. Let G be an R-group with no compact factors. Let τ : T˙ 67 → T˙ be a automorphism stabilizing RT and Φ∗ then τ preserves the Killing form. Proof. T˙ = (J˙1 T˙) + (J˙2 T˙) + (Z(J˙) T˙) ∩ ∩ ∩ 2 we know that (i) τ preserves B∗ = α (ii) the three subspaces J˙1 α Φ ∩ ∈ ∗ T˙, J˙2 T˙ and Z(J˙) T˙ are stable under τ and (iii) if W is the subgroup ∩ ∩ ∗ of Weyl group stabilizing RT then J˙1 T˙ and J˙2 T˙ are irreducible ∩ ∩ under W∗. = = Since B∗ and B are preserved by W∗, Bi CiBi∗ i 1, 2. ci , 0. = ˙ ˙ Here Bi, Bi∗(i 1, 2) are restrictions to Ji T of B, B∗ respectively. Let ˙ ˙ ∩ = B3, B3∗ are restrictions to Z(J)i T of B, B∗ respectively, then B3 B3∗. = + ∩ + Hence on T, B B3∗ C1B1∗ C2B2∗. As τ preserves B1∗, B2∗&B3∗ it also preserves B. 7.2 Lemma. Let G be an R-group without compact factors and let τ : T˙ T˙ be an automorphism stabilizing RT˙ and Φ then τ is restriction → ∗ to T˙ of an automorphism of G.˙

Proof. Let W be the subgroup of Weyl group of T generated by σα, ′ { α Φ∗ . We shall ﬁrst prove that W = W′. Given B , < β, Φ∗ >, ∈ } ∈ {△◦} 0 since G has no R-compact factors and hence G α,α Φ∗ generates { ± ∈ } G.

53 54 7. Automorphisms of Φ∗

We can ﬁnd α Φ with <β, α>< 0. ∈ ∗ 68 As 2 <β,α> σ (β) = β + q(β,α)α where q(β,α) = − > 0 α <α,α>

σσ (β) Φ∗ α ∈ 1 ∴ σσ (β) W′ but σσ (β) = σασβσ− α ∈ α α 1 ∴ σβ = σ− σσ (β)σα W′ for all β z α α ∈ ∈ {△ } ∴ W′ = W.

1 τσατ = στ(α) is a reﬂection since for α Φ τ(α) Φ − ∈ ∗ ∈ ∗ 1 τW′τ− = W′ 1 ∴ τWτ− = W.

Thus τ permutes reﬂections in W, i.e. τ permutes the set σα,α { ∈ Φ . } ∴ τΦ = Φ ∴ τ extends to an automorphisms of G˙. Chapter 8 The First Main Theorem

This section is devoted to the proof of 69 8.1 Theorem. Let G be a semi-simple analytic group with no compact factors and no center.∗ K be a maximal compact subgroup. Let X = G/K and let Γ, Γ′ be two discrete subgroups of G , isomorphic under ∗ an isomorphism θ : Γ Γ′. We assume that G Γ, G Γ′ have finite → ∗ ∗ Haar measure. Let X be the unique compact G orbit in some Satake- compactification of X.◦ Let ϕ : X X be a homeomorphism∗ such that (i) → ϕ(γx) = θ(γ)ϕ(x) γ Γ, x X: (ii) ϕ extends to a homeomorphism of ∀ ∈ ∈ X X whose restriction to X is a diffeomorphism of X , then θ extends ∪ ◦ ◦ ◦ to an automorphisms of G . ∗ [Conjecture. Condition (ii) is superfluous if G has no factors isomorphic to PSL(2, R).] For the proof of the theorem we need following lemmas. 8.2 Lemma. Let G be a connected reductive linear algebraic group. Let kT be a maximal k-split torus and T be a maximal k-torus containing kT . T Let t1, t2 be elements in k conjugate in G. Then t1, t2 are conjugate by an element in Norm (KT) Norm T. ∩ Proof. From Bruhat’s decomposition + + G = N (Norm kT)N . Suppose 1 xt1x− = t2 with x G ∈ 55 56 8. The First Main Theorem

and x = uwv u, v N+, w Norm kT ∈ ∈ 70 then

uwvt1 = t2uwv ∴ 1 = uwt1t1− [v] t2[u]t2wv.

By the uniqueness of Bruhat’s decomposition wt1 = t2w. Thus t1, t2 T 1 are conjugate by w (Norm k ). Since T and wTw− are contained in 1 Z(kT), by the conjugacy of maximal tori, λ Z(kT) such that λTλ− = 1 1 1 1 ∃ ∈ T wTw− i.e., λ− wTew− λ = T i.e., λ− w Norm (k ) Norm T. ∈ 1 ∩ It is clear that t1 and t2 are conjugate by λ− w. 8.3 Lemma. Let G be the Zariski closure of a real linear algebraic group G , let RT be a maximal R-split torus in G and T be a maximal ∗ A R-torus containing RT. W = Norm RT. P be the stabalizer in G of a point in X , P the Zariski closure of P , U∗ the unipotent radical of∗ P. ◦ ∗ We assume that P T. RU the set of roots occurring in U, RN+ the set ⊃ + of roots occurring in N , then A W (RU ) = RN+ . ± Proof. From our description of Satake compactfication in § 2, we know that P = P( ) for some R . Indeed in the notation of § 2, = Eρ △′ △′ ⊂ △ △′ where ρ is an R-irreducible representation with finite kernel, and thus P( ) contains no normal subgroup of positive dimension, equivalently, △′ the subset contains no connected component of the fundamental sys- △′ tem of restricted roots R . △ + 71 We have P( ) = G( ). N( ), U = N( ) and N = N(φ). It △′ △′ △′ △′ is easy to see that if R is connected, then RU contains a root whose △ restriction to RT has length equal to the length of any restricted root in R . We recall that the Weyl group of a connected root system permutes △ transitively all roots having the same length. Applying this observation A to each connected component of R , we find W (restriction of RU to △ RT)= all restricted roots. Hence A W (RU ) = RU+ . ± 8. The First Main Theorem 57

