F. RADULESCU

Abstract. This is a note on Radulescu’s work taken by N. Ozawa. We first review the construction of the classical Hecke operator and recall the Ramanujan–

Petersson conjecture about its spectrum. We then relate it to the study of a type II1 factor via Berezin calculus.

Part 1. Hecke algebras (Ref: [Kr]).

1. Definition of Hecke Algebras
For subsets α, β ⊂G of a groupG, we define α^{∗} and αβ ⊂G by

α^{∗} ={g^{−1} :g ∈α} and αβ ={x:∃g ∈α, ∃h ∈β such that x=gh}.

Let Γ ≤Gbe discrete groups, andρ: Gy`^{2}(Γ\G) be the quasi-regular represen-
tation defined by ρ(g)ξ(x) =ξ(xg). What isρ(G)^{00}? It suffices to knowρ(G)^{0}. Since
δ_{Γ} is ρ(G)-cyclic, it is ρ(G)^{0}-separating. Let T ∈ ρ(G)^{0}. Then, T δ_{Γ} ∈ `^{2}(Γ\G) is
ρ(Γ)-invariant. Hence, T δ_{Γ} can be regarded as a function on the double coset space
Γ\G/Γ, which is supported on those double cosets ΓgΓ which are finite unions of
right cosets.

Lemma 1.1. For Γ≤G, TFAE.

(1) Every double coset is a finite union of right cosets.

(2) Every double coset is a finite union of left cosets.

(3) For every g ∈G, one has [Γ : Γ∩g^{−1}Γg]<+∞.

Moreover, for s_{i} ∈Γ, one has ΓgΓ =F

Γgs_{i} iff Γ =F

(Γ∩g^{−1}Γg)s_{i}.

We assume that Γ≤Gsatisfies the above conditions. Such a pair Γ≤Gis called a Hecke pair. We define ind : Γ\G/Γ→N by

ind(α) =|{x∈Γ\G:x⊂α}|= [Γ : Γ∩g^{−1}Γg].

For each α ∈ Γ\G/Γ with α =Find(α)

i=1 Γg_{i}, there is an operator λ(α) ∈ ρ(G)^{0} such
that λ(α)δ_{Γ} = δ_{α} = P

iδ_{Γg}_{i}. (We freely view a function on Γ\G/Γ as a right

Date: 17 January 2013.

1

Γ-invariant function on Γ\G.) For x= Γh∈Γ\G, one has
λ(α)δ_{x} =λ(α)ρ(h)δ_{Γ}=ρ(h)X

i

δ_{Γg}_{i}x=X

i

δ_{Γg}_{i}_{h} = X

y∈Γ\G, y⊂αx

δ_{y} =χ_{αx}.
Thus, kλ(α)k ≤ind(α) and

λ(α)^{∗}δ_{Γ} = X

x∈Γ\G, Γ⊂αx

δ_{x} =λ(α^{∗})δ_{Γ}.
Let β ∈Γ\G/Γ and β =F

jΓh_{j}. Then,
λ(α)λ(β)δ_{Γ}=λ(α)X

x⊂β

δ_{x} =X

x⊂β

X

y⊂αx

δ_{y} = X

z∈Γ\G, z⊂αβ

c(α, β;z)δ_{z},
where for z = Γu one has

c(α, β;z) = |{(i, j) : Γgihj =z}|=|{j :u∈αhj}|

=|{(g, h)∈α×β :gh=u}/∼_{Γ}|.

The last one is the number of the Γ-orbits of {(g, h)∈α×β :gh=u}, where s∈Γ
acts on (g, h) by (gs^{−1}, sh). We observe that z 7→ c(α, β;z) is constant on each
double coset. For γ ∈Γ\G/Γ, let c(α, β;γ) = c(α, β;z) forz ⊂γ. It follows that

λ(α)λ(β) = X

γ∈Γ\G/Γ

c(α, β;γ)λ(γ) and

ind(α) ind(β) =X

z

|{(i, j) : Γgihj =z}|= X

γ∈Γ\G/Γ

c(α, β;γ) ind(γ).

These naturally make C[Γ\G/Γ] a ∗-algebra, C[Γ\G] a C[Γ\G/Γ]-module, and ind a homomorphism on C[Γ\G/Γ]. The trivial double coset Γ becomes the unit of the Hecke algebra C[Γ\G/Γ]. We note that for α, β ∈Γ\G/Γ, the product α·β in the Hecke algebra C[Γ\G/Γ] has the set theoretic product αβ as its support, but the coefficients of α·β are not necessarily 1. We write H =C[Γ\G/Γ] and vN(H) :=

ρ(G)^{0} = λ(H)^{00}. The proof of the last equality is easy when the vN(H)-separating
vectorδ_{Γ}is tracial (see [BC] for a general case). Indeed,λ(H)δ_{Γ} =C[Γ\G/Γ] is dense
in vN(H)δ_{Γ} which consists of ρ(Γ)-invariant functions in `^{2}(Γ\G). LetT ∈ vN(H)
and f = T δ_{Γ} on Γ\G/Γ. We write T formally as λ(f). We will write ω for the
vector state of δ_{Γ}:

ω(λ(f)) =hf, δ_{Γ}i=f(Γ).

