The Kunze Stein phenomenon

Transcription

1 The Kunze Stein phenomenon Alessandro Veca Relatori: Prof. Stefano Meda (Università Bicocca) Prof. Michael Cowling (UNSW) August 13, 2002

2 Contents 1 Preliminaries Haar measures and locally compact groups Local fields I The Kunze Stein phenomenon on algebraic groups 7 2 Preliminaries Root Systems and Weyl groups BN-pairs Spherical functions and Plancherel measures Groups of p-adic type and their structure An example Groups with an affine structure Groups of p-adic type Haar measures and integral formulas The Kunze Stein phenomenon An estimate for L 2 -functions The Kunze Stein phenomenon II The Kunze Stein phenomenon on homogeneous trees 63 5 Preliminaries Lorentz spaces Homogenous trees and their structure Homogeneous trees The group N and the boundary Ω The Kunze Stein phenomenon A maximal operator on X An endpoint estimate for the generalized Kunze Stein phenomenon On radial convolutors of L p (G) i

3 Introduction Let µ denote the Haar measure on a locally compact group G. The convolution of two compactly supported continuous functions f and g is the function f g on G defined by setting f g(x) = f(y)g(y 1 x) dµ(y), x G. G In 1960, R.A. Kunze and E.M. Stein proved in [20] that, if G is SL(2, R), then the continuous inclusion L p (G) L 2 (G) L 2 (G) (1) holds whenever 1 p < 2 (more precisely, the convolution extends to a bounded linear operator from L p (G) L 2 (G) to L 2 (G)). A group satisfying (1) for any 1 p < 2 is called a Kunze Stein group and for such a group (1) is called the Kunze Stein phenomenon for G. Following this paper, a large class of groups have been investigated. In particular, C. Gulizia proved in [12] that the group of 2 2 matrices of determinant 1 with entries in a non-discrete totally disconnected local field F is a Kunze Stein group. In 1978 M. Cowling proved in [7] that every connected real semisimple Lie group with finite center is a Kunze Stein group. This thesis is divided in two parts. After a preliminary chapter, the first part will be devoted to the proof of the following: Theorem 1. Let F be a non-discrete totally disconnected local field and let G be the group of F-rational points of a simply connected simple linear algebraic group defined over F. Then the continuous inclusion L p (G) L 2 (G) L 2 (G) holds whenever 1 p < 2. In other words, G is a Kunze Stein group. ii

4 To avoid technicalities related to linear algebraic groups, we have preferred the axiomatic definition of groups of p-adic type given in [23], based on the work of F. Bruhat and J. Tits [4] and [5]. Roughly speaking a group G of p-adic type is a triple (N, {U α } α Σ, ν) related to an affine root system Σ, such that N and U α are subgroups of G and ν is an epimorphism from N into the affine Weyl group of Σ, satisfying a certain family of axioms. It turns out (see [23]) that the group of F-rational points of a simply connected simple linear algebraic group defined over F is a group of p-adic type. In the first part of the chapter we summarize some results of groups of p- adic type, following [23]. Then we prove several theorems about the structure of certain subgroups M j which will be used later. In particular, we prove that certain subsets of these subgroups are of full measure in M j (all the groups involved will be locally compact, hence endowed with a Haar measure) and exhibit related integral formulae. These are the analogues of the integral formulae for the Bruhat decomposition in the case of semisimple Lie groups. More precisely, we prove theorems of the following form. Theorem 2. Let Ūj, Z j and U j be suitable subgroups of M j. Then they are locally compact and the product ŪjM j 1 Z j U j is open in M j. Moreover, if dg, dū, dz, du and dm are the Haar measures on M j, Ūj, Z j and U j and M j 1, then f(g) dg = f(ūmzu) (z) j dū dz dm du, M j M j 1 U j for all measurable functions f on M j. Ū j Z j By combining these results with an explicit formula for the spherical Plancherel formula, due to I.G. Macdonald [23], we will be able to obtain an estimate for L 2 -functions on G, by allowing periodicity in the imaginary variable of the class one principal series π s (where s is in C d ). Theorem 3. Let f be a function in L 2 (G). Then, γ(l(iy)) 2 π iy (f) 2 HS ψ(iy) 2 dy C f(g) 2 dg, R d G where ψ and γ are suitable analytic functions on C d, L is an automorphism of C d and dy and dg are the Haar measures on R d and G. iii

5 Then we prove Theorem 1 along the lines of [7]. In the second part we analyze the case of rank 1. In 1998 Cowling, S. Meda and A.G. Setti proved that if G is the group of isometries of a homogeneous tree X (see [9]) or a semisimple Lie group of real rank 1 (see [8]), then G satisfies a more accurate version of the Kunze Stein phenomenon. By using the Lorentz spaces L p,q, they proved that for 1 < p < 2, the continuous inclusion L p,r (G) L p,s (G) L p,t (G) (2) holds if and only if 1/t 1/r + 1/s 1. Finally in 2000, A.D. Ionescu [17] studied the validity of (2) when p 2. He proved that L 2,1 (G) L 2,1 (G) L 2, (G), (3) when G is a semisimple Lie group of rank one. Following the ideas of [17], we will prove in Chapter 7.2 the following: Theorem 4. Let G be the group of isometries of a homogeneous tree X. Then L 2,1 (G) L 2,1 (G) L 2, (G). Note that our result (and a multilinear interpolation theorem [21]) improves that of [9]. The key ingredient in Ionescu s proof is the relation between Iwasawa and Cartan decompositions for semisimple Lie groups. In Chapter 6 we introduce and study in some detail a suitable subgroup N (see [10]) which leads to a decomposition of Iwasawa type for the group G of isometries of a homogeneous tree. Although this decomposition is not a bijection, we prove a related integral formula. More precisely, we prove the following: Theorem 5. Let K be the isotropy group of a base point o in X and let τ be a one step translation in G. Then each element in G may be written as a product of the form nτ j k, where n N, j Z and k K. Moreover, for any compactly supported continuous function f on G, f(z) dz = f(nτ j k)q j dk dn, G N j Z where dg, dn and dk are the Haar measures on (the locally compact groups) G, N and K respectively. iv K

