THE CONNES-MARCOLLI GL2-SYSTEM

THE CONNES-MARCOLLI GL

-SYSTEM

MASTER THESIS

Bjarte D. Berntsen

Fifteen years ago Bost and Connes constructed a C^dynamical system with the Calois group G(Q^ab/Q) as symmetry group and with phase tran- sition related to properties of L-functions . Since then there have been nu- merous , and only partially succesful , attempts to generalize the system to arbitrary number fields . A few years ago , in order to extend that con- struction to imaginary quadratic feilds , Connes and Marcolli constructed a GL2system , an analogue of the BC-system with Q^ replaced by GL₂(Q).

They classified the KM S_states of the system for  > 2. Later Laca , Larsen and Neshveyev classified theKM Sstates for all = 0,1.

1. Proper Gruppevirkning og Gruppoide C

-Algebraer

Let G be a group and X be a set. A Group action on a set X is a homomorphism  from the group G to the group Homeo(X) of all home- omorphisms from X to itself ( Aut(X)). Thus to each g  G is associated a homeomorphism (g) : X X , which for notational simplicity we write simply as g :X X.With this notation for the map :

GX  X (g, x)  gx with conditions :

(e, x)ex=x,xX (1)

(g,(hx)) = (gh)x,xX,g, hG are equivalent to requiring to be a homomorphism.

Under these conditions , we say that X is a left G-set and we have a left group action by G on X. Similarly , one can define a right action by letting the elements of the group act on the space from the right instead.

We consider cases where the group G is countable and X is a locally compact second countable topological space.

A continous mapf :X Y is called proper if for every compactK Y , the space f^1(K) is also compact. Accordingly , an action of G on X is called proper if the map :

GX  XX (2)

(g, x)  (gx, x)

is proper. Then the space G/X , where points are identitied by the equiva- lence relation of laying on the same G- orbit {Gx} is Hausdor. Assume that G is a discrete group. Consider GX . The space X , which is a G-space is called the unit space of GX. GX has the product topology and the two maps , called the source map (s) and the range map (r) :

S, R : GX X s(g, x)  x

r(g, x)  gx

define a law of composition : ((g, y),(h, x))(GX)² (g, y)·(h, x)G X , where :

(GX)² :={((g, y),(h, x))(GX)(GX)|r(h, x) =s(g, y) = y} We see that the product onGX are defined by the formula :

(g, hx)(h, x) = (gh, x)

In this wayGX becomes a groupoid (called the transformation groupoid) , since every element has an inverse :

(g, x)^1 = (g^1, gx)

GX has stabilizer subgroup Gx ={gG|gx = x} If G has stabilizer subgroup equal to {e} for every x in X is equivalent to saying that the action of G on X is free i.e. an action whitout fixpoints for other elements of G than the identity.

The set Cc(GX) of all continous functions on GX with compact sup- port has a structure of involutive algebra given by :

(f1f₂)(g, x) = 

hG

f₁(gh^1, hx)f2(h, x) f^(g, x) = (f((g, x)^1¯)¯) = (f(g^1, gx)¯)

, where (g,x)^1= (g^1,gx) Let C0(X) be the algebra of continous functions on X that vanish at infinity. The product in C0(X) is the usual pointwise product.

If the restriction of the action to a subgroup  of G is free and proper , we can introduce a new groupoid : \G^X by taking the quotient of GX by the action of defined by :

(₁,₂)(g, x) = (₁g^₂¹,₂x)

The unit space of \G^X is \X , and the product is induced from that on G X. If the action of  is proper but not free , the quotient space \G^X is no longer a groupoid , since the composition of classes using representatives will in general depend on the choice of representatives.

Nevertheless , the same formulas for convolution and involution as in the groupoid case give us a well -defined algebra. To see this , consider the space C_c(\G^X)of continous compactly supported functions on\G^X.The elements can be considered as () -INVARIANT functions on GX.

The convolution of two such functions are defined accordingly : 1. (1.1)

(f1f₂)(g, x) = 

h\G

f₁(gh^1, hx)f2(h, x).

