1 Introduction

Conditional expectations, traces, angles between spaces and Representations of the Hecke algebras

[5mm] Florin Rădulescu

[5mm]

Abstract: In this paper we extend the results in [Ra] on the representation of the Hecke algebra, determined by the matrix coefficients of a projective, unitary representation, in the discrete series of representations of the ambient group, to a more general, vector valued case. This method is used to analyze the traces of the Hecke operators.
We construct representations of the Hecke algebra of a group , relative to an almost normal subgroup , into the von Neumann algebra of the group , tensor matrices. The representations we obtain are a lifting of the Hecke operators to this larger algebra. By summing up the coefficients of the terms in the representation one obtains the classical Hecke operators.
These representations were used in the scalar case in [Ra], to find an alternative representation of the Hecke operators on Maass forms, and hence to reformulate the Ramanujan Petersson conjectures as a problem on the angle (see e.g. A. Connes’s paper [Co] on the generalization of CKM matrix) between two subalgebras of the von Neumann algebra of the group : the image of the representation of the Hecke algebra and the algebra of the almost normal subgroup.
Keywords: Conditional expectations, Traces, Hecke Algebra, Ramanujan-Petersson conjecture
MSC2010: 11F72, 46L65

Dedicated to Professor C.T. Ionescu Tulcea, on the occasion of his 90th anniversary

footnotetext: Supported in part by PRIN-MIUR, and by PN-II-ID-PCE-2012-4-0201

## 1 Introduction

Let be a countable discrete group and be an almost subgroup. Assume is a unitary representation of into the unitary group of a separable Hilbert space and assume that is a multiple of the left regular representation of on . For simplicity, throughout the paper we assume that the groups and have infinite conjugacy classes, and hence it follows that the associated von Neumann algebras are factors, and thus have a unique trace denoted by . The Murray von Neumann theory of dimension (see e.g. [Ta], [GHJ]) associates to the representation of the group , a continuous dimension number . Here, by we denote, as customary in von Neumann algebras, the von Neumann algebra generated by . In the paper [Ra] we treated the case . (We are deeply indebted to H. Moscovici for suggesting to investigate the general case of arbitrary dimension). If is an integer or on infinity, the hypothesis on the representation of corresponds to the existence of a Hilbert subspace , such that , and .

To this data we associate a representation of the algebra of Hecke operators into the reduced von Neumann algebra of the right regular representation of , ([MvN]). (If is an integer, the algebra is , where is any Hilbert space of finite dimension ), otherwise , where is a projection in of trace , and is an arbitrary infinite dimensional Hilbert space. In the case , the representation takes values into the formal ring of infinite series in with coefficients in .

Let be the - algebra of double cosets of in . Let be the reduced von Neumann - Hecke algebra of , acting on . Recall that this the the weak operator topology closure of , viewed as a subalgebra of , (see [BC] ).

We generalize the results in [Ra], by proving that also in this more general case, the angle (see Theorem 3.2) between the algebra and the algebra , determines the Ramanujan - Petersson behavior of the representation of the (algebraic) Hecke algebra (the - algebra of double cosets) on . The representation is simply the action of the Hecke algebra on invariant elements in . The space of the - invariant elements is exactly the algebra . Here acts on by and is the commutant algebra. By the Murray von Neumann dimension theory the commutant algebra is isomorphic to .

As proved in [Ra] an essential ingredient to prove that the representations of the Hecke algebra may be used to construct Hecke operators on (which is then equivalent to the problem of determining the action of Hecke operators on Maass forms), is the fact that the representation of the Hecke algebra may be extended naturally to an operator system containing the Hecke algebra (conform Definition 2.1).

If , , is the restriction to of a representation (which could also be projective) in the discrete series of . Then, through Berezin symbol map (Cf. [Be]), one recaptures from the representation the classical Hecke operators on Maass forms.

The explanation of the fact that we are able to recapture the action of classical Hecke operators on invariant vectors is as follows: Starting with the representation of the Hecke algebra into one defines the densely defined, -algebra morphism, from into by letting be the character of (that is ).

If is contained in the domain of , then is a representation of onto (in fact ). Identifying with the -invariant vectors in the representation , we obtain in this way the representation of the classical operators Hecke operators on -invariant vectors.

