On Lie Groups and Representations of Locally Compact Groups

Lectures on Lie Groups and Representations of Locally Compact Groups

By F. Bruhat

Tata Institute of Fundamental Research, Bombay 1958 (Reissued 1968) Lectures on Lie Groups and Representations of Locally Compact Groups

By F. Bruhat

Notes by S. Ramanan

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

Tata Institute of Fundamental Research, Bombay 1958 (Reissued 1968) Introduction

We shall consider some heterogeneous topics relating to Lie groups and the general theory of representations of locally compact groups. The first part exclusively deals with some elementary facts about Lie groups and the last two parts are entirely independent of the material contained in the first. We have rigidly adhered to the analytic approach in estab- lishing the relations between Lie groups and Lie algebras. Thus we do not need the theory of distributions on a manifold or the existence of integral manifolds for an involutory distribution. The second part concerns itself only with the general theory of measures on a locally compact group and representations in general. Only a passing reference is made to distributions (in the sense of L. Schwartz), and induced representations are not treated in detail. In the third part, we first construct the continuous sum (’the direct integral’) of Hilbert spaces and then decompose a unitary representations into a continuous sum of irreducible representations. We derive the Plancherel formula for a separable unimodular group in terms of factorial representations and derive the classical formula in the abelian case.

iii

Contents

Introduction iii

I Lie Groups 1

1 Topological groups 3

2 Local study of Lie groups 9

3 Relations between Lie groups and Lie algebras - I 19

4 Relation between Lie groups and Lie algebras - II 31

II General Theory of Representations 47

5 Measures on locally compact spaces 47

6 Convolution of measures 53

7 Invariant measures 61

8 Regular Representations 73

9 General theory of representations 83

v vi Contents

III Continuous sum of Hilbert Spaces 89

10 Continuous sum of Hilbert Spaces-I 91

11 Continuous sum of Hilbert spaces - II 103

12 The Plancherel formula 115

Bibliography 127 Part I

Lie Groups

Chapter 1

Topological groups

1.1

We here assemble some results on topological groups which we need in 1 the sequel. Deﬁnition. A topological group G is a topological space with a composition law G G G, (x, y) xy which is × → → (a) a group law, and

(b) such that the map G G G defined by (x, y) x 1y is contin- × → 7→ − uous. The condition (b) is clearly equivalent to the requirement that the maps G G G defined by (x, y) xy and G G defined by × → → → x x 1 be continuous. → − It is obvious that the translations to the right: x xy are homeo- → morphisms of G. A similar statement is true for left translations also. We denote by A 1, AB the subsets a 1 : a A , ab : a A, b B re- − { − ∈ } { ∈ ∈ } spectively. It is immediate from the definition that the neighbourhoods of the identity element e satisfy the following conditions:

(V1) For every neighbourhood V of e, there exists a neighbourhood W of e such that W 1W V. (This follows from the fact that − ⊂ (x, y) x 1y is continuous). → − 3 4 1. Topological groups

(V ) For every neighbourhood V of e and for every y G, there exists 2 ∈ a neighbourhood W of e such that yWy 1 V. (This is because − ⊂ x yxy 1 is continuous). → − These conditions are also suﬃcient to determine the topology of the group. More precisely,

2 Proposition 1. Let V be a family of subsets of a groups G, such that

(a) For every V V , e V. ∈ ∈ (b) Any ﬁnite intersection of elements of V is still in V .

(c) For every V V , there exists W V such that W 1W V. ∈ ∈ − ⊂ (d) For every V V , and for every y G there exists W V such ∈ ∈ ∈ that yWy 1 V. − ⊂ Then G can be provided with a unique topology Φ compatible with the group structure such that the family V is a fundamental system of neighbourhoods at e. Supposing that such a topology exists, a fundamental system of neighbourhoods at y is given by either V y or yV . It is, therefore, a natural requirement that these two families generate the same filter. It is this that necessitates the condition (d). We may now take the filter generated by yV and V y as the neighbourhood system at y. This can be verified to satisfy the neighbourhood axioms for a topology. It remains to show that (xy) x 1y is continu- → − ous. 1 1 Let V x0− y0 be any neighbourhood of x0− y0. By (c), (d), there exist W, W V such that W 1W V and x 1y W y 1 x W. If we take 1 ∈ − ⊂ 0− 0 1 0− 0 ⊂ x x W, y y W , we have x 1y W 1 x 1y W W 1W x 1y ∈ 0 ∈ 0 1 − ∈ − 0− 0 1 ⊂ − 0− 0 ⊂ V x 1y . This shows that (x, y) x 1y is continuous. 0− 0 → − Examples of topological groups.

(1) Any group G with the discrete topology. Topological groups 5

(2) The additive group of real numbers R or the multiplicative group of non-zero real numbers R∗ with the ‘usual topology’.

(3) The direct product of two topological groups with the product 3 topology.

(4) The general linear group GL(n, R) with the topology induced by that of Rn2 .

(5) Let X be a locally compact topological space, and G a group of homeomorphisms of X onto itself. This group G is a topological group with the ‘compact open topology’. (The compact-open topology is one in which the fundamental system of neighbourhoods of the identity is given by ﬁnite intersections of the sets u(K, U) = f G : f (x) U, f 1(x) U for every x K , K { ∈ ∈ − ∈ ∈ } being compact and U an open set containing K).

1.2 Topological subgroups. Let G be a topological group and g a subgroup in the algebraic sense. g with the induced topology is a topological group which we shall call a topological subgroup of G. We see immediately that the closureg ¯ is again a subgroup. If g is a normal subgroup, so isg ¯. Moreover, an open subgroup is also closed. In fact, if g is open, G g = xg, which is open as left − x

Proposition 2. If G is a connected topological group, any neighbourhood of e generates G.

On the contrary, if G is not connected, the connected component G0 4 of e is a closed normal subgroup of G. Since G xG G 1G is contin- 0 0 → 0− 0 uous and G xG connected, G 1G is also connected, and hence G . 0 0 0− 0 ⊂ 0 6 1. Topological groups

This shows that G is a subgroup. As x yxy 1 is continuous, yG y 1 0 → − 0 − is a connected set containing e and consequently G . Therefore, G is ⊂ 0 0 a normal subgroup.

Proposition 3. Every locally compact topological group is paracompact.

In fact, let V be a relatively compact open symmetric neighbourhood ∞ n of e. G′ = V is an open and hence closed subgroup of G. G′ is n=1 countable at S and therefore paracompact. Since G is the topological ∞ union of left cosets modulo G′, G is also paracompact.

1.3 Factor groups. Let g be a subgroup of a topological group G. We shall denote byx ˙ the right coset gx containing x. On this set, we already have the quotient topology. Then the canonical map π : G G/g is open and continuous. → For, if π(U) is the image of an open set U in G, it is also the image of gU which is an open saturated set. Hence π(U) is also open. But this canonical map is not, in general, closed. The space G/g is called a homogeneous space. If g is a normal subgroup, G/g is a group and is a topological group with the above topology. This is the factor group of G by g.

1.4 Separation axiom. Theorem 1. The homogeneous space G/g is Hausdorﬀ if and only if the subgroup g is closed.

1 5 If G/g is Hausdorﬀ, g = π− (π(e)) is closed, since π(e) is closed. Conversely, let g be closed. Let x, y G/g and x , y. Since xy 1 < g ∈ − and g is closed, there exists a symmetric neighbourhood V of the identity such that xy 1V g = φ. Hence xy 1 < gV. Now choose a neighbour- − ∩ − hood W of e such that WW 1 y 1Vy. We assert that gxW and gyW − ⊂ − are disjoint. For, if they were not, γ , γ g, w , w W exist such that 1 2 ∈ 1 2 ∈ 1 1 1 γ xw = γ yw ; i.e. γ− γ x = yw w− yWW− Vy. 1 1 2 2 2 1 2 1 ∈ ⊂ Topological groups 7

Hence γ 1γ xy 1 V, or xy 1 γ 1γ V gV. 2− 1 − ∈ − ∈ 1− 2 ⊂ This being contradictory to the choice of V, gxW, gyW are disjoint or π(gxW) π(gyW) = φ. Hence G/g is Hausdorff. ∩ In particular, if g = e , G is Hausdorff if and only if e is closed. On { } { } the other hand, if e is not closed, e is a normal subgroup and G/ e is { } { } { } a Hausdorff topological group. We shall hereafter restrict ourselves to the consideration of groups which satisfy Hausdorff’s axiom.

1.5 Representations and homomorphisms. Deﬁnition . A representation h of a topological group G into a topological group H is a continuous map : G H which is an algebraic → representation. In other words, h(xy) = h(x) h(y) for every x, y G. · ∈ Obviously the image of G by h is a subgroup of H and the kernel N of h is a closed normal subgroup of G. The canonical map h¯ : G/N { } → H is a representation and is one-one. Deﬁnition. A representation h is said to be a homomorphism if the induced map h¯ is a homeomorphism.

Proposition 4. Let G and H be two locally compact groups, the former 6 being countable at . Then every representation h of G onto H is a ∞ homomorphism. It is enough to show that for every neighbourhood V of e in G/N, h¯(V) is a neighbourhood of h(e) in H. Choose a relatively compact open neighbourhood W of e such that W¯ W¯ 1 V. G is a countable union − ⊂ of compact sets and x G W x = G. Therefore, one can find a sequence ∈ x j of points such thatS G = j W x j. Since h is onto, H = j h(W x j) = { } j h(W)h(x j). H is a locallyS compact space and hence aS Baire space S(Bourbaki, Topologie générale, Ch. 9). There exists, therefore, an integer j such that h(W x j) has an interior point. h(W¯ ) being compact, h(W¯ ) = h(W). Consequently, h(W¯ )h(x j) and hence h(W) has an interior point y. There exists a neighbourhood U of e such that h(W¯ ) Uy. 1 1 1 1 ⊃ 1 Now h(V) h(W¯ W¯ ) = h((W¯ ) h(W¯ )− Uy(y U ) = UU . ⊃ − · ⊃ − − − h(V) is therefore a neighbourhood of h(e), which completes the proof of proposition 4.

Chapter 2

Local study of Lie groups

2.1 7 Deﬁnition. A Lie group G is a real analytic manifold with a composition law (x, y) xy which is → (a) a group law, and

(b) such that the map (x, y) x 1y is analytic. → − (b) is equivalent to the analyticity of the maps (x, y) xy and x 1 → → x− . Remarks. (1) A Lie group is trivially a topological group.

(2) We may replace ‘real’ by ‘complex and deﬁne the notion of a complex Lie group. We shall not have occasion to study complex Lie groups in what follows, though most of the theorems we prove remain valid for them.

(3) It is natural to inquire whether every topological group with the structure of a topological manifold is a Lie group. This problem (Hilbert’s ﬁfth problem) has been recently solved by Gleason [18] who has proved that a topological group G which is locally compact, locally connected, metrisable and of ﬁnite dimension, is a Lie group.

9 10 2. Local study of Lie groups

Examples of Lie groups.

(1) R - real numbers, C - complex numbers, T - the one-dimensional torus and Rn, Cn and T n in the usual notation are all Lie groups.

8 (2) Product of Lie groups with the product manifold structure is a Lie group. (3) GL(n, R) - the general linear group.

2.2 Local study of Lie groups. We shall assume that V is a suﬃciently small neighbourhood of e in which a suitably chosen coordinate system, which taken e into the origin, is deﬁned. The following notations will be adhered to throughout these lectures: If a V, (a ,..., a ) will denote the coordinate of a. α = (α ,..., ∈ 1 n 1 αn) is a multi-index with αi, non-negative integers. α = α + α + + α | | 1 2 ··· n [i] will stand for α with αi = 1 and α j = 0 for j , i.

α! = α1! ...αn! α α1 αn x = x1 ... xn ∂α ∂α1+ +αn = ··· ∂xα ∂xα1 ∂xαn 1 ··· n Let x, y V be such that xy V. (xy) are analytic functions of the ∈ ∈ i coordinate of x and y.

(1) (xy)i = ϕi(x1,..., xn, y1,..., yn) where the ϕi are analytic functions of the 2n variables x, y in a neighbourhood of 0 in R2n. These ϕi cannot be arbitrary functions, as they are connected by the group relations. These are reﬂected in the following equations: (x e)i = (e x)i = xi, or Local study of Lie groups 11

(2) ϕi(x, e) = ϕi(e, x) = xi. 9

The ϕi are analytic functions and so are of the form

= i α β = ϕi(x, y) λαβ x y , i 1, 2,..., n Xα,β

By equation (2), this can be written

i α β (3) ϕi(x, y) = xi + yi + α 1 λα,β x y . |β|≥1 P| |≥ By associativity,

ϕi(xy, z) = ϕi(x, yz), or

ϕi(ϕ1(x, y),...,ϕn(x, y), z) = ϕi(x,ϕ1(y, z),...,ϕn(y, z)).

These may also be written

(4) ϕi(ϕ(x, y), z) = ϕi(x,ϕ(y, z)).

One is tempted to expect another equation in ϕi, due to the existence of the inverse of every element. However, these two equations are suﬃcient to characterise locally the Lie group, and the existence of the inverse is, in a certain sense, a consequence of the associative law and the existence of the identity. To be more precise,

Proposition 1. Let G be the semigroup with an identity element e. If it can be provided with the structure of an analytic manifold such that the map (x, y) xy of GxG G is analytic, then there exists an open → → neighbourhood of e which is a Lie group.

In fact, the existence of the inverse element of x depends upon the ∂ϕi existence of the solution for y of ϕi(x, y) = 0, 1 = 1,..., n. Now = ∂x j δi j+ terms containing positive powers of the yi. If we put y = e, the latter terms vanish and

∂ϕi(x, y) = δi j. ∂x j !y=e 12 2. Local study of Lie groups

Hence 10 ∂ϕ (x, y) J = det i = 1 ∂x j !y=e J being a continuous function of x and y, J , 0 in some neighbourhood V′ of e. Therefore there exists a neighbourhood V of e every element of 1 n which has an inverse. Then the neighbourhood W = ∞ (V V ) is n=1 ∩ − a group compatible with the manifold structure. S

2.3 Formal Lie groups. Deﬁnition. A formal Lie group over a commutative ring A with unit elements, is a system of n formal series ϕi in 2n variables with coeﬃcients in A such that

ϕi(x1,..., xn, 0, 0,...) = xi = ϕi(0, 0,..., x1,..., xn)

and

ϕi(ϕ1(x, y),...,ϕn(x, y), z) = ϕi(x,ϕ1(y, z),...,ϕn(y, z)).

Almost all that we prove in the next few lectures will be valid for formal Lie groups over a ﬁeld of characteristic zero also. For a study of formal Lie groups over a ﬁeld of characteristic p , 0, one may see, for instance, [9], [10].

2.4 Taylor’s formula.

Let f be a function on an open neighbourhood of e, and let τy, σz denote respectively the right and left translates of f deﬁned by τy f (x) = f (xy); 1 σz f (x) = f (z− x) for suﬃciently small y and z. These two operators commute, i.e. τy(σz f ) = σz(τy f ) If f is analytic in V, τ f is (for y W) analytic in W, where W is a y ∈ neighbourhood of e such that W2 V. ⊂ 11 Now, Local study of Lie groups 13

ϕi f (x) = f (ϕ1(x, y),...,ϕn(x, y)) with i α β τi(x, y) = xi + yi + λα,β x y . Xα 1 |β|≥1 | |≥ If we set i α φ ui = yi + λα,β x y Xα 1 |β|≥1 | |≥ τy f (x) = f (x+u) in the usual notation. This can be expanded as a Taylor series 1 ∂α f τ f (x) = uα (x). y α! ∂xα Xα

We may now substitute for the ui in this convergent series.

uα = uα uαn 1 ··· n α1 1 γ δ = y1 + λγ,δ x y ... Xγ 1 |δ|≥1 | |≥ α α β = y + gβ (x)y βXα | |≥| | where the coeﬃcient of powers of y are analytic functions of x and α α 0 gβ (e) = 0. If here we take α = 0, u = 1, and hence gβ = 0 for every β. Thus 1 ∂α f τ f (x) = (x)(yα + gα(x)yβ). y α! ∂xα β Xα βXα | |≥| | These are uniformly absolutely convergent in a suitable neighbourhood of e, on the explicit choice of which we shall not meticulously insist. Hence the above formula can be written as 1 ∂α f 1 ∂β f τ f (x) = yα( (x) + gβ (x) (x)). y α! ∂xα α β! ∂xβ Xα βXα | |≤| | 14 2. Local study of Lie groups

This we shall denote by

1 τ f (x) = yα∆ f (x) y α! α Xα

12 where ∆α is a diﬀerential operator not depending on f . This formula is the generalised Taylor’s formula we sought establish.

2.5 Study of the operator ∆α. ∂α ∂β α! ∆ = + β β = = Now, α α gα(x) β with gα(e) 0. Since at x e, ∂x β α ∂x β! ∂α | |≤|P | ∆ = , the ∆ are linearly independent at the origin. At α = 0, ∆ is α ∂xα α α 0 the identity operator and if α , 0, ∆α is without constant term as gα = 0. ∂ ∂ Let us denote ∆[i] by Xi. Then Xi = + j ai j(x) , with ai j(e) = ∂x1 ∂x j 0. These are vector ﬁelds in a neighbourhoodP of e. Now we shall use the fact that, for every y, z G, the operators σ and τ commute. We ∈ z y have

1 α 1 α σz(τy f ) = σz( y ∆α f ) = y (σz ∆α)( f ) X α! X α! ◦ and, on the other hand,

1 α τy(σz f ) = y ∆α(σz f ). X α! Therefore, by the uniqueness of the expansion in power-series in y of σ (τ f ) = τ (σ f ), we have σ ∆α = ∆α σ , for every z in a suﬃciently z y y z z◦ ◦ z small neighbourhood of e. Otherwise stated, ∆α is left invariant in this neighbourhood. This enables us to deﬁne ∆α at every point z in the Lie group by setting ∆ f (z) = σ 1 ∆ σ f (z), so that the extended operator α z− α z remains left invariant.

Theorem 1. The linear diﬀerential operators ∆α form a basis for the algebra of left invariant diﬀerential operators. Local study of Lie groups 15

We have already remarked that the ∆α are linearly independent at e. 13 Let D be any left invariant linear differential operator on G. Then D = ∂α α r bα(x) , r being some positive integer. Define ∆ = α r bα(e) | |≤ ∂xα | |≤ P∂α P . Obviously, ∆ σ = σ ∆. At e, D = ∆, and by the left invariance ∂xα ◦ z z ◦ of both D and ∆, D = ∆ everywhere. This proves our contention that the ∆α form a basis of the algebra of left invariant differential operators. We shall hereafter denote this algebra by (G). U

2.6 The Lie algebra of a Lie group G.

Let G be the subspace of (G) generated by the X . This is the same U i as the subspace composed of vector fields which are left invariant. This is obviously isomorphic as a vector space to the tangent space at e. If there are two vector fields X, Y, then XY is an operator of order 2, as also YX. But XY YX is a left invariant vector field, as can be easily − verified. Lef [X, Y] stand for this composition law. It is not hard to see that this bracket operation satisfies

[X, Y] = 0, and [[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0.

This leads us to the following

Deﬁnition. A Lie algebra g over a ﬁeld, is a vector space with a composition law [X, Y] which is a bilinear map g g g satisfying [X, X] = 0, × → and the Jacobi’s identity, viz.

[[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0.

Example. (1) The left invariant vector ﬁelds of a Lie group form a Lie algebra.

(2) Any vector space with the composition Law [X, Y] = 0 for U every X, Y is a Lie algebra. ∈U Such an algebra is called an abelian Lie algebra. 14 16 2. Local study of Lie groups

(3) Any associative algebra with the bracket operation

[X, Y] = XY YX − is a Lie algebra.

In particular, the matrix algebra Mn(K) over a ﬁeld K and the space of endomorphisms of a vector space are Lie algebras. U Deﬁnition. A subspace J of a Lie algebra g is a subalgebra if for every x, y J , [x, y] J . ∈ ∈ A subspace of a Lie algebra g is an ideal, if for every x , U ∈ U y g, [x, y] . ∈ ∈U Example. (1) The set of all matrices in Mn(K) whose traces are zero is an ideal of Mn(K). (2) The set of all elements z g such that [z, x] = 0 for every x g is ∈ ∈ an ideal of g, called centre of g.

If is an ideal in g, the quotient space g/ can be provided with U U the structure of a Lie algebra by defining [(x+ ), (y+ )] = [x, y]+ . U U U This is called the factor algebra of g by . U 2.7 Representations of a Lie algebra. Definition . A representation of a Lie algebra g in another Lie algebra g is a linear map f such that f ([x, y]) = [ f (x), f (y)] for every x, y g. ′ ∈ It can be verified that the image of g by f and the kernel of f are subalgebras of g′ and g respectively. The latter is, in fact, an ideal and g/ ker f is isomorphic to f (g). 15 In particular, g′ may be taken to the the space of endomorphisms of a vector space V, leading us to the definition of a linear representation of g.

Deﬁnition. A linear representation of a Lie algebra g in a vector space V is a representation of g into the Lie algebra of endomorphisms of V. Local study of Lie groups 17

2.8 Adjoint representation. Let g be a Lie algebra and x g. The map g g deﬁned by y [x, y] ∈ → −→ is a linear map of g into itself. This map is denoted adx. Thus, adx (y) = [x, y].

Remarks. (1) Ad x is a derivation of the Lie algebra. We recall here the deﬁnition of a derivation in an algebra g, associative or not. A linear map D of g into itself is a derivation if for any two elements x, y g, we have D(xy) = x(Dy) + (Dx)y. In a Lie algebra ∈ derivations of the type ad x are called inner derivations.

(2) x ad x is a linear map of g into the Lie algebra of endomor- → phisms of the vector space g.

This is, moreover, a linear representation of g in g. The veriﬁcation of the relation ad[x, y] = [adx, ady] is an immediate consequence of Jacobi’s identity. This linear representation will henceforth be referred to as the adjoint representation of g.