8.4 Lemma. Let A = (RT R) and let b L.A = Z(RT)R. Then b is ◦ ∈ R-regular iff b keeps fixed exactly m/m points [here m = ♯WA and m = order of the Weyl group of G( )] and◦ on the tangent space at these◦ △′ points, the eigenvalues are different from 1 in absolute value.

Proof. Let U− denote the opposite of U. Suppose b is R-regular then the eigenvalues of b on tangent spaces at the points fixed under b are the A values of W (RU ) on b. Conversely if b LA has only finitely many − ∈ fixed points on X , then b is R-regular. The Lemma is now clear. ◦

8.5 Lemma. Let G , Γ, Γ′ be as in the hypothesis of the Theorem 8.1. Let γ be a reductive∗R-regular element of Γ. Then θ(γ) is also reductive R-regular.

Proof. Let p X and let P be the stabilizer of p in G . For g P 72 ◦ ∈ ◦ ∗ ◦ ∗ ∈ ∗ we denote byg ˆ the operation of g on G˙ /P˙ . An element g P is re- ∗ ∗ ∈ ∗ ductive R-regular iff AdN +g has eigenvalues different from 1 in absolute value; this will be true if g keeps fixed m/m points of X = G /P and on each of the tangent spaces at the fixed points,◦ takes eigenvalues◦ ∗ ∗ , 1 in absolute value.

Thus g G is reductive R-regular iﬀ it keeps ﬁxed m/m points in ∈ ◦ X and on the tangent space each point has eigenvalues , 1 in absolute value.◦ From this it will follow that if γ is R-regular then θ(γ) is also R-regular.

Remark. If G is a reductive algebraic group over any field k, then it follows immediately from definitions that an element of G is k-regular iff it keeps only a finite number of points in G/p fixed for P = P( ). ∀ △′ ′ k . It can be proved that the element is reductive iff the number △ ⊂ △ order of the Weyl group of G of fixed points is ; it is unipotent iff the order of the Weyl group of P(i.e. of C( ′)) number of fixed points is precisely 1. △

8.6 Lemma. If H = TR◦, there exists an automorphism τ of H and a Zariski dense subset Hτ of R-regular elements in H such that h H, h ∀ ∈ and τ(h) operate equivalently on X , i.e., there exists a diﬀeomorphism 1 ◦ Φ of X such that h = Φ− τ(h)Φ . ◦ ◦ ◦ ◦ 58 8. The First Main Theorem

1 Proof. Let A = a; a A α(a) > 1 α R . Let K be a maximal { ∈ ∀ ∈ △} compact subgroup of G . Recall Z(RT)R = L.A. We can assume tht K L. Let (1) be the projection∗ of 1 in X = G /K. ⊃ ∗ 73 Let p = lim an(1), a A1 and let P be the stabilizer of p . ◦ n ∗ ◦ →∞ ∈

P = P( ′) = P(Φ). △ Set V = tangent space to X at p , then V G˙ /P˙ . Let g P and ◦ ◦ ≈ ∗ ∗ ∈ ∗ letg ˆ denote the operation of g on V. If H P , Hˆ C; where C is a ⊂ ∗ ⊂ Cartan subgroup of GL(V). Let W = N(C) be the Weyl group of C. For any element γ Γ set Z(C) ∈ γ′ = θ(γ). Given a reductive R-regular element γ of Γ, there exists a g G ∈ ∗ such that g[γ] belongs to H LA′. The element θ(γ) is also reductive ∩ 1 R-regular. Therefore g′ G such that g′[γ′] H LA . ∃ ∈ ∗ ∈ ∩ Since

ϕ(γp) = θ(γ)ϕ(p)m, we can write 1 θ(γ) = ϕγϕ− (= ϕ[γ]) 1 g′[γ′] = g′[ϕ[γ]] = g′[ϕ[g− g[γ]]] 1 ∴ g′(γ′) = g′ϕg− [g[γ]] y y 1 g′[ˆγ′] = σ g[γ](ˆ σ )−

γ 1 where σ is the diﬀerential of g′ϕg− at p . ◦ Therefore there is an element τγ in W, the Weyl group of C such that

= g′[γ′] τγ(g[γ]).