6 The analogy with the Iwasawa decomposition for Lie groups is apparent. In the next chapter we define the maximal operator M on X by Mf(x) = sup r N 1 B(x, r) 1/2 y B(x,r) f(y), where B(x, r) is a ball of radius r centred in x X in the metric space X and, following [17], we will prove the following. Theorem 6. The operator M is bounded from L 2,1 (X) to L 2, (X). As a corollary, we will obtain a K-bi-invariant version of Theorem 4. The next section is devoted to the proof of Theorem 4 and, in the last section, we present some consequences (see [30]) of the Kunze Stein phenomenon by showing that there is a natural norm (see [8], [22] and [9]) which makes L p,1 (K\G/K) (i.e., the subalgebra of K-bi-invariant functions in L p,1 (G)) a commutative Banach algebra of convolutors of L p (G). We give an explicit description of the Gel fand spectrum of this subalgebra. This part of the thesis has already been published [31]. Note that, at least for some integers q, a homogeneous tree of degree q is the building related to the simply connected algebraic group SL(2, F), where F is a non-discrete totally disconnected local field (see [3]). It would have been possible to use the definitions and groups introduced in the Part I to present and prove these results. But we have preferred to change our approach in order to give a more complete panorama of the subject. v

7 Chapter 1 Preliminaries In this chapter we report some general results about locally compact groups and local fields. For the first section we refer to [11] and [1], for the second to [32]. 1.1 Haar measures and locally compact groups By locally compact group we shall mean a group G with a topology with respect to which G is Hausdorff, locally compact and the group actions are continuous. For any continuous automorphism Φ of G, we make Φ act on functions f by setting Φf = f Φ 1 and on Borel measures µ on G by setting f d(φµ) = Φ 1 f dµ, G G f C c (G). By choosing f equal to the characteristic function χ E of a measurable set E, it follows that Φµ(E) = µ(φ 1 (E)). Denote by λ(y) and ρ(y) the continuous automorphisms of left and right translation λ(y) : x yx and ρ(y) : x xy 1. 1

8 A left Haar measure on G is a non-zero positive measure on G, finite on compact sets, outer regular on all Borel sets and inner regular on open sets, such that λ(y)µ = µ for any y G. The last condition is equivalent to the requirement that µ(ye) = µ(e), y G, i.e., that µ is invariant under left translations. The definition of right Haar measure is analogous, with ρ instead of λ. The fundamental result is that on a locally compact group G there exists a unique left Haar measure up to multiplication, in the sense that if µ 1 and µ 2 are left Haar measures, then µ 1 = Cµ 2 for some positive number C. An analogous result holds for right Haar measures. When not otherwise specified, a Haar measure is understood to be a left Haar measure. Let G be a locally compact group and let µ be a Haar measure for G. If Φ is a continuous automorphism of G, then also Φ 1 µ is a Haar measure for G, because Φ 1 µ(xe) = µ(φ(xe)) = µ(φ(x)φ(e)) = µ(φ(e)) = Φ 1 µ(e). By the uniqueness of Haar measures, there exists a positive number (Φ) such that Φ 1 µ = (Φ)µ. The number (Φ) is independent of the particular choice of µ, and is called the Jacobian (or modulus) of the automorphism Φ. Then, f(φ 1 (x)) dµ(x) = f(x) d(φ 1 µ)(x) = (Φ) f(x) dµ(x). G G With the change of variable x Φ(x), we can express formally the last relation by writing dµ(φ(x)) = (Φ)dµ(x). (1.1.1) In the particular case when Φ = ρ(y), for some y G, then we write (y) for (ρ(y)). The function y (y) is a continuous homomorphism from G to R +, called modular function 1 of G. We say that G is unimodular if is equal to one. With this notation, dµ(xy 1 ) = dµ(ρ(y)(x)) = (y)dµ(x). 1 We use the same convention as [19], but this is not standard. For example, for [11] and [1] the modular function is our 1. 2 G

9 The modular function is useful to relate left and right Haar measures. Let µ be a fixed left Haar measure and define the measure µ by setting f(x) d µ(x) = f(x) (x) dµ(x), f C c (G). (1.1.2) Then G G G f(xy) d µ(x) = and µ is a right Haar measure. G f(x) (xy 1 ) dµ(xy 1 ) = G f(x) d µ(x), Theorem Let G be a locally compact group and let X and Y be closed subgroups of G such that Ω = XY is open in G. Let µ, µ X, and µ Y be Haar measures on G, X and Y and assume that X Y is compact. Finally, let G and Y be the modular functions on G and Y. Then, a function f is µ-measurable if and only if the map (x, y) f(xy) G (y) 1 Y (y) is µ X µ Y -measurable and f(ω) dµ(ω) = C f(xy) G (y) 1 Y (y) dµ X (x) dµ Y (y), Ω where C is a constant independent of f. Proof. See [1, p. 66]. X Y Let G be a locally compact group and let H be a Banach space. A representation π of G on H is a homomorphism of G H into aut(h) such that the mapping (x, ξ) π(x)ξ of G H into H is continuous. A representation π is said to be irreducible if {0} and H are the only closed subspaces of H invariant under π(g). A representation π is said to be unitary if H is a Hilbert space and π(x) is unitary for all x G. Let π be a representation of G on H such that x π(x) is bounded on G. Then, for any f L 1 (G) and ξ H, there is a unique vector π(f)ξ such that π(f)ξ, ξ = f(x) π(x)ξ, ξ dx, G 3

10 for all ξ in the dual space of H (see [11, Appendix 3]). Then, the linear operator ξ π(f)ξ on H is bounded because π(f) = sup π(f)ξ, ξ = f(x) π(x)ξ, ξ dx ξ = ξ =1 G f(x) π(x) ξ ξ dx G sup x G π(x) f 1. For example, the operator of left translation λ(x) is an isometry of L p (G) and the map x λ(x) is a representation of G on L p (G) for any 1 p <. In particular, if p = 2, then λ is unitary and it is called the left regular representation of G. If 1 p < and f is in L 1, then it is well defined the operator λ(f) and we have [λ(f)g](y) = f(x)[λ(x)g](y) dx = f(x)g(x 1 y) dx = f g(y), for any g L p (G). G G 1.2 Local fields Let F be a local field, i.e., a non-discrete locally compact commutative field and let µ be a Haar measure on the additive group of F. We denote by y the norm of y, i.e., the Jacobian of the continuous automorphism x yx, so that µ(ye) = y µ(e), (1.2.3) for any measurable subset E of F. We extend this map to the whole field by setting 0 = 0. Theorem Let F be a local field and let be as above. Then (i) xy = x y, x, y F; (ii) is a continuous function on F; (iii) the sets {x F : x m}, for m > 0, are compact and make up a fundamental system of neighborhoods of 0 in F; 4