Chapter 3

Relations between Lie groups and Lie algebras - I

3.1 Differential of an analytic representation. 16 Definition. An analytic representation of a Lie group G into a Lie group G′ is an algebraic representation which is an analytic map. Remark . It is true, as we shall see later (Cor. to Th. 4, Ch. 4.5) that any representation of the underlying topological group G in G′ is itself a representation in the above sense. We now seek to establish a correspondence between analytic representations of Lie groups and algebraic representations of their Lie algebras. As a first step, we prove the following Proposition 1. To every analytic representation h : G G there −→ ′ corresponds a map dh : (G) (G ) which is a representation of U → U ′ algebras such that ∆( f h) = (dh(∆) f ) h. ◦ ◦ Let y G and y = h(y). If f is an analytic function on G , we have ∈ ′ ′ τ ( f h)(x) = f (h(xy)) = f (h(x)h(y)) = (τ f ) h(x). y ◦ y′ ◦ We may now write down the Taylor formula for both sides of the equation and equate the coefficients of powers of y (by the uniqueness

19 20 3. Relations between Lie groups and Lie algebras - I

of development in power series). We have

1 1 α α′ y ∆α( f h) = ( y′ ∆′ f ) h. α′ α! ◦ α′! ◦ Xα Xα′

But h being an analytic map, (h(y)) = νi yα. There is no con- i α′ α α′ stant term in this summation since (h(y))i =P0 at y = e. Hence y′ = α′ α α′ 17 α α µα y , where µα are constants, and on substitution in the above | |≥| ′| equation,P we obtain 1 1 α α′ α y ∆α( f h) = ( µα y ∆′ f ) h α! ◦ α ! α′ ◦ Xα Xα ′ αXα ′ | |≥| ′| α α 1 = y ( µα ∆′ f ) h. α ! α′ ◦ Xα αXα ′ | ′|≤| ′|

the series being uniformly absolutely convergent. If Dα′ denotes α α | ′|≤| ′| α α! P µα ∆′ , which is a left invariant diﬀerential operator, then ∆α( f h) = α ! α′ ◦ (D f′ ) h. Moreover, this equation completely determines D since its α′ ◦ α′ value at e given by Dα f (e ) = ∆α( f h)(e). As the ∆α form a basis ′ ◦ for (G) in G, we may deﬁne a linear map dh : (G) (G ) by U U → U ′ setting dh(∆α) = D . It is obvious that ∆( f h) = (dh(∆) f ) h for any α′ ◦ ◦ ∆ µ(G). To complete the proof of proposition 1, one has only to show ∈ that dh(∆1∆2) = dh(∆1)dh(∆2). But this is obvious since

(dh(∆ ∆ ) f ) h = ∆ ∆ ( f h) 1 2 ◦ 1 2 ◦ = ∆ (dh(∆ ) f h) 1 2 ◦ = (dh(∆ dh(∆ ) f ) h. 1 2 ◦

α! Now, dh(∆ ) = µα′ ∆ is of order less than or equal to α α′ α α α′ | |≤| | α′! ′ that of ∆α. By linearity,P the same is also true of any operator ∆ (G). ∈U Also, dh preserves constant terms. The image of g is in g′, and by Propo- sition 1, dh restricted to g is a Lie algebra representation. This is said to be the diﬀerential of the map h. Relations between Lie groups and Lie algebras - I 21

Remarks. (1) If we have another representation h : G G , it is ′ ′ → ′′ obvious that d(h h) = dh dh. ′ ◦ ′ · (2) If h is an analytic map of a manifold V into a manifold W, then 18 we can define the differential of h at x, viz., dh : T T x x → h(x) where Tx, Th(x) are tangent spaces at the respective points. This map makes correspond to a tangent vector X at x, the vector X′ at h(x) such that X f = X( f h) for every function f analytic at ′ ◦ h(x). However, we cannot, in general, define the image a vector field. As we have seen, in the case of Lie groups, as long as one is considering only left invariant vector fields, one can talk of an image vector field. Thus we now have, corresponding to an analytic representation of a Lie group G into another Lie group G′, two notions of a differential map: the linear map of the tangent space at a into that at h(e) = e′, and the representation of the Lie algebra of G in that of G′. These two notions are essentially the same in the following sense. Let ϕ, ϕ′ be the canonical vector g g space isomorphisms of , ′ with Te, Te′ respectively. Then the diagram ϕ / g Te

dh dhe ϕ g ′ / ′ Te′ is commutative.

Proposition 2. Let G and G′ be two connected Lie groups. An analytic representation of G G is surjective if and only if the diﬀerential map → ′ is surjective.

Proposition 3. A representation h of a Lie group G in another Lie group G′ is locally injective (i.e. there axises a neighbourhood of e on which h is injective) if and only if dh is injective.

These two propositions are consequences of the corresponding prop- 19 erties of manifolds, the proofs of which we omit. 22 3. Relations between Lie groups and Lie algebras - I

3.2 Subgroups of a Lie group.

Deﬁnition . An analytic map f of a manifold U into another manifold V is said to be regular at a point x in U if the diﬀerential map d fx is injective.

Deﬁnition . A submanifold of an analytic manifold U is a pair (V,π) consisting of a manifold V which is countable at and an injective ∞ analytic map π of V into U which is everywhere regular.

Remarks. (1) The topology on π(V) is not that induced from the topology of U in general. For instance, if T 2 is the two - dimensional torus, V the space of real numbers, and π the map t (t,αt) of V into T 2, where α is irrational, it is easy to see that → π is an injective, analytic, regular map. But π cannot be a homeomorphism of V into T 2. For, every neighbourhood of (0, 0) in T 2 contains points (t,αt) with arbitrarily large values of t. Hence, the inverse image of this neighbourhood in π(U) with the induced topology can never be contained in a given neighbourhood of 0 in R.

(2) Nevertheless, it is true that locally, for every point x of V, there exist neighbourhoods W inV and W1 in U which satisfy the fol- 1 lowing : A coordinate system (x1,..., xn) can be deﬁned in W such that W is deﬁned by the annihilation of certain coordinates.

20 Deﬁnition . A Lie subgroup of a Lie group G is a submanifold (H,π), π(H) being a subgroup of G.

We deﬁne on H the group structure obtained by requiring that π be a monomorphism. Since the map π of H in G is regular, locally the analytic structure of H is induced form that of G. Hence the group operations in H are analytic in H, as they are analytic in G. H is therefore a Lie group.

Proposition 4. The Lie algebra of a Lie subgroup of a Lie group G can be identiﬁed with a subalgebra of the Lie algebra of G. Relations between Lie groups and Lie algebras - I 23

In fact, if (H,π) is the subgroup, π is a representation of H in G and dπ is injective since π is regular. We identify the Lie algebra S of H with the subalgebra dπ(S ) or g.

3.3 One-parameter subgroups. Definition. An analytic representation ρ of R into G is said to be a one- parameter subgroup of G. We know that the representation ρ gives rise to a differential map dρ d of the Lie algebra of R (spanned by ) into the Lie algebra of G. Let dt d dρ( ) = X = λ X . We now form the differential equations satisfied dt j j by the functionPρ. Let (x1, x2,... xn) be a coordinate system in a neighbourhood of e in G and Let ρ denote x ρ. Now i i ◦ ρi(t + t′) = ϕ1(ρ(t), ρ(t′)) ∂ ∂ϕ dρ (ρ (t + t )) = i (ρ(t), ρ(t )). k (t) ∂t i ′ ∂y ′ dt ′ Xk 1

Putting t′ = 0, we get dρ dρ ∂ϕ i (t) = k (0) i (ρ(t), e). dt dt ∂y Xk k 21 If Xi = ∆[i], we have ∂ ∂ X = + a (x) with a (e) = 0. i ∂x i j ∂x i j i Xj j Since (X f ) ρ = d ( f ρ) for every function analytic at e, we get ◦ dt ◦ dρ i = (Xx ) ρ by setting f = x . dt i ◦ i Hence dρ i (0) = (Xx ) ρ(0) dt i ◦ 24 3. Relations between Lie groups and Lie algebras - I

= Xxi(e)

= (λ jX j)xi(e) X = λi.

To sum up, ρ satisﬁes the system of diﬀerential equations

dρi ∂ϕi (A) = k λk (ρ(t), e) dt ∂yk P with the initial condition

(B) ρi(0) = 0. dρ (A) implies i (0) = λ . dt i Now, conversely if are given the system of differential equations (A) with the initial condition (B), then by Cauchy’s theorem on the existence and uniqueness of solutions of differential equations, there exists one and only one solution t ρ(t, λ) which is analytic in t and λ in a → neighbourhood of (0, λ). We shall now show that ρ(t + u) = ρ(t) ρ(u) · for sufficiently small values of t and u. 22 Let

σi′(t) = ϕi(ρ(u), ρ(t)). Then dσ′ ∂ϕ dρ i = i (ρ(u), ρ(t)) l (t) dt ∂y dt Xl i ∂ϕ ∂ϕ = λ i (ρ(u), ρ(t)) l (ρ(t), e) k ∂y ∂y Xk, j l k

since the ρi are solutions of (A). On the other hand, we have

ϕi(ρ(u)ρ(t), y) = ϕi(ρ(u), ρ(t)y)

= ϕi(ρ(u),ϕ(ρ(t), y)) ∂ϕ n ∂ϕ ∂ϕ i (ρ(u)ρ(t), y) = i (ρ(u), ρ(t)y) l (ρ(t), y). ∂y ∂y ∂y k Xl=1 l k Relations between Lie groups and Lie algebras - I 25

Hence, dσ′ ∂ϕ i = λ i (σ (t), e). dt k ∂y ′ Xk k i.e. σi′ is a solution of (A) with the initial condition (C):

σi′(0) = ϕi(e, ρ(u)) = ρi(u). Also, t σ(t) = ρ(t + u) is a solution of (A) since the diﬀerential → equation (A) is a invariant for translations of it. Also σ(0) = ρ(u).

Hence σi′ and σi are two sets of solutions of (A) with the same initial conditions, and therefore

σi′(t) = ρ(t + u), i.e. ϕi(ρ(t), ρ(u)) = ρi(t + u). or ρ(t)ρ(u) = ρ(t + u) for sufficiently small values of t and u. Also this map t ρ(t, X) is analytic. We assume the following → Lemma 1. Let H be a connected, locally connected and simply con- 23 nected topological and f a local homomorphism of H G (i.e. a con- → tinuous map of a neighbourhood of e into H such that f (xy) = f (x) f (y) for all x, y such that x, y, xy V). Then there exists one and only one ∈ representation f˜ of H in G which coincides with f on V. We immediately obtain (since R is simply connected), the Theorem 1. For every X g, there exists one and only one one - pa- ∈ (d) rameter subgroup ρ(t, X) such that dρ = X. The function ρ(t, X) is dt analytic in t and X. One can assign to any finite dimensional vector space over the real number field a manifold structure which is induced by that of the real numbers. In particular, The Lie algebra of a Lie group also has an analytic structure. Whenever we talk of an analytic map into of from a Lie algebra, it is to this analytic structure that we refer. Proof of the Lemma Consider the Cartesian set product Hˆ = H G. × We provide Hˆ with a topology by defining the neighbourhood system at each point (x, y) in the following way: 26 3. Relations between Lie groups and Lie algebras - I

Let W be a neighbourhood of e in H V, where V can be assumed ⊂ to be connected since H is locally connected. The fundamental system of neighbourhoods at (x, y) is given by N(W, x, y) = (x , y ) : x { ′ ′ ′ ∈ xW, y = y f (x 1 x ) . It is easily verified that this satisfies the neigh- ′ − ′ } bourhood axioms for a topology, and that Hˆ with the usual projection π : H G H is a covering space of H. Let H be the connected × → 1 component of (e, e) in Hˆ . Then H1 is a connected, covering space of H and since H is simply connected,π is a homeomorphism of H1 onto H. 24 Let η be its inverse. Define f˜(x) = π η(x) for every x in H where 2 ◦ π : H G G is the second projection. N(W, x, y) is mapped home- 2 × → omorphically by π onto xW. Hence N(W, x, y) H if W is connected ⊂ 1 and (x, y) H . It follows that f˜ is a representation which extends f . ∈ 1 3.4 The exponential map. We shall denote ρ(t, X) by exp(tX). But such a notation involves the tacit assumption that ρ(t, X) depends only on tX. In other words, one has to make sure that ρ(1, sX) = ρ(s, X) before such a notation becomes permissible. But this is obvious (d) in as much as t ρ(st, X) is a one -parameter subgroup with dρ = → d(st) (d) (d) X or dρ = sX. The one-parameter subgroup such that dρ = sX dt dt is, by definition, t ρ(t, sX). By uniqueness of the one-parameter sub- → groups, ρ(st, X) = ρ(t, sX), or in particular, ρ(s, X) = ρ(1, sX). It is easy to see that exp(tX) exp(t X) = exp(t + t )X and exp( X) = (exp X) 1. ′ ′ − − But, in general, exp Y exp Y , exp(Y + Y ). · ′ ′ Theorem 2. The map h : X exp X of g into G is an analytic iso- → morphism of a neighbourhood of 0 in g onto a neighbourhood of e in G. In fact, since h is an analytic map, it is enough to show that the Jacobian of the map h , 0 in a neighbourhood of the origin. (X1,..., Xn) form a basis for g, where Xi = ∆[i].

h y X = exp( y X )  i i i i Xi Xi     Relations between Lie groups and Lie algebras - I 27

∂h j d (0) = (exp tXk) j(t = 0) = (Xk x j)x=e = δ jk. ∂yk dt i.e. the jacobian = 1 at e. By continuity, the Jacobian does not vanish in 25 a neighbourhood of e. Now, let X1,..., Xn be an arbitrary basis of g. This can be trans- ported into a system of coordinates in G by means of the above map. For every x G suﬃciently near e, there exists one and only one system ∈ (x1,..., xn) near 0 such that x = exp( i xiXi). This system of coordinates is calledP the canonical system of coordinates with respect to any given basis. Hereafter, we will almost always operate only with a canonical system of coordinates.

Remark. Let x V, V being a neighbourhood of e in which a canonical ∈ coordinate system exists and x is suﬃciently near e. Now, if

x = exp( xiXi), Xi p x = exp( (pxi)Xi), Xi p i.e. the coordinates of x are (px1,..., pxn). Proposition 5. Let h be a representation of G in H, and dh : g its →F diﬀerential. Then the diagram

dh / g / S exp exp G / H h is commutative. ρ Consider t ′ h(exp tX). −→ This is obviously a one-parameter subgroup, and dρ = dh dρ. ′ ◦ Therefore d h(exp X) = exp(dρ ( )) ′ dt 28 3. Relations between Lie groups and Lie algebras - I

= exp(dh(X)).

It follows, therefore, that if h(G) = (e), dh is the map g (0). → 26 Conversely, if dh = C, and G connected, h(exp X) = e for every element in a neighbourhood of e, and h = e. Again, if G is connected and two representations h1, h2 of G in H are such that dh1 = dh2, then h1 = h2. Proposition 6. For every analytic function f on a neighbourhood of e, we have ∞ tn f (exp tX) = (Xn f )(e). n! Xn=0 In fact, d f (exp tX) = (X f )(exp tX). dt By induction on n, we have dn f (exp tX) = (Xn f )(exp tX) dtn or dn = n n f (exp tX) X f (e). (dt )t=0 Now, f (exp tX) is an analytic function of t and by Taylor’s formula, we have ∞ tn f (exp tX) = (Xn f )(e) n! Xn=0 α! Theorem 3. In canonical coordinates, we have ∆ = S where S α α ! α α α n α | | is the coeﬃcient of t in the expansion of ( t X ) and S α (G). i=1 i i | | ∈U P α! In fact, it is enough to prove the equality of ∆ and S at e since α α ! α both ∆α and S α are invariant. Now, | | 1 ∂α f (y) = τ f (e) = yα∆ f (e) with ∆ f (e) = f (y) y α! α α ∂yα Xα ( )y=e Relations between Lie groups and Lie algebras - I 29

y = exp( yiXi) where yi are the canonical coordinates of y. 27 ThereforeP

f (y) = f (exp( yiXi)) X ∞ 1 = ( y X )p (e) p! i i f Xp=0 n X o by Prop. 6, chapter 3, 5. Taking partial derivatives at y = e, we have

∂α α! f (y) at y = e is (S f )(e). ∂yα α ! α | | α! Hence, Delta f (e) = S f (e) which is what we wanted to prove. α α ! α | |

Chapter 4

Relation between Lie groups and Lie algebras - II

4.1 The enveloping algebras. 28 Let g be a Lie algebra, and T the tensor algebra of the underlying vector space of g. Consider the two-sided ideal I generated in T By the elements of the form x y y x [x, y]. Then the associative algebra T/I ⊗ − ⊗ − is said to be the universal enveloping algebra of the Lie algebra g.

Deﬁnition. A linear map h of al Lie algebra g into as associative algebra A is said to be a linearisation if h([x, y]) = h(x)h(y) h(y)h(x) for every − x, y g. ∈ We have obviously a canonical map of g in to T/I, which we shall denote by j.

Proposition 1. To any linearisation f of g in an associative algebra A, there corresponds one and only one representation f¯ ofT/I in such that f¯ j = f. ◦ In fact, f being a linear map, it can be lifted uniquely into a representation fˆ of the tensor algebra T in A. Obviously, the kernel of fˆ contains elements of the form x y y x [x, y] and hence contains I. ⊗ − ⊗ − Hence this gives rise to a map f¯ with the required property.

31 32 4. Relation between Lie groups and Lie algebras - II

4.2 The Birkhoﬀ-Witt theorem. With reference to the universal enveloping algebra of a Lie algebra of a Lie group, we have the following

29 Theorem 1. The algebra of left invariant differential operators is U canonically isomorphic to the universal enveloping algebra of the Lie algebra. In fact, the inclusion map of the Lie algebra g into can be lifted U to a representation of the enveloping algebra of g in by propo- U′ U sition 1. It only remains to show that this map h : ′ is an U →n U α α isomorphic. If S α is the coefficient of t in the expansion of ( tiXi)| |, i=1 it has been proved (Th. 3, Ch. 3.4) that S α form a basis for P. S α is an U operators of the form (...)Xi1 Xi α , where Xi1 Xi α are obtained αi ···αn | | ··· | | by certain permutationsP of x Xn . Let S α′ denote the element of the 1 ··· form (...)Xi1 ... Xi α , where S α = (...)Xi1 ... xi α . By definition ⊗ ⊗ | | | | of h,P we have h(S α′ ) = S α. To prove thatP h is an isomorphism, it is therefore sufficient to show that the S generate . We shall do this α′ U′ showing that the S for α r generate the space T of tensors of order α′ | |≤ r r modulo I. The statement being trivially true for r = 0, we shall ≤ assume it verified for (r 1) and prove it for r. Again, it is enough to − prove that S for α = r generate the space T of tensors of order = r, α′ | | r′ modulo I + Tr 1. Let Xi1 Xir be an element Tr′. Then − ⊗···⊗ ∈ X X ... X X X X ... + X , X X ... mod I. i1 ⊗ i2 ⊗ ⊗ ir ≡ i2 ⊗ i1 ⊗ i3 ⊗ i1 i2 ⊗ i3 ⊗ Hence, if σ is a permutation of (1, 2,..., r),

Xi1 Xi2 ... Xir Xiσ(1) ... Xiσ(r) mod (Tr 1 + I) ⊗ ⊗ ⊗ ≡ − 30 by successive transpositions. Now, S = σX ... x , where α′ σ U iσ(1) iσ(r) σ are positive integers. Therefore P U

( σ)Xi1 ... Xir S α′ mod (Tr 1 + I). − Xσ U ⊗ ⊗ ≡ Since , ( σ) 0, Xi1 ... Xir kS α mod (Tr 1 + I), − Xσ U ⊗ ⊗ ≡ Relation between Lie groups and Lie algebras - II 33 and hence the theorem is completely proved. Incidentally we have proved that the Lie algebra g can be embedded in its universal enveloping algebra by the natural map h. This is known as the Birkhoﬀ-Witt theorem, and it true in the more general case when the Lie algebra is over a principal ideal ring.

4.3 Group law in terms of structural constants. We now show that the Lie algebra of a Lie group completely charac- terises the group locally. In other words, the group laws of the Lie group can be expressed in terms of the structural constants of its Lie algebra. k k (If (X α)α A be a basis of the Lie algebra, and [Xi, X j] = k C Xk, C − ∈ i, j i, j are called the structural constants of the Lie algebra). P Theorem 2. Lie groups having isomorphic Lie algebras are locally isomorphic. If they are connected and simply connected, they are isomorphic. th Choose a basis X1,..., Xn of the Lie algebra g. If θi(x) be the i coordinates of x in the canonical of coordinates with respect to the above basis we have.

1 α ϕi(x, y) = θi(xy) = τyθi(x) = y ∆αθi(x) X α! But ∆αθi(x) = τx(∆αθi)(e) 1 = xβ∆ (∆ θ )(e). β! β α i Xβ Hence 31 1 1 ϕ (x, y) = yα xβ(∆ ∆ β )(e). i α! β! β α i Xα,β If γ γ ∆β∆α = dβ,α∆γ = dβ,α Xγ are completely known, once the Lie algebra g is given, because ∆α = α! α ! S α. Since | | ∂r ∆ θ (e) = ( ) = 1 if r=[i] r i ∂xr xi x=e 0 if r,[i] 34 4. Relation between Lie groups and Lie algebras - II

[i] dβ,α we have ϕ (x, y) = xβyα. i α!β! Xα,β [i] Thus the group law is completely determined by the constants dβ,α. [i] If two Lie groups have isomorphic Lie algebras, the constants dβ,α are the same for both, and the group operation is given locally by the above formula, which is to say the groups are locally isomorphic. By Lemma 1, Ch. 3.3, if the groups are connected and simply connected, they are isomorphic. [i] We can compute the constants dβ,α in terms of the structural constants of the Lie algebra and obtain a universal formula (i.e. a formula which is the same for all Lie groups - the Campbell–Hausdorﬀ formula). For instance, if δ is a multi-index of order 2 with 1 in the jth th 1 and k indices and 0 elsewhere, it can easily be seen that ∆δ = 2 S δ = 1 i 2 (X jXk + XkX j) and if C j,k are the structural constant of the Lie algebra, 1 1 X X = X , X + (X X + X X ) j k 2 j k 2 j k k j 1 1 = Ci X + (X X + X X ) 2 j,k i 2 j k k j Xi 1 and hence d[i] = Ci . [ j],[k] 2 j,k We have therefore

1 i ϕi(x, y) = xi + yi + C j,k x j xk + terms of order 3. 2 X ≥ 32 Again, if x = exp X, y = exp Y (xi, yi begin canonical coordinates) and xy = exp Z, we have 1 Z = X + Y + [X, Y] + terms of order 3. 2 ≥

4.4 We have proved (Prop. 4, Ch. 3.2) that the Lie algebra of a Lie subgroup can be identiﬁed with a subalgebra of the Lie algebra. We now establish the converse by proving the following Relation between Lie groups and Lie algebras - II 35

Theorem 3. To every subalgebra J of a Lie algebra g of a Lie group G, there corresponds one and only one connected Lie subgroup H, having it for its Lie algebra.

Let X1,..., Xn be a basis of the Lie algebra g such that X1,..., Xr is a basis of J . Let ϑ be the subalgebra generated by X with i r in i ≤ U (G). We assert that the subspace ϑ of U (G) generated by S α with α = (α ,..., α ) is the same as ϑ. By deﬁnition of S α, it is evident that 1 n ◦··· V V . It is enough to show that elements of the form X X U ′ ⊂ i1 ··· is ∈ ′ if 1 i r. We prove this by induction on the length of the product. ≤ k ≤ Now,

Xi1 Xi2 ... = Xi2 Xi1 ... + Xi1 , Xi2 Xi3 ... and since X , X ϑ, J being a subalgebra, we have, by induction i1 i2 ∈ assumption X X ... X X ... mod ϑ′. i1 i2 ≡ i2 i1 If σ is any permutation of (1, 2,..., s), we have

X ... X X ... x ( mod ϑ′). i1 is ≡ iσ(1) iσ(s)

It follows (as in Th. 1), that ϑ = ϑ′. Now, let U be a symmetric neighbourhood of e in which the system 33 of canonical coordinates with respect to X1,..., Xn is valid. Let N denote the subset of U consisting of points for which x + = = x = 0. r 1 ··· n N is obviously a closed submanifold of U. Let x, y N suﬃciently near ∈ e. 1 1 (xy) = xβyαd[i] i α! β! β,α Xα,β

We now show that (xy)i = 0 for i > r. In the summation, unless β α both α and β are oﬀ the form (α1,...,αr, 0 ...)x y = 0. If both α and β are of the above form, ∆α, ∆β ϑ and ϑ being generated by ∆γ, γ of [i] ∈ the same form dβ,α = 0 for i > r. Hence (xy)i = 0 for i > r. Thus, x, y N xy N and x x 1 N for x, y suﬃciently near e. ∈ ⇒ ∈ ∈⇒ − ∈ Finally, let H be the subgroup algebraically generated by the connected component N1 of e in N. Then H can be provided with an analytic structure such that the map H G is everywhere regular. We → 36 4. Relation between Lie groups and Lie algebras - II

deﬁne neighbourhoods of e in H by intersecting neighbourhood of e in G with N1. This system can easily be seen to satisfy the neighbourhood axioms for a topological group (Prop 1, Ch. 1.1). For every x H, ∈ the neighbourhood xN1 of x can be provided with an analytic structure induced by that of G, since xN is a closed submanifold of xU. For x, y H, these analytic structure agree an xN1 yN1 because those of ∈ ∩ xU and yU agree on xU yU. H of course has J as its Lie algebra. ∩ We now prove the uniqueness of such a group. Let H1 be another connected Lie subgroup having J for its Lie algebra. Exp J is open 34 in H1, as the map h exp h is open (Th. 2, Ch. 3.4). But exp J H. → ⊂ Hence H is open in H1. As H is open, it is also closed (Ch. 1.2) and therefore =H1. This completes the demonstration of Theorem 3.

Remark. We have incidentally proved that if a Lie subgroup has J for its Lie algebra, it contains H as an open subgroup.