One still has

(λ(f)λ(g))(γ) = X

α,β∈Γ\G/Γ

c(α, β;γ)f(α)g(β).

Moreover,

kT δ_{Γ}k^{2} =X

α

ind(α)|f(α)|^{2} and kT^{∗}δ_{Γ}k^{2} =X

α

ind(α^{∗})|f(α)|^{2}.
Thus, δ_{Γ} is a trace vector for vN(H) iff ind is symmetric on Γ\G/Γ.

A map σ: G → H between groups is called an anti-homomorphism if σ(gh) = σ(h)σ(g) for all g, h∈G.

Lemma 1.2. Let σ be an anti-automorphism on G such that σ(Γ) = Γ. Then, σ
extends to an anti-automorphism on H. Suppose moreover that σ(α) = α for all
α ∈Γ\G/Γ. Then, His commutative andδ_{Γ} is a separating trace vector for vN(H).

Proof. We define σ(α) = {σ(g) : g ∈ α} ∈ Γ\G/Γ. Then, for α, β, γ ∈ Γ\G/Γ and u∈γ, one has

c(α, β;γ) =|{(g, h)∈α×β :gh =u}/∼_{Γ}|

=|{(h^{0}, g^{0})∈σ(β)×σ(α) :h^{0}g^{0} =σ(u)}/∼_{Γ}|

=c(σ(β), σ(α);σ(γ))

It follows that σ(α·β) = σ(β)·σ(α) and σ extends to an anti-homomorphism on H
such that ind(σ(α)) = ind(α^{∗}). If σ(α) =α for all α ∈Γ\G/Γ, then

α·β =σ(α·β) = σ(β)·σ(α) =β·α

and ind(α) = ind(σ(α)) = ind(α^{∗}).

Lemma 1.3. Let g ∈G be in the normalizer of Γ. Then, for everyh∈G, one has (ΓgΓ)·(ΓhΓ) = ΓghΓ and (ΓhΓ)·(ΓgΓ) = ΓhgΓ.

Proof. We only prove the fist equality. Since (ΓgΓ)(ΓhΓ) = ΓghΓ as a set, it remains to show c(ΓgΓ,ΓhΓ; ΓghΓ) = 1. This reduces to show ind(ΓghΓ) = ind(ΓhΓ).

Observe now that for hj ∈G, one has ΓhΓ =F

Γhj iff ΓghΓ =F

Γghj.

2. Actions of Hecke algebras

Let Γ≤G be a Hecke pair and V be a vector space on whichG acts. Then, the
Hecke algebra H acts on the spaceV^{Γ} of Γ-invariant vectors by

αv=X
g_{k}v
for v ∈ V^{Γ} and α = F

g_{k}Γ∈ Γ\G/Γ. Note that we are using left cosets here. It is
easy to see that αv is indeed Γ-invariant. Moreover, if α = F

g_{k}Γ and β = F
h_{l}Γ
are elements in Γ\G/Γ, then for every γ ∈Γ\G/Γ and u∈γ, one has

|{(k, l) :u∈g_{k}h_{l}Γ}|=|{(g, h)∈α×β :gh =u}/∼_{Γ}|=c(α, β;γ).

It follows that (α, v) 7→ αv is indeed an action of the Hecke algebra H. If V is a
Hilbert space and GyV is unitary, then for every v ∈H^{Γ} and g ∈G, one has

αv=P_{H}^{Γ}

ind(Γg^{−1}Γ)

X

k=1

g_{k}v = ind(Γg^{−1}Γ)P_{H}^{Γ}gv,
since P_{H}^{Γ}s=sP_{H}^{Γ} =P_{H}^{Γ} for any s∈Γ.

Example 2.1. LetV be the space of all functionsξ: G→Cand define theG-action
onV by (gξ)(h) =ξ(g^{−1}h). Then, V^{Γ}is identified as the space of functions on Γ\G.

Suppose x∈Γ\G and viewδ_{x} also as the characteristic functionχ_{x} on G. Then,
αδx =χ^{F}g_{k}x =X

k

δg_{k}x =χαx =λ(α)δx.

3. The Hecke algebra associated with SL(2,Z)≤GL^{+}(2,Q).

Let Γ = SL(2,Z) and G = GL^{+}(2,Q). We will show Γ ≤G is a Hecke pair and
studies the structure of its Hecke algebra. Let M(l) = {A ∈ M(2,Z) : detA = l}

and M=S∞

l=1M(l)⊂G.

Theorem 3.1. Let Γ act on Z^{2} and on M from the left. Then,
Z^{2} =

∞

G

α=0

Γ (^{α}_{0}) and M=G

{Γ (^{a b}_{0} _{d}) :a, d∈N, b∈N^{0}, 0≤b < d}.

Proof. The first assertion is easy: (^{a}_{c}) ∈ Γ (^{α}_{0}) iff α = gcd(a, c). By the first
assertion, every Γ-orbit inMhas a representative of the form (^{a b}_{0}_{d}) wherea, b, d∈N0

and ad > 0. By multiplying appropriate ^{1}_{0 1}^{f}

from the left, one may assume the
representative (^{a b}_{0} _{d}) satisfies 0 ≤ b < d in addition. Now, suppose that (^{a b}_{0}_{d}) and

α β 0 δ

belong to the same Γ-orbit. Then,
(^{a b}_{0}_{d}) ^{α β}_{0} _{δ}−1

=_{a/α}_{(αb−aβ)/αδ}

0 d/δ

∈Γ,

which implies a=α, d=δ, and b=β.