11 (iv) x + y A max{ x, y } for every x, y in F, where A = sup{ x + 1 : x 1}. For the proof see [32, Chap. I]. It is clear that 1 A <. Theorem Let F be a local field and let A be as above. If A > 1, then either F = R or F = C. If A = 1, then F is a discrete subgroup of R +. For the proof see [32, Chap. I and IX]. We shall now assume that A = 1, so that F satisfies the so called ultrametric condition x + y max{ x, y }, x, y F, (1.2.4) and F is a discrete subgroup of R +. This fact implies that F is generated by some positive number q, so that the compact sets {x F : x q j }, j Z, are open, and F is totally disconnected in view of (iii) of Theorem The assumption A = 1 is then equivalent to the assumption that F is a totally disconnected local field (Theorem 1.2.2). For each x in F, we will write v(x) for the integer such that x = q v(x). (1.2.5) The map v( ) is called the (non-archimedean) discrete valuation of F. The ultrametric condition (1.2.4) implies that the subset U, given by U = {x F : v(x) 0}, is a subring of F. Moreover, it is compact and contains all the elements x such that x j = x j is bounded. It is therefore a maximal compact subring of F. Let now U 0 = {x F : v(x) = 0} and P = {x F : v(x) > 0}. Note that P is an open maximal ideal in U, so that U/P is a finite field, called the residue field of F. 5

12 and Consider an element p 0 in P such that p 0 = q 1. Then P = p 0 U µ(p) = µ(p 0 U) = p 0 µ(u) = q 1 (U : P)µ(P), where (U : P) denotes the index of P in U. It follows that q is equal to the index of P in U, i.e., to the number of elements in the residue field. In particular q is an integer, and actually a power of a prime. We conclude this section by illustrating the field of p-adic numbers Q p. In view of the classification theorem for locally compact fields (see [32]), Q p may be considered the prototype of a totally disconnected local field of characteristic zero. Example 1. Let p be a fixed prime. Any non-zero rational number r may be written uniquely as r = p m r, where m is an integer and r is a rational number whose numerator and denominator are not divisible by p. Then we define the p-adic norm of r, denoted by r p, to be p m, and we set 0 p = 0. The completion of Q with respect to the metric d p (r 1, r 2 ) = r 1 r 2 p is a totally disconnected local field, called the field of p-adic numbers. The elements in Q p are the series m c jp j, where c j {0,... p 1}. If c m 0, then the norm is given by c j p j = p m, j=m which coincides with p, when restricted to Q. Then, { } { } U = c j p j and P = c j p j, j=0 where c j {0,... p 1}, so that the residue field U/P is (isomorphic to) the cyclic group of p elements Z p. j=1 6

13 Part I The Kunze Stein phenomenon on algebraic groups 7

14 Chapter 2 Preliminaries In this chapter we summarize some standard results which will be used in the rest of Part I. We refer to [2] and [23] for the first section, to [6] for Section 2.2 and [11] and [14] for Section Root Systems and Weyl groups Let a be a real vector space of dimension d endowed with a scalar product,. Then, induces a natural isomorphism between a and its dual space a and a scalar product on the latter also denoted by,. For each non-zero functional a a, let a be the image of 2a/ a, a under this isomorphism and denote by h a the kernel of a. Also, write w a for the orthogonal reflection (with respect to, ) in the hyperplane h a. We make w a also act on a by setting w a (b) = b wa 1, b a. A root system Σ 0 in a is a subset Σ 0 of a satisfying the following axioms: (RS1) Σ 0 is finite, spans a and does not contain 0; (RS2) w a (Σ 0 ) = Σ 0 for any a Σ 0 ; (RS3) b(a ) Z for any a and b in Σ 0. A root system Σ 0 is called reduced if a = ±b whenever a and b are proportional and irreducible if Σ 0 admits no trivial decomposition Σ 0 = Σ 0 Σ 0 with every member of Σ 0 orthogonal to every member of Σ 0. The elements 8

15 of Σ 0 are called roots. The group W 0 generated by the reflections w a, when a Σ 0, is called the Weyl group of Σ 0. Then W 0 is a finite group of isometries of a. The connected components of the complement of a Σ 0 h a, are called the chambers of Σ 0, and the choice of a chamber c 0 identifies the subset Σ + 0 of those roots which are positive when evaluated on the vectors of c 0. The elements in Σ + 0 are called the positive roots relative to c 0. It also determines the set Π 0 of those positive roots which are not the sum of two positive roots. The elements in Π 0 are called the simple roots relative to c 0 and satisfy the following conditions: (i) they form a basis for a ; (ii) every root in Σ 0 is a linear combination of roots in Π 0 with integer coefficients which are either all non-negative or all non-positive. A subset of Σ 0 satisfying (i) and (ii) is called a fundamental system of roots. Then each chamber identifies a fundamental system and, on the other hand, each fundamental system Π 0 identifies the chamber c 0 = a Π 0 {x a : a(x) > 0}. This correspondence is a bijection. For any subset (Π 0 ) of a fundamental system Π 0, let (a ) be the subspace spanned by (Π 0 ), let (Σ 0 ) = Σ 0 (a ) and let (W 0 ) be the subgroup of W 0 generated by the reflections w a, when a (Π 0 ). Proposition The set (Σ 0 ) is a root system on (a ), (Π 0 ) is a fundamental system of roots and the Weyl group of (Σ 0 ) is (W 0 ). Proof. See [6, p. 27]. 9

16 In particular, if c 0 is a fixed chamber and Π 0 = {a 1,..., a d } is the corresponding fundamental system of the simple roots relative to c 0, then, for j = 1,..., d, the subset (Π 0 ) j = {a 1,..., a j } of Π 0 is a fundamental system of roots for the corresponding root system on (a ) j = span R {a 1,..., a j }. (Σ 0 ) j = Σ 0 span R {a 1,..., a j } Lemma Let Σ 0 be irreducible. Then it is possible to give an order to the set of simple roots Π 0 such that (Σ 0 ) j is irreducible for all j = 1,..., d. Proof. The irreducibility of the root system Σ 0 implies that Π 0 cannot be written as a disjoint union of orthogonal subsets (see [2, Corollary 5, p. 162]). It is therefore possible to give an order to the simple roots such that a 1 is chosen arbitrary and a j is not orthogonal to span R {a 1,..., a j 1 }, 1 < j d. Assume now that Π 0 (and then (Π 0 ) j ) is given such an order, and write (Σ 0 ) j as a disjoint union (Σ 0 ) j = Σ 0 Σ 0, where every element in Σ 0 is orthogonal to every element in Σ 0. Since (Π 0 ) j is contained in (Σ 0 ) j, then a 1 belongs to, say, Σ 0. Since a 2 is not orthogonal to a 1, it has to belongs to Σ 0 too and so on. It follows that (Π 0 ) j Σ 0, so that Σ 0 = (Σ 0 ) j, and irreducibility follows. Consider a root a of a root system Σ 0. For each k Z, we define an affine function a + k on a by setting (a + k)(x) = a(x) + k, x a. (2.1.1) 10