It has already been proved (Prop. 1, Ch. 3.1) that if f : G H is a → representation of Lie groups, there exists a representation d f : g J → of Lie algebras. Now, we establish the converse in the form of a

Corollary. Let G and H be two Lie groups having g and J as their Lie algebras. If G is connected and simply connected, to every representation π of g in J , there corresponds one and only one representation f of G H such that d f = π. → If there exists one such representation, by Prop. 5, Ch. 3.4, it is unique. We shall now prove the existence of such an f . We first remark that if f is a representation of G in H, K the graph of f in GxHviz. the set (x, f (x)), x G , and λ the restriction to K of { ∈ } the projection of GxH G, then λ is an analytic isomorphism. Con- → versely, to every subgroup of GxH the first projection from which is an isomorphism to G, there corresponds one and only one representation of G in H. gxJ is evidently the Lie algebra of GxH. Now, Let π be a representation of g in J . Let be the subset K (x,π(x)), x g of g J . It can easily be seen that is a subalgebra. { ∈ } × K Then there exists (Th. 3) a connected Lie subgroup K of GxH whose 35 Lie algebra is isomorphic to . Let λ be the restriction to K of the K Relation between Lie groups and Lie algebras - II 37 projection of G H G. Then dλ is the map g defined by × → K → dλ(x,π(x)) = x. Obviously dπ is an isomorphism. Hence λ is a local isomorphism of K in G. K is therefore a covering space of G and G being simply connected, λ is actually an isomorphism. To this there corresponds (by our remark above) a representation f of G in H,the graph of whose differential is , i.e. d f = π. K 4.5 Theorem 4 (E. Cartan). Every closed subgroup of a Lie group is a Lie subgroup. For proving this, we require the following Lemma 1. Let G be a Lie group with Lie algebra g. Let g be the direct sum of vector subspaces ℘, δ. Then the map f : (A, B) exp A exp B of → g G is a local isomorphism. → It is obvious that f is an analytic map. To prove that it is a local isomorphism, it is enough to show that the Jacobian of the map , 0 in a neighbourhood of (0, 0). Let (X ,..., X ) be a basis of and 1 r U Xr+1,..., Xn, a basis of δ. Let (y1,..., yn) be the canonical coordinate n system with respect to (X1 ..., Xn) and yi = fi. Then, if X = yiXi, ◦ i=1 P r n f X = exp ( y X ) exp ( y X ) { } i i j j Xi=1 jX=r+1 ∂ fk d (0) = ( exp tXl)k (t = 0) = (Xl.xk)x=e = δk,l. ∂yl dt Hence Jacobian , 0 at (0, 0) and by continuity, , 0 in a neighbourhood of the origin. This completes the proof of the lemma. Let H be a closed subgroup of a Lie group G. We first construct a 36 subgroup J of G and prove that the Lie subgroup H1 of G with J as its Lie algebra is relatively open in H. Then H is the topological union of cosets of H modulo H1 and is hence a closed submanifold of G. Let J be the set X g : exp tX H for every t R . We assert { ∈ ∈ ∈ } that J is a Lie subalgebra of g. To prove this, we have to verify 38 4. Relation between Lie groups and Lie algebras - II

(i) X J αX J for every α R ∈ ⇒ ∈ ∈ (ii) X, Y J X + Y J ∈ ⇒ ∈ (iii) X, Y J [X, Y] J . ∈ ⇒ ∈ (a) is a trivial consequence of the deﬁnition. (b) Let now X, Y J . We have seen that (Ch. 4.3) ∈ 1 exp X exp Y = exp(X + Y + [X, Y] + ) 2 ··· tX tY X + Y 1 t2 1 Hence (exp exp )n = (exp t + [X, Y] + 0( )n n n { n 2 n2 n2 } t2 1 = exp t(X + Y) + [X, Y] + 0( ). 2n n tX tY tX But exp , exp H and H is a subgroup. Therefore (exp n n ∈ n tY tX tY exp )n H and since H is closed, lim (exp exp )n = n ∈ n n n exp t(X + Y) (by the above formula) H.→∞ Hence X + Y J . ∈ ∈ (iv) As before,

tX tY tY tY 2 X, Y J lim (exp exp exp − exp − )n H. n n n n n ∈ ⇒ →∞ ∈ The right hand side in this case tends to exp t2[X, Y] as n . →∞ Hence exp t[X, Y] H for positive values of t, and since exp( t ∈ − [X, Y]) = (exp t[X, Y]) 1 for all values of t, i.e. [x, y] J . − ∈ 37 Let K be the connected Lie subgroup of G having J for its Lie algebra (Th. 3, Ch. 4.4). We now show that K is open in H. It is obviously suﬃcient to prove that K contains a neighbourhood of e in H. If is a vector subspace of g supplementary to J , by Lemma 1, U there exists a neighbourhood V of 0 in g such that the map λ : (X, A) → exp X exp A, X J , A is an isomorphism of V onto λ(V) = W. ∈ ∈ U Suppose that K does not contain any neighborhood of e in H. Then, we can ﬁnd a sequence of points a H W which are not in K and n ∈ ∩ Relation between Lie groups and Lie algebras - II 39

which tend to e. There is no loss of generality in assuming an to be of the form exp An, An V, An , 0. For, If an = exp Xn exp An, then 1 ∈ U ∩ 1 (exp Xn)− an = exp An < K. Let V be a compact neighborhood of 0 in g V/2. For suﬃciently large n, A V1. Let r be the largest integer ⊂ n ∈ n for which r A V1. i.e. (r + 1)A < V1. n n ∈ n n But (r + 1)A = r (A ) + A V/2 + V/2 = V arn W1 = λ(V1) n n n n n ∈ · n ∈ rn+1 1 1 and an < W but W. Since W is compact, we may assume (by ∈ rn 1 taking a suitable subsequence) that an converges to an a W . Now, r +1 ∈ r we assert that a , e. For, if a = e, a n = a rn a e. But a n cannot n n n → n tend to e. Hence a , e W1. Therefore a = lim a r = exp A with ∈ n n A , 0 and A V1. ∈ U ∩ We shall now show that A J , which will imply that J , (0) ∈ ∩U and hence will give the contradiction we were seeking. It is enough to show that exp p/qA H for every rational number p/q. Now let ∈ pr /q = s + t /q, s an integer and 0 t q. n n n n ≤ n ≤ pA prn exp = lim exp( An) q n q →∞ tnAn = lim exp snAn exp( ) n q →∞ tn s Now, exp An e as n , and lim exp snAn = lim an n H 38 q n n → → ∞ →∞ →∞ ∈ as H is closed. Hence A J , and Theorem 4 is completely proved. ∈ Remark. The theorem is not true in the case of complex Lie groups. For instance, the space of real numbers is a closed subgroup of the complex plane, but is not a complex Lie group. Corollary 1. Every continuous representation f of the underlying topological group of a Lie group G into that of another Lie group H is an analytic representation. In fact, the graph K of f is a closed subgroup of the Lie group GxH, and hence is a Lie subgroup. Then f is the composite of the maps G → K, and K H, and both of them can be seen to be analytic. As an → immediate consequence, we have the following 40 4. Relation between Lie groups and Lie algebras - II

Corollary 2. Lie groups with isomorphic underlying topological group structures are analytically isomorphic.

4.6 Some examples. We have seen that the general linear group GL(n, R) is a Lie group, and by Theorem 4, every closed subgroup, and in particular, the orthogonal and symmetric groups are Lie groups. A group matrices deﬁned by some polynomial identities in the coeﬃcients of the matrices is a Lie group. 39 We proceed to study GL(n, R) in greater detail. If x GL(n, R) is ∈ the matrix (ai j), xi, j = ai, j δi, j is a coordinate system which takes the − 2 unit matrix to origin in the space Rn . Now,

ϕi, j(x, y) = xi, j + yi, j + xi,kyk, j Xk

Setting ui, j = yi, j + k xi,kyk, j, we have P τy f (x) = f (x + u) x ∂ f k,i ∂ f = f (x) + y ( + ) i, j ∂x ∂x Xi, j i, j Xk k, j The left invariant diﬀerential operators of order 1 are therefore gen- n ∂ erated by xi, j = ak.i . The Xi, j form a basis of the Lie algebra k=1 ∂xk, j Pλ of GL(n, R).Y = i, j Xi, j is a generic element of the Lie algebra. We associate the matrixP Yˆ = (λi, j) with this element Y. we now have the Proposition 2. The map Y Yˆ of the Lie algebra of GL(n, R) into the → algebra of all n-square matrices Mn(R) is a Lie algebra isomorphism. Let Y be an element of the Lie algebra of GL(n, R). We show that the map t exp tY assigns to t the usual exponential matrix exp tY. It → has been proved (Ch. 3.3) that x = exp tY satisﬁes

∂xi. j ∂ϕi, j = λ (x(t), e) ∂t k,l ∂y Xk,l k,l Relation between Lie groups and Lie algebras - II 41

= λk.lδ j,l(δi,k + xi,k) Xk,l

= λk, j(δi,k + xi,k) Xk d i.e. x is a matrix satisfying x(t) = x(t)Yˆ with x(O) = I. These two dt conditions can easily be seen to be satisfied by exp tYˆ. By the uniqueness theorem on differential equations, exp tY = exp tYˆ, where the latter 40 exponential is in the sense of the exponential matrix. Let Y, Z g the ∈ Lie algebra of GL(n, R). The map X Xˆ is trivially a vector space → isomorphism of g onto Mn(R). Now, 1 exp Y exp Z = exp(Y + Z + [Y, Z] + ) by Ch. 4.3. 2 ··· Yˆ n Zˆm But exp Y exp Z = X n! X m! Yˆ 2 = 1 + Yˆ + Zˆ + + 2! ··· and 1 1 exp(Y + Z + [Y, Z] + ) = 1 + Yˆ + Zˆ + (Yˆ 2 + yˆZˆ + ZˆYˆ + Zˆ2) + 2 ··· 2 ··· Comparing the coefficients, we get

[Y,ˆZ] = [Yˆ, Zˆ]. d Remarks. (1) Y = (exp tY) . dt t=0 (2) The Lie algebra of a closed subgroup H of GL(n, R) is simply the Lie algebra of matrices Y such that exp tY H for every t R (by ∈ ∈ Theorem 4, Ch. 4.5).

(3) Let B(a, b) be a bilinear form on Rn.

Then, the set of all regular matrices x which leave B(a, b) invariant is a Lie subgroup H of GL(n, R). Then the Lie algebra of this Lie Group 42 4. Relation between Lie groups and Lie algebras - II

consists of matrices Y such that B(Ya, b) + B(a, Yb) = O for every a, b ∈ Rn. In fact, if Y is in the Lie algebra, exp tY H, B being invariant under ∈ H, B (exp tYa, exp tYb) = B(a, b). But

tp+q B(exp tYa, exp tYb) = B(Y pa, Yqb). p!q! Xp,Q

Hence B(Ya, b) + B(a, Yb) = 0.

41 Conversely, if Y satisﬁes this condition, by induction it can be seen tp+q that B(Y Pa Yqb) = 0, which proves that B(exp tYa, exp p+q=n p!q! · tYb) =PB(a, b), i.e. exp tY H for every t R. This proves that Y ∈ ∈ is in the Lie algebra of H.

4.7 Group of automorphisms. Let G be a connected Lie group. Then the set of automorphisms of G (continuous representations of G onto itself), form a group. We shall denote this group by Aut G. Let α Aut G. This gives rise to a map dα : g g where dα is an ∈ → automorphism of the Lie algebra. We thus have a map; Aut G Aut g. → This map is one-one, and, if G is simply connected, onto. We have, in this case, an isomorphism of Aut G Aut g, for d(α1,α2) = dα1 dα2 1 1 → ◦ and dα− = (dα)− . Now, Aut g GL(g). Aut g is actually a closed k ⊂ subgroup of GL(g). For, if Ci, j be the structural constants of g, A k ∈ Aut g A(xi), A(x j) = Ci, jA(xk) for every i, j and A GL(g), xi ⇔ k ∈ { } being a basis of g. Since AutP g is determined by these n2 equations, it is a closed subgroup of GL(g). Hence, Aut g is a Lie group.

Proposition 3. Let Γ be the Lie algebra of Aut g. Then X Hom(g, g) ∈ (which is the Lie algebra of GL(g)) is in Γ if and only if exp tX Aut g ∈ for every t in R. This is obvious from the proof of Theorem 4, Ch. 4.5.

Proposition 4. X Hom(g, g) is in Γ if and only if X is a derivation. ∈ Relation between Lie groups and Lie algebras - II 43

42 By Proposition 3,

(exp tX)[Y, Z] = [exp tX Y, exp tX Z] · · tnXn tp+q i.e., [Y, Z] = [xpY, XqZ] n! p!q! X Xp,q n! Xn[Y, Z] = [X pY, XqZ] p!q! pX+q=n for every n. In particular

X[Y, Z] = [XY, Z] + [Y, XZ]

X is therefore a derivation. Conversely, n! X[Y, Z] = [XY, Z] + [Y, XZ] Xn[Y, Z] = [X pY, XqZ] p!q! ⇒ pX+q=n by induction on n exp tX[Y, Z] = [exp tX Y, exp tX Z]. ⇒ · · Hence X Γ. ∈ In other words, the Lie algebra of Aut g is only the Lie algebra of derivations of g (it is a trivial verification to see that the derivations of g form a Lie algebra and the set of inner derivations (Remark 1, Ch. 2.8) form an ideal in that algebra). Now, corresponding to every y G, there exists an inner automor- ∈ phism ρ : x yxy 1 of G. Obviously y ρ is an algebraic represen- y → − → y tation of G in Aut G. ρy induces an automorphism dρy of g. We denote this by ady. y ady is an algebraic representation of G in Aut g. We now show → that this is an analytic representation. By Corollary to Theorem 4, Chap- ter 4.5, it is enough to show that this is continuous, i.e. if y e then → adyX X for every X g. Since G and g are locally isomorphic, it → ∈ suffices to prove that as y e, y exp Xy 1 exp X but this is obvious. 43 → − → This analytic representation of G in Aut G = Aut g is called the adjoint representation of G. Let θ be the differential of the representation y ady. We now show that this is actually the adjoint representation → (Ch. 2.8) of the Lie algebra g. 44 4. Relation between Lie groups and Lie algebras - II

Theorem 5. θ(X) = ad X for every X g. ∈ By Remark 1, Prop. 2, Ch. 4.6, d θ(X) = (exp t θ(X)) = {dt }t 0 d = (ad exp t x) = dt t 0 by deﬁnition of exponential. We have now to show that θ(X)Y = [X, Y] d for every Y g. Let x = exp tX θ(X)Y = (ad Y) = . But (adx Y f ) ∈ · dt t 0 ◦ ρ = Y( f ρ ) where f is any analytic function on G. It follows that x ◦ x

σ 1 τ 1 adx Y f = Yσ 1 τ 1 f x− ◦ x− x− ◦ x− ◦ or (adx Y) τx = τx σxYσ 1 ◦ ◦ x− = τxY since Y is left invariant

adx Y = τ Yτ 1 . x x− But tnXn τ f (e) = f (exp tx) = f (e) (Prop. 6, Ch. 3.4) x n! Xn tmXm ( t)nXn Hence adx Y = − m! n! Xm,n tm+n = ( 1)n XmYXn m!n! Xm,n − Therefore d θ(X)Y = (adx Y) = = XY YX = [X, Y]. {dt }t 0 − Corollary . Let H be a connected Lie subgroup of G. H is a normal subgroup if and only if its Lie algebra J is an ideal in g. Relation between Lie groups and Lie algebras - II 45

44 In fact, if H is a normal subgroup, ρ (H) H for every y G, i.e. y ⊂ ∈ adyJ J for every y G. Let X g. Then ad exp tXJ J . ⊂ ∈ ∈ ⊂ d (ad exp tX) = J = ad XJ J dt t 0 ⊂ i.e. J is an ideal. tnadXn Reciprocally, let J be an ideal. ad XJ J ( )J ⊂ · n! ⊂ J . But exp tad X = ad exp tX (ad exp tX)H H Pfor every X in a · ⊂ neighborhood of e. Since G is connected, H is normal.

4.8 Factor groups. Theorem 6. Let H be a closed subgroup of a Lie group G. The homogeneous space G/H is an analytic manifold in a canonical way. The operations by G on G/H are isomorphisms. If H is a normal subgroup, is a Lie group, and its Lie algebra is isomorphic to g/J .

Let J be the Lie algebra of H, and let be a vector subspace U of g supplementary to J . We have seen in the proof of Theorem 4, Ch. 4.5, that there exists a neighborhood V1 of (0, 0) in J such ×U that the map λ : (X, A) exp X exp A is an isomorphism of V1 onto → a neighborhood W1 of e in G. Let U and V be neighborhoods of 0 in J and respectively such that U V V1 and WW 1 W1 with U × ⊂ − ⊂ W = λ(U V). We now show that L = exp V is a cross-section of the × canonical map η : G G/H in the neighborhood W˙ = η(W) of η(e). → In other words, L Hx contains one and only one element for every ∩ x W. For, we have x = exp X exp A with X U and A V and ∈ ∈ ∈ exp A L Hx. On the other hand, if exp A1 and exp A2 belong to Hx ∈ ∩ 1 1 (with A1, A2 V), then exp A1(exp A2)− H W ; hence there exists 45 1 ∈ ∈ ∩ an X V J such that exp A1 = exp X exp A2 and this implies X = 0, ∈ ∩ 1 1 A1 = A2 because λ is an isomorphism from V onto W . We can, therefore, provide W˙ with a manifold structure induced from that of . This can be extended globally by translating that on U W. It is easily seen that on the overlapsx ˙W˙ ,y ˙W˙ , the analytic structures agree because the analytic structure on W˙ is induced from that of . By U 46 4. Relation between Lie groups and Lie algebras - II the deﬁnition of the manifold G/H, it is obvious that the operations by G on G/H are analytic isomorphisms. If H is normal subgroup, G/H has also a group structure and is a Lie group with the above manifold. By Theorem 6, Ch. 4.8, J ia an ideal of J and if is the Lie algebra of G/H, the map η : G/H gives rise K → to a representation dη : J . The kernal of this map is J since → K is isomorphic as a vector space to the tangent space at e of L which K is . Hence H /J is isomorphic to as a Lie algebra also, i.e. G/H U K has its Lie algebra isomorphic to g/J . Corollary. Let f be a representation of a Lie group G which is countable at in another Lie group H. Then the image f (G) is a Lie sub- ∞ π f¯ group. If N is the kernel of f , then f can be factored into G G/N −→ −→ H where π is the canonical map and f¯ an injective regular map.

The proof is an immediate consequence of the isomorphism theorem on Lie algebras and Theorem 6. Part II

General Theory of Representations

Chapter 5

Measures on locally compact spaces

1.1 Deﬁnition of a measure.

In this chapter and the following, we shall give a brief summary of cer- 46 tain results on measure theory, a knowledge of which is essential in what follows. Let X be a locally compact topological space and CX the algebra of continuous complex-valued functions on X with compact support. Let K be a compact subset of X and C the subset f : support of f K of K { ⊂ } CX. Then CK is a Banach space under the norm f = sup f (x) . || || x K | | ∈ Deﬁnition. A measure on X is a linear form on CX such that the restriction to CK is continuous for every compact subset K of X. A measure µ is said to be positive if µ( f ) 0 for every f 0. ≥ ≥ Proposition 1. Every positive linear form on CX is a measure on X. In fact, if K is any compact subset of X, there exists a continuous function f on X which = 1 in K, = 0 outside a compact neighbourhood of K and 0 f 1. If g is a function belonging to C , obviously ≤ ≤ K g f g g f and hence −|| || ≤ ≤ || || g µ( f ) µ(g) g µ( f ). −|| || ≤ ≤ || || 47 48 5. Measures on locally compact spaces

i.e., µ(g) g µ( f ), | | ≤ || || which shows that µ is continuous when restricted to CK. On the other hand, if µ is any real measure (i.e. a measure µ such that µ( f ) is real whenever f is real) it can be expressed as the diﬀerence of 47 two positive measures. Moreover, a complex measure µ can be uniquely decomposed into ν + iν′ where ν and ν′ are real. Hence a measure can alternatively be deﬁned as a linear combination of positive linear forms on CX. For a real measure ν, there exists a unique ‘minimal’ decomposition µ µ (µ ,µ positive) in the sense that if ν = µ µ be any 1 − 2 1 2 1′ − 2′ other decomposition, we have µ1′ = µ1 + π, µ2′ = µ2 + π with π positive.

1.2 Topology on CX.

The space CX = CK where K runs through all the compact subsets of K X can be providedS with the topology obtained by taking the direct limit of the topologies on CK. This topology makes of CX a locally convex topological vector space. The fundamental property of this space is that a linear map of CX in a locally convex space is continuous if and only if its restriction to each CK is continuous. (Bourbaki, Espaces vectoriels topologiques, Chapter 6). A measure is, by definition, a continuous linear form on CX with its topology of direct limit. The space MX of measures is none other than the dual of CX. One can provide MX with several topologies, as for instance, the weak topology in which µ 0 → ⇔ for every f C , µ( f ) 0. ∈ X → 1.3 Support of a measure. Definition. The support of a measure µ is the smallest closed set S such that for every function f C whose support is contained in X S, ∈ X − µ( f ) = 0. Let M c be the space of measures with compact support. If f is a continuous function on X, for every µ M c, we can define µ( f ) = ∈ 48 µ(α f ) where α is a function 1 on a neighbourhood of the support K of µ, and 0 outside a compact neighbourhood of K. It is obvious that the Measures on locally compact spaces 49

value of µ( f ) does not depend on α. We shall denote by EX◦ the space of continuous function on X. EX◦ with the topology of compact convergence is a locally convex topological vector space. µ defined on EX◦ in the above manner is continuous with respect to this topology. Conversely, let µ be a continuous form on EX◦. The topology on CX is finer than that induced from the topology of EX◦. Hence µ restricted to CX is again continuous and is consequently a measure. We now show that this has compact support. Since µ is a continuous function on EX◦, we can find a neighbourhood V of 0 such that µ( f ) < 1 for every f V. V may | | ∈ be taken to be of the form f : f < ǫ on K as the topology on E is { | | } X◦ the topology of compact convergence. Let g C be a function 0 on ∈ X K. Then µ(g) < 1. If λ is any complex number, µ(λg) = λµ(g), and | | µ(λg) < 1. Hence µ(g) = 0, i.e. the support of µ is contained in K. | |

1.4 Bounded measures.

Let µ ba a measure M . We deﬁne a positive measure µ in the ∈ X | | following way: µ f = sup µ(g) for every positive function f and extend it by | | 0 g f | | ≤| |≤ linearity to all functions C . If µ is a real measure with the minimal ∈ X decomposition (Ch. 1.1) µ = µ µ , then µ = µ + µ . 1 − 2 | | 1 2 Deﬁnition . A measure µ is bounded if and only if there exists a real number k such that µ( f ) k f with f = sup f (x) . | |≤ || || || || x X | | ∈ Obviously µ is bounded if and only if µ is bounded, µ is continuous 49 | | for this norm and can be extended to the completion C X (which is only the space of continuous functions tending to zero at ). C is actually ∞ X the adherence of CX in the space of all continuous bounded functions. The space of bounded measures is a Banach space under the norm µ = µ( f ) || || sup | |. µ is the smallest number k such that µ( f ) k f . It can f CX f || || | | ≤ || || ∈ || || proved that every bounded continuous function is integrable with respect to a bounded measure, and we have still the inequality µ( f ) µ f | | ≤ || || || || for bounded continuous functions f . 50 5. Measures on locally compact spaces

1.5 Integration of vector valued functions. We introduce here the notion of integration of a vector valued function with respect to a scalar measure, a use of which we will have frequent occasions to resort to in the sequel. Let K be a compact space and E a locally convex quasi-complete topological vector space (i.e. every closed bounded subset is complete). We shall provide the space C (K, E) of continuous functions of K into E with the topology of uniform convergence. Theorem 1. Corresponding to every measure µ on K, there exists one and only one continuous linear map µ˜ of C (K, E) in E such that µ˜( f a) = · µ( f ) a for every continuous complex valued function f on K and a E. · ∈ µ can obviously be lifted to a linear mapµ ¯ of C E in E by setting ⊗ µ¯(c e) = µ(c)e and extending by linearity. Also, C E can be identiﬁed ⊗ ⊗ 50 with a subset of the space C (K, E) of continuous functions of K into E. We will now show thatµ ¯ is continuous with respect to the induced topology on C E and that C E is dense in C (K, E). We will in fact ⊗ ⊗ prove more generally that every function f C (K, E) is adherent to a ∈ bounded subset of C E. ⊗ Let V be a convex neighbourhood of 0 in E. Then there exists a neighbourhood A of each point x K such that f (y) f (x) V for every x ∈ − ∈ y A . Now the A cover the compact space K and let A ,..., A be ∈ x x x1 xn a ﬁnite cover extracted from it. Let ϕi be positive continuous functions n

on K such that ϕi = 1 and the support of ϕ Axi . If g = ϕi f (xi), i=1 ⊂ then g C E,P and we have P ∈ ⊗

g(y) f (y) = ϕi(y) f (xi) ϕi(y) f (y) − X − X = ϕi(y)[ f (xi) f (y)] X − V since V is convex. ∈ By allowing V to describe fundamental system of neighbourhoods of 0, we see that f is adherent to the set of such functions g. This set is a bounded subset of C (K, E). For, ϕi(y) f (xi) is in the convex envelope of f (K) for every y K and theP convex envelope of a compact set is ∈ Measures on locally compact spaces 51

bounded. It follows that ϕi f (xi) are uniformly bounded and hence form a bounded subset of CP(K, E). This proves, in particular,that C E ⊗ is dense in C (K, E). Now let g = g a be a function C E tending to zero in the i i ∈ ⊗ topology of C (K,PE). Then giai, a′ 0 uniformly on the compact h i → set K and on any equicontinuousP subset H of the dual of E.