In particular applying the above construction to the representation acting on , one obtains (as it was also explained in [Ra]) the classical Hecke operators on the Hilbert space of - invariant vectors in . This latest Hilbert space is canonically identified to the Hilbert space associated to the von Neumann algebra .

We use the following construction of the Hilbert space of the associated - invariant vectors. We assume the representation , denoted in the following by , acts on the Hilbert space . To define a Hilbert space of - invariant vectors we use a ”scale for -invariant vectors” (as it is done in the Petersson scalar product).

Thus, we assume that is contained in a larger Hilbert space , and we assume that there exists a representation of into the unitary group of , such that invariates , and such that

 π(g)|H0=π0(g), g∈G.

We assume , is a subspace (which will be the scale mentioned above) such that (thus ). We define the Hilbert space of -invariant vectors (as in the Petersson scalar product) by letting be the subspace of densely defined forms, on , such that is finite (that is such that the densely defined map on associating to a vector in the domain the value extends to a continuous form on ).

Then is the classical Hecke operator representation on (see Section 3). We obtain in this way another representation of the Hecke operators on invariant vectors associated to the action of on . The Hecke operators are represented as operators acting on a subspace of . These representation can be used then to analyze the traces of the Hecke operators.

Finally, in the last section we determine a precise formula for the absolute value of the matrix coefficients of the 13-th element in the discrete series of , when restricted to .

This proves that this coefficients are absolutely summable on cosets of (a fact that is needed to prove that the map defined above is densely defined).

## 2 The representation of the Hecke algebra

In this section, we introduce a representation of the Hecke algebra associated to an almost normal subgroup of a discrete group . This representation of the Hecke algebra is canonically associated to a projective, unitary representation of the larger group .

Let be a projective, unitary representation of the group into the unitary operators on the Hilbert space . We assume that is multiple representation of the left regular representation of the group (which, as was observed in [Ra]) implies that the associated Hecke algebra of double cosets is unimodular, i.e. , with , for all .

In addition throught this section we assume that there exists a Hilbert subspace of such that is generating, and - wandering for (that is is orthogonal to for all in , and the closure of the linear span is equal to ). Here by we denote the identity element of the group .

Then is identified to the Hilbert space , and is identified to , and the representation is identified to the left regular representation of on , tensor the identity of .

In the sequel we denote the right regular representation of (respectively ) by (respectively ) on (respectively ). When no confusion is possible we simply denote this by . By we denote the algebra of formal series

with coefficients , , that have the additional property that for every , the coefficient

defined by

is an element of , the von Neumann algebra of right, bounded convolutors on .

With the above definitions, we consider the following linear operator system (see [Pi] for the definition of an operator system):

###### Definition 2.1.

By we denote the tensor product . Here is the Hecke algebra of double cosets and is embedded both in and , simply by writing a double coset as a sum of right, respectively left, cosets (see e. g. [BC]).

There is an obviously operation on (whence the notation). Then , and are canonically identified as subspaces of . We have a canonical bilinear map from onto . This bilinear map, when restricted to the double cosets in the Hecke algebra , becomes the Hecke algebra multiplication.

As observed in ([Ra]), the operator system is isomorphic to the linear span of cosets:

 C{σ1Γσ2|σ1,σ2∈G}=C{(σ1Γσ−11)σ2|σ1,σ2∈G},

where the bilinear operation maps into , for all .

There is an obviously completion of this system, which is still a operator system, which we denote as . This operator system is, by definition, , where is the reduced von Neumann algebra of the Hecke algebra, that is the weak closure of acting on (respectively on by right multiplication).

A priori, there is no canonical Hilbert space structure on . Using the construction in this paper, we define at the end of the paper, a canonical Hilbert structure on that is compatible with the action of . Thus we construct a scalar product on cosets

that depends only on , . By cyclically rotating the variables, the scalar product corresponds to a bilinear form on . (Note that in [Ra] we also constructed a scalar product, with positive values on the generators, such that the corresponding bilinear form is again positive definite).

With the above definitions we have the following.