17 Let Σ = {a + k : a Σ 0, k Z}. The elements in Σ are called affine roots and are denoted by Greek letters. For each affine root α in Σ, let h α be the affine hyperplane on which α vanishes and let w α be the reflection in h α. The group W generated by the w α as α runs over W is called the affine Weyl group of Σ. Then W 0 is the isotropy subgroup of 0 a. The affine Weyl group acts on Σ by the rule where w W and α Σ. w(α) = α w 1, (2.1.2) Theorem The translations belonging to W form a free abelian group T of rank d, and W is the semidirect product of T and W 0. For each a Σ 0, consider the translation t a = w a w a+1. Then {t a : a Π 0 } is a basis for T. Moreover, t a (0) = a, and the map t t(0) maps T isomorphically onto the lattice spanned by the a, when a runs over Σ 0. Proof. See [2, Proposition 1, p. 173]. The connected components of the complement of α h α are called the affine chambers of Σ. We conclude this section with a few examples of abstract root systems: in the examples we denote by {x i } the standard basis of R n, by {e i } its dual basis and by, the usual scalar product. Example 2 (Root system of type A d ). Let and a = {x R d+1 : x x d+1 = 0}, Σ 0 = {e i e j, i j}. Then Σ 0 is an reduced irreducible abstract root system, is called the root system of type A d and its Weyl group W 0 acts by permutating the coordinates x 1,... x d+1 of x a (i.e., it is isomorphic to the symmetric group S d+1 ). The positive chamber c 0 may be taken to be c 0 = {x a : x 1 > x 2 > > x d+1 }, 11

18 so that sets of positive roots Σ + 0 and and simple roots Π 0 are given by Σ + 0 = {e i e j, 1 i < j d + 1} Π 0 = {a j = e j e j+1, 1 j d}. Note that (e j e i ) = x j x i, so that the lattice spanned by the a j, isomorphic to the group of translations T of the associated affine Weyl group W, is given by a Z d+1. Example 3 (Root systems of type C d and BC d ). Let a = R d, and Σ 0 = {±2e j, ±e i ± e k : 1 j d, 1 i < k d}, Σ 1 = {±e j, ±2e j, ±e i ± e k : 1 j d, 1 i < k d}. Then Σ 0 and Σ 1 are irreducible abstract root systems. The former is called the root system of type C d and is also reduced, while the latter is not reduced and is called the root system of type BC d. The Weyl groups related to these root systems coincide and it is the group W 0 of signed permutations of e 1,..., e d. The positive chamber c 0 may be taken (for both root systems) to be c 0 = {x a : x 1 > x 2 > > x d > 0}, so that sets of positive roots Σ + 0 and Σ + 1 are given by and Σ + 1 Σ + 0 = {2e j, e i ± e k, 1 j d, 1 i < k d} = {e j, 2e j, e i ± e k, 1 j d, 1 i < k d}, and the set of simple roots are given by Π 0 = {2e d, e i ± e i+1, 1 i d 1} Π 1 = {e d, 2e d, e i ± e i+1, 1 i d 1}. Note that (e i e k ) = x i x k, e j = 2x j and that (2e j ) = x j. It follows that the lattice spanned by the a, when a runs over Π 0, and lattice spanned by {(2e j ), 1 j d} coincide and are equal to Z d. In other words, the group of translations T of the affine Weyl group W, generated by the translations t a when a runs over Π 0, is also generated by the translations {t 2ej, 1 j d}. 12

19 2.2 BN-pairs Let G be a group, B and N be subgroups of G and R be a subset of the coset space N/(B N). The pair (B, N) is called a BN-pair if it satisfies the following set of axioms (BN1) B and N generate G and H = B N is normal in N; (BN2) the group W = N/H is generated by R and each r R has order 2; (BN3) rbw BwB BrwB; (BN4) rbr B for each r in R. Proposition Let (B, N) be a BN-pair and W = N/(B N). BwB = Bw B then w = w, and hence G = BwB. w W If Proof. See, for example, [25, p. 56]. Example 4 (The BN-pairs of SL(2, F)). Let Z be the diagonal group in SL(2, F), N be its normalizer and let B 0 be the full upper diagonal group (i.e., diagonal included). Then Z = B 0 N is normal in N and W 0 = N/Z is the group of permutations of two elements and (B 0, N). Moreover, (B 0, N) is a BN-pair for G = SL(2, F). Note incidentally that the finite group W 0 is the Weyl group of a root system of type A 1. If F is a totally disconnected local field, there is another BN-pair which will play a crucial role in the following chapter. Let N be as above and let B be the group of matrices of the form ( ) x1 y b =, z x 2 where v(x i ) = 0, v(y) 0 and v(z) 1. Then (B, N) is a BN-pair for G. In this case, B N = M, where M is the group of diagonal matrices with entries in the unit sphere of F. Then M is compact and open in N and W = N/M is infinite (N is not compact because it contains Z), and is isomorphic to the affine Weyl group of an affine root system of type A 1. 13

20 Remark The BN-pairs have been introduced by Tits in the study of the structure of algebraic groups. If the group W of (BN2) is the affine Weyl group of a root system Σ 0 on an Euclidean space a, then it is possible to define an affine building (see [4], [5] and [3]) associated to the group G. Roughly speaking, this is obtained by gluing together several copies of a. For example, in the case when G = SL(2, F), the associated root system is of type A 1, the Euclidean space is R and the affine building is a homogeneous tree of degree q + 1, where q is the order of the residue field of F. 2.3 Spherical functions and Plancherel measures In this section K is a compact subgroup of a locally compact unimodular group G. We say that (G, K) is a Gel fand pair if the algebra C c (K\G/K) of compactly supported K-bi-invariant continuous function on G is commutative. A continuous function ω is called a spherical function for the pair (G, K) if is K-bi-invariant, ω(1) = 1 and satisfies the functional equation ω(xky) dk = ω(x)ω(y), x, y G. K A spherical function is said to be positive definite if n ω(x 1 i x j )s i s j 0, i,j=1 for all finite sets x 1,... x n of elements in G and complex numbers s 1,..., s n. The set of positive definite spherical functions is denoted by Ω + and plays an important role in the harmonic analysis of K-bi-invariant functions. The spherical fourier transform of a function f L 1 (K\G/K) is the map ω ˆf(ω), where ˆf(ω) = f(x)ω(x 1 ) dx. G The set Ω + is given the weakest topology for which all the spherical Fourier transforms are continuous. Then there exists a unique measure µ (called spherical Plancherel measure) such that f(x) 2 dx = ˆf(ω) 2 dµ(ω), G Ω + 14