µ g, a′ = µ giai, a′ h i hX i = µ( gi ai, a′ ) = µ(gi) ai, a′ X h i X h i = µ(gi)ai, a′ = µg, a′ hX i h i 51 Since g, a 0 uniformly on K H, µ g, a 0 and hence h ′i → × h ′i → µ , a 0 uniformly on any equicontinuous subset H of E. Conse- h g ′i → quently µg 0. This shows that µ is continuous on C E. → ⊗ Therefore µ can be extended uniquely to a continuous linear map of C (K, E) in the completion Eˆ of E. But if f C (K, E), it is adherent to a ∈ bounded set B and µ(B) is also bounded in E. By the quasi-completeness of E, the closure of µ(B) in Eˆ and E are the same. Hence µ(f) µ (B) ∈ µ(B) E. Thus we have extended µ to a continuous linear mapµ ˜ ⊂ ⊂ of C (K, E) E and it is obvious this is unique. Now by Theorem → 1, if G be any locally compact space and µ a measure on G, we can deﬁne f (x)dµ = µ˜( f ) for every continuous function f from G to E with compactR support.

Remark. The measure with this extended meaning is factorial in character in the following sense: Let E and F be two locally convex spaces and f a continuous map of a compact space K into E. If A is a continuous map of E in F, we have A f C (K, F) and µ satisﬁes µ(A f ) = Aµ( f ). ∈

Chapter 6

Convolution of measures

2.1 Image of a measure 52 Definition. Let X, Y be two locally compact topological spaces and π a map X Y. Let µ be a positive measure on X. Then Π is said to be → µ proper if for every function f C , f π is integrable with respect − ∈ Y ◦ to µ. The value µ( f π) depends linearly on f and is therefore a linear ◦ form on CY . In other words, µ( f π) defines a positive measure on ◦ = Y, which we denote by π(µ). We have, by definition, Y f (y)dπ(µ)(y) f π(x)dµ(x). R X ◦ R If µ is not positive, but is equal to (µ µ ) + i(µ µ ), µ , µ , µ , 1 − 2 3 − 4 1 2 3 µ positive, and if π is µ - proper, we can define the image measure 4 | | π(µ) = π(µ ) π(µ ) + iπ(µ ) iπ(µ ). 1 − 2 3 − 4 Examples.

(1) A continuous proper map of X Y (i.e. a map such that inverse → image of every compact set is compact) is µ - proper for every µ. 1 In fact, f CK f π Cπ 1(K) and π− (K) is compact. ∈ → ◦ ∈ − (2) Let π be a continuous map G H and let µ have compact support → K. Then π is µ-proper; the support of π(µ) π(K) and is hence ⊂ compact.

53 54 6. Convolution of measures

If f is a continuous function on H with compact support K, f π ◦ is continuous and hence µ-integrable (Ch. 1.3). This shows that π is µ-proper and if f = 0 on π(K), then f π = 0 on K and therefore ◦ µ( f π) = 0, i.e. support of π(µ) π(K). ◦ ⊂ 53 (3) More generally, when π is continuous and µ bounded, π is µ- proper. Also π(µ) is bounded and π(µ) µ . || || ≤ || || In fact, f π is bounded and in view of the remark in Ch. 1.4, f π ◦ ◦ is integrable with respect to µ. Moreover,

πµ(g) µ(g π) π(µ) = sup | | = sup | ◦ | µ || || g CY g g CY g π ≤ || || ∈ || || ∈ || ◦ ||

2.2 Convolution of two measures. Let G, H be two locally compact topological spaces and µ, ν measures on G, H respectively. Then there exists one and only measure λ on G H × such that if f , g be functions with compact support respectively on G, H we have

f (x)g(y)dλ(x, y) = ( f (x)dµ(x))( g(y)dν(y)). Z Z Z λ shall be called the product measure of µ and ν. If µ and ν are two measures on a locally compact group G, we denote the product measure by µ ν and, if the group operation π : G G G ⊗ × → deﬁned by (x, y) xy is µ ν - proper, its image in G by µ ν. The latter → ⊗ ∗ is said to be the convolution product of µ and ν. The most general class of measures for which convolution product can be deﬁned are those for which f (xy) is integrable with respect to the product measure for every function f C . The following cases are the particular interest to us: ∈ G (1) If µ and ν are bounded, the convolution product exists and is bounded. This is almost obvious, π being continuous and µ ν bounded ⊗ (Example 3, Ch. 2.1). Convolution of measures 55

54 (2) If µ and ν are measures on G with compacts K, K′ respectively, µ ν exists and has compact support. In fact, µ ν has support ∗ ⊗ K K . Hence, convolution product exists, and has support ⊂ × ′ KK . ⊂ ′ (3) If either µ or ν has compact support, µ ν exists. Let f be a ∗ continuous function on G with compact support K and let µ have compact support K′. Obviously

f (xy)dµ(x)dν(y) = dµ(x) f (xy)dν(y) " 1 ZK′ ZKK′−

Hence f (xy) is integrable with respect to µ ν. Consequently, µ ν ⊗ ∗ exists. 1 c 0 We denote as usual by M , M , EG the spaces of bounded measures, the space of measures with compact support and the space of all continuous functions on G respectively. Let λ, µ, ν be three measures on G such that either all three are bounded or two of them have compact support. In any case the function (x, y, z) f (xyz) is integrable with → respect to λ µ ν and hence Fubini’s theorem can be applied. ⊗ ⊗ f (xyz)dλ(x)dµ(y)dν(z) = dν(z) f (xyz)dλ(x)dµ(y) $ Z Z Z = dν(z) f (tz)d(λ µ)(t) Z Z ∗ = f (tz)d(λ µ)(t)dν(z) " ∗ = (λ µ) ν f { ∗ ∗ } = λ (µ ν) f { ∗ ∗ } by a similar computation. This shows that m1 with the convolution product is an associative algebra and that M c acts on M on both sides and 55 makes it a two-sided module. Moreover, M 1 is actually a Banach algebra under the usual norm, since we have µ ν µ ν . || ∗ || ≤ || || || || Remarks. (1) It is good to point out here that the associativity does not hold in general. Take,for instance, R to be the locally compact 56 6. Convolution of measures

group and λ the Lebesgue measure. Let µ be ǫ ǫ (ǫ begin the 1 − 0 a Dirac measure at a - see Ch. 2.5 and ν = ϕ(x)dx where ϕ is the Heaviside function viz. ϕ = 0 for x < 0 and =1 for x 1. Then ≥ (λ µ) ν = 0 and λ (µ ν) = dx. Again λ (µ ν) may exist ∗ ∗ ∗ ∗ ∗ ∗ without λ µ begin well deﬁned. Let R be the locally compact ∗ group, λ and µ Lebesgue measures on R and ν = ǫ ǫ . Then 1 − 0 λ (µ ν) = 0 but λ µ is not deﬁned. However, when f (xyz) is ∗ ∗ ∗ integrable with respect to λ µ ν then the convolution product ⊗ ⊗ is associative.

(2) The formula for the integration of functions with respect to the convolution of two measures is valid also for vector-valued function. Thus we have

f (x)d(µ ν) = f (xy)dµ(x)dν(y). ZG ∗ "

2.3 Continuity of the convolution product.

That the convolution product is continuous in M 1 is trivial in virtue of our remark that it is a Banach algebra. Regarding the continuity of the convolution product in the other cases, we have the following

Lemma 1. Let f be a continuous function and µ a measure on G, one of them having compact support. Then (i) the function g(x) = f (xy)dµ(y) 56 is continuous; (ii) the map f g is a continuous linear mapR of CG in 0 → EG (with the usual topologies); (iii) if µ has compact support, then the above map f f (xy)dµ(y) is also continuous from E E and → G◦ → G◦ from C C . R G → G (1) Let H and K be two compact subsets of G. Then H K is also × compact and f (xy) is uniformly continuous on H K. For every × ǫ > 0, and for every x H, there exists a neighbourhood U of x ∈ such that f (x y) f (xy) < ǫ for every x U H and y K. | ′ − | ′ ∈ ∩ ∈ If f has compact support S , we choose a compact neighbourhood H of x and K such that HS 1 K. If y < K, then xy, x y < S . − ⊂ ′ Convolution of measures 57

Hence

f (x′y) f (xy) dµ(y) = 0. Z − G K − So, we have

g(x′) g(x) f (x′y) f (xy) d µ (y) 6 ǫ µ (K). | − |≤ ZK − | | | | This shows that g is continuous. If however µ has compact support C, we take K = C and the same inequality as above results. (2) Again, as in (i) if we assume that f has compact support S ans HS 1 K, it is immediate that − ⊂ g(x) f (xy) d µ (y) for every x H | |≤ ZK | | | | ∈ 1 f (y) d µ (x− y) ≤ ZK | | | | 1 sup f µ (H− K). ≤ | || | 1 Hence we have sup g(x) sup f µ (H− K). x H | |≤ | || | ∈ It follows that whenever f 0 on C , g(x) 0 uniformly on → S → the compact set H. A similar proof holds when µ has compact support.

(3) Let now C be the support of µ, and f has compact port K; obvi- 57 ously g C 1. Since the map C E is continuous, so also ∈ KC − G → G◦ is the map CK C 1 and by the properly of the direct limit → KC− topology, C C is continuous. An analogous proof holds for G → G the other part.

2.4 Duality and convolution products

Let E be a locally convex topological vector space an E′ its dual. Then E′ can be provided with several interesting topologies (Bourbaki, Es- paces vectoriels topologiques, Chapter 8). The following three are of fundamental importance: 58 6. Convolution of measures

(i) The weak topology, in which x E 0 if and only if x , x ′ ∈ ′ → h ′ i→ 0 for every x E. ∈ (ii) The convex compact topology, in which x E 0 if and only ′ ∈ ′ → if x , x 0 uniformly on every convex compact subset, and h ′ i→ (iii) The strong topology, where x E 0 if and only if x , x 0 ′ ∈ ′ → h ′ i→ uniformly on every bounded set.

In general, these topologies are distinct. If E is a Banach space, the usual dual is the E′ with the strong topology. However, the convex compact topology is often the most useful, in as much as it shares the ‘good’ properties of both the weak and the strong topologies. To mention but one such, (E′)′ = E is true for the weak, but not for the strong, topology. The convex compact topology possesses this property. We shall almost always restrict ourselves to the consideration of this topology. c 58 In particular, the spaces M , M being duals of CG and EG◦ respectively, they can be provided with the convex compact topology. With reference to the convolution map we have the

Proposition 1. The convolution map (µ, v) µ ν is continuous in each → ∗ variable separately in the following situations:

M c M c M c; M c M M ; M M c M . × → × → × → In fact, let µ be ﬁxed in M c and ν 0 in M . Then →

µ ν( f ) = f (xy)dµ(x)dν(y) ∗ " = dν(y) f (xy)dµ(x). Z Z

Denoting by f (y) the function f (xy)dµ(x) the map f f is con- 0 → 0 tinuous from C C (Lemma 1,R Ch. 2.3). The image of a convex G → G compact subset being again a convex compact subset, µ ν 0 uni- ∗ → formly on a convex compact subset. All other assertions in the proposition can be demonstrated in an exactly similar manner. Convolution of measures 59

2.5 Convolution with the Dirac measure

If x is a point of G, the Dirac measure ǫx is deﬁned by ǫx( f ) = f (x). This is trivially a measure with compact support. Let ν be any arbitrary measure. Then

ǫ ν( f ) = f (yz)dǫ (y)dν(z) x ∗ " x = f (xz)dν(z) Z = ν(σ 1 f ). x − In a similar manner, ν ǫ ( f ) = ν(τ f ). We may deﬁne left and right ∗ x x translation of a measure by setting d(τxν)(y) = dν(yx) and d(σxν)(y) = 1 dν(x− y). It requires a trivial veriﬁcation to establish that τxν( f ) = 59 ν(τ 1 f ) and σ ν( f ) = ν(σ 1 f ). Hence we have x− x x−

1 ǫ ν = σ ν, and ν ǫ = τ− ν. x ∗ x ∗ x x In particular, ǫ ǫ = σ ǫ = ǫ . In other words, the map x ǫ x ∗ y x y xy → x is a representation in the algebraic sense of the group G into the algebra M c or M 1. As a matter of fact, this can be proved to be a topological isomorphism (Bourbaki, Int´eration, Chapter 7).

Chapter 7

Invariant measures

3.1 Modular function on a group. 60 We assume the fundamental theorem relating to measures on locally compact groups, namely the existence and uniqueness (upto a positive constant factor) of a right invariant positive measure. If µ is such a measure, we have

(ǫ µ) ǫ = ǫ (µ ǫ ) = ǫ µ. y ∗ ∗ x y ∗ ∗ x y ∗ Hence ǫ µ is also a right invariant positive measure. By our remark y ∗ above, ǫy µ = kµ where, of course, k depends on y. We shall denote k by 1 ∗ ∆(y)− where ∆(y) is a positive real number. It is immediate that ∆(yz) = ∆(y)∆(z). ∆ is therefore a representation of G in the multiplicative group 1 f (y− x)dµ(x) of R+. In fact, the continuity of ∆(y) = follows at once R f (x)dµ(x) 1 from that of f (y− x)dµ(x) (Lemma 1, Ch.R 2.3). This representation ∆ of a locally compactR group is said to be its modular function.

Proposition 1. If a right invariant positive measure on G is denoted by 1 1 dx, then the following identity holds: dx− = ∆(x− )dx

1 In fact, if dµ stands for ∆(x− )dx, we have

1 1 1 dµ(yx) = ∆(x− y− )d(yx) = ∆(x− )dx = dµ.

61 62 7. Invariant measures

1 Hence dµ is left invariant. So also is dx− for,

1 1 1 1 f (yx)dx− = f (x)d(y− x)− = f (x)d(x− ). Z Z Z

1 1 So kdx− = ∆(x− )dx where k is a constant. We now prove that 1 k = 1. When x is near e, ∆(x− ) is arbitrarily near 1 and if we take g(x) = 1 61 f (x) + f (x− ), f being a positive continuous function with suﬃciently small support, we have

1 1 1 g(x)dx = g(x)dx− = g(x)∆(x− )dx Z Z k Z

1 k being a ﬁxed number and ∆(x− ) arbitrarily near 1; it follows that k = 1.

Deﬁnition . A locally compact group G is said to be unimodular if its modular function is a trivial map which maps G onto the unit element of R.

a11...0 The group of triangular matrices of the type ...... can be proved ...ann to be non-unimodular. Examples of unimodular groups.

(1) A trivial example of unimodular groups is that of commutative groups.

(2) Compact groups are unimodular. This is due to the fact that ∆(G) is a compact subgroup of R+ which cannot but be (1).

(3) If in a group the commutator subgroup is everywhere dense, then the group is unimodular. This again is trivial as ∆ maps the commutator subgroup and consequently the whole group onto 1.

(4) A connected semi-simple Lie group is unimodular. (A Lie group G is said to be semi-simple if its Lie algebra g has no proper abelian ideals. Consequently, it dose not have proper ideals such that the quotient is abelian). The kernel N of the representation Invariant measures 63

d∆ of the Lie algebra into the real number is an ideal in g such that g/N is abelian and is therefore the whole Lie algebra. It follows that the map d∆ maps the Lie algebra onto (0). This shows that group is unimodular.

3.2 Haar measure on a Lie group. 62 Let G be a Lie group with a coordinate system (x1,..., xn) in a neighbourhood of e. We now investigate the form of the right invariant Haar measure on the Lie group. By invariance of a measure µ here we mean 1 that f (xy− )dµ(x) = f (x)dµ(x) for y which are suﬃciently near e andR for f whose supportsR are suﬃciently small. We set dµ(x) = λ(x)dx ... dx and enquire if integration with respect to this measure 1 ∧ ∧ n is invariant under right translations. For invariance, we require

∂ϕi(x′, y) f (x′)λ(x′y) det x1′ ... dxn′ Z ∂x′j ∧ ∧

= f (x)λ(x)dx1 ... dxn, Z ∧ ∧ ∂ϕ or still λ(x) = λ(xy)J(x, y) with J(x, y) = det i (x, y) . ∂x j

1 For this it is obviously necessary and suﬃcient to take λ(x) = J− (e, x). This gives an explicit construction of the Haar measure in the case of Lie groups.

3.3 Measure on homogeneous spaces. If G/H is the quotient homogeneous space of a locally compact group G by a closed subgroup H, we denote the elements of G by letters x, y,... those of H by ξ, η, . . . the respective Haar measure by dx, dy,... dξ, dη... and the respective modular functions by ∆, δ. Also π is the canonical map G G/H. Let f be a continuous function on G with compact → o = ξ ξ support K. Then f (x) H f ( x)d is a continuous function on G (as in lemma 1, Ch. 2.3) and weR have f o(ζx) = f o(x) for every ξ H. There- ∈ fore f o may be considered as a continuous function on G/H. Obviously it has support π(K) which is again compact. 64 7. Invariant measures

63 Proposition 2. The map f f 0 is a homomorphism (in the sense of N. → Bourbaki) of CG onto CG/H. We prove this with the help of

Lemma 1. There exists a positive continuous function f on G such that for every compact subset K of G the intersection of HK and the support S of f is compact and such that f (ξx)dξ = 1 for every x G. H ∈ R A locally compact group is always paracompact (Prop. 3, Ch. 1.2. part I) and using the fact that the canonical map is open and continuous, we see that G/H is also paracompact. Let U be an open relatively compact neighbourhood of e in G. π(Ux) is a family of open subsets

covering G/H. Let (Vi), (Vi′) be two locally finite open refinements of this covering such that V¯ V . Then there exist open relatively compact i ⊂ i′ subsets (W ), (W ) such that W W and π(W ) = V . In other words i i′ i ⊂ i′ i i these are families of subsets such that each point in G has a saturated neighbourhood which intersects only a finite number of the subsets. We can moreover say that for every compact subset K of G, HK intersects

only a finite number of Wi′. Now, let us define continuous functions gi such that gi = 1 on Wi and 0 outside Wi′, and set g = gi. This last i summation has a sense as the summation is only over aP finite indexing set at each point.This is continuous, as every point in G has a neighbourhood in which g is the sum of a finite number of continuous functions. Let S be the support of g and K any compact subset of G. Then HK S ∩ is the union of a finite number of Wi and is hence compact. 0 = Now let g H g(ξx)dξ > 0. 64 This inequalityR is strict as at each point x, xH intersects some Wi. Obviously f = g/g0 is a continuous function of G with S as its support. Trivially, f 0 = 1 and the proof of the lemma is complete. Proof of the Proposition 2: That the map f f 0 is continuous from 0 → CG EG has already been proved (Lemma 1, Ch. 2.3) and it is easy → 0 to see that this implies that the map ϕ : f f of C C / is also → G → G H continuous. We now exhibit a continuous map ψ : C / C such G H → G that ϕ ψ = Identity. For this one has only to define for every g C , ◦ ∈ GH ψ(g) to be ψ(g)(x) = g(π(x)) f (x) where f is the function constructed in Invariant measures 65 the lemma. This has support HK S where K is a compact subset of G ∩ canonical image in G/H is the support of g.

(ψ(g))0(x) = g(π(ξx)) f (ξx)dξ ZH = g(π(x)) f (ξx)dξ ZH = g(π(x)) by the construction of f . Hence ϕ(ψ(g)) = g ψ is of course continuous. · Every measure ν on G/H gives rise to a measure ν0 on G in the following way ν0( f ) = V( f 0) for every continuous function f on G with compact support. 0 Corollary to Proposition 2: The image of M / under the map ν ν G H → is precisely the set of all measure on G which vanish on the kernel N 0 of the map f f of C C / . → G → G H This is an immediate consequence of the proposition.

Proposition 3. A measure µ on G is zero on N if and only if dµ(ξx) = δ(ξ)dµ(x) for every ξ H. ∈

By the above corollary µ is of the form ν◦ where ν is a measure on 65 G/H. Hence

1 0 0 1 f (ξ− x)dν (x) = f (ξ− x)dν(x) ZG Z G/H

1 = f (ξ− ηx)dηdν(x) Z Z G/H H

= δ(ξ) f (ηx)dηdν(x) Z Z G/H H = δ(ξ) f 0(x)dν(x) = δ(ξ) f (x)dν0(x) Z ZG G/H 66 7. Invariant measures

It follows that dµ(ξx) = δ(ξ)dµ(x). Conversely let dµ(ξx) = δ(ξ)dµ(x). Let f , g be any two continuous functions on G with compact support. Then

µ(g0 f ) = f (x)dµ(x) g(ξx)dξ ZG ZH = f (ξ 1 x)g(x)dµ(ξ 1 x)dξ " − − = f (ξ 1 x)g(x)δ(ξ 1)dµ(x)dξ " − − = f (ξx)g(x)δ(ξ)dµ(x)δ(ξ 1)dξ " − (by Prop. 1, Ch. 3.1)

= µ( f ◦g).

If f is in N , one can choose g such that g0 = 1 on the support of f . Then µ( f ) = µ( f g0) = µ( f 0g) = 0. Hence µ = 0 on N . If there exists an invariant measure ν on G/H, then ν0 must be the Haar measure and conversely if the Haar measure is of the form ν0 then ν is an invariant measure on G/H. Hence δ(ξ) = ∆(ξ) is a necessary and suﬃcient condition for the existence of a right invariant measure on G/H.

3.4 Quasi-invariant measures. 66 Deﬁnition. Let Γ be a transformation group acting on a locally compact space E. We say that a positive measure µ on E is quasi-invariant by Γ if the transform of µ by every γ Γ is equivalent to µ in the sense that ∈ there exists a positive function λ(x, γ) on E Γ which is bounded on × every compact subset and measurable for each γ such that dµ(γ, x) = λ(x, γ)dµ(x).