74 For any element w W, letHw denote the subset of H LA′ G [Γ] ∈ γ ∩Γ ∩ ∗ on which the map γ τ is constant.ÿ Since H LA′ G [ ] is Zariski → ∩ ∩ ∗ dense in H and W is ﬁnite, there exists a τ W such that Hτ is Zariski ∈ dense in H. Denoting Zariski closure by superscript , we can write ∗ = Hτ∗ H∗ 8. The First Main Theorem 59 since

τ(Hτ) H ⊂ ∴ τ(H∗) = H∗ and therefore τ(H) = H.

Thus τ induces an automorphism of H, and by deﬁnition, h and τ(h) operate equivalently on x for all h Hτ. ◦ ∈ Proof of the Theorem 8.1. = = R Let S 1 Hw and let S S 1∗ non -regular ele- w W ∪ Hw not Zariski∈ dense ments in H. Then clearly S ∗ , H∗. Let τ be an automorphism of H given by the previous lemma. Then τ permutes the roots Φ∗, that is

α(h); h Hτ,α Φ∗ = α(τ(h)); α Φ∗, h Hτ { ∈ ∈ } { ∈ ∈ }

By the lemma 8.3 τ permutes Φ. Hence Tr Adg[γ] = Tr Ad(g′[γ′]) 75 that is, Tr Adγ = Tr Adγ′ g Γ G [Hτ]. It follows that Tr Adγ = ∀ ∈ ∩ ∗ Tr Adγ′ for all γ Γ G [H S 1]. ∈ ∩ ∗ − Since G is without center we can identify it with AdG. Given γ Γ and S H with S , H and n any positive inte- ∈ ⊂ ∗ ∗ ger, by Lemma 4.4. γ Γ G[H S ] such that γ ,γ2,...γn, γγ , ∃ ◦ ∈ ∩ − ◦ ◦ ◦ ◦ γγ2,...γγn G[H S ]. ◦ ◦ ∈ − Let n = dim G. Then Tr (γγm) = Tr θ(γγm) = Tr θ(γ)θ(γ )m for ◦ 2 ◦ n ◦ m = 1,... n. We can write 1 = c1γ + c2γ + + cnγ = f (γ ) by ◦ ◦ ◦ ◦ setting the characteristic polynomial of γ equal to zero. ◦ Then

n n Tr γ = Tr γ f (γ ) = Tr (cmγγ ) ◦ m=0 ◦ n m = cm Tr θ(γ)θ(γ ) = ◦ m0 60 8. The First Main Theorem

= Tr θ(γ) f (θ(γ )). ◦ But Tr γm = Tr θ(γ )m for m = 1,..., n and thus γ and θ(γ ) have the same characteristic◦ polynomial,◦ by Newton’s formulae.◦ ◦ Hence f (θ(γ )) = 1 and Tr γ = Tr θ(γ) for all γ Γ. ◦ ∈

Suppose Cγγ = 0). Then 0 = Tr Cγγ dγ γ∗ dγ C ∗ ∗ γ Γ γ Γ ∀ ∈ ∈ ∈ γ Γ. This will imply Ñ é ∀ ∗ ∈

Tr Cγθ(γ) dγ θ(γ∗) = 0 ∗ γ Γ γ Γ ∈ ∗∈ Ñ é 76 Let denote the C linear span of Γ. Clearly is an associative E E matrix algebra. By the density theorem (that the Zariski closure of Γ is G), the linear span of Γ s linear span of G . Thus Tr Cγθ(γ)e = 0 for ∗ γ Γ all e . ∈ ∈E We can (and we will) assume that AdG is self adjoint. The we can assert t Tr ( Cγθ(γ)) Cγθ(γ) = 0 t This implies that Cγθ(γ) = 0. Therefore θ induces a linear isomorphism of onto , since Cγγ = 0 implies Cγθ(γ) = 0. Clearly E E θ is an R-algebra automorphism θ(Γ ) R = (θ(Γ)) R implies that ∗ ∩E ∗ ∩E G = G◦R = (G R)◦ = θ((G R)◦) = θ(G ), since Γ and θ(Γ) are ∗ ∩E ∩E ∗ Zariski dense in G. Thus we have proved that θ extends to an automorphism of G . ∗ Chapter 9 The Main Conjectures and the Main Theorem

Let G be a real analytic semi-simple group with no center and no com- 77 pact factors, and let K be a maximal compact subgroup. Let X = G/K and let Γ, Γ′ be two discrete subgroups of G, isomorphic under an isomorphism θ : Γ Γ . We assume that G/Γ, G/Γ have ﬁnite Haar mea- → ′ ′ sure. Let ϕ : X X be a homeomorphism such that ϕ(γx) = θ(γ)ϕ(x) → to γ Γ and x X. Then ∀ ∈ ∈ Conjecture 1. θ extends to an analytic automorphism of G provided G contains no factor locally isomorphic to SL(2, R).