21 for any K-bi-invariant function f L 1 (G) L 2 (G). Moreover, the map f ˆf extends to an isomorphism of Hilbert spaces between L 2 (K\G/K) and L 2 (Ω + ) (Plancherel-Godement Theorem). The spherical functions arise in a natural way in the study of unitary representations of G. A unitary representation π of G on the Hilbert space H π is said to be of class one if there exists a non-zero vector ξ H π which is fixed by the action of π(k), for any k K. We denote by Ĝ the set of equivalence classes of irreducible unitary representations of G and by Ĝ1 the set of equivalence classes of representations of class one. If π is of class one and (G, K) is a Gel fand pair, then the dimension of the subspace invariant under π(k) is equal to one: if 1 π is a vector of norm one which is invariant under the action of π(k), then the function ω π, defined by setting ω π (x) = 1 π, π(x)1 π, is a positive defined spherical function for (G, K) and it only depends on the class of equivalence of π [14]. Note that ˆf(ω π ) = f(x)ω π (x 1 ) dx = π(f)1 π, 1 π, (2.3.3) for all f L 1 (G) L 2 (G). G Theorem Let G be unimodular and assume that (G, K) is a Gel fand pair. Then the map [π] ω π (2.3.4) is a bijection between Ĝ1 and Ω +. Proof. See [14, Theorem 3.7]. Finally, if G is second countable, there is a canonical way to define a σ-algebra on Ĝ (Mackey Borel structure [11]). From now on, we assume that G is second countable and that there is a Plancherel formula for G, i.e., we assume that there exists a measure µ on Ĝ such that f 2 L 2 (G) = π(f) 2 HS d µ([π]), (2.3.5) Ĝ for all f L 1 (G) L 2 (G). Here, π(f) HS denotes the Hilbert Schmidt norm of the operator π(f), i.e., the square root of the trace of the operator π(f) π(f). 15

22 Remark If F is a totally disconnected local field and G is the group of F-rational point of a simply connected simple algebraic group, then F induces on G a topology which makes G a second countable locally compact group. The existence of the Plancherel measure µ has been proved (in a more general case) by Harish-Chandra [13] (see also [26] and [27]). We will see later that if K is suitably chosen, then (G, K) is a Gel fand pair [23] so that the results of this section hold for G. We want to show now that, when restricted to Ĝ1, the full Plancherel measure µ and the spherical Plancherel measure µ coincide, i.e., that µ([π]) = µ(ω π ), [π] Ĝ1. Fix a function f C c (K\G/K) and let π be a irreducible unitary representation on G. Then, π(f)ξ = f(x)π(x)ξ dx = f(xk)π(x)ξ dx dk G K G [ ] = f(x)π(x) π(k 1 )ξ dk dx (2.3.6) G K [ ] = π(f) π(k )ξ dk. Note that the vector K K π(k )ξ dk is invariant under π(k) for all k in K and ξ in H π, hence equal to zero whenever π is not of class one. Then, in this case, π(f) = 0. On the other hand, assume that π is of class one. Then K π(k )ξ dk is a multiple of 1 π, π(k )ξ dk = c(ξ)1 π, where, K c(ξ) = c(ξ)1 π, 1 π = = K K = ξ, 1 π. 16 π(k)ξ, 1 π dk ξ, π(k 1 )1 π dk

23 By (2.3.6), we see that π(f)ξ = ξ, 1 π π(f)1 π, so that the operator π(f) is a composition of the orthogonal projection on the one dimensional space spanned by 1 π and a multiplication by π(f)1 π, 1 π. Then { π(f) 2 π(f)1 π, 1 π 2 = HS = ˆf(ω π ) 2 if [π] Ĝ1 0 if [π] Ĝ\Ĝ1, so that We conclude that G f(x) 2 dx = π(f) 2 HS d µ([π]) Ĝ = π(f) 2 HS d µ([π]). Ĝ 1 = ˆf(ω π ) 2 d µ([π]). Ω + µ Ĝ1 ([π]) = µ(ω π ), (2.3.7) by the arbitrariness of f and the uniqueness of the spherical Plancherel measure µ. Proposition Let Ω + denote the set of positive definite spherical functions on G relative to K and µ the spherical Plancherel measure on Ω +. Then, for any compactly supported continuous function f on G, Ω + π(f) 2 HS dµ(ω π ) f 2 L 2 (G). Proof. It is a consequence of formulae (2.3.5) and (2.3.7): f 2 L 2 (G) = π(f) 2 HS d µ([π]) π(f) 2 HS d µ([π]) Ĝ Ĝ 1 = π(f) 2 HS dµ(ω π ). Ω + 17

24 Chapter 3 Groups of p-adic type and their structure Consider the group of F-rational points of a simply connected, simple, linear algebraic group defined over a totally disconnected field F. One of the main results of the theory of Bruhat and Tits (see [4] and [5]) is that there exist a reduced irreducible root system Σ 0, a family of subgroups {U α } (where α runs over the associated affine root system Σ), a subgroup N and a homomorphism ν from N onto the affine Weyl group W satisfying a certain set of axioms. Following [23], we give an axiomatic definition of group of p-adic type as an abstract group G, endowed with a triple (N, ν, {U α } α Σ ) satisfying that set of axioms. In the first section we illustrate the ideas by describing the triple (N, ν, {U α } α Σ ) when G is SL(d, F). In the next section we report some results (from [23]) and, as in [7], we introduce certain subgroups M j which will play a crucial role in the proof of the Kunze-Stein phenomenon. In the last two sections we prove some topological properties and integral formulae related to the Bruhat decomposition of these subgroups: the resemblance between the results here obtained (which are new, as far as I know), and the analogous formulae for Lie groups is apparent (see [14] or [7]), but the techniques involved in their proof quite different. 3.1 An example Let G = SL(d + 1, F) be the group of matrices of determinant one with entries in a field F. This group is associated to the root system Σ 0 of type A d 18

25 described in Section 2.1. For each a = e i e j in Σ 0, define a map u a : F G by setting u a (x) = I + xe ij, where I is the unit matrix and E ij is the matrix with 1 in the (i, j) th -place and 0 elsewhere. Then we can consider the subgroups U (a) := u a (F) which are isomorphic to the additive group of F. Let Z be the diagonal subgroup of G and denote by N denote its normalizer. Then there is a canonical epimorphism ν 0 from N and the Weyl group W 0 associated to the root system Σ 0. This is the group of permutation of r + 1 elements. If F is a totally disconnected non-discrete local field with discrete valuation v( ), then, for any a Σ 0, the filtration {x F : v(x) k} k Z of the additive group of F induces a filtration in the groups U (a), related to the affine root system Σ associated to Σ 0. For any affine root α = a + k Σ, let U α := {u a (x) : v(x) k}. It is clear that k>0 U α+k = {1} and that U α+1 U α. Moreover, k Z U a+k is the group previously denoted by U (a). The connection between the affine root structure and G is much deeper; the map ν 0 has a unique extension to an homomorphism ν from N onto the affine Weyl group W. The restriction of ν to Z is described as follows: for any z = diag(x 1,..., x r+1 ), let ν(z) be the translation in W which maps the origin 0 to ( v(x 1 ),..., v(x r+1 )) a. The kernel of ν is therefore the group M of diagonal matrices with entries in U 0 = {x F : v(x) = 0} and the diagonal group Z is given by Z = ν 1 (T ), where T is the subgroup of the translations in W. Moreover, T = ν(z), so that T is isomorphic to the abelian free group Z/M Z d (according with Theorem 2.1.3). 19