If under the above conditions λ(x, γ) is independent of x, the measure is said to be relatively invariant. Invariant measures 67

Proposition 4. There always exists a quasi-invariant measure on the homogeneous space G/H. We prove this by making use of Lemma 2. There exists a strictly positive continuous function ρ on G such that ρ(ξx) = δ(ξ)/∆(ξ)ρ(x) for every x G and ξ H. ∈ ∈ Let f be the function on G constructed in Lemma 1, Ch. 3.3. Deﬁne 1 δ(ξ) − ρ(x) = f (ξx)dξ. H ∆(ξ)! R δ(η) Then it is an immediate veriﬁcation to see that ρ(ηx) = ρ(x) ∆(η) and that ρ is positive continuous. Proof of Proposition 4: Let µ be the measure ρ(x)dx on G, where ρ is the function of Lemma 2. Then δ(ξ) dµ(ξx) = ρ(x)∆(ξ)dx ∆(ξ) = δ(ξ)ρ(x)dx = δ(ξ)dµ(x). By proposition 1, Ch. 2.4, there exists a measure ν on G/H such that dµ(x) = dν0(x). Now 67 ρ(xy) dµ(xy) = dµ(x) ρ(x) and hence we get ρ(xy) dν(π(xy)) = dν(π(x)), ρ(x) ρ(xy) depending only on the coset of x modulo H. ν is therefore quasi- ρ(x) invariant. This incidentally gives also the following relation between the Haar measure on G and the quasi-invariant measure on G/H, viz.

f (x)ρ(x)dx = dν(π(x)) f (ξx)dξ. ZG Z Z G/H H 68 7. Invariant measures

If we relax the condition of continuity on ρ, then we can assert that all the quasi invariant measures on G/H can be obtained this way [6]. So ν is relatively invariant if and only if there exists a positive function ρ on G Such that ρ(xy)/ρ(x) = ρ(y)/ρ(e). If we take ρ(e) = 1, we have ρ(xy) = ρ(x) ρ(y) with ρ(ξ) = δ(ξ)/∆(ξ) for every ξ H. In other · ∈ words, the one dimensional representation ξ δ(ξ)/∆(ξ) of H can be → extended globally to a representation of G.

3.5 Some applications. Let G be the group product of two closed subgroups A and B such that the map (a, b) ab of A B G is a homeomorphism. Then the → × → homogeneous space G/A is homeomorphic to B. We deﬁne a function on G by setting ρ(ab) = δ(a)/∆(a). To this function, there corresponds a quasi-invariant measure on G/A such that

f (ab)δ(a)/∆(a)dx = dµ(b) f (ab)da. ZG ZB ZA 68 = = = If x ab′, we have ρ(xb)/ρ(x) ρ(ab′b)/ρ(ab′) 1 by deﬁnition. Hence dµ(b) is right invariant, and dµ(b) = db. Let dr x, dl x denote respectively the right and left Haar measures. = Then G f (ab)δ(a)/∆(a)dr x drb f (ab)dra, or again R R R ∆(a) f (x)dr x = f (ab) dradrb ZG " δ(a) AxB = f (ab)∆(a)d ad b " l r AxB Thus we have got the right Haar measure on G in terms of the Left and right Haar measures on A, B respectively and the modular function on G. This dependence on the modular function can be done away with if we restrict ourselves to unimodular groups.Thus in the case of a unimodular group G, we have the simple formula

f (x)dr x = f (ab)dladrb ZG " AxB Invariant measures 69

Again when A is a normal subgroup, we have δ(a) = ∆(a) or A and hence

f (x)dr x = f (ab)dradrb. ZG " AxB

3.6 Convolution of functions Deﬁnition. A function f is said to be locally summable with respect to a measure µ if for every continuous function ϕ with compact support, ϕ f is µ-integrable.

This has the property that for compact set K, χ(K) f is µ-integrable. If f is locally summable, the map ϕ ϕ f dµ is a continuous → linear form on CG and hence deﬁnes a measureR denoted by µ f . Let now f be locally summable with respect to the Haar measure on G and ν another measure on G. We shall assume that µ ν exists. Then for 69 f ∗ every continuous function g with compact Support, we have

µ ν = g(xy) f (x)dxdν(y) f ∗ (g) " = dν(y) g(xy) f (x)dx Z Z 1 = dν(y) g(x) f (xy− )dx Z Z

Now the map (x, y) (xy 1, y) obviously preserves the product → − measure dxdν(y) on GxG because for continuous functions u with compact support we have

1 u(xy− , y) dxd ν(y) = u(x, y)dxdν(y). " Z Z

1 Hence f (xy− g(x) dxd ν(y) exists and the theorem of Lebesgue- 1 Fubini can! be applied. It therefore results that f (xy− )dν(y) exists for 1 almost every x and g(x) f (xy− )dν(y) is integrable,R In other words, 1 1 f (xy− )dν(y) is locally summable.R If we denote by h(x), f (xy− )dν R R 70 7. Invariant measures

(y), this can be expressed by µ ν =µ . We can now define the con- f ∗ h volution of a measure and a locally summable function f by putting h(x) = f ν(x).One can similarly define a convolution ν µ =µ where ∗ ∗ f k k(x)=ν f (x)= f (y 1 x)∆(y 1)dν(y). This is unsatisfactory in as much ∗ − − as it is necessaryR to choose between left and right before one can identify functions with measures. Thus the notion of convolution of a function and a measure is not very useful in groups are not unimodular. Let now f , g be two locally summable functions on G. Then we can define convolution of f and g such that µ µ = µ by setting f ∗ g f ∗g

1 f g(x) = f (xy− )g(y)dy ∗ Z 1 1 = f (y)g(y− x)∆(y− )dy Z

1 1 But we have in Prop. 1, Ch. 3.1 that ∆(y− )dy = dy− . 70 Thus the convolution of f and g can be satisfactorily defined even if the group G is not unimodular. Note that the convolutions of two measures and of a measure and a function are uniquely defined, whereas the convolution of two functions is defined only upto a constant factor, as it depends on the particular Haar measure we consider.

If f and g are integrable, µ f , µg are bounded and so is µ f ∗g. Conse- quently f g is also integrable. Thus the map f µ is an imbedding ∗ → f of L1 in M 1 as a closed subspace. It is linear and one-one and also preserves metric, for,

f 1 = f (x) dx = dµ f µ f and, || || Z | | Z | | ≤ || ||

on the other hand, µ f , trivially. Actually L1 is a Banach sub- || f || ≤ || ||1 algebra of M 1. In M 1, the Dirac measure at the unit element acts as the unit element of the algebra but L1 does not possess any unit element, unless the group is discrete. Invariant measures 71

3.7 Convolutions of distributions We close this chapter with a brief discussion of convolutions of distributions on a Lie group. A detailed account of distributions may be found in Schwartz’s ‘Théorie des Distributions’ and de Rham’s ‘Variétés différentiables’. Let G be Lie group and the space of indefinitely differentiable DG functions on G which have compact support. Let be the subset of DK consist ring of functions whose supports are contained in the com- DG pact set K. One can provide with the topology of uniform conver- DK gence of each derivative, and with the topology of direct limit of DG those on . The topology on can be characterised by the fact that 71 DK DK f 0 on if for every differential operator D on G with continuous → DK coefficients, D f 0 uniformly on K. It is enough to consider only the → left invariant differential operators, or still the ∆α alone (Ch. 2.4, Part I). This topology makes of a Fréchet space (i.e. a locally convex DK topological vector space which is metrisable and complete).

Definition. A distribution on G is a continuous linear form on . DG As in the case of measures, one can define the notion of the support of a distribution, distributions with compact support, etc. Let T, S be two distributions on G. If one of them has compact support we define the convolution product as for measures:T S (ϕ) = ϕ(xy)dT(x)dS (y). ∗ Let ξ′ be the space of distributions with compact support.! It is an algebra with convolution as product and the space of distributions is a module over ξ . We denote by ξ the space of distributions with support = e . G′ e′ { } Let (x1,... xn) be a coordinate system at e. Then T ξe′ implies the ∂αϕ ∈ existence of λα C such that T(ϕ) = λα (e), λα being zero except ∈ ∂xα for a finite number of terms. If f isP a locally summable function, we can identify f with the distribution f (x)dx and we can define the notion of convolution f T of f and distribution T under some assumptions ∗ on f and T. But this product even when it is defined, is not in general a distribution of the form g(x)dx; however, if f is indefinitely differentiable with compact support (or if f is indefinitely differentiable and T has compact support) f T is a distribution of the form g(x)dx where ∗ 72 7. Invariant measures

g(x) is an indefinitely differentiable function. This function is, in virtue 1 of the above identification, given by g(x) = f T(x) = f (xy− )dT(y). α α ∗ ∂ ∂ R 72 = = 1 = If T λα α (e), f T(x) λα α f (xy− ) y e. α ∂x ∗ α ∂y The mapP f f T is a differentialP operator which is left invariant. → ∗ We have already seen that ǫ µ = σ (µ) (Ch. 2.5). Hence y ∗ y (σ f ) T = ǫ f T y ∗ y ∗ ∗ = ǫ ( f T) y ∗ ∗ = σ ( f T) y ∗ In other words, T commutes with the left translation. At x = e we α ∂ 1 have f T(e) = αλα f (y ) i.e. every left invariant differential ∗ ∂yα − y=e operator is obtainedP in the above manner. This leads us to the

Proposition 5. The algebra (G) is canonically isomorphic to the alge- U bra of distributions with support e with convolution as multiplication. { } Chapter 8

Regular Representations

4.1 General notions 73 Let G be a locally compact group and E a locally convex topological vector space. Deﬁnition. A continuous representation of G in E is a map x U of G → x into Hom(E, E) such that this is a representation in the algebraic sense (i.e. U = U U and U = identity) and such that the map (a, x) U a xy x y e → x of E G E is continuous. × → The latter of these conditions, which we denote by R, is equivalent to the following: R : For every compact subset K of G, the set U : x K is 1′ { x ∈ } equicontinuous, and R : for every a E, the map x U a of G E 2′ ∈ → x → is continuous. In fact, R R trivially. Let V be a neighbourhood of 0 in E. For ⇒ 2′ every x K, there exists an neighbourhood A of x in G and W of 0 ∈ x x in E such that for every y A and a W , U a V. Since K is ∈ x ∈ x y ∈ compact, we can choose Ax1 ,... Axn which cover K and let W = Wx j . Now, x K, b W Uxb V. Hence the set Ux is equicontinuous.T ∈ ∈ ⇒ ∈ { } We proceed to prove the converse; in fact, we show more generally that

R1′ with the following condition R2′′: There exists a dense subset F of E such that for every a F, the map x U a of G E is continuous, ∈ → x → implies R. It is required to show that the map (x, a) U a is continuous → x 73 74 8. Regular Representations

at any point (x, a). Let V be any convert neighbourhood of 0 in E. We seek a neighbourhood W of 0 in E and a neighbourhood A of x in G such that b a + W, y A U b U a + V. Let K be a compact ∈ ∈ ⇒ y ∈ x neighbourhood of x in G. Then there exists a neighbourhood V1 of 0 such that y K, b V U b V/4 (by R ). Let b (a + V ) F. ∈ ∈ 1 ⇒ y ∈ 1′ ∈ 1 74 Then we can ﬁnd a neighbourhood A of x in C contained in K suchT that U b U b V/4 for every y A (by R ). Now, if c a + V , y A, we y − x ∈ ∈ 2′′ ∈ 1 ∈ have

U c U a = U (c a) + (U U )b + (U U )(a b) V. y − x y − y − x y − x − ∈ This completes the proof of the equivalence. Moreover, if E is a barrelled space (or in particular a Banach space) then axiom R itself R. For, the map x U is continuous from 2′ ⇒ → x G to Hom(E, E) with the topology of simple convergence and hence the image of a compact subset is again compact and, E being barrelled, equicontinuous.

4.2 Examples of representations

(i) Unitary representations. Let U be a representation of G in a Hilbert space H such that U is a unitary operator, i.e. U 1 = U for x x − x∗ every x G. Then V is called a unitary representation. ∈ (ii) Bounded representations. A representation U of G in a Banach space B is said to be bounded if there exists M such that U < M || x|| for every x G. It should be remarked here that in general rep- ∈ resentations in Banach spaces are not bounded as for instance the representation x ex. Id of R in itself. However, such represen- → tations are bounded on every compact subset.

(iii) Regular representations. Left and right translations in G, as we have seen before, give rise to representations of G in the space p CG, L etc. In fact they give rise to representations of G in any function space connected with G with a reasonably good deﬁni- tion and a convenient topology. Regular Representations 75

The space CG with the usual topology can easily be seen to be barrelled. Therefore, in order to verify that τ is a continuous representation, we have only to prove that y τ f is continuous. 75 → y It is again sufficient to establish continuity at the point y = e. The function f is uniformly continuous and hence when y e, → τ f f uniformly having its support contained in a fixed com- y → pact set. In the case of Lp(1 p ) with respect to the right Haar ≤ ≤ ∞ measure, τ = 1 by the invariance of the measure and hence || y|| the τ form an equicontinuous set for y G. Also the map y ∈ y τy f is continuous for the topology on CG which is finer → p p than that of L and since CG is dense in L is a representation of G in Lp. On the other hand, if we consider σ , we have σ f = y || y || ( f (y 1 x) pdx)1/p = (∆(y))1/p f and the continuity of y | − | || ||p → σRy f follows as a consequence of the continuity of ∆(y). Note that the proof is not valid when p is infinite as CG is not dense in L . ∞ In fact the map y τy f of G L is continuous if and only if → → ∞ f is uniformly continuous on G.

(iv) Induced representations. Let H be a closed subgroup of a topological group G, and L a continuous representation of H in a locally convex space E. Let C L be the space of functions on G with values in E which are continuous with compact support modulo H (i.e. their supports are contained in the saturation of a compact set), and which satisfy the following equality: 1 δ(ξ) 2 f (ξx) = Lξ f (x) for every x G and ξ H. The factor ∆(ξ)! ∈ ∈ 1 δ(ξ) 2 , it will be noted, occurs purely for technical reasons and ∆(ξ)! can without much trouble be done away with. The above equality, in essence, expresses only the condition of covariance of f with respect to left translations by ξ. On this space C L we can, as we have more than once done before, introduce the topology of L direct limit of those on CK , the latter being the space of functions 76 in C L whose supports are contained in HK, with the topology of 76 8. Regular Representations

uniform convergence on K. This is again a locally convex space and right translations by elements of G give rise to a (regular) representation of G in C L. This is called the representation of G induced by L.

(v) Let us again assume L to be a unitary representation of H in a Hilbert space E. Let C L be the space of continuous functions on G with values in E having compact support modulo H such that 1 δ(ξ) 2 f (ξx) = Lξ f (x). Naturally one tries to introduce a scalar ∆(ξ)! product in C L in the usual way, but the possibility of f (x) not being square integrable (which it is not in general) foils the attempt. However, though f (x) may not be square integrable, we are in a position to assert that f (x) 2(ρ(x)) 1dx < (ρ of course G/H || || − ∞ is the function deﬁnedR in Lemma 2, Ch. 3.4 and dx the quasiin- variant measure on G/H). In fact, the function f (x) 2(ρ(x)) 1 is || || − invariant modulo H and consequently can be considered as a continuous function on G/H with compact support. Hence we can de- 2 2 1 L ﬁne f = f (x) ((ρ(x)))− dx. Let H be the completion || ||L G/H || || of C L underR this norm. As usual, H L is the space of measurable functions f which satisfy the condition of covariance and are such that f (x) 2(ρ(x)) 1dx < . This is a Hilbert space in which G/H || || − ∞ the scalarR product is given by f, g = f (x), g(x) (ρ(x)) 1dx. h i h i − GR/H The right regular representation of G in H L is unitary. For,

2 2 1 τy f = f (xy) ((ρ(x)))− dx˙ || || ZG/H || || 1 ρ(xy 1) = f (x) 2 (ρ(xy 1)) − − dx by quasi invariance − ρ(x) ZG/H || || 2 1 = f (xy) ((ρ(x)))− dx˙ ZG/H || || = f 2 for every f H L || ||L ∈

77 The same proof as in (iv) gives the continuity of the representation. Regular Representations 77

Thus one can define induced representations in many ways, in each case the representation space being so chosen as to reflect the particular properties of the representation one wishes to study. We shall not dwell on induced representations any longer, but only give the following general definition which, it is needles to say, includes the last two cases.

Deﬁnition. A representation U of G in F is said to be induced by representation L of H in E if there exists a linear continuous map η of C L into F such that

(i) η is injective with its image in F everywhere dense, and

(ii) η commutes with the representation in the sense that U η = η τ x◦ ◦ x for every x G. ∈ 4.3 Contragradient representation Let U be a continuous representation of G in a locally convex space E. For every x G consider t Hom(E , E ) which is continuous for any ∈ Ux ∈ ′ ′ ‘good’ topology on E′ (weak, strong of convex-compact). We denote by Uˇ the map x tU 1. Regarding this map we have the following → x− Proposition 1. If U is a continuous representation of G in a quasi complete locally convex space E, then Uˇ is also a continuous representation of G in EC′ (convex compact topology). We need here the following for- mulation of Ascoli’s theorem. Let X be a locally compact topological space, F a uniform Hausdorﬀ space and C (X, F) the space of continuous functions from X F. Let Λ be an equicontinuous subset of → C (X, F) such that the set λ(x) : λ Λ is relatively compact in F { ∈ } for every x X. Then (i) Λ is relatively compact in C (X, F) with the ∈ topology of compact convergence, and (ii) on Λ the topology of com- 78 pact convergence coincides with every Hausdorﬀ weaker topology (in particular, with the topology of simple convergence).

Proof of proposition 1. Let K be a compact of G. The set U : x K { x ∈ } is equicontinuous and for every a E, U a : x K is compact as the ∈ { x ∈ } map x U a is continuous. Hence by Ascoli’s theorem, the topology → x 78 8. Regular Representations

of simple convergence and the topology of compact convergence are one and the same on this subset of C (E, E) i.e. x e U a a → ⇒ x → uniformly for a in a compact subset H of E. Let a be an element of E . We wish to prove that x Uˇ a is ′ C′ → x ′ continuous. It is enough to prove the continuity at the unit element e. Let x e. Then U 1 a a uniformly on a compact set H and hence → x− → U 1 a, a′ = a, Uˇ xa′ a, a′ uniformly on H. This shows that h x− i h i → h i x Uˇ a is continuous for the convex-compact topology on E . We → x ′ ′ have to show moreover that the set Uˇ : x K is equicontinuous. If { x ∈ } H be a convex compact subset of E, we seek to prove the existence of another convex compact set H such that a (H )0 Uˇ a H0 ′ ′ ∈ ′ ⇒ x ′ ∈ for every x K where A0 denotes the polar of A. But Uˇ a H0 for ∈ x ′ ∈ every x K if and only if U 1 a, a′ 1 for every x K and a H. ∈ h x− i ≤ ∈ ∈ Let H be the closed convex envelope of the compact set description by ′ 1 0 U 1 a. It is obvious that a (H′) b, a′ 1 for every b H′ x− ∈ ⇒ h i ≤ ∈ ⇒ U 1 a, a′ 1 for every x K and a H. It only remains to show h x− i ≤ ∈ ∈ that H′ is compact. But H′ is precompact, and being a closed bounded

set, also complete. This shows that H′ is compact.

This representation in E′ is called the contragradient of U.

Remark. It will be noted that we have used the quasi completeness of 79 the space E only to prove that the closed convex envelope of a compact set is also compact. Hence the proposition is valid for the more general class of locally convex spaces which satisfy the above condition.

Example. We have seen that the right and left translations give represen- 0 tations of G in the function spaces CG, E , C¯G, etc. By our proposition above,we see that σ is continuous in M , M C, M 1 (with the convex compact topology).

Remark. The regular representation of G in M 1 is not continuous with respect to the strong topology. For, τ ǫ = ǫ and ǫ ǫ = 2 if x , e. x e x || x − e|| Hence as x e, ǫ does not tend to ǫ . → x e Regular Representations 79

4.4 Extension of a representation to M C Let U be a continuous representation of G in a locally convex quasi complete space E. Let µ be any measure on G with compact support. Then we write Uµa = U adµ(x). (The function x U a is a vector- G x → x valued function and RUxadµ(x) has been deﬁned in Ch. 1.5). R Theorem 1. (1) Uµ is a linear continuous function of E in itself.

C (2) µ Uµ is an algebraic representation of M in E. → (3) If U is a bounded representation in a Banach space, then this representation can be extended to a continuous representation of the Banach algebra M 1 in to the Banach algebra Hom(E, E).

(1) In fact, as a 0, U a 0 uniformly on the compact support of → x → µ and hence Uµ is continuous. Its linearity is trivial.

(2) Again the linearity of the map µ Uµ is obvious. →

Uµ νa = Uxad(µ ν)(x) ∗ Z ∗ = U adµ(x)dν(y) (see Remark 2, Ch. 2.2) " xy

= dµ(x)Ux(Uνa)(Remark, Ch. 1.5) Z = UµUνa.

(3) If µ M C, we have ∈

Uµa = Uxadµ(x) || || || Z ||

Uxa d µ ≤ Z || || | | k a µ ≤ || || || || 80 8. Regular Representations

C This proves that the map (a,µ) Uµa of E M E is con- → × → tinuous. Since M 1 is only the completion of M C with the topology of the norm, this map can be extended to a continuous map E M 1 E, × → which proves all that was asserted.

4.5 Convolution of measures We can get, in particular,representations of the space M of measures on a group G by considering regular representations of G. We deﬁne σµν = σx(ν)dµ(x). M is the dual of the barrelled space CG and is hence quasiR complete. We therefore have

f,σµ(ν) = f,σx(ν) dµ(x) h i Z h i 1 = dµ(x) f (y)dν(x− y) Z Z = f (xy)dµ(x)dν(y) " = f,µ ν h ∗ i

In other words, σµ(ν) = µ ν and τµ(ν) = ν µˇ where dµˇ(x) = 1 ∗ ∗ dµ(x− ). We have not imposed any conditions on ν, and µ has been assumed to have compact support. Thus convolution of two measures could have been deﬁned as σµ(ν) = σx(ν)dµ(x) straightaway. R 4.6 81 Proposition 2. Let E be a subspace of M with a ﬁner topology such that

(a) E is invariant by τ;

(b) τ restricted to E is continuous;

C Then for every µ M and a E, we have a µ E and a µ = τµa. ∈ ∈ ∗ ∈ ∗ ˇ If moreover τ is bonded on E, then this true for µ M 1. ∈ Regular Representations 81

The proposition is immediate in view of our remarks in Ch. 4.5. In particular, if a LP(p < ), µ M 1, then a µ Lp, and ∈ ∞ ∈ ∗ ∈ a µ a µ . Again, we may take an integrable function f instead || ∗ ||p ≤ || ||p|| || of µ and get a f Lp and a f a f . Otherwise stated, L1 is ∗ ∈ || ∗ ||p ≤ || ||p|| ||1 represented as an algebra of operators in the Banach space Lp(p < ). ∞ Another case is that of E = CG. If f is a function and µ a measure both with compact supports, then f µ C . ∗ ∈ G 4.7 Process of regularisation. Let V be any neighbourhood of e in G. Let A be the set f C : f 0 V ∈ v ≥ and f (x)dx = 1 . As V describes the neighbourhood filter at e, AV alsoR describes a filter Φ in the function space CG. Proposition 3. If U is a continuous representation of G in a quasi complete space E, then U a a following Φ for every a E. f → ∈ In fact U f a = Uxa f (x)dx. If W is a closed convex neighbourhood of 0 in E, then byR the continuity of U f a one can find a neighbourhood V of e such that U a a + W for every x V. Now U a a = x ∈ ∈ f − (U a a) f (x)dx W whenever f A by convexity of W. x − ∈ ∈ V R Remark. U f a has certain properties of continuity stronger than that of 82 a. For instance if we take for U the regular representation of G in M , τ µ E 0. When Φ e, τ µ µ. This is a process of approxi- f ∈ → f → mation of a measure, as it were by continuous functions. If G satisfies the first axiom of countability, we can find a sequence f of continu- { n} ous functions generating the filter Φ. In particular, if G is a Lie group, we have thus an approximation of measures by sequences of continuous functions. Finally we remark in passing that the same procedure can be adopted in the case of Lie groups for distributions instead of measures. Thus a distribution on a Lie group can be approximated by a sequence of indefinitely differentiable functions.