Conjecture 2. Let X be the unique compact G-orbit of a Satake com- pactiﬁcation of X. Then◦ ϕ extends to a homeomorphism of X X . 1 ∪ ◦ Let ϕ be the restriction to X of the extension, then ϕ Gϕ− = G as transformation◦ of X , provided◦ G has no factor locally◦ isomorphic◦ to SL(2, R). ◦

It is not diﬃcult to see that Conjecture 2 implies Conjecture 1. In- deed we remark ﬁrst that G operates faithfully on X , since G has no compact factors and no center. Since X is topologically◦ dense in X X ∪ ◦ we have ϕ (γx) = θ(γ)ϕ (x) for all x X and all γ Γ; that is, θ(γ) = 1 ◦ ◦ ∈ ◦ 1 ∈ 1 ϕ γϕ− as transformations of X . If ϕ Gϕ− = G, then g ϕ gϕ− ◦ ◦ ◦ ◦ ◦ → ◦ ◦ is a continuous automorphism of G with respect to the compact open

61 62 9. The Main Conjectures and the Main Theorem

topology of G as a transformation group of X . As is well-known; this 1 ◦ implies that g ϕ gϕ− is a continuous automorphism of the analytic → ◦ ◦ group G and hence an analytic automorphism. 78 The following example shows that SL(2, R)/ 1 violates the con- ± jecture.

9.1 Example. Let G = SL(2, R)/ 1, K = S O(2, R)/ 1. Then X is ± ± the upper half plane with G operating as linear fraction transformations z az+b . Alternatively, we may identify X with the interior of the unit → cz+d ball in the plane.

Let S and S ′ be two compact Riemann surfaces of genus > 1 which are diffeomorphic but not conformally equivalent. Let Γ = π1(S ) and Γ = π1(S ) be the fundamental groups of S and S . Let ψ : S S ′ ′ ′ → ′ be a diffeomorphism, let θ : Γ Γ be the induced isomorphism of → ′ fundamental groups, and let ϕ : X X be the lift of ψ to the simply → connected covering spaces of S and S ′; by uniformization theory, the latter may be identified with X. Then ϕ(γx) = θ(γ)ϕ(x) for all γ Γ, x ∈ 1∈ X. As transformation groups on X we can therefore write Γ′ = ϕγϕ− . 1 However G , ϕGϕ− unless ϕ is a Mobius transformation of X. Pursuing the example further, the map ϕ is a so-called quasiconfor- mal map (cf. next sections for definitions and properties) and therefore induces a homeomorphism ϕ of the boundary X of the unit ball. Then 1 ◦ ◦ ϕ Γ′ϕ− =Γ′ as transformations of X since X is dense in X X . How- ◦ ◦ 1 ◦ ∪ ◦ ever G , ϕ Gϕ− unless ϕ is a Moebius transformation of the circle X . ◦ ◦ ◦ ◦ 79 The following trivial example serves to illustrate that once θ is given ϕ is uniquely determined by contrast with ϕ which is not unique; and ◦ 1 1 that ϕGϕ− = G is not necessary even when ϕ Gϕ− = G. ◦ ◦

9.2 Example. Let Γ=Γ′, θ = Identity, ψ a homeomorphism which is the identity map except on some small neighbourhood of X/Γ. Then 1 ϕGϕ− , G since otherwise ϕ would have to be the identity map. How- 1 ever ϕ is the identity map and in particular ϕ Gϕ− = G. ◦ ◦ ◦ In these lectures we prove a slightly modified form of conjecture 2 9. The Main Conjectures and the Main Theorem 63 for the group G = 0(1, n)/ 1 where n > 2. More precisely ± 9.3 Theorem. Let G = 0(1, n)/ 1, n > 2, and let X be the associated ± Riemannian space. Let Γ, Γ′ be discrete subgroups such that G/Γ and G/Γ have finite Haar measure. Let ϕ : X X be a homeomorphism ′ → and θ : Γ Γ an isomorphism such that ϕ(γx) = θ(γ)ϕ(x) for all γ Γ, → ′ ∈ x X. Assume that ϕ is quasi-conformal (cf. below for definition) then ∈ ϕ induces a diffeomorphism ϕ of the boundary component X of the ◦ 1 ◦ Satake compactification of X and moreover ϕ Gϕ− = G. ◦ ◦ Note. The condition that ϕ be quasi-conformal is automatically fulfilled if G/Γ and G/Γ′ are compact and ϕ is diffeomorphism. The proof of this theorem is based on the theory of quasi conformal mappings cf. [17]. In the following section we present a summary of our proof.

Chapter 10 Quasi-conformal Mappings

Deﬁnition. Mobius¨ n-space is the one point compactiﬁcation of eucli- 80 dean n-space Rn, it will be denoted by Rn . ∪ {∞} GM(n) the Mobius¨ group of Mobius¨ n-space is the group of transformations generated by “inversion” in the sphere S n

2 2 2 η + η + + η + = 1. 1 2 n 1 If we set η = yi (i = 1,..., n+1) then S n is realized as the projective i yi 2 2 2 = variety y y1 yn+1 0, and one can prove that GM(n) becomes ◦ − −−−−− identified with 0(1, n + 1)/ 1([17]p.57). ± 10.1 Theorem. The subgroup G′ of GM(n) which stabilizes the hemisphere S (ηi+1 < 0) is isomorphic to GM(n 1) under the restriction − − homomorphism into its action on the equatorial n 1 sphere ηn+1 = 0. − Moreover G′ operates transitively on S and keeps invariant a positive definite quadratic differential form dS 2.− Under stereographic projection 2 + + 2 from (0, 0,..., 0, 1), S maps onto the unit ball x1 xn < 1 and its 2 − dx2 invariant metric dS upto a constant factor becomes 1 x 2 , where dx is usual euclidean metric. (loc.−| cit.| pp. 58-59)

2 The unit ball x < 1 with metric dx where dx is euclidean met- | | 1 x 2 ric, is a Riemannian space called the hyperbolic−| | n-space, the isotropy subgroup at a point is 0(n) (In this realization of hyperbolic space, the isometries of hyperbolic metric preserve euclidean angles).