26 This link between the group and the root system is extremely useful for describing the relations between various subgroups of G. For instance, consider an element n = diag(x 1,..., x r+1 ) Z N and a generic element u a (x) U a+k (we assume here that v(x) = k). a = e i e j, then nu a (x)n 1 = u a (x i x 1 j x) U β, where β = e i e j + v(x i x 1 j x) = e i e j + v(x i ) v(x j ) + k. Note that, for any y a If β(y) = (e i e j + v(x i ) v(x j ) + k)(y) = (e i e j )(y) + v(x i ) v(x j ) + k = (e i e j + k)(ν(n) 1 (y)) = (ν(n)a)(y), according to (2.1.2) and (2.1.1). We have proved a particular case of the more general property (I) below. (I) For any n N and α Σ, nu α n 1 = U ν(n)α. In particular, M = ker ν normalizes each U a. As we have already noted, the triple (N, {U α } α Σ, ν} also satisfies the following. (II) If k is a positive integer, then U α+k U α and U α+k = {1}. k>0 In particular, k Z is a group, which is denoted by U (a). U a+k 20

27 It follows that Z = ν 1 (T ) normalizes each U (a). Consider now the groups and U = U (a), a Σ + 0 Ū = U (a), a Σ + 0. For G = SL(d + 1, F), they are the upper and lower diagonal groups. Since Z normalizes each U (a), we also have that B 0 = ZU is a group. This is the full upper diagonal group. The following is therefore clear: (III) Let U and Ū be the groups generated by U (a) with a Σ + 0 Σ + 0. Then Ū ZU = {1}. and a There are other nice properties satisfied by the triple (N, {U α } α Σ, ν) (see proof, see [29]). (IV) If β = α + k with k > 0, then (V) For any α Σ, U β, M, U α = U α MU β. U α, M, U α = (U α ν 1 (w α )U α ) (U α MU α+1 ). For α and β in Σ let [α, β] denote the set of all affine roots of the form la + mβ with l and m positive integers, and let [U a, U β ] be the subgroup of G generated by all commutators uvu 1 v 1 with u U a and v U β. The following is called Chevalley s commutator formula. (VI) If α cannot be written as β + k for any k Z, then [U α, U β ] U [α,β]. (VII) G is generated by N and the U α, with α Σ. 3.2 Groups with an affine structure We use here the notation introduced in Section 2.1. In particular a is an Euclidean space of dimension d, Σ 0 is an irreducible reduced root system in a, Σ is the associated affine root system and W 0 and W are the related Weyl groups. Moreover, c 0 and c are the chosen positive chambers, Σ + 0 and Π 0 are the positive and simple roots corresponding to c 0. Motivated by the example of the previous section, we give the following axiomatic definition. 21

28 Definition Let Σ 0 be a reduced irreducible root system and let Σ and W be the associated affine root system and affine Weyl group. An affine root structure of type Σ 0 on a group G is a triple (N, ν, {U α } α Σ ), where N and U α are subgroups of G and ν is a homomorphism of N onto W satisfying the axioms (I) (VII) of the previous section. From now on G will be a group endowed with an affine root structure. According to the definitions given in (I) (VII), we have the subgroups U (a) = k Z U a+k, a Σ 0, the subgroups U and Ū, generated by the U (a) when a runs through the positive and negative roots, the subgroup Z = ν 1 (T ), where T is the subgroup of the translations in the affine Weyl group W, the group M = ker ν and the group N of the Definition Proposition Let U and Ū be as defined in Axiom (III). Then, the positive roots can be arranged in a sequence (a 1,..., a n ) such that U = U (a1 ) U (an) and Ū = U ( an) U ( a1 ), with uniqueness of expression, and U (ai )U (ai+1 ) U (ar) and U ( ar) U ( ai+1 )U ( ai ) are normal in U and Ū, for i = 1,..., r. Moreover, U = U (a) and Ū = U ( a), a Σ + 0 a Σ + 0 with uniqueness of expression, the positive roots being taken in any fixed order. Proof. See [23, Proposition 2.6.3]. 22

29 We want now to construct other families of subgroups of G by considering groups generated by U α when α is in suitable subsets of Σ. Let Π 0 = {a 1,..., a d } be the set of simple roots relative to the chamber c 0 and ordered as in Lemma For any j = 1,..., d, consider the root system (Σ 0 ) j on (a ) j = a span R {a 1,..., a j } defined in Section 2.1. We will write (Σ) j for the associated affine root system and (W 0 ) j and (W ) j for the relative Weyl groups. In view of Proposition 2.1.1, (Π 0 ) j = Π 0 (a ) j is a fundamental system for (Σ 0 ) j, so that (Σ + 0 ) j = Σ + 0 span R {a 1,..., a j } is the corresponding set of positive roots and (Σ 0 ) j = Σ 0 span R {a 1,..., a j } = (Σ + 0 ) j (Σ + 0 ) j. For any j = 1,..., d, let T j be the group generated by the translation t aj associated to the simple root a j and set T j = T 1 T j. Then T j is the group of translations in (W ) j. Let Z j = ν 1 (T j ), j = 1,..., d and Z j = ν 1 (T j ), j = 1,..., d. Then M = ker ν is normal in Z j, and Z j for all j = 1,..., d and Z j = Z 1 Z j, j = 1,..., d. Let M 0 = M and M j = M, {U (a) : a (Σ 0 ) j }, j = 1,..., d. 23

30 Finally, for j = 1,..., d, let (Σ + 0 ) j = (Σ + 0 ) j /(Σ + 0 ) j 1 (where (Σ + 0 ) 0 = ) and define U j and U j by setting U j = U (a) : a (Σ + 0 ) j and U j = U (a) : a (Σ + 0 ) j. Analogously, let Ū j = U (a) : a (Σ + 0 ) j and Ū j = U (a) : a (Σ + 0 ) j. In view of Proposition we have that U and Ū decompose as U = d U j and Ū = j=1 with uniqueness of expression. Example 5. We give an explicit expression for the groups above when G = SL(d + 1, F). As already noticed, Z is the diagonal group in G and M is the diagonal group with entries in the unit sphere of F. Write I j for the identity matrix in SL(j, F). Then, d j=1 Ū j Z j = {mz j (x) : m M, x F }, where z j (x) is the matrix given by I j z j (x) = 0 x x I r j. Next, U j and Ūj are the groups whose elements u j (x) and ū j (x) are defined by setting I j x 0 I j 0 0 u j (x) = and ū j (x) = x 1 0, x F j, 0 0 I r j 0 0 I r j and, finally, M j is the group generated by M and the subgroup of matrices of the form ( ) A 0, 0 I d j with A SL(j + 1, F). 24