Corollary 3.2. The pair Γ≤G is a Hecke pair.

Proof. Let A ∈ G and choose m ∈ N so that mA ∈ M. Then for B_{i} ∈ G, one has
ΓAΓ =F

ΓB_{i} iff ΓmAΓ =F

ΓmB_{i}. Since ΓmAΓ⊂M(l) forl= det(mA) andM(l)
has finitely many Γ-orbits by Theorem 3.1, one has |Γ\ΓAΓ|<+∞.

For A∈M(2,Z), let δ(A) be the g.c.d. of its entries. Then, one has det(U AV) = detA and δ(U AV) =δ(A)

for every A∈M and U, V ∈Γ. The pair of these values is a complete invariant for the double coset in M.

Theorem 3.3. ForA∈M, one hasA∈Γ (^{a}_{0} ^{0}_{d}) Γiffa=δ(A)andd= (detA)/δ(A).

In particular,

M=G

{Γ (^{a}_{0} ^{0}_{d}) Γ :a, d∈N, a|d}.

Proof. Since A is a 2-by-2 matrix, detA is divisible by δ(A)^{2} and a=δ(A) divides
d = (detA)/δ(A). It remains to show that every A∈M is equivalent to (^{a}_{0} ^{0}_{d}) with
a|d. We may assume thatδ(A) = 1. By Theorem 3.1, we may further assume that
A= (^{a b}_{0}_{d}). By the Chinese Remainder Theorem, there ism∈Zsuch thatb+md≡1
mod p for all prime divisors p of a which do not divides d. Since gcd(a, b, d) = 1,
one has gcd(a, b+md) = 1. Replacing A with (^{1}_{0 1}^{m})A, we may now assume that
gcd(a, b) = 1. Then, for x, y ∈Z such thatax−by = 1, one has A _{−y a}^{x} ^{−b}

= (^{1}_{∗ ∗}^{∗}).

By elementary transformation over Z, it is equivalent to (^{1 0}_{0} _{∗}).

Corollary 3.4. The Hecke algebra of Γ≤G is commutative.

Proof. The transpose σ is an anti-automorphism of G such that σ(Γ) = Γ. Let A∈G and choose m∈N so that mA∈M. Then, by Theorem 3.3, one has

m(ΓAΓ) = Γ(mA)Γ = Γσ(mA)Γ =m(Γσ(A)Γ).

Hence, ΓAΓ = Γσ(A)Γ. By Lemma 1.2, we are done.

By Lemma 1.3, for any r ∈ Q+, the element Γ (^{r}_{0} ^{0}_{r}) Γ is in the center of H and
satisfies (Γ (^{r}_{0}^{0}_{r}) Γ)·(ΓAΓ) = Γ(rA)Γ. for every A∈G.

Lemma 3.5. If gcd(k, l) = 1, then for every A ∈ M(k) and B ∈ M(l), one has δ(AB) =δ(A)δ(B) and (ΓAΓ)·(ΓBΓ) = ΓABΓ.

Proof. By the remark preceding this lemma, we may assume that δ(A) = 1 =δ(B).

We first prove that δ(AB) = 1. In light of Theorem 3.3, it suffices to show that for
every U = (^{a b}_{c d}) ∈ Γ one has δ (^{1 0}_{0}_{k})U(^{1 0}_{0} _{l})

= 1. Suppose that a prime number
p divides the entries of (^{1 0}_{0}_{k})U(^{1 0}_{0} _{l}) = (_{kc kdl}^{a} ^{bl} ). Then p | a and p | kc, but since
U ∈ Γ, this implies p| k. Likewise p| l. A contradiction. This proves δ(AB) = 1.

Now, Theorem 3.3 implies that (ΓAΓ)(ΓBΓ) = ΓABΓ as a set. It remains to show
c(ΓAΓ,ΓBΓ; ΓABΓ) = 1. Since this value is independent of the representatives, we
may assume that A and B are of the formsA= (^{1 0}_{0}_{k}) andB = (^{1 0}_{0} _{l}). According to
Theorems 3.1 and 3.3, one has

ΓBΓ =G

{(^{a b}_{0} _{d}) :ad=l, 0≤b < d and gcd(a, b, d) = 1}=G
ΓB_{j}
It follows that

c(ΓAΓ,ΓBΓ; ΓABΓ) = |{j :AB∈ΓAΓB_{j}}|.

But, the matrixABB_{j}^{−1} =

1/a−b/ad 0 kl/d

is integral only ifa= 1,b = 0, or equivalently
only if B_{j} =B. Therefore, c(ΓAΓ,ΓBΓ; ΓABΓ) = 1.

4. The Hecke algebra associated with PSL(2,Z)≤PGL^{+}(2,Q).
The structure of the Hecke algebra of PSL(2,Z)≤PGL^{+}(2,Q) is easier than that
of SL(2,Z)≤GL^{+}(2,Q).