Chapter 9

General theory of representations

5.1 Equivalence of representations 83 Deﬁnition. A representation U of a topological group G in locally convex space E is said to be equivalent to another representation U′ in E′ if there exists an isomorphism T of E onto E′ such that TUx = Ux′ T for every x G. ∈ This is evidently a very strong requirement which fails to characterise as equivalent certain representations which are equivalent in the intuitive sense. However, we are interested in the case of unitary representations in Hilbert spaces and the deﬁnition is good enough for our purposes.

Deﬁnition . Two representations U in H, U′ in H′ are unitarily equivalent if there exists a unitary isomorphism T : H H such that → ′ TU = U T for every x G. x x′ ∈ Proposition 1. Two equivalent unitary representations are unitarily equivalent. In fact, TT ∗Ux′ = TUxT ∗ = Ux′ TT ∗ i.e. Ux commutes with the positive Hermitian operator TT ∗ and hence also with H = √TT ∗. 1 It can be easily seen that H− T is a unitary operator which transforms U into U′.

83 84 9. General theory of representations

5.2 Irreducibility of representations

Deﬁnition 1 (algebraic irreducibility). A representations U of a group G in a vector space E is said to be algebraically irreducible if there exists no proper invariant subspace of E.

Deﬁnition 2 (topological irreducibility). A representation U of a topological group G in a locally convex space E is said to be topologically irreducible if there exists no proper closed invariant subspace.

84 Deﬁnition 3 (complete irreducibility). A representation U of a topological group G in a locally convex space E is said to be completely irreducible if any operator in Hom(E, E) (with the topology of simple convergence) can be approximated by ﬁnite linear combinations of the Ux.

It is at once obvious that (i) (ii) and that (iii) (ii). It can be ⇒ ⇒ proved that when E is a Banach space, (i) (iii) (Proof can be found in ⇒ Annals of Mathematics, 1954, Godement). For unitary representations, (ii) and (iii) are equivalent (due to von Neumann’s density theorem, Th. 2. Ch. 5.6). Finally, all the three deﬁnitions are equivalent for ﬁnite dimensional representations ((ii) (iii) due to Burnside’s theorem, Th. ⇒ 1, Ch. 5.5).

5.3 Direct sum of representations

Deﬁnition. A representation U of G in E is said to be the direct sum of representations Ui of G in Ei if Ei are invariant closed subspaces of E such that the sum Ei is direct and is everywhere dense in E, and if Ui is the restriction ofP U to Ei. Moreover, if U is a unitary representation in Hilbert space, U is said to be the Hilbertian direct sum of the Ui if Ei is orthogonal to E j whenever i , j.

Deﬁnition . A representation is completely reducible if it can be expressed as a direct sum of irreducible representations. General theory of representations 85

5.4 Schur’s lemma*

We give here two formulations (Prop. 2 and 3) of Schur’s lemma, the ﬁrst being trivial and the second more suited to our purposes.

Proposition 2. Let U and U′ be two algebraically irreducible representations in E, E respectively. If T is a linear map: E E such ′ → ′ that TU = U T for every x G, then either T = 0 or an algebraic 85 x x′ ∈ isomorphism.

From this, we immediately deduce the following

Corollary . Let U be an algebraically irreducible ﬁnite - dimensional representation of a group G in E. The only endomorphisms of E which commute with all the Ux are scalar multiples of the identity.

In fact, if λ is an eigenvalue of T, T λI is not an isomorphism and − is, by Schur’s lemma, = 0.

Proposition 3. Let U, U′ be two unitary topologically irreducible representations in H, H′ respectively. If T is a continuous linear map H H such that TU = U T for every x G, then either T = 0 → ′ x x′ ∈ or an isomorphism of Hilbert spaces.

In fact, T ∗ is a continuous operator with UT ∗ = T ∗U′. H = T ∗T is a Hermitian operator commuting with every Ux. Hence Ux commutes with every Eλ in the spectral decomposition H = λdEλ and consequently leaves every spectral subspace invariant. TherefoR re, the spectral sub-spaces reduce to 0 or E. i.e. H is a scalar = λI. Similarly { } H′ = TT ∗ = λI. Hence T is either 0 or an isometry up to a constant. The proof of Prop. 3 implicitly contains the following

Corollary. Let U be a unitary topologically irreducible representation of a group G in a Hilbert space E. The only operators of E which commute with all the Ux are scalar multiples of the identity. This is immediate since any operator can be expressed as a sum of Hermitian operators for which the corollary has been proved in prop. 3. 86 9. General theory of representations

5.5 Burnside’s theorem

Theorem 1 (Burnside). Let U be an algebraically irreducible represen- 86 tation of G into E of ﬁnite dimension. Then every operator in E is a linear combination of the Ux.

Consider the algebra of finite linear combinations of U . Let B A x be the subset of Hom(E, E) consisting of elements B such that Tr(AB) = 0 for every A . Obviously it is enough to show that B = 0 . Now ∈ A { } we define a representation V of G in B by defining V (B) = U B x x ◦ for every B B. This is not in general an irreducible representation. ∈ However, if B , 0 , we can find a non-zero irreducible subspace C { } of B. Now the map λ : B B of C into E is a linear map which a → a transforms the representation V into U. For,

λ V (B) = λ U B = U Ba = U λB for every B C . a ◦ x a ◦ x ◦ x ◦ x ◦ ∈

Hence by Schur’s lemma (prop. 2, Ch. 5.4), λa = 0 or is an isomorphism. If λ = 0 for every a E, B = 0 for every B C and hence a ∈ ∈ C = 0 . But this is contradictory to our assumption that C is non-zero. { } Therefore, there exists a E such that λ , 0. So, λ is an isomor- 1 ∈ a1 a1 phism of C onto E. Let (a , a ...) be a basis of E. Then λ 1 λ is an 1 2 a−1 ◦ a2 operator on C . This obviously commutes with every Vx. Hence by cor. to prop. 2, Ch. 5.4 λ 1 λ = µ I where µ is a scalar. We shall thus a−1 ◦ a2 2 2 write λa j = µ jλa1 with µ1 = 1. Now, one can introduce a scalar product in E such that Tr(UxB) = UxBa j, a j = 0 for every x E. But j h i ∈ P

U Ba , a = µ U Ba , a h x j ji jh x 1 ji Xj Xj = U Ba , µ a = 0 h x 1 j ji Xj

Since the UxBa1 generate E, µ ja j = 0 or again µ j = 0 for every j 87 j. But µ = 1. This gives us a contradictionP and it follows that B = 0 . 1 { } General theory of representations 87

5.6 Density theorem of von Neumann Let U be a unitary representation of a topological group G in a Hilbert space H. We denote by .the subset of the set Hom(H, H) of operators A on H consisting of finite linear combinations of the Ux. This is a self adjoint subalgebra of Hom(H, H), but is not in general closed in it. Let be the set of operators which commute with every element of . A′ A Obviously is also self-adjoint. A′ Proposition 4. is weakly closed in Hom(H, H). A′ In fact, if A = lim A (in the weak topology) with A , then i i ∈ A′ BAx, y = lim BA x, y = lim A Bx, y h i h i i h i i = ABx, y for every B . h i ∈ A Hence A . ∈ A′ Let be the commutator of the algebra . Then is a weakly A′′ A′ A′′ closed self-adjoint subalgebra containing and hence it contains the A weak closure of . We can in fact assert A Theorem 2 (von Neumann). = weak closure of . A′′ A We actually prove a stronger assertion, viz. Let (xn) be a sequence 2 of elements in E such that xn < and T an operator in ′′. Then || || ∞ 2 A for every ǫ > 0, there existsPA such that T xn Axn < ǫ. (This ∈ A || − || in particular implies that ′′ = strong closureP of or even the strongest A A closure of , in the sense of von Neumann). A We first show that for every x E, T x is in the closure of Ax : A ∈ { ∈ . In fact, F = Ax is a closed invariant subspace of E and let F be 88 A} { } ⊥ its orthogonal complement. F⊥ is also invariant under the self adjoint algebra and hence the orthogonal projection P of E onto F commutes A with every element of . P therefore belongs to and T leaves F A A′ invariant. Since x F, T x is also in F. ∈ Now consider the space E = E E (Hilbertian sum). Every 1 ⊕ ⊕······ element x E is of the form (x ,... x ) with x 2 < . Let A˜ ∈ 1 1 n ··· || n|| ∞ be the operator on E1 defined by Ax = (Ax1,...PAxn,...). The map A A˜ is an isomorphism of Hom(H, H) into Hom(H , H ). Denote the → 1 1 88 9. General theory of representations

image of by ˜. If B is any operator on E , it can be expressed in the A A 1 form B(x1,... xn ...) = (y1,... yn ...) where yn = bn,p xp and bn,p is p P an operator on E. We now show that B ( ˜) if and only if b , ∈ A ′ n p ∈ A′ for every n and p. For, BA˜(x1,... xn,...) = AB˜ (x1,... xn,...) for every x E implies that b , A = Ab , by taking all x , n , p to be zero. ∈ 1 n p n p n Conversely, if this is satisﬁed, AB˜ (x1,..xn..)

= A˜(..., bn,p xp,...) = (..., bn,pAxp,...) Xp Xp

= B(Ax1,... Axn,...) = BA˜(x1,... xn,...)

Again, B (A˜) if and only if the diagonal elements of the inﬁnite ∈ ′′ matrix b , are equal and in , and the rest of the elements are zero. n p A′′ In fact, if C (A˜) , we have (BC) , = (CB) , for every B , ∈ ′′ m n m n ∈ A′ or bm,pcp,n = cm,qbq,n for every bi, j ′. Putting bi, j = δn,iδn, j, p q ∈ A P P 89 we get ci, j = 0 if i , j and cm,m = cn,n for every m and n. So we have (A˜) = (A ). Therefore there exists A such that T x Ax 2 < ǫ. ′′ ′′ ∈ A || n− n|| Moreover, in theorem 2, if T is Hermitian we can ﬁnd a Hermitian f P operator A such that T x Ax 2 < ǫ. As before it is enough to prove || n − n|| this for one vector xP. In other words, we have to show the existence of Hermitian operator A such that T x Ax < ǫ. We know that T is the || − || strong limit of A . Hence T = T ∗ is the weak (and not strong, in ∈ A A + A A + A general) limit of A or again the weak limit of ∗ . Now ∗ is ∗ 2 2 a Hermitian operator in . Hence T is weakly adherent to this convex A set. In this case, weak adherence is the same as the weak adherence in the sense of topological vector spaces, but weak and strong topologies are the same in a convex space. In particular, if we have a unitary topologically irreducible representation, by Schur’s lemma (Prop. 3, Ch. 5.4) = λI and hence A′ { } = Hom(E, E). Therefore every operator is strongly adherent to . A′′ A This is the analogue of Burnside’s theorem (Th. 1, Ch. 5.5) for unitary representations. Part III

Continuous sum of Hilbert Spaces

Chapter 10

Continuous sum of Hilbert Spaces-I

1.1 Introduction 90 In the general theory of unitary representations of a locally compact group, two main problems are (i) to determine all the irreducible unitary representations of a group, and (ii) to decompose a given unitary representation into irreducible ones. The ﬁrst of these has been completely solved in certain cases (e.g. abelian groups, compact, certain semisimple Lie groups), but it is to the latter that we address ourselves in the following pages. We start by giving some Examples. Let U be the regular representation of the one dimensional torus T 1 in the space L2 of square summable functions. If f belongs to 2 inz it L , it can be expressed in Fourier series ane . If x = e , then inz inz σx f = σxane = (ane )e and we haveP decomposed a unitary n n R representationP into a directP sum of irreducible representations. If we take R instead of T 1 and F L2, then fˆ(y) = f (x)eixydx also ∈ belongs to L2. By the inversion formula, when f is suRﬃciently regular 1 (we do not enter into these details) we have f (x) = fˆ(y)e ixydy. 2π − Hence R

1 i(x+z)y σz f = fˆ(y)e− dy 2π Z

91 92 10. Continuous sum of Hilbert Spaces-I

1 izy ixy = fˆ(y)e e− dy 2π Z 1 ixy = (σˆf )(y)e− dy. 2π Z

izy Therefore (σˆz f )(y) = e fˆ(y). The regular representation has again been decomposed into one-dimensional representations but this is not a discrete sum but a ‘continuous sum’ - a concept which we shall deﬁne presently. Before proceeding with the formal deﬁnitions, we give one more example which is more akin to the theory we are to develop. Let E 91 be a locally compact space with a positive measure µ and H a Hilbert space. In the space C (E, H ) of continuous functions f : E H with → compact support, we introduce a semi-norm

1 2 f = f (z) 2dµ(z) < . || || Z || || ! ∞

Let L 2(H ) be the completion of the Hausdorﬀ space associated with C (E, H ). We have also a scalar product in this space given by f, g = f (z), g(z) dµ(z). h i h i If µ Ris discrete (i.e. is a linear combination of Dirac measures at certain points), L 2(H ) becomes a discrete sum of Hilbert spaces associated to each of those points, all the Hilbert spaces being isomorphic to H . These considerations motivate some kind of a continuous sum of Hilbert spaces indexed by points of a locally compact space. We therefore assume the following data to start with:

(1) , a locally compact space (which will be assumed for simplicity Z to be countable at ) with a positive measure µ; ∞ (2) For every z , a Hilbert space H (z). In other words, we ∈ Z assume given at each point a ‘tangent space’ which is a Hilbert space. For instance, in a Riemannian manifold, the metric assigns a scalar product to the tangent space at each point of . Of course Z in this case the spaces are of ﬁnite dimension. Having in mind the Continuous sum of Hilbert Spaces-I 93

example above, we seek to find an analogue of the concept of functions on . This is served by the notion of a vector field (in Z exactly the same sense as in manifolds) which is an assignment to each point, of an element of the associated Hilbert space. We would like to have a notion analogous to that of continuous functions in our example. To this end, we introduce a fundamental family of vector fields with reference to which the continuity of 92 an arbitrary vector field will defined. Thus we suppose given

(3) A fundamental family of vector ﬁelds which satisﬁes the follow- ∧ ing conditions:

(a) forms a vector space under the ‘usual’ operations. ∧ (b) For every vector ﬁeld X , the real valued function X(z) ∈ ∧ || || is continuous. This in particular implies that the map z → X(z), Y(z) is continuous for every X, Y . h i ∈ ∧ (c) For every z, the vectors X(z) for X are everywhere dense ∈ ∧ in H (z). This only ensures that the system is suﬃciently ∧ large. Sometimes the fundamental family will be sup- ∧ posed to satisfy the following stronger condition: (c ) There exists a countable subset Λ = X of Λ such that ′ 0 { n} for every z , X (z) are dense in H (z). In particular, ∈ Z n this implies that all the Hilbert spaces H (z) are separable. (We will always assume that the stronger condition (c′) is valid though some of the results remain true without this supposition).

1.2 Notion of continuity We proceed to construct the continuous sum of the spaces H (z). In our axioms relating to the fundamental family, we have not imposed any restrictions on its behaviour at . Consequently it cannot be asserted ∞ that the X(z) are square summable. Moreover, the class Λ may be too || || small (as they will be if we take them to be constants in our example) to be dense in L 2(H ). This necessitates the extension of this family to the class of continuous vector ﬁelds by means of the 94 10. Continuous sum of Hilbert Spaces-I

93 Definition. A vector field X is continuous at a point ζ0 if given an ǫ > 0 there exists Y Λ and a neighbourhood V of ζ such that X(ζ) Y(ζ) < ∈ 0 || − || ǫ for every ζ V. ∈ Remark . When we take Λ to be constants in the example, this corresponds to the usual continuity of functions. Proposition 1. A vector field X is continuous if an only if X is contin- || || uous and X, X is continuous for every X Λ . h ni n ∈ 0 If X is continuous, trivially X is continuous. Also if X and X are || || ′ continuous, then X, X is continuous: in fact, for every ǫ > 0 and every h ′i ζ , there exist a neighbourhood V of ζ and Y, Y Λ such that ∈ Z ′ ∈ X Y < ǫ, X Y < ǫ in V. || − || || ′ − ′|| Hence

X, X′ Y, Y′ X Y, X′ + Y, Y′ X′ |h i − h i| ≤ |h − i| |h − i| Mǫ in V, where M is some constant. As Y, Y is continuous, it ≤ h ′i follows that X, X is also continuous. To prove the converse, it is h ′i enough to show that X(ζ) X (ζ) is continuous. But X(ζ) X (ζ) 2 = || − n || || − n || X 2 2Rl X, X + X 2, all continuous by our assumption. || || − h ni || n|| A continuous vector field can be multiplied by a scalar continuous function without affecting its continuity. Proposition 2. The vector fields ϕ (ζ)Y (ζ) with ϕ C ( ) and Y i i i ∈ Z i ∈ Λ are dense in the space of continuousP vector fields with the topology of uniform convergence on compact sets.

At each point x in a compact set K, there exists a neighbourhood Ax in which X Y < ǫ for some Y Λ. We can extract a ﬁnite cover || − x|| x ∈ 94 A from A and take the partition of unity with respect to this cover. { xi } { x} Hence there exist continuous functions φ such that X φ Y < ǫ i || − i xi || on k. P

1.3 The space L2 ∧ Our next step is to construct the space L2 of square summable vector ﬁelds. We shall say that X belongs to L2 if∧ given an ǫ > 0 there exists a ∧ Continuous sum of Hilbert Spaces-I 95 continuous vector ﬁeld Y with compact support such that

∗ X(ζ) Y(ζ) 2dµ(ζ) < ǫ. Z || − || (We do not know a priori whether X(ζ) Y(ζ) 2 is measurable or not || − || and hence we can only consider the upper integral). In this space, we can deﬁne

X 2 = X(ζ) 2dµ(ζ) and || || Z || ||

X, X′ = X(ζ), X′(ζ) dµ(ζ). h i Z h i By passing to the quotient space modulo vector fields of norm 0, we get a Hilbert space (which again we denote by L2 ). As in the case of the theory of integration, we have of course to prove∧ the completeness of L2 , but there is no trouble in imitating the proof of the Riesz-Fisher theorem∧ in this case. L2 is the continuous sum of the H (ζ) that we wished to construct. ∧ 1.4 Measurablility of vector fields Definition. A vector field X is said to be measurable if for every compact K and positive ǫ, there exists a set K K such that µ(k k ) < ǫ and 1 ⊂ − 1 X is continuous on K1. Proposition 3. A vector field X is measurable if and only if X, X is h ni measurable for every X Λ . n ∈ 0 If X is measurable, X, X is continuous on K and hence X, X 95 h ni 1 h ni is measurable. Conversely, let X, Xn be measurable. Then X = X, Xn h i X, Xn || || sup | h i | is also measurable. (Here we put h i = 0 if Xn(ζ) = Xn Xn || || 0). But|| X,||X are continuous outside a set K K|| ||of measure < ǫ/2n. h ni { − n} If K = Kn, it is obvious that µ(K K ) < ǫ and all the functions ∞ ∩ − ∞ X, Xn are continuous on K . Hence X is continuous on K . By h i ∞ || || ∞ prop. 1, Ch. 1.2, prop. 3 follows. In particular, this shows that strong measurablility and weak measurablility are the same in a separable Hilbert space. 96 10. Continuous sum of Hilbert Spaces-I

Proposition 4. A vector ﬁeld X belongs to L2 if and only if X is measurable and X(ζ) 2dµ(ζ) < . ∧ || || ∞ R The proof is exactly similar to that for ordinary integration of scalar functions.

1.5 Orthogonal basis We now find an orthogonal basis for the space of vector fields by Schmi- dt’s orthogonalisation process. We can of course define

X X X , e e e = 1 , e = 2 − h 2 1i 1 ,... 1 X 2 X X , e e || 1|| || 2 − h 2 1i 1|| where we put e1 = 0 whenever the numerator is zero [20]. However, the process is defective as the basic elements are not continuous, and the following orthogonalisation seems preferable: Put e = X ; e = X X , e e , 1 1 2 2 − h 2 1i 1 en = xn - orthogonal projection of Xn on the space generated by

e1,... en 1 − 96 These are of course continuous vector ﬁelds. At each point ζ, the nonzero e(ζ) from an orthogonal basic for H (ζ).

1.6 Operator fields Let X be a square summable vector field and A(ζ) an operator on H (ζ). Then we may define (AX)(ζ) = A(ζ)X(ζ) and get another vector field AX. The assignment to each ζ of an operator of H (ζ) is called an operator field. However, in order that A may act as operator on L2 , we have to make sure that AX is also square summable. Obviously∧ some restrictions an A(ζ) will be necessary to achieve this. In the particular case when all the spaces H (ζ) are the isomorphic, A(ζ) is a map of Z into the set of operators of the Hilbert space H . Now, Hom (H , H ) can be provided with uniform, strong or weak topologies and we may Continuous sum of Hilbert Spaces-I 97 restrict A to be continuous for one of these topologies. However, the first topology is too strong and consequently the space ‘good operator fields’ will become too restricted to be of any utility. Therefore, in this particular case, we may require the map A to be continuous in the weak or strong topology as the case may be. To transport this definition to the general case, we have to reformulate this in a suitable way. Take for instance the requirement of strong continuity. This is equivalent to requiring that (a) A(ζ) is locally bounded (i.e. bounded on compact sets) and (b) for every continuous function f (ζ) of with values in H , Z A(ζ) f (ζ) is continuous. This motivates the following

Definition. An operator field ζ A(ζ) is said to be strongly continuous 97 → if (a) A(ζ) is locally bounded and (b) for every continuous vector field X(ζ),A(ζ)X(ζ) is also continuous.

Similar considerations for weak continuity give us the following

Definition. An operator field A is said to be weakly continuous if (a) it is locally bounded, and (b) for any two continuous vector fields X(ζ), Y(ζ), the map ζ A(ζ)X(ζ), Y(ζ) is continuous. → h i Proposition 5. An operator field A is strongly continuous if and only if A is locally bounded and ζ A(ζ)X (ζ) is continuous for every X . → n n ∈ ∧0 This follows straight from the definition. The strongly continuous operator fields form an algebra which is not however self adjoint, while the weakly continuous operator fields do not even form an algebra. In order to ensure that our definitions are good enough, we should know if there exist sufficiently many non-scalar continuous operator fields. This is answered by the following

Theorem 1. Let K be a compact subset of and Y ,... Y , Z ,... Z , Z 1 n 1 n 2n continuous vector ﬁelds such that Yi(ζ) are linearly independent for every ζ K. Then there exists a continuous operator ﬁeld A such that ∈ A(ζ)Y (ζ) = Z (ζ) for every ζ K and such that A(ζ) is also continu- j j ∈ ∗ ous. 98 10. Continuous sum of Hilbert Spaces-I

We first remark that we can as well assume that K = . For, Y (ζ) Z i are linearly independent if and only if ∆ = det Y (ζ), Y (ζ) , 0. This h i j i being a continuous function of K, there exists a compact neighborhood V of K such that ∆ α > 0 on V. If the theorem were true for a | | ≥ compact space, there exists an operator A on V satisfying the conditions ◦ 98 above. Set A = ϕA0 where ϕ is a continuous function 1 on K, 0 outside V; then A verifies all the conditions. Now let P(ζ) be the space spanned by Yi(ζ). We can then define A(ζ) = 0 on the orthogonal complement of P(ζ) and A(ζ)Yi(ζ) = Zi(ζ) and extend A by linearity. If π is the projection of H (ζ) onto P(ζ), we have X, Y = πX, Y and if πX = ξ Y , then X, Y = ξ Y , Y h ji h ji k k h ji kh k ji with det Y j, Yk = ∆ α> 0. OnP solving the linear equationsP for ξk, | h i| | |≥ we get ξk = ∆k/∆. Since the functions occurring in the linear equations ∆ are continuous, ξ are continuous functions and we have ξ = k k | k| | ∆ | ≤ M X where M is a constant. Hence A(ζ)X(ζ) = ξ (ζ)Z (ζ) is contin- || || k k uous for every continuous vector field X. A is locallyP bounded by virtue of the above remark and hence A is a continuous operator field. It is obvious that A∗ is also locally bounded. Now, A∗ maps the whole of H (ζ) onto P(ζ) and hence A∗(ζ)X(ζ) = ηk(ζ)Yk(ζ) where the η(ζ) are given by P

ηk(ζ) Yk, Y j = ηk(ζ)Yk, Y j X h i hX i = A∗(ζ)X(ζ), Y (ζ) h j i = X(ζ), A(ζ)Y (ζ) h j i = X(ζ), Z (ζ) h j i The Gram determinant in this case also is ∆. Hence by the same argument as before, A∗ is a continuous operator ﬁeld.