31 Proposition Let N j = M j N and let ν j be the restriction to N j of the map ν, and let (Σ) j be the affine root system associated to (Σ 0 ) j. Then (M j, {U α } α (Σ) j, ν j ) is a group with an affine structure of type (Σ 0 ) j. Proof. In view of Lemma 2.1.2, the root system (Σ 0 ) j is irreducible for all 1 j r. It is also reduced because Σ 0 is reduced. Then we have to prove that (M j, {U α } α (Σ) j, ν j ) satisfies the axioms of the definition. Axiom (III) is clear because Ū j Z j U j Ū ZU = {1}. Axioms (I) (V) follow immediately because G is a group with an affine structure of type Σ 0. Next, note that (Σ 0 ) j is closed in Σ 0, i.e., a + a (Σ 0 ) j whenever a and a are in (Σ 0 ) j and a+a Σ 0, so that (VI) also follows immediately. All we have to check is Axiom (VII). If a (Σ 0 ) j, then a + k (Σ) j, so that U (a) = k Z U a+k {U α } α (Σ) j, a (Σ 0 ) j. On the other hand, It follows that M N M j = N j. N j, {U α } α (Σ) j contains the generators of M j, hence M j, and Axiom (VII) is proved. Theorem Let G be a group endowed with a structure affine of type Σ 0. Denote by ν 0 the epimorphism N W 0 obtained by composing ν : N W with the epimorphism W W 0 defined by the semidirect decomposition W = T W 0. Then the following hold: (i) let B 0 be the group generated by Z = ν 1 (T ) and those U (a) such that a is positive on c 0. Then (B 0, N) is a BN-pair on G. We have that B 0 N = Z, so that ν 0 induces an isomorphism of N/(B 0 N) onto the Weyl group W 0. The distinguished generators of N/(B 0 N) correspond under this isomorphism to the reflections in the walls of the chamber c 0 ; (ii) let B be the group generated by M = Ker(ν) and those U α such that α is positive on c. Then (B, N) is a BN-pair on G. We have that 25

32 B N = M, so that ν induces an isomorphism of N/(B N) onto the affine Weyl group W. The distinguished generators of N/(B N) correspond under this isomorphism to the reflections in the walls of the affine chamber c. Proof. See [23, Theorem and 2.6.9]. Note that Z normalizes each U (a) in view of Axiom (I) and that, by definition, a root a is positive on c 0 if and only if a Σ + 0. Therefore the group B 0 in the theorem above is given by B 0 = ZU. The map ν 0 : N W 0 of Theorem identifies N/Z with W 0. Let w W 0 and ẇ ν0 1 (w), so that ν0 1 (w) = ẇz. Then B 0 ẇb 0 = ZUẇZU = B 0 ν 1 (w)b 0, so that B 0 ẇb 0 does not depend on the particular choice of ẇ in ν 1 0 (w). We will therefore write B 0 wb 0 instead of B 0 ẇb 0. Corollary Let G be a group with affine structure of type Σ 0. Then G = w W 0 B 0 wb 0, where the union is disjoint. Proof. This is a direct application of Proposition and the first part of Theorem Proposition The groups G and M d coincide, and Z j is contained in M j for j = 1,..., d. Proof. Let α and w α be an affine root and the corresponding reflection in W. In view of Axiom (V), ν 1 (w α ) U α, U α, M, α Σ. (3.2.1) 26

33 This implies that M d contains ν 1 (w α ) for any α in (Σ) d = Σ, hence N = ν 1 (W ). Therefore M d contains N and {U α } α Σ, which generate the whole group by Axiom (VII), so that G = M d. The other part is similar: consider a root a (Σ 0 ) j and the related translation t a. Since ν is surjective, then ν 1 (t a ) = ν 1 (w a w a+1 ) = ν 1 (w a )ν 1 (w a+1 ). The inclusion (3.2.1) now implies that ν 1 (t a ) M j, a (Σ 0 ) j. The proof is complete because the group T j of the translations in (W ) j is generated by elements of type t a = w a w a+1 with a (Σ 0 ) j and Z j = ν 1 (T j ). Corollary Let B j 0 = Z j U j. Then (B j 0, N j ) is a BN-pair for M j. Proof. We already know that (N j, (U α ) α (Σ) j, ν j ) is a group with an affine structure of type (Σ 0 ) j. We have to show that B j 0 is that of part (i) of Theorem Since U j is the group generated by the U (a) when a runs through the positive roots (Σ + 0 ) j, all we have to prove is that Z j = (ν j ) 1 (T j ). But this is obvious because Z j M j, Z j = ν 1 (T j ) N and ν j is the restriction of ν to N M j. Proposition For j = 1,..., d, Z normalizes M j and M j 1 normalizes U j and Ūj. Proof. The first part is clear because Z normalizes each U (a) and M is normal in Z. Then Z normalizes the subgroups generating M j, hence the whole group. For the second part we shall just consider the case involving U j, the other being similar. We have to show that, if u U j and m M j 1, then mum 1 U j. (3.2.2) 27

34 Since, by definition, and U j = {U (a) : a (Σ + 0 ) j = (Σ + 0 ) j \(Σ + 0 ) j 1 } M j 1 = M, {U (a) : a (Σ 0 ) j 1 }, it is enough to prove (3.2.2) when u is in U (a), with a (Σ + 0 ) j, and m is in M or U (a ), with a (Σ 0 ) j 1. If m is in M, then (3.2.2) is true because M Z normalizes each U (a), a Σ 0. Next, if m is in U (a ) with a (Σ 0 ) j 1, write j 1 a = k ia i i=1 to express the expansion of a with respect to {a i : i = 1,..., j 1}. Analogously, if a (Σ + 0 ) j = (Σ + 0 ) j \(Σ + 0 ) j 1, write a as j a = k i a i, i=1 and note that k i 0 for all 1 i < j and k j > 0 (because a (Σ + 0 ) j but a / (Σ + 0 ) j 1 ). For any positive integers l and l such that j 1 l a + la = (l k i + lk i )a i + lk j a j i=1 is a root, l a + la has to be in (Σ + 0 ) j (because lk j > 0). By Axiom (VI), we conclude that mum 1 u 1 U j, and the proposition is proved. Corollary For all j = 1,..., d, M j 1 Z j U j is a group. Proof. Let mzu and m z u be in M j Z j U j. Then mzum z u = mzm z ( (m z ) 1 um z ) u = m(zm z 1 )zz u u = mm zz u u so that M j 1 Z j U j is closed under multiplication. Analogously, (mzu) 1 = u z m = z m u = m z u, so that M j 1 Z j U j is also closed under taking inverses. 28