65 66 10. Quasi-conformal Mappings

Hence the spaces have constant curvature. 81 We introduce following notations: Let V, W be two Riemannian spaces and let ϕ : V W be a home- → omorphism. Let

Lϕ(p, r) = sup d(ϕ(p),ϕ(q)) d(p,q)=r

lϕ(p, r) = inf d(ϕ(p),ϕ(q)) d(p,q)=r Lϕ(p, r) Hϕ(p) = lim r 0 l (p, r) → ϕ Lϕ(p, r) Iϕ(p) = lim . r 0 r → m(ϕ(Tr(p))) Jϕ(p) = lim r 0 (mT (p)) → r m is the Hausdorﬀ measure. *** and *** where for any subset E of V, Tr(E) denotes the tubulor neighbourhood of E of radius r.

TrE = v; v V d(v, E) r { ∈ ≤ } Definitions. *** is said to be quasi-conformal iff there exists a constant B with Hϕ(p) B p V. ≤ ∀ ∈ A quasi-conformal is said to be k-quasi-conformal iff Hϕ(p) k for ≤ almost all p V. ∈ The foregoing definition is not well-suited for proving some of the basic theorems concerning quasi-conformal mappings. The develop- ment below leads to an alternative definition of quasi-conformal mapping in terms of the modulus of a shell.

82 Deﬁnitions. A shell D in Mobius¨ n-space Rn is an open connected ∪{∞} set whose complement consists of two connected components C and n ◦ C1. A shell not containing the point is called a shell in R . The com- ∞ ponent C1 of its complement which contains is the un bounded com- ∞ ponent and the other component C will be referred to as the bounded component. ◦ 10. Quasi-conformal Mappings 67

For a shell D in Mobius-¨ n space, we deﬁne its conformal capacity

C(D) = inf ▽u ndD u | | D 1 where u varies over C -functions with u(c ) = 0u(c1) = 1 C , C1 being connected components of the complement◦ of D. We will call◦ such a function u a smooth admissible function. It is easy to see that C(D) is invariant under conformal mapping, since the integral ▽u ndD is D | | invariant. Let Cn 1 denote the area of the surface of the unit n-ball. Then we deﬁne − 1 Cn 1 n 1 mod D = − − . C(D) n 10.2 Example. If Da,b = m x, x R a < x > b then { ∈ Å | |ã } (n 1) b − − b C(Da,b) = Cn 1 log and mod Da,b = log . − a a

Proof. Let u be a smoothÅ admissibleã function for Da,b then 83 b b n 1 n 1 − − 1 ▽u dr = ▽u r n r− n dr. ≤ | | | | a a

By Holder’s¨ inequality

b b 1/n n 1 b −n n n 1 1 1 ▽u dr < ▽u r − dr r− dr ≤ | | | | a a a Ñ é raising to the power n, and integrating over allÇ rays å

(n 1) n b − cn 1 ▽u dD log − ≤ | | a D (n 1) b − − ∴ C(Da,b) Åcn 1 log ã Å . ã ≥ − a

Å ã 68 10. Quasi-conformal Mappings

On the otherhand by taking smooth admissible approximations of the function 0 x a | |≤ = log x log a u  log b−log a a x b  − ≤ | |≤  1 b x ≤ | |  we get  n b n n b − 1 n 1 C(Da,b) ▽u dD = Cn 1 log r − dr ≤ | | − a r a (n 1) b − − = Cn 1 Ålog ã Å ã − a (n 1) b − − ∴ C(Da,b) = Cn 1 Ålog ã . − a

84 Therefore Å ãb mod (D ) = log . a,b a

Deﬁnition. Let D, D′ be two shells with C′ C and C1′ C1 then ◦ ⊃ ◦ ⊃ we say “D separates the boundary of D”. Clearly in this case C(D ) ′ ′ ≥ C(D) and mod D mod D. ′ ≤ n 1 10.3 Lemma. Let S r = x x R , x = r and let u be a C function on { | ∈ | | } S r then there exists a constant A depending only on n such that

n n (CS CS u) A.r ▽u dS r r ≤ | | S r (For a proof see p.69 [17]). 10.4 Lemma (Loewner). Let D be a shell in M¨obius n-space and let C , C1 denote the connected components of the complement of D, then ◦ C(D) > 0 if neither C nor C1 consists of a single point. ◦ n Proof. Choose a point p in R such that S r, the sphere with center at p and radius r meets C and C1 for all r with 0 < r1 < r < r2 then ◦

▽u ndD = ▽u ndx ▽u ndx D | | n | | ≥ D | | r1,r2 10. Quasi-conformal Mappings 69

r2 n = ▽u dσdr where dσ is n 1 measure on S r. | | − r1 Sr

By the previous lemma 85

n 1 1 n 1 1 ▽u dσ A− r− (CS CS u) = A− r− . | | ≥ r Sr

r2 Thus ▽u ndD A 1 dr = A 1 log r2 for all smooth admissible − r − r1 D | | r1 1 r2 functions u. Hence C(D) A− log > 0. ≥ r1 Deﬁnition. A continuous function f on the interval 0 x b is called d f ≤ ≤ absolutely continuous if its derivative dx exists almost everywhere and x1 d f = is integrable and dx dx f (x1) f (x ) for all a x , x1 b. x − ◦ ≤ ◦ ≤ ◦ A function u on an open subset D of Rn is called ACL in D, if in any closed ball lying in D it is absolutely continuous on almost all lines in the ball parallel to the coordinate axes.