To see that the convolution is well-defined :

Assume the support of f_i is contained in (  )({g_i}  U_i) , where g_i  G andU_i is a compact subset of X.(i=1,2). Let {₁, ....,_n} be the set of elements  such thatg₂U₂U₁ =.This set is finite since the action of  is assumed to be proper.

Iff₂(h, x)= 0 , then there exist such thath^1 g2 andxU₂. Since the number of ´s such thatxU₂ is finite , the above sum must be finite. If furthermoref₁(gh^1, hx)= 0 , thengh^1 =_ag₁^_b¹ for some_a,_b

, since (gh^1, hx) is

contained in the support off₁ . We can replace h by another representa- tive of the right coset h . If we replace h by_bh , then gh^1 =_ag₁ g1, and alsohxU₁.If nowh^1 = ˜g₂with˜ , we gethx= ˜g₂x˜g₂U₂. Hence˜must be equal to_i , for some i , and thereforeg g1h=g1_ig₂.

Thus the support of f₁ f₂ is contained in ⁱ()({g₁_ig₂}U₂). Thus the set of representativesg_i giving a nonzero contribution to the above sum are finite and independent of the choice of   . The support of f₁f₂ is contained in a compact set , sof₁f₂ C_c(\G^X), and the latter space becomes a well-defined algebra. The convolution is also associative :

(f1(f2f₃))(g, x) = 

t\G

f₁(gt^1, tx)(f2f₃)(t, x) = 

t,h\G

f₁(gt^1, tx)f2(th^1, hx)f3(h, x) ((f1f₂)f₃)(g, x) = 

h\G

(f1f₂)(gh^1, hx)f3(h, x) = 

t,h\G

f₁(gh^1t^1, thx)f2(t, hx)f3(h, x) =tth^1



t,h\G

f₁(gt^1, tx)f2(th^1, hx)f3(h, x)

1. (1.2) Define also aninvolution onC_c(\G^X) by : f^(g, x) =f((g, x)^1¯) = f(g^1, gx¯)

If the support of f is contained in ()({g0}U) for g0  G and compact U X , then the support of f^ is contained in :

(()({g₀}U))^1 = ()({g₀}U)^1 = ()({g^₀¹}g₀U) , which is a compact set in (\G^X)and thereforef^ C_c(\G^X)for every f C_c(\G^X).

For each xX , define a representation : 1. (1.3)

_x : C_c(\G^X)B(l²(\G))

_x(f)h = 

g\G

f(gh^1, hx)g

Here _g denotes the characteristic function of the coset g . Consider

_g as a one of the unit basis vectors in the (standard) orthonormal basis {_g}^g\G forl²(\G).

Lemma 1 1.1 For each f C_c(\G^X) the operators_x(f) , xX , are uniformly bounded.

Proof. For₁,₂ l²(\G) we have :

|_x(f)·₁,₂|  

g,h\G

f(gh^1, hx)· |₁(h)| · |₂(g)|





 

g,h\G

f(gh^1, hx)· |₁(h)|²





1 2

·



 

g,h\G

f(gh^1, hx)· |₂(g)|²





1 2

.

(Applying Hølders inequality.)

Thus if we denote byfI the quantity :

max



 sup

xX,hG



g\G

f(gh^1, hx), sup

xX,gG



h\G

f(gh^1, hx)



, we get_x(f)  fI for any x  X, so it suces to show that f_I is finite. Replacing x by h^1x and g by gh in the first supremum above , we see that this supremum equals :

fI,s:= sup

xX



g\G

|f(g, x)|

Asf^(hg^1, gx) = (f((hg^1, gx)^1))^ = (f(gh^1, hg^1gx))^ = (f(gh^1, hx)^ , we see that f(gh^1, hx) = (f^(hg^1, gx))^. Then the second supremum must be equal to f^I,s. Therefore fI = max

fI,s,f^I,s



. Now , the claim is thatfI,s is finite for everyf Cc(\G^X).If this claim is true , the Lemma is proved.

Proof of Claim :

We may assume without loss of generality that the support off is con- tained in ()({g₀}U) for some g₀  G , and compact U  X. Since

the action of  is proper , there exists n  N such that the sets _iU , i = 1, ..., n+ 1 have trivial intersection for any dierent ₁, ...,_n+1  .