###### Proposition 2.2.

Let and as above. Then there exists a canonical representation

that is a morphism in the sense that where is viewed as an element of , for all . In this assertion, we implicitly require that the elements , may be multiplied in , for all .

By restricting to , we obtain a representation of the Hecke algebra. When is of finite dimension, this representation may be extended to (and hence to ). Let be the the projection from onto .

The formula for is

 S(A)=∑θ∈Aρ(θ−1)⊗PLπ(θ)PL

where is any of the cosets or for.

###### Proof.

In the summation, with as in the statement, we have to take , for , since the right regular representation has the property that , for .

The multiplicativity of the map follows from the fact that

 ∑γ∈ΓPLπ(θ1γ)PLπ(γ−1θ2)PL=PLπ(θ1θ2)PL.

This formula is a consequence of the fact that (this is also valid for projective representations). This is indeed the coefficient of in a product for all .

The fact that the matrix coefficients of belong indeed to , is observed as follows. Fix in .

Then

 ∑θ∈Γσρ(θ−1)⟨π(θ)l1,l2⟩=∑γ∈Γρ((γσ)−1)⟨π(γ)π(σ)l1,l2⟩=
 =∑γ∈Γρ(σ−1)ρ(γ−1)⟨π(σ)l1,π(γ−1)l2⟩

(since are orthogonal).

The fact that may be extended by linearity and continuity to in the case that is finite dimensional is proved as follows: in this case the matrix entries , , are, when composed with , the matrix entries of a representation of , into , which is however trace preserving, and thus extendable by continuity to . In particular, if is finite dimensional, then for all , we have that , belong to .

###### Remark 2.3.

We define formally

 Eπ(Γ)′(A)=∑γ∈Γπ(γ)Aπ(γ)−1,

for in . If the above sum is so convergent, then this sum represents the conditional expectation onto .

Moreover if (i.e. is a trace class operator), then the sum from the statement is simply the restriction of the state to . Here the linear functional is the trace on the trace class operators .

In particular if belongs to (or even ) then is clearly equal to .

In particular, the preceding formula for

 S(σΓ)=∑γρ((σγ)−1)⊗PLπ(σ)π(γ)PL

might be interpreted as .

Let be the densely defined character on defined as the sum of coefficients. Then we define on (here is densely defined) with values in by the formula

 ˜ε(∑ρ(g)⊗Ag)=∑gAg.

For , the condition for to be in the domain of is that the sum be so-convergent.

Then , if , is a representation of into .

###### Remark 2.4.

Assume that is an infinite measure space and and assume that acts by measure preserving transformation on . Let be a fundamental domain for the action on (implicitly we assume here that the action of is so that such a domain exists). Let be the Koopman representation (see [Ke]) of on . Let be any element of . Let be the characteristic function of , viewed as a projection in . Then

 S(ΓσΓ)=∑θ∈ΓσΓρ(θ−1)⊗χFπKoop(θ)χF,σ∈G.

Then we have that

 ˜ε(S(ΓσΓ))=∑χFπKoop(θ)χF,σ∈G.

This is exactly the classical Hecke operator associated to the double coset that is acting on the space associated to the fundamental domain.

###### Remark 2.5.

In the previous more general setting of Remark 2.3, we identify the - wandering subspace of , with a subspace of invariant vectors in (here by - invariant vectors we understand densely defined, - invariant functionals on ).

Thus to every vector , we associate the - invariant vector . Then clearly

 ˜ε(S(ΓσΓ))l=˜ε(S(ΓσΓ))(PL∑γ∈Γπ(γ)l)=∑θ∈ΓσΓPLπ(θ)l,l∈L,

This is the further equal to

,

and this is exactly the action of the classical Hecke operator associated to the double coset on (which is ). Here ; are cosets representatives so that ). In particular we obtain an extension from classical Hecke operators, which are a representation of to a representation of .

One example, in which is well defined, is the case of a tensor product of two representations of the type of those considered above.

###### Proposition 2.6.

Let be as above and let be the diagonal representation. We identify with , the Schatten von Neumann class of operators from into . In this identification, the representation is the adjoint representation, mapping into .

For simplicity we assume that has infinite conjugacy non-trivial classes, so that its associated von Neumann algebra has a unique trace .