Proposition 4.1. The pair PSL(2,Z) ≤PGL^{+}(2,Q) is a Hecke pair and the quo-
tient map from GL^{+}(2,Q) onto PGL^{+}(2,Q) induces a surjective homomorphism
between their Hecke algebras.

Proof. Let G = GL^{+}(2,Q), Γ = SL(2,Z), G^{0} = PGL^{+}(2,Q) and Γ^{0} = PSL(2,Z).

The quotient map π: G→G^{0} extends to a surjection from Γ\G/Γ onto Γ^{0}\G^{0}/Γ^{0}.
Claim 4.2. If ΓAΓ =F

ΓA_{i}, then Γ^{0}π(A)Γ^{0} =F

Γ^{0}π(A_{i}).

Proof. We only prove that Γ^{0}π(A_{i})’s are mutually disjoint. Suppose the contrary.

It follows that there are i 6= j and B ∈ Γ such that A^{−1}_{i} BA_{j} ∈ kerπ. But, since
det(A^{−1}_{i} BA_{j}) = 1 and kerπ ={(^{r}_{0}^{0}_{r})}, one has A^{−1}_{i} BA_{j} ∈Γ. A contradiction.

We proceed to prove that π is a homomorphism between Hecke algebras. Let α = ΓAΓ =F

ΓAi, β = ΓBΓ =F

ΓBj and γ = ΓCΓ⊂αβ. To see
c(α, β;γ) =|{(i, j) :C ∈ΓA_{i}B_{j}}|

=|{(i, j) :π(C)∈Γ^{0}π(A_{i}B_{j})}|=c(π(α), π(β);π(γ)),

it suffices to show π(C)∈Γ^{0}π(A_{i}B_{j}) impliesC ∈ΓA_{i}B_{j}. Ifπ(C)∈Γ^{0}π(A_{i}B_{j}), then
there exists D ∈ Γ such that DA_{i}B_{j}C^{−1} ∈ kerπ. But det(DA_{i}B_{j}C^{−1}) = 1, this
implies DA_{i}B_{j}C^{−1} ∈Γ. This completes the proof.

Theorem 4.3. Let A ∈ GL^{+}(2,Q) and r ∈ Q be such that rA ∈ M(2,Z) and
δ(rA) = 1; and let k = det(rA) ∈ N. Then, A belongs to the same double coset as
(^{1 0}_{0}_{k}) in PSL(2,Z)\PGL^{+}(2,Q)/PSL(2,Z). Moreover,

PSL(2,Z)\PGL^{+}(2,Q)/PSL(2,Z) =

∞

G

k=1

PSL(2,Z) (^{1 0}_{0}_{k}) PSL(2,Z).

Proof. LetA,randkbe as in the statement. Then,A =rAinG^{0}andAis equivalent
in Γ^{0}\G^{0}/Γ^{0} to (^{1 0}_{0}_{k}) by Theorem 3.3. Moreover, if (^{1 0}_{0}_{k}) and (^{1 0}_{0} _{l}) belong to the
same double coset, then there are A, B ∈ Γ such that A(^{1 0}_{0} _{k})B(^{1 0}_{0} _{l})^{−1} ∈ kerπ.

Hence, there is r ∈ Q such that A(^{1 0}_{0}_{k})B = (^{r}_{0}_{lr}^{0}). It follows that r ∈ Z and, by

Theorem 3.3, r= 1 andk =l.

Let H be the Hecke algebra of PSL(2,Z) ≤ PGL^{+}(2,Q) and, for every prime
number p, Hp be the subalgebra generated by { ^{1 0}_{0}_{p}^{l}

: l ∈ N^{0}}. By Theorem 4.3
and Lemma 3.5, one has

H ∼= O

p prime

H_{p}.

We investigate the structure of H_{p}. We write T_{p} for the element in H represented
by ^{1 0}_{0}_{p}

. We note that T_{p}^{∗} =Tp.

Theorem 4.4. One has T_{p}^{2} =T_{p}^{2} + (p+ 1) and T_{p}·T_{p}k =T_{p}k+1+pT_{p}^{k−1} for k ≥2.

In particular, H_{p} is generated by T_{p}.
Proof. By Theorem 3.3, one has (Γ ^{1 0}_{0}_{p}

Γ)(Γ ^{1 0}_{0}_{p}

Γ) = Γ ^{1 0}_{0}_{p}^{2}

ΓtΓ ^{p}_{0}^{0}_{p}
Γ as a
subset of GL^{+}(2,Q). By Theorem 3.1, one has

Γ ^{1 0}_{0}_{p}

Γ = Γ ^{p}_{0 1}^{0}
t

p−1

G

b=0

Γ ^{1}_{0}_{p}^{b}
.
We note that

Γ ^{p}_{0 1}^{0} _{p}_{0}

0 1

= Γ ^{p}_{0 1}^{2} ^{0}
,
Γ ^{p}_{0 1}^{0} _{1}_{b}^{0}

0 p

= Γ

p pb^{0}
0 p

= Γ ^{p}_{0}^{0}_{p}

for all b^{0},
Γ ^{1}_{0}_{p}^{b} _{p}_{0}

0 1

= Γ ^{p b}_{0}_{p}

for all b,
Γ ^{1}_{0}_{p}^{b} _{1}_{b}^{0}

0 p

= Γ_{1} _{bp+b}0

0 p^{2}

for all b and b^{0}.
It follows that T_{p}^{2} =T_{p}^{2} + (p+ 1)Γ ^{p}_{0} ^{0}_{p}