Now if f(g, x)= 0 for some g andx,there exists such that g^1 g0

and xU. Since the number of ´s such that x U is at most n , we see that for each xX the sum in the definition offI,s is finite .

To see that

is a representation , one has to check :

) _x(f^) = (x(f))^

) _x(f1f₂) = _x(f1)·_x(f2)

_x(f)·_h = 

g\G

f(gh^1, hx)·_g Consider f  C_c(\G^ X). As f = 

s\Gf(s, x)·_s observe that the vector  = _e in l²(\G) is both cyclic and tracial for operators in B(l²(\G))

U_g_e,_e = 1 if g =e , 0else , therefore , for all g_i  Gwe have :

U_g1U_g2_e,_e = U_g1U_g2_e,_e

)

(x(f)h,_t) = 

g\G

f(gh^1, hx)(g,_t) =f(th^1, hx) similarly ,

(h,_x(f^)t) = f^(ht^1, tx¯) =f(th^1, hx).

Hence

_x(f)^ =_x(f^).

) Can checked similarly to associativity of the convolution.

But let me see this from another perspective :

For each x X , f can be thought of as a vector in l²(\G) ) . Let U_g be the unitary operator on l²(\G) defined by U_g_h = _hg1. Expanding on the cyclic and tracial vector _e gives :

f = ( 

g\G

f(g, x)U_g^)e.

For eachxX ,f can be thought of as a vector inl²(\G)) , expanding its adjoint on the cyclic and tracial vector _e gives :

f^ = ( 

g\G

U_g(f^(g, x)))e= 

g\G

U_g(f^(g, x)U_g^U_g_e= 

g\G

U_g(f^(g, x))U_g^_g1

Proof. Letf₁and f₂be two functions inC_c(\G^X). Then , for arbitrary h \G :

(x(f1f₂))h = 

g\G

(f1f₂)(gh^1, hx)·_g = 

g\G



t\G

f₁(gh^1t^1, thx)·f₂(t, hx)·_g

= 

g\G



t\G

U_hf₁(gt^1, thx)·f₂(t, hx)·_g = 

g\G



t\G

U_hf₁(gt^1, thx)·U_h^1·f₂(th^1, hx)·_g

= 

g\G



t\G

U_hf₁(gt^1, thx)·U_h^·f₂(th^1, hx)·_g = 

t\G



g\G

U_hf₁(gt^1, thx)·U_h^·f₂(th^1, hx)·_g

= 

t\G

U_h(hx(f1))U_h^ ·_t·f₂(th^1, hx) = (x(f1)·_x(f2))·_h so)is checked in this way of thinking . Hencex is a representation for every (fixed) xX.

Definition 2 We denote by C_r^(\G^X)the completion of C_c(\G^X) in the norm defined by the representation :

(^xX_x) :C_c(\G^X)B(^xXl²(\G) , that is ,

f= sup

xX_x(f)

Remark 3 As we observed above , for everys Gand its associated unitary U_s B(l²(\G)) such that U_s_h =_hs1 , f C_c(\G^X) and _x the representation defined above , we have

U_s_x(f)U_s^ =_sx(f).

Proof. Observe first that , for every g , s andh in\G , we have : U_s_x(f)U_s^·_h = _x(f)·_hs =U_s· 

g\G

f(gs^1h^1, hsx)·_g = 

g\G

U_s·f(gs^1h^1, hsx)·_g



g\G

f(gs^1h^1, hsx)·_gs1 = ^gl=gs1 

l\G

f(lh^1, hsx)·_l

 U_s_x(f)U_s^·_h = 

g\G

f(gh^1, hsx)·_h =_sx(f)·_h . Hence

U_s_x(f)U_s^ =_sx(f) Therefore

_x(f)=_sx(f) , and so

Remark 4

f= sup

xG\X_x(f)

Closely related is the notion of aC^dynamical system (A, G,), where Ais aC^-algebra,Ga locally compact group andis a homomorphism from G into Aut(A). A covariant representation of ( A, G , ) is a pair (, U) , where  is a *-representation of A on a HilbertspaceH and

sU_s

is a unitary representation ofGon the same H such that : U_s(A)U_s^ =(s(A)) ,

for all a A , sG.