Then the - invariant vectors in are identified with the - space of - intertwiners of the representations . We denote the space of -intertwiners by . If then this space is simply the commutant algebra . The - space associated to the space of intertwiners is canonically determined by the trace on the von Neumann factor .

In general the - space of , denoted by , is obtained by first fixing one of the two spaces, which ever has greater equal Murray von Neumann dimension. Assume, for example,that this space is . Then the - norm of is .

The classical Hecke operator on invariant vectors associated to the representation is then the extension to the space of intertwiners, of the operator defined by the following construction: for , assuming that the disjoint decomposition of the coset is , we let

 Ψ(ΓσΓ)(X)=∑iπ2(siσ)Xπ1(σ−1s−1i),X∈IntΓ(π1,π2).

This is obviously a bounded operator. We identify with and we identify the associated Hilbert space with . Let , for , be the representation of the Hecke algebra associated with the representations constructed in Proposition 2.2. Let be the representation of the Hecke algebra , given by Proposition 2.2, associated to the representation .

Then, for every double coset , the operator is equal to . In particular, the construction in Proposition 2.2 for the representation , yields the classical Hecke operators on . Moreover, we have the following formula for :

 Ψ(ΓσΓ)(X)=ER(G)⊗B(L1,L2)R(Γ)⊗B(L1,L2)(S2(ΓσΓ)XS1(ΓσΓ)),

for all

 X∈IntΓ(π1,π2)≅R(Γ)⊗B(L1,L2).

Here is the canonical conditional expectation.

###### Proof.

We identify with . Then we have that

 H1⊗¯¯¯¯¯H2=l2(Γ)⊗¯¯¯¯¯¯¯¯¯¯¯l2(Γ)⊗L1⊗¯¯¯¯L2,

and hence, a possible wandering subspace for is . This space is identified with as follows:

Fix vectors in . Fix in . Let be the identity element of the group . Consider the following 1-dimensional operators:

 A0=⟨e⊗l1,⋅⟩(γa⊗l2),B0=⟨e⊗m1,⋅⟩(γb⊗m2).

Let

 A=∑π1(γ)A0π2(γ)−1,B=∑π1(γ)B0π2(γ)−1.

Then

 τ(AB∗)=Tr(AB0).

We obtain that

 Ψ(ΓσΓ)(A)=∑θ∈ΓσΓπ2(θ)A0π1(θ)−1.

The matrix coefficients are

 ∑θ∈ΓσΓTr(π2(θ)[⟨e⊗l1,⋅⟩(γa⊗l2)]π1(θ)−1(⟨e⊗m1,⋅⟩γb⊗m2)∗)=
 =∑θ∈ΓσΓTr([⟨π1(θ)−1(e⊗l1),⋅⟩π2(θ)(γa⊗l2)][⟨e⊗m1,⋅⟩(γb⊗m2)]∗)=
 =∑θ∈ΓσΓ⟨π1(θ)−1(e⊗l1),γa⊗l2⟩⟨π2(θ)(e⊗m1)(γb⊗m2)⟩.

But this are exactly the matrix coefficients of where is the representation of associated to the representation on the vectors in the - wandering subspace

 l2(Γ)⊗L1⊗¯¯¯¯L2≅Ce⊗¯¯¯¯¯¯¯¯¯¯¯l2(Γ)⊗L1⊗¯¯¯¯L2,

corresponding to the vectors and .

To verify the formula for the conditional expectation, note that it is sufficient to check that we get the same values for the matrix entries, when evaluated at elements as above.

The trace on is . Thus we have to compute for ,

 τR(Γ)⊗B(L1)(ER(G)⊗B(L1,L)R(Γ)⊗B(L1,l2)(S2(ΓσΓ)AS1(ΓσΓ)B∗))=
 =τR(Γ)⊗B(L1)(S2(ΓσΓ)AS1(ΓσΓ)B∗)=
 =τ((∑θ1∈ΓσΓρ(θ−11)⊗PLπ2(θ1)PL)(Rγa⊗⟨l1,⋅⟩l2)
 (∑θ2∈ΓσΓρ(θ−12)⊗PLπ1(θ2)PL)Rγ−1b⊗⟨m1,⋅⟩m2)=
 =∑θ1,θ2∈ΓσΓθ1γaθ2γ−1b=eTr((PLπ2(θ1)PL)(⟨l1,⋅⟩l2)PLπ1(θ2)PL⟨m1,⋅⟩m2)=

(as )

 =∑θ∈ΓσΓ⟨PLπ2(θ)PLl2,m2⟩¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⟨PLπ1(γaθ−11γb)PLl1,m1⟩=
 =∑θ∈ΓσΓ⟨π2(θ)l2,m2⟩¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⟨π1(γ−1aθ1γ−1b)l1,m1⟩

which is exactly the previous formula.