Notations.

n E+ = x; x R xn > 0 + ∈ S = S r E+. r ∩ 10.5 Lemma. If u is an ACL function on E+ then

b OS C n dr n + u 2A ▽u dx. S r r ≤ E+ | | a

This is a slight generalizationÄ ä of Lemma 10.3, for a proof see pp. 72-73 [17]. 70 10. Quasi-conformal Mappings

86 10.6 Lemma.

n n Iϕ(p) (Hϕ(p)) Jϕ(p) n ≤ n and I 1 (ϕ(p)) (Hϕ(p)) J 1 (ϕ(p)). ϕ− ϕ− Proof. The ﬁrst inequality comes from

L (p, r) n L (p, r) n l (p, r) n ϕ = ϕ ϕ . r lϕ(p, r) r

The proof of theÅ second inequalityã Ç is similar.å Å ã Remark. It can be proved that if ϕ is diﬀerentiable at p then

n n 1 I (p) (Hϕ(p)) − Jϕ(p). ϕ ≤ 10.7 Lemma. Let ϕ be a quasi-conformal mapping then ϕ exists almost everywhere. Proof. By the previous lemma

n n I (p) (Hϕ(p)) Jϕ(p). ϕ ≤

By hypothesis Hϕ(p) < B p. By Lebesgue’s theorem (Saks [19] p. ∀ 115)

Jϕ(p) < a.e. ∞ ∴ Iϕ(p) < a.e. ∞ ϕ(q) ϕ(p) i.e. lim − < a.e., q p q p ∞ → | − | By the Radamacher-Stepnoﬀ theorem ([19] pp. 310-312)ϕ ˙ exists a.e.,

n 87 10.8 Lemma. Let D, D′ be open in R and let ϕ : D D′ be homeo- n → morphism of D into D′. Let p be a hyperplane in R , if Hϕ(p) < k for p D p, then ϕ in ACL on D and ϕ 1 is ACL on ϕ 1(D). (See [17] ∈ − − − for a proof.) 10. Quasi-conformal Mappings 71

Deﬁnition. Given a shell D; a continuous function u on D¯ , ACL in D is said to be admissible if u(L D¯ ) = 0 and u(C1 D¯ ) = 1, C , C1 being ◦ ∩ ∩ ◦ connected components of the complement of D.

10.9 Lemma. C(D) = inf ▽u ndD u admissible | | D (See [17] pp. 64 for a proof).

10.10 Lemma. Let ϕ : D D′ be a homeomorphism of shells, if n n 1 → is ACL and I ϕ k Jϕ almost everywhere, then mod ϕ(D) k ≤ − ≤ mod D.

Proof. Given u an admissible function on D, set u = u ϕ 1 then u ′ ◦ − ↔ u′ is bijective correspondence between admissible functions on D and D′.

u(q) u(p) ▽(u)(p) = lim | − | q p q p → | − | u(q) u(p) ϕ(q) ϕ(p) = − | − | ϕ(q) ϕ(p) q p | − | | − | = ▽(u′)ϕ(p) Iϕ(p). | | n n 1 ∴ C(D) (▽(u′))(ϕ(p)) k − Jϕ(p). | | n 1 n = k − ▽u′ dD | | D′ n 1 ∴ C(D) k − C(D′) ≤ ∴ mod D′ k mod D. ≤ We now deﬁne the spherical symmetrization of a shell for the pur- 88 pose of obtaining a rough quantitative estimate for the modulus of a shell. Let L denote the ray (t, 0,... 0) < t 0 in Rn, and let E be a n { −∞ ≤ } n set, open or closed, in R . For each sphere S r = x, x R , x = r place { ∈ | | } along S ra spherical cap (of dimension n 1) with center at S r E. Take − ∩ 72 10. Quasi-conformal Mappings

the cap open if E is open closed if E is closed, and equal to S r if S r E. ⊂ The resulting set is denoted by E∗. Clearly E∗ is open resp. closed, resp. connected) if E is open (resp. closed, resp. connected).

Deﬁnition. Let D be a shell in Rn. The spherical symmetrization of D is the set D◦ = (D C )∗ C∗. ∪ ◦ − ◦ where C is the bounded component of D. It is clear that D is a shell. ◦ ◦

10.11 Theorem. C(D) C(D ) ≥ ◦ The proof of this theorem makes use of the isoperimetric inequalities for both euclidean and spherical space (cf. Mostow, loc. cit, p.87). Intuitively the result is plausible because the spherical symmetrization 89 of D is a “smoothing” of D and hence admissible function for D◦ need to be “twist less”, accordingly C(D ) C(D). ◦ ≤ In the proof of next lemma, we will estimate the modules of a shell by comparing it with a special shell which generalizes a special slit plane domain considered by Teichmuller.