Γ in Γ\G/Γ. Passing to the quotient, one
sees T_{p}^{2} =T_{p}^{2} + (p+ 1) in Γ^{0}\G^{0}/Γ^{0}. Next, observe that

(Γ ^{1 0}_{0}_{p}

Γ)(Γ ^{1 0}_{0}_{p}^{k}

Γ) = Γ ^{1}_{0}_{p}^{k+1}^{0}
ΓtΓ

_{p} _{0}

0p^{k}

Γ

as a subset of GL^{+}(2,Q). Indeed, ifU = (^{a b}_{c d})∈Γ, then gcd(a, c) = 1 and the value
δ ^{1 0}_{0}_{p}

U ^{1 0}_{0}_{p}k

=δ

a bp^{k}
cp dp^{k+1}

can only be 1 or p. We note that
Γ ^{1 0}_{0}_{p}k

Γ =G {Γ

p^{k−i} b^{0}
0 p^{i}

: 0≤i≤k,0≤b^{0} < p^{i}, δ

p^{k−i} b^{0}
0 p^{i}

= 1}

and

Γ ^{p}_{0 1}^{0}

p^{k−i} b^{0}
0 p^{i}

= Γ

p^{k−i+1} [pb^{0}]
0 p^{i}

, where [pb^{0}]≡pb^{0} mod p^{i},
Γ ^{1}_{0}^{b}_{p}

p^{k−i} b^{0}
0 p^{i}

= Γ

p^{k−i} bp^{i}+b^{0}
0 p^{i+1}

.

Observe that Γ_{p} _{0}

0 p^{k}

appears only in the first case and only ifi=k and p^{k−1} | b^{0},
because δ _{p} _{0}

0p^{k−1}

6= 1. It follows that T_{p} ·T_{p}^{k} = T_{p}^{k+1}+pΓ_{p} _{0}

0p^{k}

Γ in Γ\G/Γ.

Passing to the quotient, one sees Tp·T_{p}^{k} =T_{p}^{k+1}+pT_{p}^{k−1} in Γ^{0}\G^{0}/Γ^{0}.
Let N = (p+ 1)/2 and χ_{k} = P

|g|=kλ(g) ∈ vN(FN). Then, {T_{p}^{k} : k ∈ N0}
satisfies the same relations as {χ_{k} :k ∈N0} and ω(T_{p}k) = δ_{k,0} =τ(χ_{k}). Hence, the
von Neumann subalgebra vN(H_{p}) of vN(H) is naturally∗-isomorphic to vN(χ_{1}). In
particular, the spectrum of λ(T_{p}) is [−2√

p,2√

p] by Kesten’s theorem.

Lemma 4.5. Letpbe a prime number and consider the Hecke algebra ofSL(2,Z)≤ SL(2,Q[√

p]). Then, the elements T˜_{p}k represented by

p^{−k/2} 0
0 p^{k/2}

are self-adjoint
and satisfy the same recursion formula as T_{p}^{k} in Theorem 4.4.

Proof. Since ^{a}_{0}_{a}−1^{0}

−1

= (^{0}_{1 0}^{−1}) ^{a}_{0}_{a}−1^{0}

(_{−1 0}^{0 1}), the elements ˜T_{p}^{k} are self-adjoint.

Moreover, one has ˜T_{p}k =p^{−k/2}T_{p}k in the Hecke algebra of SL(2,Z)≤GL^{+}(2,Q[√
p]).

It follows from the proof of Theorem 4.4 that
T˜_{p}^{2} = 1

pT_{p}^{2} = 1
p

T_{p}^{2} + (p+ 1)Γ ^{p}_{0} ^{0}_{p}
Γ

= ˜T_{p}^{2} + (p+ 1)
and

T˜_{p}·T˜_{p}^{k} = 1

pp^{k+1}T_{p}·T_{p}^{k} = 1
pp^{k+1}

T_{p}^{k+1}+pΓ_{p} _{0}

0p^{k}

Γ

= ˜T_{p}^{k+1} +pT˜_{p}^{k−1}.
5. Classical Hecke operators and the Ramanujan–Petersson

conjecture

(Ref: [Iw].) LetH={x+iy:y >0}be the upper half plane and PGL^{+}(2,R)y H
be the action given by

a b c d

: z 7→ az+b cz+d.