Denote by _g the automorphism (g) for g in G. The Cross Product , A^Gof a C^-algebraAand a groupGis the universal C^-algebra generated by A and unitariesv_g ,g G such that :

1)v_gav^_g = _g(a)

2)g  v_g is a homomorphism , g  G

IfGis countable and discrete , the spaceC_c(A, G)of continous compactly supported A-valued functions onG is the algebra of all finite sums :

f =

tG

A_t·v_t with coecients in A.

One defines aC^-norm by :

f= sup

 (f) , as  runs over all *-representations ofC_c(A, G).

The supremum is always bounded by :

f1 =

tG

At

The supremum is always taken over a nonempty family of representations because certain representations can be explicitly constructed. Let be any

*-representation of A on a Hilbertspace H.Then one can always construct the representation :

˜

 : A^GB(Hl²(G) =B(H) ¯B(l²(G))

˜

(a)(g) = (^_g¹(a))(g)

˜

(vg)(_h) = _gh , for  H and g, hG.

Due to constuction , this representation is covariant :

˜

(vg)·(a)˜ ·(˜(vg))^(_h) = ˜(vg)·(a)(˜ _g1h) = ˜(vg)(h^1g(a)_g1h)

= (_h1g(a))(_h) = ˜(g(a))(_h) hence

˜

(vg)·(a)˜ ·(v˜ g)^ = ˜(g(A))

The Reduced Cross-Product , A ^r G is defined to be : = (A ^ G))/Ker(˜), where  is any faithful representation ofA.

The functionsf C_c(\G^X) can be considered as()-invariant functions onGX . Define an action ofG onC_c(\G^X) by :

_g(f) = f(h,(g^1x)).

Define for eachg G the following unitaries v_g on C_c(\G^X): v_gf(s, x) = f(sg, g^1x)

f(s, x)v_g^ = f(s, x)v_g1 =f(sg^1, x) For these

v_gf(s, x)v_g^ =f(s, g^1x) ,

and as we have seen , C₀(\X) can be considered as a subalgebra of Cc(\G^X) , so we have aC^dynamical system (C0(\X), G,).

Now , for eachxX , define a map :

˜

_x : C_c(\G^X)^GB(l²(\G)l²(G))B(l²(\G))B(l²(G))

˜

_x(f)(h_g) = _x(g^1(f))h_g

˜

_x(vg)(l_h) = _l_gh

By the calculation above , this representation is covariant for any Hilbertspace H to which C₀(\X)  C_c(\G^ X) can be represented on, so also for l²(\G). With U_s the unitary operator defined above , observe that :

(Us_x(f)U_s^1)(_s) = (sx(f)1)(_s) = (x(^_s¹(f))1)(_s) = ˜_x(f)(_s) , for f C₀(\X)C_c(\G^X)and l²(\G) . Then we have :

˜

_x(f)(_g) = _x(g^1(f))(_g) = (Ug_x(f)U_g^)_g = (Ug 1)(x(f)1)(U_g^1)(_g)

˜

_x(vg)(_h) = (1v_g)(_h) = _gh and we get :

˜

_x(vg)·˜_x(f)·(˜_x(vg))^(l_h) = ˜_x(vg)·˜_x(f)·(˜_x(vg))^(l_h)

= (1v_g)·˜_x(f)·(1v_g^)(l_h)

= (1vg)·˜x(f)(l_g1h) = (1vg)(Ug^1h1)(x(f)1)(Uh^1g1)(l_g1h)

= (1v_g)·(Ug^1h_x(f)Uh^1g1)(l_g1h)

= (x(h^1g(f))1)(l_h) = (x(h^1(g(f)))1)(l_h)

= ˜_x(g(f))(l_h)

From this we conclude that˜_x , for everyxX,U˜_g = (1v_g)and hence also(^xX˜_x) := ˜,U˜ is a covariant representation of(C0(\X), G,).(C0(\X) can be considered as a subalgebra ofCc(\G^X). The embedding :X  G X , x  (e, x). In this way \X is an open subset of \G^ X ,

and then the algebra C₀(\X)is a subalgebra of C_r^(\G^X).) Then , by the Universal Property of the Crossed Product , there exists a representation

 of C₀(\X)^ G into C^(˜(C0(\X)),U˜_g, g  G) obtained by setting

(f) = ˜(f(s, x))·U˜_s , forf =f(s, x)·_s =f(s, x)·U_s^ ·_e.