Deﬁnition. The Teichmuller shell D+(b) is the shell in Rn whose com- plementary components consist of the segment 1 x1 0, x2 = = − ≤ ≤ xn = 0 and the ray b x1 < , x2 = x3 = xn = 0 where b > 0. ≤ ∞ − n 10.12 Lemma. ϕ : R R′ be a homeomorphisms of domains in R , → k assume mod ϕ(D) k mod D, then Hϕ < C , where C depends only ≤ on n.

Proof. For p R, we consider the spherical shell Dl (p,r),L (p,r) centered ∈ ϕ ϕ at ϕ(p). Let D = ϕ 1(D ) then log Lϕ(p,r) = − lϕ(b,r),Lϕ(p,r) lϕ(p,r) mod Dl(p,r),L(p,r) k mod D k mod D (by 10.11) [D is spheri- ≤ ≤ ◦ ◦ cal symmetrization of D]

k mod Dτ(1). ≤

Since D◦ separates the boundaries of Dτ(1). 10. Quasi-conformal Mappings 73

Set C = mod Dτ(1). Then Lϕ(p,r) = Ck lϕ(p,r)

k ∴ Hϕ(p) C p R. ≤ ∀ ∈

Note. The idea of comparing mod D with mod Dτ(1) is due to A. Mori cf. his posthumous paper in the Transaction of the AKS V. 84 (1957) pp. 56-77.

Putting together Lemmas 10.8, 10.10 and 10.12, we can now assert 90

10.13 Theorem. Let ϕ : E E be a homeomorphism of domains in → ′ Rn n. Then ϕ is quasi-conformal iﬀ

(1) ϕ is ACL in E.

(2) For all shells D E, k 1 mod D mod ϕ(D) k mod D, for ⊂ − ≤ ≤ some constant k.

We now prove two theorems that are of central importance for our main theorem.

10.14 Theorem. Let ϕ be a quasi-conformal mapping of an open ball in Rn onto itself. Then ϕ extends to a homeomorphism of the closed ball.

Proof. Mapping the domain of ϕ onto the upper half space X = (x1 ... xn), xn > 0 via a Mobius¨ transformation, the theorem is seen { } to be equivalent to the assertion a quasi conformal mapping ϕ : X → Y = y : y < 1 extends to a continuous mapping at any point x of the { | | } boundary of X. for convenience, we take x = 0.

The proof is by contradiction. If lim ϕ(p) (p X) does not ex- p 0 ∈ → ist, we can ﬁnd two sequences pk and qk in X approaching 0 with { } { } lim ϕ(pk) = p′, lim ϕ(qk) = q′ and q′ p′ = a > 0. Denoting by pq k k | − | →∞ →∞ the line segment joining two points p and q, we select points p′ and q′ = ◦ =◦ in Y such that d(p′ pk′ , q′ qk) > a for all large k, where pk′ (pk), qk′ ◦1 ◦ 1 (qk). Set p = ϕ− (p′ ), q = ϕ− (q′ ). Then for sup( pk , qk ) < r < ◦ ◦ ◦ + = ◦ = | | | | inf(p , q ), the hemisphere S r x; x r, xn > 0 meets the curves 91 ◦ ◦ { | | 74 10. Quasi-conformal Mappings

1 1 ϕ− (p′ pk′ ) and ϕ− (q′ qk′ ). For each such r at least one of the coordinate ◦ ◦ functions of ϕ(x) = (ϕ1(x),...ϕn(x)) satisﬁes

OS C + ϕ > a/ √n. S r i Hence ∞ OS C n dr + + ϕ = . S r i i r ∞ ◦ By Lemma (10.8), ϕ is ACL in X. Applying Lemma 10.5, we get i Ä ä for each i = 1,..., n

∞ OS C n dr ▽ n n S + ϕi 2A ϕi dx 2A Iϕdx r r ≤ X | | ≤ X ◦ n 1 n 1 2A K − Jϕdx 2AK − dy. ≤ ≤ Ä ä X Y This yields a contradiction.

10.15 Theorem. Let ϕ be a k-quasi conformal mapping of an open ball Bn in Rn, onto itself, n 2, and let ϕ denote the boundary home- ≥ ◦ omorphism induced by ϕ. Then ϕ is Ck-quasi conformal where c = ◦ mod Dτ(1) depends only on n.

Proof. By mapping Bn onto upper half space E+ via Mobius¨ transformation we can replace Bn by E+ in the theorem. By previous theorem ϕ 92 extends to the boundary. Let ϕ also denote its extension by symmetry to n n R . ϕ is k-quasi conformal in R -hyperplane xn = 0. Hence ϕ is ACL in Rn by Lemma 10.8.

n We have Hϕ(p) k a.e. in R . ≤ By Lemma 10.6 and the remark following it

n n 1 (Iϕ(p)) k − Jϕ(p) a.e. ≤ Therefore for any shell D in Rn mod ϕ(D) k mod D by Lemma ≤ 10.10. Applying Lemma 10.12, we get that ϕ is Ck-quasi conformal. The following two Lemmas round out prerequisites◦ for our main theorem 10. Quasi-conformal Mappings 75

10.16 Lemma. Let ϕ : S n S n be a 1-quasi conformal map then ϕ is → aMbius¨ transformation if n > 2 (See [17] pp. 101-102.)