Hence, g ∈ PGL^{+}(2,R) also acts on the functions on H by (gξ)(z) = ξ(g^{−1}z). We
note that

=

az+b cz+d

= (=z)(ad−bc)

|cz+d|^{2}

and the measure dµ_{0} = y^{−2}dx dy is PGL^{+}(2,R)-invariant. Let F be a PSL(2,Z)-
fundamental domain ofH, and viewL^{2}(F, µ_{0}) as the space of those functions which

are PSL(2,Z)-invariant and square-integrable on F. Recall the discussion in Sec-
tion 2 and consider the action of the Hecke algebra of PSL(2,Z)≤ PGL^{+}(2,Q) on
L^{2}(F, µ_{0}). Since

Γ ^{1 0}_{0}_{p}

Γ = Γ ^{p}_{0 1}^{0}
t

p−1

G

b=0

Γ ^{1}_{0}_{p}^{b}
,

it is given by

(T_{p}f)(z) =f( ^{p}_{0 1}^{0}
z) +

p−1

X

b=0

f( ^{1}_{0}^{b}_{p}

z) = f(pz) +

p−1

X

b=0

f(z+b p ).

This is the classical Hecke operator. The constant function1 is an eigenvector with
eigenvalue p+ 1. “Hecke’s trivial estimate” says kT_{p}k ≤p+ 1, and the Ramanujan–

Petersson conjecture for Maass forms asserts that kT_{p}|_{L}^{2}

0(F,µ_{0})k ≤ 2√

p. Namely,
T_{p} restricted to the orthogonal complement L^{2}_{0}(F, µ_{0}) := (C1)^{⊥} of the constant
functions has norm ≤2√

p.

Consider the hyperbolic Laplacian

∆ = −y^{2} ∂^{2}

∂x^{2} + ∂^{2}

∂y^{2}

on L^{2}(F, µ_{0}). It is an unbounded, (essentially) self-adjoint and positive operator.

The spectral resolution of ∆ has both discrete and continuous parts:

L^{2}(F, µ_{0}) = C1⊕M

Cu_{j}⊕
Z +∞

0

E(·,1

2+ it)dt.

The constant function1has eigenvalue 0. The other eigenvectorsu_{j} are called Maass
(cusp) forms. Selberg’s conjecture asserts that the eigenvalues of Maass forms are
all ≥ ^{1}_{4}. The function z 7→ E(z,^{1}_{2} + it) belongs to the Eisenstein series. It is an
eigenvector for ∆ with the eigenvalue ^{1}_{4}+t^{2}, but a care is needed because these func-
tions do not belong to L^{2} (and this is why they do not constitute discrete spectra).

Since the hyperbolic Laplacian on Hcommutes with PGL^{+}(2,R)y H, every Hecke
operators on L^{2}(F, µ_{0}) commutes with ∆ also. Thus u_{j}’s and Eisenstein series are
eigenvectors forT_{p} as well. The eigenvalues ofT_{p} at the Eisenstein seriesE(·,^{1}_{2}+ it)
are all≤2√

pand pose no problem. (The Ramanujan–Petersson conjecture in more
general setting only asserts that the eigenvalues of T_{p} at cusp forms are ≤2√

p.) Part 2. Berezin transform: From the Ramanujan–Peterson conjecture

to operator algebras (Ref: [Ra1]).

Throughout this chapter, let Γ = SL(2,Z) (or any other lattice in SL(2,R)), and F be a Γ-fundamental domain of H.

6. Discrete series representations of SL(2,R)

(Ref: Section IX in [La] and Sections 16 and 17 in [Ro].) Throughout this paper, let m≥2 be an integer.

Let µ_{m} = y^{m}µ_{0} and H_{m} = H^{2}(H, µ_{m}) be the space of all L^{2}(µ_{m})-holomorphic
functions on H. The space H_{m} is closed in L^{2}(H, µ_{m}). For g = (^{a b}_{c d})^{−1} ∈SL(2,R),

(πm(g)f)(z) = f(g^{−1}z)(cz+d)^{−m}

defines a unitary representation πm on Hm. Indeed, multiplicativity is directly checked and since

=(g^{−1}z) = =z

|cz+d|^{2},
one has

kπ_{m}(g)fk^{2}_{2} =
Z

H

|f(g^{−1}z)|^{2} |=z|^{m}

|cz+d|^{2m} dµ_{0}(z) =kfk^{2}_{2}.

The representation π_{m} is irreducible and square integrable, i.e., for everyu, v ∈H_{m},
the function c^{u}_{v}: g 7→ hπ_{m}(g)u, vi belongs to L^{2}(SL(2,R)). (Proofs omitted.) For
a fixed non-zero v, the map H_{m} 3 u 7→ c^{u}_{v} ∈ L^{2}(SL(2,R)) defines a (bounded)
intertwiner between π_{m} and the right regular representation ρ. By irreducibility,
H_{m} can be regarded as a subrepresentation ofρ_{SL(2,R)}.

Lemma 6.1. There is d >0, called the formal dimension of π_{m}, which satisfies for
every u, v, u^{0}, v^{0} ∈H_{m} that

hc^{u}_{v}, c^{u}_{v}0^{0}i_{L}2(G) = 1

dhu, u^{0}ihv, v^{0}i.