35 3.3 Groups of p-adic type Let G be a group with affine root structure (N, ν, {U α } α Σ ) of type Σ 0 and let B be as in part (ii) of Theorem Suppose in addition that G is a topological group such that (VIII) N and U α are closed subgroups of G; (IX) B is a compact open subgroup of G. Definition A topological group with affine root structure of type Σ 0 satisfying (I) (IX) will be called a simply connected group of p-adic type. The term p-adic type in the definition above is motivated by the following reason: if F is a totally disconnected local field (e.g., F is the field of p-adic numbers), then the group of the F-rational points of a simply connected simple linear algebraic group defined over F inherits a topology from F and an affine structure which satisfy Axioms (I) (IX) (see [23, p. 35] and [5]). For the rest of the section we shall assume that G is a simply connected group of p-adic type. We shall present some consequences of (VIII) and (IX) after the following lemma. Lemma Let 1 j d. Then M j ŪZU = Ū j Z j U j. Proof. In view of Axioms (IV), (V) and (VI), in order to write an element in Z as a product of elements in M and in the U α, where α Σ, one has to multiply elements in M, U (a) and U ( a). In this case, one gets the group generated by ν 1 (t a ) (i.e., Z j when a = a j for some simple root a j ), where as usual t a is the translation associated to the root a. When considering M j we are restricted to deal with those U α such that α (Σ) j : it follows that the intersection between M j and Z i is equal to M whenever i > j. On the other hand Z j M j, so that M j Z = Z j. Now let m = ūzu be in M j ŪZU. Then mu 1 z 1 = ū = ū d ū 1 is in the group generated by M j, U and Z, i.e., it is a product of elements in U (a), with a (Σ 0 ) j Σ + 0, and Z. It follows that, by Axioms (IV), (V) and (VI), ū d = ū r 1 = = ū j+1 = 1, 29

36 so that m may be written as m = ū j ū r 1 ū 1 zu. Arguing in a similar manner, one shows that, if u = u 1 u d, then Therefore and the lemma follows. u j+1 = = u d 1 = u d = 1. z = (ū j ū 1 ) 1 m(u 1 u j ) 1 M j Z = Z j Theorem Let G be a simply connected group of p-adic type. Then (i) M is a compact subgroup; (ii) for each a Σ 0, the group U (a) is closed in G and, for each α of the form α = a + k, U α is a compact open subgroup of U (a) ; (iii) U, Ū and B 0 = ZU are closed subgroups of G; (iv) M j is closed. Proof. By the second part of Theorem 3.2.4, M = B N. Then part (i) follows from Axioms (VIII) and (IX). For part (ii) and (iii), see [23, p. 36]. For part (iv), consider B M j. In view of [23, Prop 2.6.4], B may be written as B = U (c) MU+ (c), where U (c) and U + (c) are suitable compact subgroups of Ū and U. In particular, B ŪZU. (3.3.3) Then B M j = B ŪZU M j = B Ū j Z j U j = (U (c) Ū j )M(U + (c) U j ), where the last equality follows from the fact that an element in ŪZU may be written in a unique way as ūzu. Then B M j is both compact, M and U ± (c) being compact, and open M j, B being an open subgroup of G by (IX). It follows that M j, being a group, is locally compact, hence close (see [15]). 30

37 3.4 Haar measures and integral formulas Lemma Let 1 j d and let be a continuous homomorphism from M j to R +. Then = 1. Proof. The only compact subgroup of R + is {1}. Since a continuous homomorphism maps compact subgroups into compact subgroups, it follows that the kernel of must contain all compact subgroups of M j. In particular is trivial on M and U α, for any α Σ. Since U (a) is the union of some U α and M j is generated by some U (a) and M, it follows that is trivial on the whole group. In view of Proposition 3.2.8, M j 1 acts by conjugation on Ūj for all j = 1,..., d. Since each action leads to a continuous homomorphism, the following is automatic. Corollary For all j = 1,..., d, the action of M j 1 on Ūj and U j is unimodular and M j is itself unimodular. In particular, G = M d is unimodular. Observe now that each U α is compact, hence unimodular. It follows also that U (a), U j, U j, Ū j, U and Ū, being union of compact subgroups, are unimodular. On the other hand B 0 and B j 0 are not unimodular. Lemma Let du and dū be the Haar measures on U and Ū and, for each a Σ 0, let du a be the Haar measure on U (a). Then du = du a and dū = du a. (3.4.4) a Σ + 0 a Σ + 0 Proof. It follows from Proposition (see [23, p. 40]). Recall that Z normalizes U, and denote by the Jacobian of the transformation u zuz 1. In symbols, if du is the Haar measure on the locally compact unimodular group U, (z) = d(zuz 1 ). (3.4.5) du 31

38 Denote now by d l (zu) and d d (zu) the left and right Haar measures on B 0 = ZU, by dz the Haar measure on Z and by B0 the modular function on B 0. Since Z and U are unimodular, and d l (zu) = dz du d r (zu) = B0 (zu)d l (zu) = B0 (zu) dz du. Lemma The modular function B0 is trivial on U and is equal to. Proof. Since U is union of compact subgroups, B0 is trivial on U, so that B0 only depends on Z. For a continuous compactly supported function f, f(z u ) d r (z u ) = f(z u z) d r (z u ) B 0 B 0 = f(z zz 1 u z) B0 (z ) dz du Z U = f(z (z 1 u z)) B0 (z z 1 ) dz du Z U = B0 (z) 1 (z) f(z u ) B0 (z ) dz du B 0 = B0 (z) 1 (z) f(z u ) d r (z u ). B 0 It follows that and the lemma is proved. B0 (zu) = (z), Theorem Let dg, dū and du denote Haar measures on G, Ū and U. Then ŪZU is an open subset of full measure in G and, for any measurable function f on G, f(g) dg = f(ūzu) (z) dū dz du. (3.4.6) G Ū Z U Proof. Consider the subgroups B 0 = ZU and Ū. In view of formula (3.3.3), ŪB 0 contains the subgroup B, which is an open neighborhood of the identity in G,by Axiom (IX). For any ūb ŪB 0, ūbb ūūb 0b = ŪB 0 32