Observe thatf:= sup_xX_x(f)= sup_xX_x(g(f)= sup_xX_x(f(·, g^1x) by replacing x by gx , since for every x  X : _gx(f) = U_g_x(f)U_g^ ,

where U_g is the unitary operator on l²(\G) : U_g_h = _hg1. Therefore , and since_x(f)=˜_x(f), for everyxX , i conclude that the kernel of the representation˜_x is isomorphic toG , since s _s is a homomorphism andker ˜= 

xX

ker ˜_x = 

xX

G=G.

By the universal property ofC^(˜(C0(\X)),U˜_g, g G) := A , there is a Homomorphism H from this algebra onto C₀(\X)^G taking (f˜ ) B(^xX(Cxl²(\G)l²(G))) tof C0(\X) andU˜g to U_g^.

(The point is that . . . . the composed map C_c(\G ^ X)  C₀(\X)^rGextends to an isomorphism : C_r^(\G^X)C₀(\X)^rG . I will import a diagram above here to clarify this . )

C_r^(\G^X) is the completion of C_c(\G^X) with respect to the norm defined by the representation = (^xX_x) , f = sup_xX _x(f)l²

. Then by the first iso thm , C_r^(\G^X)C₀(\G)^rG.

For the special case when={e} , we have the following :

Claim 5 C_r^(GX) is isomorphic to C₀(X)^rG.

Proof. For each xX , define a map :

x : C_c(GX)B(l²(G)l²(G))B(l²(G))B(l²(G))

x(f)(h _g) = _x(g^1(f))h_g

x(vg)(l_h) = _l_gh

whereg(f)(x) =f(g^1x), for f C0(X).

Lemma 6 1.2

There exists a conditional expectation

E :C_r^(\G^X)C₀(\X) such that :

E(f)(x) = f(e, x) , for f C_c(\G^X) .

Proof. If B A are C^Algebraes , a map E : A B is called a Condi- tional Expectation if : ) E is a projection onto B.i.e. (E(x) =x , x B)

) E is Bbilinear : E(xy) = E(x)y and E(yx) = yE(x) , for all x  A , y B and ) E is Positive.

For each xX define a state_x on C_r^(\G^X) by :

_x(a) = (x(a)·_,_).

Then the function E(a) on X defined by : E(a)(x) = _x(a)

is bounded by a. As E(f)(x) = f(e, x) , for f  C_c(\G ^ X) ( since _x(f) = (x(f),_) = (

s\Gf(s, x) ·_s,_e) = f(e, x) ) , we conclude that E(a)C₀(\X) for every aC_r^(\G^X).Thus E is such a conditional expectation.

The Boxproduct

Let Y  X be a -invariant clopen subset (Y  Y).Then the charac- teristic function 1\Y of the set\Y is an element of the multiplier Algebra of C_r^(\G^X).See this by using the embeddingX GX , x(e, x) , to consider \X as an open subset of \G^ X , and then the algebra C0(\X)as a subalgebra of C_r^(\G^X).

Denote by\G^Y the quotient of the space : {(g, x) , g G, xY , gxY}

• by the action of :

(₁,₂)(g, x) = (₁g^₂¹,₂x)

Then

1\YC_c(\G^X)1\Y =C_c(\G^Y).

Therefore the algebra1\YC_r^(\G^X)1\Y , which we denoteC_r^(\G^ Y)is a completion of the algebra of compactly supported functions on\G^ Y with convolution product given by :

(f1f₂)(g, y) = 

h\G:hyY

f₁(gh^1, hy)·f₂(h, y) ,

and involution :

f^(g, y) =f(g^1, gy¯)

Observe that _x(1\Y) is the projection onto the subspace l²(\G_x) , where the subset G_x of Gis defined by :

G_x={g G| gxY} Then , forf C_c(\G^Y)andh G_x we have :

_x(f)h = 

g\Gx

f(gh^1, hx)g

So ifx /GY ,_x(f) = 0 in particular. We saw above that the represen- tations _x and_gx are unitarily equivalent for anyg  G.Therefore we can conclude thatC_r^(\G^Y)is the completion ofC_c(\G^Y)in the norm

f= sup

yY _y(f).