1.7 Measurablility of operator fields 99 Definition. An operator field A is said to be measurable if (a) it is almost everywhere locally bounded, and (b) for every compact K and positive ǫ, there exists K K such that µ(K K ) < ǫ and A(ζ) is continuous 1 ⊂ − 1 on K1. Continuous sum of Hilbert Spaces-I 99

If (ep) form an orthogonal basis (Ch. 1.5), then any operator can be expressed by means of a matrix with respect to this base. The matrix coefficients are only Ae , e . We then have h p qi Proposition 6. A locally bounded operator field A is measurable if and only if the matrix coefficients of A(ζ) are measurable functions on . Z That a measurable operator field satisfies the above condition is a trivial consequence of the definition. Conversely, if Ae , e is measur- h p qi able for every q, by prop. 3, ch. 1.4, Aep is measurable for every p. As in Prop. 3, Ch. 1.4, we can find a compact set K1 such that the Aep are continuous on k and µ(K K ) < ǫ. 1 − 1

1.8 Decomposed operators Let A(ζ) be a measurable operator field bounded almost everywhere. For every X(ζ) L2 , A(ζ)X(ζ) is also a measurable vector field and ∈ A(ζ)X(ζ) A(ζ)∧ X(ζ) . Hence A(ζ)X(ζ) 2dµ exists and we || || ≤ || ||∞|| || || || have AX L2 A(ζ) X L2 . In otherR words, A is a continuous opera- || || ≤ || ||∞|| || tor on L2 and∧ A A(ζ) ∧. || || ≤ || ||∞ If A∧is an operator in L2 which arises from an operator field, we say that it is a decomposed operator∧ and write A A(ζ). In particular if ∼ we take A(ζ) = f (ζ). Identity where f L (µ) weRZ obtain a decomposed ∈ ∞ operator M f (ζ) Id. This is called a scalar decomposed operator 100 f ∼ on L2 . We denoteR the space of all such operators by M . This is a self adjoint∧ subalgebra of Hom(L2 , L2 ). For, ∧ ∧

M f X, Y = f (ζ)X(ζ), Y(ζ) dµ(ζ) h i Z h i = X(ζ), f (ζ)Y(ζ) dµ(ζ) Z h i = X, M ¯Y h f i

Hence M∗f = M f¯. We now give a characterisation of decomposed operators in terms of this algebra M by means of 100 10. Continuous sum of Hilbert Spaces-I

Theorem 2. The set of decomposed operators is precisely the commutator of M . IfA A(ζ), then A = A(ζ) . ∼ || || || ||∞ R In fact, if A is a decomposed operator, it commutes with all the elements of M and hence belongs to M ′. 2 Conversely let A be an operator in L which commutes with M f for every f L (µ). Let K be a compact∧ subset of . Then χ e (ζ) ∈ ∞ Z K n ∈ L2 , e being the orthogonal basis (Ch. 1.5). Let H be a compact { n} nighbourhood∧ of K. Then

A(χK en) = A(χK χH en) = AM (χ e ) χK H n = M A(χ e ) by assumption χK H n

= χK A(χH en) almost everywhere.

It is obvious that at the intersection of any two compact sets K, K1,

the vector ﬁelds A(χK en) and A(χ1 en) coincide almost everywhere. K Hence there exists a vector ﬁeld A(en) such that χK (en)A(en)(ζ) =

A(χK en)(ζ) almost everywhere. Thus we have a countable family of rela-

tions and hence there exists a set N of measure 0 such that χK A(en)(ζ) = < 101 A(χK en)(ζ) for every ζ N. Since the Xn in the fundamental sequence are ﬁnite linear combinations of the e we have χ A(X )(ζ) = A(χ ∧0 n K n K X )(ζ) for every ζ < N. Also A(X (ζ)) 2dµ(ζ) = A(χ X) 2 n K || n || || K || ≤ A 2 X 2µ(ζ) for every compactR set K. Hence the set of elements || || K || n|| ζ suchR that A(X )(ζ) is strictly greater than A X (ζ) is of measure || n || || || || n || zero. We have thus proved that the algebra of all decomposed operators is M and we know that M M . We can moreover assert ′ ⊂ ′′ Theorem 3. M is a weakly closed algebra.

2 In fact, let (en) be an orthogonal basis of L . Then we deﬁne for any ∧ two integers p, q, a decomposed operator Hp,q by setting

Hp,q(ζ)en(ζ) = 0i f n , p or q 2 H , (ζ)e (ζ) = e (ζ) e (ζ). p q p || p || q Continuous sum of Hilbert Spaces-I 101

and H (ζ)e (ζ) = e (ζ) 2e (ζ). pq q || q || p If B commutes with every A M , in particular, it commutes with ∈ ′ all the Hp,q therefore B(ζ) commutes with Hp,q(ζ) for every ﬁxed p, q and almost every ζ. Since the Hp,q are countable, B(ζ) commutes with Hp,q(ζ) for almost every (ζ) and for all p, q. Hence outside a set of measure zero, we have

Bep, Hn,q, en B(ζ)e , e = h i for every p, q. p p 2 h i en || || en = BH , e , (since the H , are Hermitian) h n q p e 2 i n q || n|| = 0 if p , q.

On the other hand, 102 e ζ , = , n B( )ep ep BHn,pep 2 h i h en i 2 || || ep = || || Be , e e 2 h n ni || n|| This shows that B(ζ) is a scalar operator for almost every (ζ). Hence B = B(ζ)d(ζ) is a scalar decomposed operator and hence ǫM . This showsR that M = M ′′ and by Th. 2, Ch. 5.6, Part II, is weakly closed.

Chapter 11

Continuous sum of Hilbert spaces - II

2.1 103 In the last chapter, given a family H (ζ) of Hilbert spaces indexed by elements ζ of a locally compact space , we constructed the continuous Z sum H = L2 . Now we shall decompose a Hilbert space H into a continuous sum∧ with reference to a given comutative, weakly closed *- subalgebra m of Hom (H , H ). M satisﬁes Gelfand’s conditions and is hence isomorphic and iso- metric to the space C (Ω) of continuous complex valued functions on a compact space Ω which is called the spectrum of M . By this isomorphism, every continuous linear form on M is transformed into a continuous liner form on C (Ω) or, what is the same, a measure on the space Ω. In particular, the continuous linear form Mx, y where x, y H h i ∈ gives rise to a measure which we shall denote by dµx,y i.e. we have Mx, y = Mˆ (χ)dµ , (χ). This measure is called the spectral measure h i Ω x y associatedR to x, y. This depends linearly on x and anti linearly on y.

Let M ′ be the commutator of M . We now assume that there exists an element a H such that the set Aa : A M is dense in H . ∈ { ∈ ′} This assumption however is not a real restriction on our theory. For otherwise, we can decompose H into a discrete sum of Hilbert spaces

103 104 11. Continuous sum of Hilbert spaces - II

each of which satisﬁes the above condition. Let a1 be any element of H and H the closed subspace generated by Aa , A M . H is 1 1 ∈ ′ 1 104 invariant under both M and M ′. If H1⊥ is the orthogonal complement of H1, we can carry out the same process for H1⊥, and so on. Thus H = H H .. where all the H satisfy the above condition. 1 ⊕ 2⊕ i Now we show that the spectral measure dµa,a has Ω as its support. In fact, if f is a positive continuous function on Ω such that its integral 2 2 is 0, then f = 0. We can write f as Mˆ and we have Mˆ dµ(a,a) = 2 |2 | | | M Ma, a = Ma . Hence if Mˆ dµ , = 0, Ma =R0 or AMa = 0 h ∗ i || || | | (a a) for every A M , or again M(AaR ) = 0. Since the As are dense in H , ∈ ′ it follows that M = 0. Moreover, this shows that the measure dµa,a is positive. Proposition 1. Corresponding to any operator A M there exits one ∈ ′ and only one continuous function ϕA on Ω such that dµ(Aa,a) = ϕAdµa,a and we have ϕ A . || A|| ≤ || || In fact, it is enough to prove the proposition when A is a positive Hermitian operator since any operator is a ﬁnite linear combination of them. Under this assumption we have

2 Mˆ dµAa,a = M∗MAa, a Z | | h i = AMa, Ma h i = A Ma, Ma ≤ || ||h i 2 = A M dµa, a || || Z | | b Now dµ , is positive and dµ , k dµ , . Therefore, by Lebesgue Aa a Aa a ≤ a a - Nikodym theorem, there exist a measurable function ψ L (µ , ) such ∈ ∞ a a that dµAa,a = ψAdµa,a. We also have ψA A . || ||∞ ≤ || || The proof is complete if we prove the

105 Lemma. For every bounded measurable function ψ on Ω there exists one and only one continuous function ϕ such that ϕ = ψ a.e.

By the deﬁnition of µ , , we have µ , x y . Therefore x y || x y|| ≤ || || || || ψ(χ)dµx,y(χ) ψ x y . ≤ || ||∞|| || || || R Continuous sum of Hilbert spaces - II 105

Thus ψ(χ)dµx,y(χ) is a sesquilinear map which is continuous in each of theR variables x, y. As a consequence of Riesz representation theorem, there exits a linear operator T such that ψ(χ)dµ , (χ) = T x, y . x y h i We now show that T commutes with everyR element of M ′. For, if M M , we have ∈ ′

MT x, y = T x, M∗y h i h i = ψ(χ)dµX,M y(χ) Z ∗

= ψ(χ)dµMx,y(χ)( since M M ′) Z ∈ = T Mx, y h i Since M is weakly closed, T M . ∈ We have ψ(χ)Mdˆ µa,a = Tˆ Mdˆ µa,a and hence Tˆ = ψ(χ)a.e - This proves the lemma.R R The function ϕ C (Ω) thus constructed satisﬁes the following A ∈ properties:

(a) ϕλA+µB = λϕA + µϕB. For,

ϕλA+µBdµa,a = (λA + µB)a, a = λ Aa, a + µ Ba, a Z h i h i h i

= (λϕA + µϕB)dµa,a Z

= (b) ϕA∗ ϕ¯ A. For,

ϕA dµa,a = A∗a, a = Aa, a Z ∗ h i h i

= ϕ¯ Adµa,a. Z

ϕMAdµa,a = MAa, a Z h i

= Mdˆ µAa,a = Mˆ ϕAdµa,a Z Z

(d) ϕ 0 A∗A ≥ In fact, if f is a positive function, we can express it as Mˆ 2. | |

2 2 Mˆ ϕA Adµa,a = AMa 0. Z ∗ || || ≥

(e) ϕ A | A| ≤ || || This has been proved in the Prop. 1, Ch. 2.1.

2.2 In order to get a decomposition of H into a continuous sum, we need the following

Lemma . Let be any *-subalgebra of Hom(H , H ) which is uni- B formly closed. Let ϕ be a positive continuous linear form on such B that ϕ(A ) = ϕ(A),ϕ(A A) 0 and ϕ(A) k A . To ϕ we can make ∗ ∗ ≥ | | ≤ || || correspond a canonical unitary representation of the algebra . B

In fact, because of the conditions we have imposed on ϕ, ϕ(B∗A) is a positive Hermitian form on B. Hence we have by the Cauchy-Schwarz inequality ϕ(B A) 2 ϕ(B B)ϕ(A A). Therefore, ϕ(B B) = 0 if and | ∗ || ≤ ∗ ∗ ∗ only if ϕ(B A) = 0 for every A Hom(H , H ). Hence ϕ(B B) = ∗ ∈ ∗ 0 ϕ((AB) AB) = 0. It follows that the set N of elements B such that ⇒ ∗ ϕ(B∗B) = 0 is a left ideal. On the space B/N, ϕ is transformed into a positive deﬁnite Hermitian from and consequently ϕ gives rise to a scalar product. The completion of this space under this norm shall be denoted 2 107 Hϕ. The canonical map B Hϕ is continuous since ϕ(B B) k B . → ∗ ≤ || || On the other hand we have also a map f : B Hom(Hϕ, Hϕ) deﬁned ˙ → by f (A) = UA where UA(B˙) = AB. We show that this as also continuous. Consider B A 2B B A AB = B ( A 2 A A)B; ( A 2 A A) is positive ∗|| || − ∗ ∗ c∗ || || − ∗ || || − ∗ Continuous sum of Hilbert spaces - II 107

1 2 Hermitian and hence so is H = B∗ A B B∗A∗AB. H 2 is the uniform || || − 1 limit of polynomials in H and since B is uniformly closed, H 2 B. ∈ Therefore we have ϕH = ϕ 1 1 0 by assumption. We have now H 2 H 2 ≥ proved that ϕ(B A AB) A 2ϕ(B B). This shows that the map U ∗ ∗ ≤ || || ∗ A is continuous. This can be extended to an operator of H . We have also shown that U A . It remains to prove that this is a unitary || A|| ≤ || || representation. Let A, B, C B. Then U B˙, C˙ = ϕ((A C) B) = ∈ h A i ∗ ∗ B˙, U C . Hence U = U which shows that this is unitary. h A ∗ i A∗ A In fact all the above considerations hold for a Banach algebra with involution.

2.3 After this lemma in the general set-up, we revert to our decomposition of H into a continuous sum. For every fixed χ Ω, ϕ (χ) is a pos- ∈ A itive continuous linear from on M ′ which satisfies all the conditions of the lemma. Hence we have a unitary representation of M ′ in the M = ′ = = Hilbert space Hχ where Nχ A : ϕA∗A(χ) 0 . In other Nχ { } words, to each point χ Ω we have assigned a Hilbert space Hχ. If ∈ M M , since ϕ = Mˆ ϕ , we have U (χ) = Mˆ (χ) identity. For, ∈ MA A M U (χ)B˙, C˙ = ϕ (χ) = Mˆ (χ)ϕ (χ) = Mˆ (χ) B˙, C˙ . We have now h M i C∗ MB C∗ B h i all the data necessary for the construction of a continuous sum except 108 the fundamental family of vector fields. We have so far operated with M ′, but, in practice, M ′ is very large. For instance it is not in general separable in the norm. So we assume given a subalgebra of M such A ′ that

(a) is uniformly closed. A (b) There exists a sequence A such that is generated by the n ∈ A A A and M . n A ∩ (c) There exists a H such that Aa : a H is dense in H . It is ∈ { ∈ } actually this algebra which is in general given and the problem A 108 11. Continuous sum of Hilbert spaces - II

will then be to ﬁnd an M such that the above conditions are ⊂ A′ satisﬁed.

We have a map Hom(Hχ, Hχ). However it is possible that A → there are more functions ϕ than would be absolutely necessary. That is, there may exist elements χ, χ′ in Ω such that ϕA(χ) = ϕA(χ′) for every A . In this case, we have two points χ, χ in the base space ∈ A ′ Ω which are in some sense equivalent with respect to . Therefore we A introduce an equivalence relation R in Ω by setting χ χ whenever ∼ ′ ϕ (χ) = ϕ (χ ) for every A . This is a closed equivalence relation A A ′ ∈ A and Ω/R = is a compact Hausdorff space. The same procedure for Z Z and as for Ω and M gives a Hilbert space ζ at each point ζ A ′ H ∈ Z and a continuous representation of the algebra in Hom(Hζ, Hζ ). The A image of the measure dµ , by the canonical map Ω is denoted by a a →Z µ. At each point ζ we have a map ζ and hence for a fixed A A→H ∈ A we obtain a vector field. This family of vector fields is the fundamental family we sought to construct. In fact, (a) They constitute a vector space, since is an algebra. A 1 109 (b) XA(ζ) = ϕA A(ζ) 2 and hence XA(ζ) is continuous. || || ∗ || || (c) For each ζ, XA(ζ) in everywhere dense since Hζ is only the completion of the space of XA(ζ). (c ) Consider M B where M M and B is a finite product ′ i i ∈ A ∩ Ai1Ai2 ...PAip of the An. Then X Mi Bi = UMi XBi with UM being P scalars. XBi is only a countable family andP the vectors X Mi Bi (ζ) { } are dense in H (ζ). Consequently the countable family XP αi Bi (ζ) where the αi are complex numbers with real and imaginaryP parts rational is also dense in H (ζ). Thus we have now all the data for the construction of a continuous sum L 2. of course we have still to establish that L2 = H . In fact, since ∧ is compact, every continuous vector field is square∧ summable. Z Therefore, we have

2 2 2 XA = XA(ζ) dµ(ζ) = ϕA A(χ)dµa,a(χ) = Aa || || Z || || Z ∗ || || Continuous sum of Hilbert spaces - II 109

Also Aa = 0 implies X = 0. Therefore, the map Aa X is an A → A isometry of a dense subspace of H and hence can be extended to an isometry J : H L2 . It remains to prove that this map is surjec- → tive. We have seen (Prop.∧ 2, Ch. 1.2) that vector fields of the form 2 ϕi(ζ)Yi(ζ) where ϕi(ζ) are continuous functions of ζ are dense in L . ∧ ThereforeP it is enough to prove that J(H ) contains all such elements 2 (the image of J(H ) being closed in L ). Let M0 be the self adjoint subalgebra of M consisting of elements M∧ such that Mˆ is constant on cosets modulo the equivalence relation R. Consequently, Mˆ may be considered as a continuous map on . But we have M Aa 2 = Mˆ X 2dµ Z || i i || || i Ak 1 a ˆ RZ and therefore the map J : MiAi PMiXAi is an isometryP which → ˆ coincides with J on the elementsP Aa. ThisP shows that MiXAi J(H ) 110 2 ∈ and hence L = J(H ). P Hereafter∧ we shall identify L2 with H . We now assert that is A contained in the space of decomposed∧ operators on H . In fact, we will show that A = UA(ζ). We have already proved (Ch. 2.2) that U (ζ) A and UR (ζ) is hence bounded. Also U (ζ)X (ζ) = X (ζ) || A ||| ≤ || || A A B AB is again a continuous vector field and by Prop. 5, Ch. 1.6, UA(ζ) is a continuous operator field. If now A¯ = UA(ζ), then ABa¯ = UA(ζ)XB(ζ) = X (ζ) = ABa by our identificationR for every B . Since Ba : B AB ∈ A { ∈ is dense in H , we have A¯ = A. In other words, every operator A} A . is decomposable into a continuous operator field. ∈ A Now, we have another algebra M of operators on H . It is natural to expect then M consists of scalar decomposed operators. It is of course true, but the proof is not obvious. As before, let M0 be the subalgebra of M composed of elements M such that Mˆ C ( ). We first prove ∈ Z that M consists of scalar decomposed operators. Let B, C . Then 0 ∈ A 110 11. Continuous sum of Hilbert spaces - II

= = ¯ MBa, Ca MC∗Ba, a M(χ)dµC∗ Ba,a(χ) h i h i ZΩ

= Mˆ (χ)ϕC B(χ)dµa,a(χ) Z ∗

= Mˆ (ζ)ϕC B(ζ)dµ(ζ) (since the continuous Z ∗ Z functions are constant on the equivalence classes)

′ = Mˆ (ζ) XB(ζ), XC(ζ) dµ(ζ) by deﬁnition of the norm. Z h i Z

111 That is to say that M = Mˆ (ζ). Identity. We now extend this result to every element M M . SinceRZ we know that the space of scalar oper- ∈ ators is weakly closed, it suﬃces to prove that M M . Again by the ⊂ 0′′ Hahn-Banach theorem, it is enough to show that any weakly continuous

linear form which is zero on M0 (and hence on M0′′) is also zero on M . But any weakly continuous linear form on HomS (H , HW ) is of n the form U UXi, Yi . if MXi, Yi = 0 for every M M0, → i=1h i h i ∈ P P then Mˆ (ζ) Xi(ζ), Yi(ζ) dµ(ζ) = 0. Hence Xi(ζ), Yi(ζ) = 0 Z h i h i for almostP R every ζ in , or again Xi(π(χ)), YPi(π(χ)) = 0a.e. on Z h i Ω where π is the canonical map Ω P . Therefore, MXi, Yi = → Z h i Mˆ (X) Xi(x), Yi(x) dµa,a = 0 for every M µ. ThisP completes the Ω h i ∈ Rproof of ourP assertion.

2.4 Irreducibility of the components - Mautner’s theorem Finally it remains to show that the unitary representations of the algebra in the Hζ are irreducible. The algebra M is at our choice and we are A interested in taking it as large as possible. Thus we assume that M is a maximal commutative subalgebra of and obtain the A′ Theorem 1 (Mautner). Let be any uniformly closed *-subalgebra of A Hom (H , H ) such that Continuous sum of Hilbert spaces - II 111

(a) there exists a sequence A which generates ; n A (b) there exists an element a H such that the set Aa : a is ∈ { ∈ A} dense in H . Let M be any maximal commutative *-subalgebra of . Then in A′ the decomposition of H into the continuous sum of the Hζ with respect to M and , almost every representation U (ζ) of in H (ζ) is 112 A A A irreducible. Let e be the orthogonal basics given in Ch. 1.5 with respect to { n} the fundamental sequence and let M be the subset M M : Mˆ ∧0 0 { ∈ ∈ C ( ) of M . If B be the algebra generated by and M , then M = Z } A 0 B . In fact, we have seen that M M (Ch. 2.3) and therefore ′ ⊂ 0′′ M = M . Hence B = M = M and M = M , M ′ 0′ ′ A′ ∩ 0′ A′ ∩ ′ A′ ∩ ′ being a maximal subalgebra. We deﬁne for any two integers p, q as in theorem 3, Ch. 1.8, Hermitian decomposed operators Hp,q on H such that

Hp,q(ζ)en(ζ) = 0 if n , p or q; 2 H , (ζ)e (ζ) = e (ζ) e (ζ), and p q p || p || q 2 H , (ζ)e (ζ) = e (ζ) e (ζ). p q q || q || p We have already seen (Ch. 1.8) that any operator which commutes with all the Hp,q(ζ) is a scalar operator. Now, Hp,q is bounded since we have H , (ζ) e (ζ) e (ζ) e e . H , is continuous since || p q || ≤ || p || || q || ≤ || p|| || q|| p q it transforms every vector ﬁled of type e j(ζ) into another continuous vector ﬁeld (Prop. 5, Ch. 1.6). Now Hp,q = Hp,q(ζ)dµ(ζ) and this commutes with every element of M . ThereforeRZ

Hp,q = Hp,q(ζ)dµ(ζ) M ′ = B′′. Z ∈ Z Let Y = e / e 1/n. Then we have Y 2 < and by theorem n n || n|| || n|| ∞ 2, Ch. 5.6 Part II, there exist HermitianP operators Bk B such that ∈ n 2 2 en H , Y B Y < 1/k . Therefore H , e B e || || 0 n || p q n − k n|| || p q n − k n|| ≤ k → 2 Pas k ., i.e., H , (ζ)Y (ζ) B (ζ)Y (ζ) dµ(ζ) 0 as k . 113 →∞ n || p q n − k n || → →∞ R P 112 11. Continuous sum of Hilbert spaces - II

As in Riesz-Fisher theorem, we can ﬁnd a subsequence Bki such that H , Y B Y 0 as k outside a set of measure zero. Since the p q n − ki n → i →∞ Hp,q are only countable in number, we can pass to the diagonal sequence and get a sequence B such that for every p, q, H , Y B Y 0 as k j p q n − k j n → k outside a set N measure zero. j →∞ Let ζ < N and L be a subspace invariant under (ζ) or again under A (ζ) . Let S be any element of (ζ) . Then S commutes with every A ′′ A ′ U (ζ) where B B (where U (ζ) = M (ζ)U (ζ) whenever B = B ∈ B i i A i MiAi). P P So,

SH , (ζ)e (ζ), e (ζ) = lim SU (ζ), e (ζ), e (ζ) h p q n m i h Bk n m i = lim U (ζ), S e (ζ), e (ζ) h Bk n m i = lim S (ζ), U (ζ), e (ζ) h en Bk m i = S (ζ), H , (ζ), e (ζ) . h en p q m i or H , (ζ) commutes with every S (ζ) for almost every ζ. Hence p q ∈ A ′ (ζ) consists only of scalar operators for almost every ζ. Hence the A ′ only invariant subspaces of (ζ) are the trivial ones and the represen- A ′′ tation A U (ζ) of in H (ζ) is irreducible. → A A The following corollary is more or less immediate: Corollary. Let U be unitary representation of a separable, locally compact group G in a Hilbert space H such that there exists a H with ∈ the minimal closed invariant subspace containing a = H . Then U is a continuous sum of unitary representations which are almost all irreducible.