Proof. Fix v, v^{0} and let T u = c^{u}_{v} and T^{0}u^{0} = c^{u}_{v}0^{0}. Then, T^{∗}T^{0} commutes with
π_{m}(SL(2,R)) and hence is a constant α_{v,v}^{0}. Thus hc^{u}_{v}, c^{u}_{v}0^{0}i = γ_{v,v}^{0}hu, u^{0}i for all
u, u^{0}. Likewise there exists β_{u,u}^{0} which satisfies hc^{u}_{v}, c^{u}_{v}0^{0}i = β_{u,u}^{0}hv, v^{0}i for all u, u^{0}.
Therefore, γ_{v,v}^{0}hu, u^{0}i=β_{u,u}^{0}hv, v^{0}i and d=hu, u^{0}i/β_{u,u}^{0} is independent of u, u^{0}.
Every lattice in SL(2,R) is essentially ICC; namely, there are only finitely many
conjugacy classes that are finite. Note that πm|Γ isstably unitary equivalent to the
subrepresentation ρ^{χ} of the right regular representation. Here χ= (πm|Z(Γ))^{−1} is a
character on the center Z(Γ) of Γ and, more generally for any central subgroup Z
of Γ and a character χ on Z, we define ρ^{χ} to be the right regular representation
restricted to the subspace

`^{χ}_{2}Γ :={ξ ∈`_{2}Γ :∀s∈Γ, ∀z ∈Z ξ(z^{−1}s) = χ(z)ξ(s)}.

Likewise for λ^{χ}. Note that λ^{χ}(z) = χ(z)∈C1 for z ∈Z. We define
A_{m} =π_{m}(Γ)^{0} ⊂B(H_{m}).

Note that Am is stably isomorphic to the factor L^{χ}Γ := λ^{χ}(Γ)^{00} = p^{χ}LΓ, where
p^{χ} =|Z|^{−1}P

g∈Zχ(g¯ )λ(g) is a central projection in LΓ. It is also isomorphic as the
group von Neumann algebra of Γ/Z twisted by the 2-cocycle associated withχ. We
view χas a function on Γ which is supported on Z. Then vector ˆ1^{χ}=|Z|^{−1/2}χ¯ is a
cyclic separating tracial vector for L^{χ}Γ and

τ^{χ}(λ^{χ}(s)) :=hλ^{χ}(s)ˆ1^{χ},ˆ1^{χ}i=χ(s)

for all s ∈Γ. A calculations shows that the formal dimension of π_{m} is ^{m−1}_{4π} , which
depends on the choice of a Haar measure of SL(2,R). We use the standard Haar
measure dµ_{0} _{2π}^{dθ}, and in this case, vol(SL(2,Z)\SL(2,R)) = ^{π}_{6}. A further calculation
([GHJ, 3.3.d]) shows

dim_{π}_{m}_{(SL(2,Z))}^{00}H_{m} = m−1
12 .
7. Berezin Calculus

The Hilbert space H_{m} is a reproducing kernel Hilbert space. Namely, for every
z ∈ H, the map Hm 3 f 7→ f(z) ∈C is bounded and hence there is ez ∈Hm such
that

∀f ∈H_{m} hf, e_{z}i=f(z).

The kernel K(z, ζ) = he_{ζ}, e_{z}i = e_{ζ}(z) is called the reproducing kernel. It is known
that for some constant c_{m}>0 (probably c_{m} = (m−1)/4) one has

K(z, ζ) = c_{m}

((z−ζ)/2i)¯ ^{m} and in particular K(z, z) = c_{m}
(=z)^{m}.

(See Section 1.1 in [HKZ], where they work with the disk model, but it is inter-
changeable with the upper half plane model by using formulae in Section IX.3 in
[La].) Note that dµm(z) = cm·K(z, z)^{−1}dµ0(z). We define

u(z, ζ) = |z−ζ|^{2}
4(=z)(=ζ).

Recall the relation between uand the hyperbolic distance d_{H} onH:
coshd_{H}(z, ζ) = 1 + 2u(z, ζ) = |z−ζ|¯^{2}

2(=z)(=ζ) −1.

In particular,uis invariant under the diagonal action of SL(2,R). We further define δ(z, ζ) =

1 1 +u(z, ζ)

m

=

4(=z)(=ζ)

|z−ζ|¯^{2}
m

= |K(z, ζ)|^{2}
K(z, z)K(ζ, ζ).

Note that cm

R

Hδ(z, ζ)dµ0(ζ) = K(z, z)^{−1}R

|ez(ζ)|^{2}dµm(ζ) = 1. Also note that
every f ∈H_{m} is expressed as the weak integral

f = Z

H

f(ζ)e_{ζ}dµ_{m}(ζ).

Indeed, hf, ezi=R

f(ζ)ez(ζ)dµm(ζ) = R

f(ζ)heζ, ezidµm(ζ).

Let A∈B(H_{m}) be given. Since {e_{z}} is a total subset of H_{m}, the kernel
A(z, ζˆ ) =hAe_{ζ}, e_{z}i/K(z, ζ)

on H×H determines A. The kernel ˆA is sesqui-holomorphic, i.e., it is holomorphic in the first variable and anti-holomorphic in the second variable. It follows from the Cauchy–Riemann equations that if F(z, ζ) is a sesqui-holomorphic function and f(z) =F(z, z), then

(∂f

∂z)(z) = (∂F

∂z)(z, ζ)

ζ=z and (∂f

∂z¯)(z) = (∂F

∂ζ¯)(z, ζ) ζ=z.