## 3 The case on non-integer Murray and von Neumann dimension

In this section we extend the results in the previous section to the case of non-integer Murray von Neumann dimension. We assume that there exists a representation as in the previous section, but we do not assume that there exists a splitting .

Instead we assume that there exists a representation on a larger Hilbert space , containing , which has a splitting , with , and such that if is the canonical orthogonal projection, then commutes with and . Also we assume is trace class. Denote by the representation of constructed in the previous section.

We will prove that also commutes with , and that the correspondence

 ΓσΓ→pS(ΓσΓ)p,  σ∈G

extends to a representation of into , with the required properties. Note that this representation of takes values into .

###### Lemma 3.1.

Let as above, and let the subspace , where is a projection such that , for all in . Let . We assume also that is trace class .

Then commutes with , for in . The explicit expression for is

 ∑ρ(γ)⊗PLπ0(γ−1)PL=E(π0(Γ))′(pPL)=Eπ0(Γ)′(pPLp).

In this case, since is trace class, the above expression belongs to .

Then

 pS(Γσ)=∑θ∈Γσρ(θ−1)⊗PLπ0(θ)PL∈p(R(G)⊗B(L)),

and

 pS(ΓσΓ)=S(ΓσΓ)p=∑θ∈ΓσΓρ(θ−1)⊗PLπ0(θ)PL.

Hence determines a representation of , while determines a representation o with

 S(σ1Γ)pS(Γσ2)=∑θ∈σ1Γσ2ρ(θ−1)⊗PLπ0(θ)PL.
###### Proof.

The expression for follows from the statement of Proposition 2.2 in Section 2.

Then all the statements are a consequence of the formula

 ∑γPLπ0(γ)PLπ(γ−1θ)PL=PLπ0(θ)PL.

Indeed, for example, when doing

 pS(Γσ)=(∑γ∈Γρ(γ−1)⊗PLπ0(γ)PL)(∑θ∈Γσρ(θ−1)⊗PLπ(θ)PL)=
 =∑θ1∈Γσρ(θ−11)⊗∑θ∈Γσγθ=θ1PLπ0(γ)PLπ(θ)PL=
 =∑θ1∈Γσρ(θ−11)⊗∑γ∈ΓPLπ0(γ)PLπ(γ−1θ1)PL=
 =∑θ1∈Γσρ(θ−11)⊗PLπ0(θ1)PL.

To check that is indeed a representation of we have to verify that it implements the Hecke operators on .

Here is described as the transformation mapping, for in , where is a double coset. As we mention above, extending by continuity to the - space we obtain the composition of with the Hecke operators for .

Here is isomorphic (as belongs to ) to

 pπ0(Γ)′p=p(R(Γ)⊗B(L))p.

We have

###### Theorem 3.2.

Let be as above. Let be the representation constructed in the previous proposition.  Then we have

 Ep(R(G)⊗B(L))pp(R(Γ)⊗B(L))p(Sp(ΓσΓ)XSp(ΓσΓ))=Ψ[ΓσΓ](X),

for   all     in   , .

###### Proof.

Let be the orthogonal projection onto . Let and assume that , a disjoint union. Then

 Ψ[ΓσΓ](X)=∑iπ0(σsi)−1Xπ0(σsi).

An element in has the following representation in

 ∑ρ(γ−1)⊗PLπ0(γ)X′PL.

We compute first the product

 ∑θ1,θ2∈ΓσΓγ∈Γ[ρ(θ−11)⊗PLπ0(θ1)PL][ρ(γ−1)⊗PLπ0(γ)X′P