114 In fact, in Mautner’s theorem, we have only to take for the uni- A form closure of the algebra generated by elements of the form U , x G. x ∈ 2.5 We have already said that the space Ω could have been used in much the same way as the space . We now give an illustration to explain our Z remark that Ω is too large for practical purposes and that in the decomposition with respect to Ω the same representations may repeat ‘too often’ Continuous sum of Hilbert spaces - II 113

(which is what we sought to avoid by our equivalence relation). Let H 2 2 be the space L (R) and U the regular representation σx of R in L (R). 1 Let be the algebra of operators σ f for f L . It can easily be seen A 2 ∈ that contains all operators σµ on L with µ, a bounded measure. A′′ A and are of course commutative. Therefore . We say take A′′ A′ ⊃ A′′ m to be itself. If Ω is the spectrum of M it contains the spec- ′ A′′ ′ trum of M 1. This is the much larger than the spectrum of L1, whereas a ‘good’ decomposition of L2(R) into a Fourier transform is given by 1 f fˆ(y) = f (x)eixydx. The Fourier in version formula will be → √2π 1 R f = fˆ(y)eixydy. L2 is the direct integral of Hilbert spacers of √2π R dimension 1. If M 1 denotes the set of bounded measures, then L1 is an ideal in M 1 and is hence contained in a maximal ideal of M 1. Thus two diﬀerent characters of M 1 give rise to the same character of L1. Thus a decomposition with Ω consists of unnecessarily repeated representations while that with (spectrum of L1 in our example) economises Z them and reduces the decomposed representations to a minimum.

2.6 Equivalence of representations 115 The decomposition into continuous sum is obviously not unique, because the process depends on the choice of a H such that Aa : A ∈ { ∈ is dense in H and on this choice of M . The question therefore A} arises whether all these decompositions are equivalent in some sense.

Definition . Two decompositions L2 , ( ,µ , ) and L2 ( ,µ , ) 1 Z1 1 ∧1 2 Z2 2 ∧2 of H are said to be equivalent if∧ there exists a measurable∧ one-one map t : and a map Uζ of H (ζ) onto H (t(ζ)) such that, the Z1 → Z2 correspondence to every vector field X on of a vector field on Z1 Z2 defined by Y(t(ζ)) = U X(ζ), is an isomorphism of L2 onto L2 . ζ 1 2 ∧ ∧ It is almost immediate that if the vector a is changed, we get equivalent decompositions. However, it is not true that if M is chosen in different ways the corresponding decompositions are equivalent.

Chapter 12

The Plancherel formula

3.1 Unitary algebras 116 Consider the regular representation of a locally compact group G in the space L2 and decompose this into a continuous sum of irreducible representations. Then we have an isometry f X of L2 onto L2 . By → f the definition of the norm in L2 , we have f 2dλ = X 2d∧µ( ). G | | || f || Z The map f X is in a sense∧ the FourierR transformRZ for G, and the → f above equality, the Plancherel formula. As a matter of fact, we do get the classical Plancherel formula from this as a particular case when G is commutative. However, we have had many choices to make in the decomposition and as such this definition of a Fourier transform is not sufficiently unique and consequently uninteresting. We now proceed to obtain a Plancherel formula which is unique. Let G be a separable, locally compact, unimodular group. We have seen (Ch. 8, Part II) that the regular representation of G in L2 gives rise to a representation of L1 in L2. In fact we have the formula for every f L1 and g L2. ∈ ∈

1 g f (x) = g(xy− ) f (y)dy ∗ Z 1 = τy(a) whereg ˇ(x) = f (x− )

If we take x = e, g f (e) = e(y) fˇ(y)dy (Since the group is uni- ∗ R 115 116 12. The Plancherel formula

modular) 1 = g, f ∗ where f ∗(x) = f (x ) h i − By associativity of convolution product, we see that

g f, h = g f h∗(e) h ∗ i ∗ ∗ = g (h f ∗)∗(e) ∗ ∗ 1 2 = g, h f ∗ for every f, g, h L L h ∗ i ∈ ∩ 117 The operation we have deﬁned is an involution. Moreover L1 acts ∗ on L2 or, what is the same, L2 is a representations space for L1. The mapping f T where T f (g) = g f is unitary. Thus L1 is a self → f − ∗ adjoint algebra of operator on L2 = H . = L1 L2 is a subalgebra A ∩ H with an involution . This satisﬁes the following axioms: ∗ (a) x, y = y , x for every x, y h i h ∗ ∗i ∈ A For,

f, g = f (x)g(x)dx h i Z 1 1 = g(x ) f (x− )dx Z −

= g∗(x) f (x)dx Z ∗ 1 2 = g∗, f ∗ for every f, g, L L . h i ∈ ∩ (b) x, yz = y x, z , or equivalently h i h ∗ i yx, z = y, zx for every x, y, z . h i h ∗i ∈ A (c) is dense in H . A As a consequence the operators V (y) = yx on can be extended x A to operators on H .

(d) The identity operator is the strong limit of that Vx. This is an immediate consequence of prop. 3, Ch. 4.7, Part II, The Plancherel formula 117

Definition . Let be subspace of the Hilbert space H . If is an A A associative with involution satisfying conditions (a), (b), (c) and (d), A is said to be a unitary algebra (Godement). (Ambrose with a slightly different definition calls it an H∗- algebra).

3.2 118 Associated with a given unitary algebra, we have a representation Ux(y) = xy and an antirepresentation Vx(y) = yx. Axiom (b) asserts that these two representations are unitary. The map x x is an isom- −→ ∗ etry by a (a) and consequently can be extended to a map S : H H . → The U and the V are related by means of the relations V = SU S x x x x ∗ for every x . In fact, if y, z , we have ∈ A ∈ A V y, z = yx, z h x i h i = z∗, x∗y∗ h i = z∗, U y∗ h x∗ i = SUx S y, z h ∗ i Hence V = SU S on and hence on H . We shall denote by , x x ∗ A U the uniformly closed algebras generated by the U , V respectively. V x x Let be the uniformly closed algebra generated by both the U and the R x Vx.

Deﬁnition . An element a H is said to be bounded if the the linear ∈ map x V a of H is continuous. → x A→

The mapping shall be denoted Ua, and the set of bounded elements B.

Remark. To start with, one should have deﬁned right -boundedness and left-boundedness of elements in H . But if a is bounded in the above sense, a trivial computation shows that U x = V S a for every x . So a∗ x ∈ A S a is bounded and US a = Ua∗. Now we have Uxa = SVx S a = SUa∗S x and the map x U a is continuous and hence deﬁnes∗ a continuous → x operator Va and we have Va = SUa∗S . 118 12. The Plancherel formula

Proposition 1. M = U : a B is a self adjoint ideal which is 119 { a ∈ } weakly dense is . V′ In fact, for everyx, y , U V y = U (yx) = V a = V V a = ∈ A a x a yx x y V U y; therefore U commutes with V ; hence U . Moreover if x a a x a ∈ V′ T , we have TU x = TV a = V Ta; hence Ta is bounded and ∈ V′ a x x U = TU and consequently M is an ideal in . Since U = U , Ta a V′ a∗ S a it is self-adjoint. Since is weakly closed, it only remains to show V′ that M , or again that T , X M implies that TX = XT. V′ ⊂ ′′ ∈ V′ ∈ ′ But we have seen that, for x , TU M . Hence TU X = XTU ∈ A x ∈ x x and we may now allow Ux to tend to 1 in the strong topology to obtain TX = XT. From this follows at once the

Theorem 1 (Godement-Segal). In the notations, = , or equiva- U′ V′′ lently = . V′ U′′ In fact, since , . We have only to show that V ⊂U′ V ⊃U′′ . In other words, we have to prove that every element of V′ ⊂ U′′ commutes with every element of . Since the U , a B and sim- V′ U′ a ∈ ilarly V , a B are dense in , respectively, it suﬃces to establish a ∈ V′ U′ the commutativity of V , V , a, b B. First we assert that U b = V c a b ∈ c b for every c B. ∈ For,

U b, x = b, U x = b, V S c h c i h S c i h x i = V∗b, S c = c, SV∗b h x i h x i = c, U S b = c, V x = V c, x h x i h S b i h b i

Now, UaVb x = UaUxb = UUaxb = Vb(Ua x) by the above calculation and the proof of theorem 1 is complete.

3.3 Factors 120 A weakly closed self-adjoint subalgebra of operators on a Hilbert space H is said to be a factor if its centre reduces to the scalar operators. The Plancherel formula 119

If in the above discussion we assume R to be irreducible, then R′ = = = Centre of . Since R is irreducible, by cor. to U ∩ V′ U′ ∩U′′ U′′ Schur’s lemma (Ch. 5.4, Part II), R = scalar operators. Hence is a ′ U′′ factor.

Example. (1) The set of all bounded operators on H is a factor.

(2) The set of bounded operators on H which is isomorphic to Hom(H1, H1) where H1 is another Hilbert space is also a factor. This is said to be a Factor of type I. If H1 is of dimension n, this is said oe be if type In.

3.4 Notion of a trace If we consider only the operators on H which are of finite rank, then we have the notion of a trace defined by Te , e where the e form nh n ni n an orthonormal basis. In the general case,P we may define trace axiomat- ically in the following way:

Deﬁnition. If P is the set of positive operators on H , trace is a map of P into [0, ] satisfying ∞ 1 (a) Tr(UPU− ) = Tr P for every unitary operator U, and

(b) If P is a positive operator = Tα, where the Tα are also positive operators, and the series inP strongly convergent, then Tr P = Tr Tα. P In particular, (b) implies that for every positive λ, Tr(λP) = λ Tr P. It 121 is obvious that this is true if λ is rational and since the rational numbers are dense in R, by (b), it is also true for all λ R+. If A is any operator ∈ on H with a minimal decomposition into positive operators, then trace can be deﬁned on A by extending by linearity. Now, instead of Hom(H , H ), we may consider any -subalgebra ∗ F of Hom(H , H ) and deﬁne the notion of a trace as above. However, for arbitrary -subalgebras, neither the existence of a non-trivial trace ∗ nor its uniqueness is assured. For instance, if H = H1 + H2 is the direct sum of the Hilbert spaces H1 and H2 and F is the subalgebra 120 12. The Plancherel formula

Hom(H1, H1) + Hom(H2, H2) of Hom(H , H ), then the function ϕ deﬁned by ϕ(T1 + T2) = λ1 Tr T1 + λ2 Tr T2 (where λ1 and λ2 are arbitrary positive constants) is a trace. However, when F is a factor, the nontrivial trace, if it exists, is unique. Those factors which do not possess a nontrivial trace are said to be of type III. A nontrivial trace on F can be proved to have the following properties:

1. If H F P, TrH = 0 of and only if H = 0; and ∈ ∩ 2. For every positive H F, there exists H1 F such that 0 < H1 ∈ ∈ ≤ H and Tr H1 < . ∞ Definition . An element A of a -algebra F operators on H , is said to ∗ be normed (or of Hilbert-Schmidt type) with respect to a trace on F, if Tr(A A) < . ∗ ∞ 122 Let F0 be the set of operators of finite trace and F1 the set of normed operators in F. Than if A, B F , we have B A F and Tr(B A) is a ∈ 1 ∗ ∈ 0 ∗ scalar product on F1. In the case of the algebra , one can prove U′′ Theorem 2. There exists on one and only one trace such that U′′ (a) A is normed if and only if A = U for some a B. ∈U′′ a ∈ (b) If A = U and B = U are normed, then Tr(B A) = a, b . a b ∗ h i The proof may be found in [23] or [12], Ch. I, 6, n 2. § ◦ In particular, if R is irreducible, then the factor is not of type III. U′′ 3.5 We now assume that two more conditions are satisfied by , viz. A (1) is separable i.e. there exists a -subalgebra everywhere dense A ∗ in , which has a countable basis (in the algebraic sense). A (2) There exists an element e such that e = e and R is dense in ∈ A ∗ . A The Plancherel formula 121

Condition (2), however, is, as in the case of general decomposition, not a real restriction, and if it is not satisfied, can be split up into a A discrete sum of algebras each of which satisfies this condition. We shall now perform the decomposition of H into a continuous sum of Hilbert spaces with reference to the uniformly closed algebra R of operators on H . Let be the set Re : R R . A1 { ∈ } The fundamental family of vector fields in L2 is given by a = Re 123 → a˜(ζ) = X (ζ) H (ζ). We have already seen∧ (Ch. 2) that this map R ∈ is an isometry. Consider for every ζ the set (ζ) = a˜(ζ) : a . A { ∈ A1} U , a is a decomposed operator = U (ζ)dµ(ζ). a ∈ A a Our object now will be to put on (RζZ) the structure of an algebra and A an involution with respect to which (ζ) becomes a unitary algebra, A To this end, we define for ξ = a˜1(ζ) and η = a˜2(ζ), a1, a2 1, ξ = ∈ A · η Ua1 (ζ)ã2(ζ) and ξ∗a˜1∗(ζ). Of course we have to prove that these definitions are independent of the particular a1, a2 we choose. In other words we have to verify that ifa ˜1(ζ) = a1′ (ζ) anda ˜2(ζ) = a2′ (ζ), then U (ζ)ã (ζ) = U (ζ)a and thata ˜ (ζ) = a˜ (ζ). In order to prove the a1 2 a1 2′ 1∗ e 1′ ∗ e former, we show that U (ζ)ã (ζ) = V (ζ)ã (ζ). If a = R e and a = ea1 2 a2 1 1 1 2 R2e, we have

Ua1 (ζ)˜a2(ζ) = (Ua1 a2)(ζ) = XU a1 R2(ζ) = b˜(ζ) where b = U R e g a1 2

= (R1e)(R2e) = Va2 (R1e)

˜ Hence b(ζ) = XVa2R1 (ζ) = Va2 (ζ)ã1(ζ). Therefore, we have proved that the vector fields Ua1 (ζ)ã2(ζ) and Va2 (ζ)ã1(ζ) are equal and so we have U (ζ)ã (ζ) = V (ζ)ã (ζ) for almost every ζ. But since has a a1 2 a2 1 A1 countable basis, we can find a set N of measure zero such that for every a , a , we have U (ζ)ã (ζ) = V (ζ)ã (ζ) for ζ < N. Now the left 1 2 ∈ A1 a1 2 a2 1 hand side is unaltered if we replace a2 by a2′ while the right hand side remains the same if we replace a1 by a1′ . 124 It only remains to prove that ξ∗ is well-defined for almost all ζ′. It is enough to prove that for every a , a , the set of ζ such that a (ζ) 1 2 ∈ A1 1 = a2(ζ) and a1 (ζ) , a2 (ζ) is of measure zero. Let E = ζ : a1(ζ) = ∗ ∗ { e e g g e 122 12. The Plancherel formula

a (ζ) . The operator ﬁeld A deﬁned by 2 } e Identity if ζ E A(ζ) = ∈ 0 if ζ < E   is a Hermitian scalar decomposed operator in H and we have Aa1 = Aa2. Hence we have Aa1∗ = Aa2∗, i.e. A(ζ) a1∗(ζ) = A(ζ)a2∗(ζ) almost everywhere and a (ζ) = a (ζ) for almost every ζ E. 1∗ 2∗ e ∈ e We shall now prove that with the above operations, A(ζ) is a unitary e e algebra.

(a) ξ, η = η ,ξ h i h ∗ ∗i We have, in our usual notation, ξ, η = ϕR R (ζ). a∗ = SR1S e = h i 2∗ 1 1 SR1S e (since e∗ = e) and a2∗ = SR2S e. We have now to prove

that ϕR R1 (ζ) = ϕ(SR1S ) (SR2S )(ζ). We assert that M∗ = S MS for 2∗ ∗ every M R. In fact, if x , U M = MU = U and ∈ ∈ A x x Mx UM S x = M∗US x = U(Mx) = US Mx. Since the map x Ux is ∗ ∗ → one-one, we have M∗SX = S Mx or M∗ = S MS . Therefore

M(SR ) (SR S )e, e = MSR S e, SR e h 1 ∗ 2 i h 2 1 i = R e, R SMSe h 1 2 i = R e, R M∗e . h 1 2 i

In other words,

M(SR S ) (SR )e, e = MR∗R e, e h 1 ∗ 2 i h 2 1 i for every M A. By the deﬁnition of the spectral measure, ∈

dµ(SR S ) (SR2S )e, e = dµR R e,e 1 ∗ 2∗ 1

125 and hence ϕ(SR1S ) (SR2S ) = ϕR R . ∗ 2∗ 1 (b) ξ ξ ,ξ = ξ ,ξ ,ξ . h 1 2 3i h 1 3 2∗i We have to show that ϕ = ϕ which is obvious. R3∗R1R2 (R3R2∗)∗R1 The Plancherel formula 123

(d) Regarding the existence of suﬃciently many operators, we cannot assert that is true for all ζ. However, this is true for almost all ζ For this we need a

Lemma 1. In a self adjoint algebra of operators on H , the set Ax : A { A , x H is dense in H if and only if the Identity is the strong ∈ A ∈ } limit of A . ∈ A In fact, if x is an element of H , we denote Ax by F. Let F { } ⊥ be its orthogonal complement. If x is not in F, let x = x1 + x2 with x F, x F . Then Ax = Ax + Ax with Ax F, Ax F 1 ∈ 2 ∈ ⊥ 1 2 1 ∈ 2 ∈ ⊥ (since is a self-adjoint algebra). But Ax F. Hence Ax f A ∈ 2 ∈ ⊥ as well as F. Since the sum is direct, Ax = 0 for every A . If 2 ∈ A x , 0, this contradicts the assumption that Ax : A , x H is 2 { ∈ A ∈ } dense in H . For, space E = x : Ax = 0 for every A is non- { ∈ A} zero. E is invariant under and therefore E is invariant under . A ⊥ A Consequently a : a is contained in E . Now, U y : x, y {A ∈ H} ⊥ { x ∈ A} is dense in and hence xy : x, y is dense in 1. On the other H { ∈ A∞} A hand, there exists a sequence Y is such that the Y˜ (ζ) are dense in n A1 n (ζ) for every ζ. Each Y can be approximated by a sequence X , Y , 126 H n n p n p with Xn,p, Yn,p A1. Hence we have Y˜n(ζ) = lim UX (ζ)Yn˜,p(ζ) for ∈ p n,p almost every ζ and for each n, since we have only→∞ a countable family Y . Therefore ξη : ξ, η A(ζ) is dense in (ζ) for almost every ζ. n { ∈ } A Thus we have shown that the algebra (ζ) is unitary for almost ev- A ery ζ. By Mautner’s theorem, almost all these algebras are irreducible. Thus U(ζ)′′ is a factor. We can apply Theorem 2 to this factor. Thus the scalar product in the space of bounded elements is given by a trace. More precisely, we have a trace function on U(ζ) such that a˜ (ζ), ′′ h 1 a˜ (ζ) = Tr(U (ζ)U )(ζ)). But we know that the correspondence 2 i a2 ∗ a1 a a˜ is an isometry and hence one gets →

a1, a2 = Tr ua∗ (ζ)Ua1 (ζ) dµ(ζ) h i Z 2 Z 124 12. The Plancherel formula

This is the Plancherel formula which is in a certain sense unique. However this is obviously not absolutely unique as the trace function is unique only upto a constant multiple. In the case of a locally compact, separable, unimodular group G, we take = L1 L2 and H = L2. In this case Plancherel formula can be A ∩ rewritten as

f (x)g(x)dx = Tr(Ug∗(ζ)U f (ζ))dµ(ζ). ZG Z Z

Or again g∗ f (e) = Tr(Ug f (ζ))dµ(ζ). ∗ ∗∗ If we write g f = hR,Z we get ∗ ∗

h(e) = Tr(Uh(ζ)dµ(ζ) Zz This is the generalisation of the Fourier inversion formula. At any point x, the value of h(x) is given by

h(x), = Tr(Uh(ζ)Ux(ζ))dµ(ζ) Z Z 127 This is of course true not for all functions, but only for function of the type g f with g, f L1 L2 as in the classical case. ∗ ∗ ∈ ∩ 3.6 A particular case We have obtained a Plancherel formula in terms of the factorial representations of the group G and it would be more desirable to have a formula in terms of the irreducible representations of the group. This is however possible only in the following particular case.

Deﬁnition. A locally compact group is said to be type I if every factorial representation of the group is of type I.

This deﬁnition implies that every factorial representation is a discrete multiple of an irreducible representation. In fact, if F is the factor corresponding to a factorial representation of G, then F is isomorphic (algebraically) to Hom (H2, H2) where H2 is a Hilbert space. If F is The Plancherel formula 125 of type I, it can be proved (see, for instance [1], 8, Ch. I) that F is § ′ also of type I. by the isomorphism between Hom (H1, H1) and F′, we therefore get that for every projection P in F′, there exists a minimal projection < P. in other words, every invariant subspace , 0 of H contains a minimal invariant subspace. The restriction of the operators of F to any such minimal invariant subspace gives rise to an irreducible unitary representation and since the minimal projections in a Hilbert space are conjugate by unitary isomorphisms, these irreducible unitary representations are equivalent. On the other hand, the family of invariant subspaces of H which are direct sums of minimal invariant subspace, partially ordered by inclusion, is obviously inductive. By Zorn’s lemma, 1 1 1 there exists in it a maximal element say H . If H , H , H ⊥ is 128 1 nonempty and consequently contains a minimal invariant subspace H1 . Then H 1 H 1 again belongs to the family, thereby contradicting the ⊕ 1 maximality of H 1. Thus if x , the operator of the factorial representation is decom- ∈ A posed into irreducible U0 which are all equivalent. The map U U0 x x → x is an isomorphism. Hence in a group of type 1, we have the formula

f (x)g(x)dx = Tr(Ug∗(ζ)U f (ζ))dµ(ζ) ZG Z Z where the U(ζ) are irreducible representations and not merely factorial representations, and the trace is the usual trace. The deﬁnition of a group of type I seems a little inoccuous but is of importance since all semisimple lie groups are of type I. The problem remains however to give an explicit Plancherel measure, etc. It is known in the case of complex semisimple lie groups (see [26]) and in the case of SL(2, R) ([2], [25]), but not in the general case.

3.7 Plancherel formula for commutative groups

Let Ω be the spectrum of R′ in this case. We have however to pass to a quotient by means of an equivalence relation. It can be proved that Z is actually the one point compactiﬁcation of the spectrum of L1. We Z now assert that every representation of L1 in a space E arises from a representation of the group G. In fact if a = U b, with b H , f L1, f ∈ ∈ 126 12. The Plancherel formula

we put Uxa = Uǫx f b. It can be proved (see for instance [22], [6]) that ∗ the Ux are well deﬁned. 129 This establishes a one-one correspondence between characters of G and one dimensional representation of L1. Hence the spectrum of L1 is only the character group of GU 0 to compactify it. In this case, { } every factor consists only of scalar operators and hence any factorial representation is a discrete multiple of irreducible representations of dimension 1. If the character group of G is denoted by Gˆ, we have, since Tr χ( f ) = χ( f ) = f (x)χ(x)dx, R f 2dx = χ( f ) 2dµ(χ). ZG || || ZGˆ | | It only remains to prove that the Plancherel measure in this case is the Haar measure on Gˆ. But this is obvious since each character is of norm 1 and multiplication of χ( f ) by another character leaves the integral invariant. Thus in this case, we have the classical plancherel formula f 2dx = χ( f ) 2 dχ. ZG || || ZGˆ | | Again the Fourier inversion formula becomes in this case

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