Hecke Pairs

Consider the algebraC_r^(\G^X).Our next goal is to show that under an extra assumtion on the pair(G,), the multiplier algebra contains other interessting elements in addition to the -invariant functions onX.

The pair(G,)is called a Hecke pair if andgg^1 are commensurable for any g  G. That (, gg^1) are commensurable means that 

gg^1

  is a subgroup of finite index. Equivalently , every double coset of  contains finitely many right ( and left ) cosets of , i.e. :

R_(g) := |\g|< , for any g G.

If(G,)is a Hecke pair , the spaceH(G,)of finitely supported functions on \G/ is a-algebra with product :

(f1f₂)(g) = 

h\G

f₁(gh^1)f2(h), and involution :

f^(g) =f(g^1¯).

We can consider the functionsf H(G,) as bounded operators on the Hilbertspace l²(\G) represented as :

f ·_h = 

g\G

f(gh^1)·_g

The corresponding completion is called the reduced HeckeC^-algebra of (G,)and denoted byC_r^(G,).Denote by[g] the characteristic function of the double coset g, considered as an element of the Hecke algebra.

The elements of H(G,) may be considered as continous functions on

\G^X.Although these functions are not compactly supported in general , the formulas defining thealgebra structure and the regular representation of H(G,)coincide with (1.2)-(1.4).

Moreover , the convolution of an element of H(G,) with a compactly supported function on \G ^ X gives a compactly supported function : If f₁ = [g1] , and the support of f₂  C_c(\G ^ X) is contained in ()({g₂}U) for a compact U  X , then the support of f₁ f₂ is contained in()(g1g2U).Since \g1g2 is finite , we see thatf₁f₂ is compactly supported on\G^X.Therefore , we have :

Lemma 7 1.3

If (G,) is a Hecke pair , then the reduced Hecke C^algebra C_r^(G,) is contained in the multiplier algebra of theC^algebra C_r^(\G^X).

2. Dynamics and KMS-states

Assume as above that we have an action ofG onX such that the action of   G is proper , and Y  X is a -invariant ( Y  Y) clopen set.

Assume now that we are given a homomorphism : N :GR^+ = (0,+)

such that  is contained in the kernel of N. We define a one-parameter group of automorphisms of C_r^(\G^X) by :

_t(f)(g, x) = N(g)^it·f(g, x)

, forf C_c(\G^X).

More precisely :

N¯ ·_g =N(g)·_g

Since N¯ is selfadjoint ( easy to check) , then by applying functional calculus for bounded operators on Hilbertspace with f_t(z) = z^it , the operator N¯^it

 B(l²(\G)) is unitary , implementing the dynamics _t spatially by its associated unitary operator (^xXN¯^it)on (^xXl²(\G)).

In other words ,

_x(t(a)) = ¯N^it_x(a) ¯N^it

for all xX . See this by considering the operatoraction as represented on l²(\G):

_x(t(f))·_h = 

g\G

_t(f)(gh^1, hx)·_g

= 

g\G

N(gh^1)^it·f(gh^1, hx)·_g = 

g\G

N(g)^itN(h^1)^itf(gh^1, hx)·_g

= 

g\G

N(g)^itN(h)^itf(gh^1, hx)·_g = 

g\G

N(g)^itf(gh^1, hx)N(h)^it·_g

= 

g\G

N¯^itf(gh^1, hx)N(h)^it·_g = ¯N^it_x(f) ¯N^it·_h.

A semifinite invariant weight  is called a KM S_weight if , or equivalently , it satisfies the  KM S condition at inverse temperatures

 R if :

(aa^) =(i/2(a)^_i/2(a)),

for any analytic element a.( An element is called analytic if the map R  C_r^(\G^ X) , t  _t(a) extends to an analytic map C  C_r^(\G^X) . A map f :CC_r^(\G^X) is called analytic iff is an analytic function for any (C_r^(\G^X))^.)