10.17 Lemma. Let ϕ be a quasi conformal mapping of a domain of Rn Rn = into , n > 1. Then m(ϕ(E)) E Jϕdx for any measurable set E in the domain of ϕ. (cf. loc. cit. p. 94). Now we prove theorem 9.3.

Theorem. Let G = 0(1, n)/ 1, n > 2 and let X be the associated sym- ± metric Riemannian space. Let Γ, Γ′ be discrete subgroups such that G/Γ and G/Γ have finite Haar measure. Let ϕ : X X be a homeomor- ′ → phism and θ : Γ Γ an isomorphism such that ϕ(γx) = θ(γ)ϕ(x) for → ′ all γ Γ, x X. Assume that ϕ is quasi-conformal. Then ϕ induces 93 ∈ ∈ a diffeomorphism ϕ of the boundary component X of the Satake com- ◦ 1 ◦ pactification of X and moreover ϕ Gϕ− = G as transformations of X . ◦ ◦ ◦ Proof. The symmetric space X is the hyperbolic n-space which we iden- n n 2 dx 2 tify with the open unit ball B : x < 1 in R with metric ds = | | . | | H 1 x 2 the Satake compactification of X then can be identified with the closed−| | n 1 unit ball and X is its bounding sphere S − . ◦

Quasi-conformality of ϕ with respect to dS H implies that ϕ is quasi- conformal with respect to dx . so in view of Theorem 10.14, ϕ extends | | to a homeomorphism of the closed ball. Let ϕ be the restriction of this n 1 ◦ extension to the boundary X = S − . By Theorem 10.15 and Lemma 10.7, ϕ is almost everywhere◦ diﬀerentiable. Furthermore since X is dense in◦ X X , ϕ (γx) = θ(γ)ϕ (x), γ Γ and x X . Also note that ∪ ◦ ◦ ◦ ∀ ∈ ∈ ◦ G which is the full group of isometriese of X acts canonically on X and conversely from the identiﬁcation of GM(n 1) with G (cf. [17] p.◦ 57 − n 1 and p. 98), it follows that each Mobius¨ transformation of S − extends to n 1 a unique isometry of X. We replace X by R − via stereographic ◦ ∪ {∞} projection. Let ψ denote the homeomorphism of Rn 1 onto itself − ∪ {∞} induced by ϕ . Let A be the 1-parameter subgroup of G corresponding to ◦ n 1 the 1-parameter subgroup of Mobius¨ transformations of S − obtained from the homotheties x λx (with λ R+, x Rn 1) and of → ∈ ∈ − ∞ → ∞ Rn 1 . − ∪ {∞} 76 10. Quasi-conformal Mappings

Let p be a point at which the diﬀerential ψ exists; we can assume n 1 94 that p = 0 for convenience. We identify the tangent space to R − at 0 n 1 with R − in the usual way. Deﬁne

n 1 n 1 f : G HomR(R − , R − ) by → f (g) = (ψg)(0)

t 1/m Set F(g) = (t f (g) f (g))(det f (g) f (g)) whose m = n 1. Since a − − linear map L is conformal iﬀ

< L(x), L(y) > = < x, y > L(x) L(y) x y || || || || || || || || For any two orthogonal unit vectors x, y we deduce from < x+y, x − y >= 0 that 0 =< L(x y), L(x + y) >= L(x) 2 L(y) 2. Thus L maps − || || − || || the unit ball into a ball and

t 1/m LL = (det tLL) Id where m = dimension of the vector space. t t 1/m Thus if L is conformal we have ( LL)(det LL)− = Id. Moreover L is K-quasi-conformal iﬀ the ratio of largest to the smallest eigenvalue of tLL is K2. From the above it follows that f (g) is a conformal mapping of the tangent space at 0 iﬀ F(g) = identity. One can check that F(ga) = F(g), a A and F(γg) = F(g) for γ Γ. Moreover F is a measurable ∀ ∈ n 1 n 1∈ 2 mapping of G into Hom(R − , R − ). It has bounded (by K for some K) entries almost everywhere since ψ is k-quasi-conformal for some 2 n 1 n 1 K. Therefore F gives rise to an element of L (G/Γ, Hom(R − , R − )); 2 n 1 which we again denote by F. For an element L (G/Γ, Hom(R − , Rn 1 2 = t ∧ ∈ − )) let norm G/Γ Tr ( (g) (g))d. || ∧L ||2 Γ R∧n 1 R∧n 1 = 95 G operates on (G/ , Hom( − , − )) via (Z. f )(g) f (gz) uni- tarily and we have A.F = F. Hence by Lemma 5.2, G.F = F i.e. F is constant almost everywhere. In particular

F(gk) = F(g), k 0(n 1) ∀ ∈ − 10. Quasi-conformal Mappings 77

t 1 i.e., the group of rotations about 0. k = k− implies by the special choice of F that 1 F(g) = F(gk) = k− G(g)k. Since F(g) commutes with 0(n 1), we conclude − F(g) = const. Id.

Since the matrix F(g) is positive deﬁnite and of determinant 1, the constant must equal 1. Therefore ψ is 1-quasi-conformal and therefore ψ is Mobius¨ transformation by Lemma 10.16. 1 In particular ϕ Gϕ− = G. ◦ ◦

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