If is finite , then theKM Scondition is equivalent to

(xy) = (yi(x)) , for any analytic x, y. This follows from

(yi(x)) =(_i/2(y)i/2(x)) =(i/2(y^)^_i/2(x)) and the identity :

xy= 1

4((x+y^)(x+y^)^(xy^)(xy^)^+i(x+iy^)(x+iy^)^i(xiy^)(xiy^)^).

The following result will be the basis of our analysis ofKM Sweights.

Proposition 8 2.1 Assume the action of G on X is an action without fix- points (free action) , so that in particular \G^Y is a genuine groupoid.

Then for any   R , there exists a one-to-one correspondence between

KM S_ weights onC_r^(\GY)with domain of definition containing C_c(\Y) Radon measuresµ onY such that

µ(gZ) = N(g)^µ(Z) ,

for every g  G and every compact subset Z  Y such that gZ  Y.

Namely , such a measure µisinvariant , so it determines a measure  on

\Y such that :



Y

f(y)dµ(y) =



\Y



 

yp^1({t})

f(y)



d(t)

for f C_c(Y) , wherep:Y \Y is the quotient map , and the associated weight  is given by

(a) =



\Y

E(a)(x)d(x) , where E is the conditional expectation from Lemma 1.2.

Proof. For={e}the result is well-known , see e.g. [19, Proposition II.5.4]

. For arbitrary  , a way to argue is as follows :

Since the action of  on Y is free , the quotient space \G Y is an etale groupoid. In fact it is an etale equivalence relation on \Y , or an r-discrete principial groupoid in the terminology of [19].To veryfy this , we have to check that the isotropy group of every point in \Y is trivial , that is , if g  G is such that gy  Y and p(gy) = p(y) , for some y  Y , then (g, y) belongs to the () - orbit of (e, y). But on the other hand , if p(gy) = p(y) , there exist    such that gy = y. Then g = e , since the action of G is free , and therefore (g, y) = (^1, e)(e, y). Then by [19,Proposition 11.5.4] ,

KM S_weights with domain of definition containingC_c(\Y)on the C^algebra C_r^(\G^ Y) of the etale equivalence relation are in ono-to- one correspondence with Radon measures on \Y with Radon-Nikodym cocycle (p(y), p(gy))N(g)^.

This means that :

If we assumeY₀ is an open subset ofY such that the mapp:Y \Y is injective on Y₀ , and g G is such thatgY₀ Y. Define an injective map

˜

g : p(Y0)p(gY0) by ˜g(p(y))  p(gy)

foryY₀ , and let ˜g_ be the push-forward of the measure under the map

˜

g , which again means that : g˜_(Z) =(˜g^1(Z)) , forZ p(gY0). Then : d˜g_

d =N(g)^ onp(gY0).

Therefore ; if we denote byµtheinvariant measure onY corresponding to  via (2.2 below) , then to say that the Radon-Nikodym cocycle of  is (p(y), p(gy))  N(g)^ is the same as saying that µ satisfies : µ(gZ) = N(g)^µ(Z) , for every g  G and every compact subset Z  Y such that gZ Y.( the scaling condition).

Recall that a Radon measure on Y is a Borel measure which is finite on compact sets , outer regular (*) on all Borel sets , and inner regular(**) on all open sets. Then , by The Riesz Representation Theorem , for each positive linear functional , and hence also for eachKM S_weight with domain of definition containingC_c(\Y) on the C^algebra C_r^(\G^Y) , there exist an unique Radon measure  on\Y such that (f) =

f d , for f C_c(\Y) ,  a KM S_weight. This establishes theone-to-one correspondence above.

The next lemma is about extension of the Radon measureµ from Y to GY :

Lemma 9 2.2. Ifµis a measure onY as in Proposition 2.1 , then it extends uniquely to a Radon measure on GY  X satisfying (2.1) forZ  GY and g G.

Proof. We can choose Borel subsetsY_i Y and elements g_i G such that GY = ⁱg^_i ¹Y_i , where  denotes disjoint union. There is only one choice for a measure extendingµ and satisfying (2.1) onGY , namely , for a Borel subset Z GY let

µ(Z) = 

i

N(gi)^µ(giZY_i).

To show that µ(Z) is independent of any choices and that the extension satisfies (2.1) , assumeGY =^jh_jZ_j for someh_j Gand Borel Z_j Y. Let g G. Then :



i

N(gi)^µ(gigZY_i) = 

i

N(gi)^·

j

µ(gigZY_ig_igh^_j¹Z_j)

= 

i

N(gi)^·

j

N(gigh^_j¹)^µ(hjZ)h_jg^1g^_i ¹Y_iZ_j)

= N(g)^

j

N(hj)^

i

µ(hjZh_jg^1g^_i ¹Y_iZ_j)

= N(g)^

j

N(hj)^µ(hjZ Z_j).

Taking g =e we see that the extension of µ to GY is well-defined. But then for arbitrary g the above identity reads as :

µ(gZ) =N(g)^µ(Z).

Lemma 10 2.4. Let Y0 be a invariant Borel subset of Y such that : (ı) if gY₀ Y₀ = for some g G , then g  ;

(ıı) for anyy Y , there exists g G such that gyY₀.

Then any invariant Borel measure on Y0 extends uniquely to a Borel measure on Y satisfying the scaling condition from Proposition 2.1.

Proof. Letµ₀ be ainvariant measure onY₀.Since the assumptions imply that Y is a disjoint union of translates of Y₀ by representatives of the right cosets of , that is , Y =^h:\G(h^1Y₀Y), there is only one choice for a measure µextending µ₀ and satisfying Proposition 2.1 , namely ,

µ(Z) = 

h:\G

N(h)^µ₀(hZ Y₀).

Since µ₀ is invariant , µ(Z) is independent of the choice of represen- tatives , so all we need to check is that Proposition 2.1 holds : Let g  G.

Then

µ(gZ) = 

h:\G

N(h)^µ₀(hgZY₀) =N(g)^ 

h:\G

N(hg)^µ₀(hgZY₀) =N(g)^µ(Z) , which proves the Lemma.

Although the condition for a measureon\Y to define a KMS-weight is easier to formulate in terms of the corresponding invariant measure onY , it will also be important to work directly with.For this we introduce the following operators on functions on \X. We shall often consider functions on \X as invariant functions onX.

Definition 11 2.5. LetGact on a setX and suppose(G,)is a Hecke pair.

The Hecke operator associated to g G is the operator T_g on invariant functions on X defined by :

(Tgf)(x) = 1 R_(g)



l\g(finite)

f(lx).

Clearly Tgf is again invariant. Recall that [g^1] denotes the charac- teristic function of the double coset g^1 considered as an element of the Hecke algebra. The map :

g^1

R_(g)Tg

is a representation of the Hecke algebraH(G,)on the space ofinvariant functions.

Notice that for X = G , this is exactly the way we defined the regular representation of H(G,) on l²(\G) : For f  H(G,) , considered as operator on l²(\G) , we defined its action by :

f ·_h = 

l\G

f(lh^1)·_l

Indeed , for f = [g^1] , using the regular representation ( on l²(\G) ) we get :

g^1

·h = 

l\G

g^1

(lh^1)·l = 

l\G

_g1(lh^1)·l

so ([g^1] ·_h)(s) = 1 , if sh^1  g^1  s  g^1h and = 0 otherwise.

On the other hand ,

using the representation  : C_r^(G,) B(l²(\G)) defined as above by [g^1]R_(g)Tg , we get :

( g^1

)·_h(s) = R_(g)Tg(h)(s) = 

l\g

_h(ls) ,

so (([g^1]) ·_h)(s) = {1 , if h  gs  s  g^1h ,and = 0 otherwise.

By decomposing an arbitraryX intoGorbits one can obtain that[g^1]

R_(g)Tg is a representation without any computations.

The following three lemmas will be our main computational tools : Lemma 12 2.6. Supposeµis as in Proposition 2.1 and thatis the measure on \Y determined by (2.2). Assume further that Y = X , the action of G on X is free and that (G,) is a Hecke pair with modular function

(g) := ^R_R^(g1)

(g) . Then for any positive measurable function f on \X and g G , we have :



\X

T_gf d =(g)·N(g)^·



\X

f d.

Proof. Let us first prove the following claim :