In particular, the restriction F 7→ f is a one-to-one map for sesqui-holomorphic functions. Thus ˆA(z) := ˆA(z, z) determines A. This abuse of notation should not cause any confusion. The function ˆA is called the symbol or the Berezin transform of A.

Proposition 7.1. Let A, B ∈B(H_{m}). Then,
(1) sup_{z∈H}|A(z)| ≤ kAk,ˆ

(2) Ac^{∗}(z, ζ) =A(ζ, z),¯ˆ
(3) AB(z, ζd ) =R

H

K(z,η)K(η,ζ)

K(z,ζ) A(z, η) ˆˆ B(η, ζ)dµm(η).

Proof. We only prove the third identity:

hABe_{ζ}, e_{z}i=hA
Z

(Be_{ζ})(η)e_{η}dµ_{m}(η), e_{z}i=
Z

H

hAe_{η}, e_{z}ihBe_{ζ}, e_{η}idµ_{m}(η).

Lemma 7.2. Forg = (^{a b}_{c d})∈SL(2,R)andz ∈H, one has π_{m}(g)e_{z} = (c¯z+d)^{−m}e_{gz}.
Proof. he_{ζ}, π_{m}(g)e_{z}i= (π_{m}(g^{−1})e_{ζ})(z) =e_{ζ}(gz)(cz+d)^{−m} = (cz+d)^{−m}he_{ζ}, e_{gz}i.

Proposition 7.3. For A∈B(H_{m}) and g ∈SL(2,R), the symbol of π_{m}(g)^{−1}Aπ_{m}(g)
is A(gz, gζ). In particular,ˆ A∈ A_{m} iff Aˆis Γ-invariant.

Proof. We note that

K(gz, gζ) = he_{gz}, e_{gζ}i= (c¯z+d)^{m}(cζ +d)^{m}K(z, ζ)

It follows that

hπm(g^{−1})Aπm(g)eζ, ezi/K(z, ζ) = hAegz, egζi(c¯z+d)^{−m}(cζ +d)^{−m}/K(z, ζ)

=hAe_{gz}, e_{gζ}i/K(gz, gζ).

LetA∈ A_{m}. Then, the symbol ˆA(z) is a bounded Γ-invariant function on H and
hence A is determined by ˆA|F. Recall that µ_{0} is the SL(2,R)-invariant measure on
H given by dµ_{0} =y^{−2}dx dy.

Theorem 7.4. Let τ:B(H_{m})→C be the positive linear functional defined by
τ(A) = 1

µ_{0}(F)
Z

F

A(z)ˆ dµ_{0}(z)
for A∈B(H_{m}). Then, τ is the unique tracial state on A_{m}.

Proof. Observe thatτ is indeed a state onB(H_{m}). LetA∈ A_{m}. By Proposition 7.1,
one has

τ(A^{∗}A) = 1
µ_{0}(F)

Z

z∈F

Z

ζ∈H

K(z, ζ)K(ζ, z)

K(z, z) Ac^{∗}(z, ζ) ˆA(ζ, z)dµ_{m}(ζ)dµ_{0}(z)

= c_{m}
µ_{0}(F)

Z

z∈F

Z

ζ∈H

δ(z, ζ)|A(z, ζˆ )|^{2}dµ_{0}(ζ)dµ_{0}(z).

The last integration is over a Γ-fundamental domain of H×H (w.r.t. the diagonal
action) and since the measure µ0 ×µ0 is Γ-invariant, we may replace it with the
integration over another fundamental domain (z, ζ)∈H× F. Therefore, the above
integration is invariant under the swap z ↔ζ. This means that τ(A^{∗}A) =τ(AA^{∗})

and hence τ is tracial.

We note another expression of τ(A^{∗}A):

τ(A^{∗}A) = 1
µ_{0}(F)

Z

F

h(A^{∗}A)e_{z}, e_{z}i

K(z, z) dµ_{0}(z) = 1
µ_{0}(F)

Z

F

kAe_{z}k^{2}

ke_{z}k^{2} dµ_{0}(z).

8. Toeplitz operators

For f ∈ L^{∞}(H), we define the associated Toeplitz operator Tf on Hm by Tf =
P Mf|_{H}^{2}_{(H,µ}_{m}_{)}, where Mf is the multiplication operator by f and P is the orthog-
onal projection from L^{2}(H, µ_{m}) onto H_{m} = H^{2}(H, µ_{m}). One calculates the symbol

Tˆf(z, ζ) of Tf as

Tˆ_{f}(z, ζ) = hf e_{ζ}, e_{z}i/K(z, ζ) =
Z

H

f(w)e_{ζ}(w)e_{z}(w)

K(z, ζ) dµ_{m}(w)

=c_{m}
Z

H

f(w)K(w, ζ)K(w, z)

K(z, ζ)K(w, w)dµ_{0}(w).

In particular,
Tˆ_{f}(z) =c_{m}

Z

H

f(w) |K(w, z)|^{2}

K(z, z)K(w, w)dµ_{0}(w) =c_{m}
Z

H

δ(z, w)f(w)dµ_{0}(w).

Since δ andµ0 are SL(2,R)-invariant, Proposition 7.3 implies that forg ∈SL(2,R),
f ∈L^{∞}(H) and (g·f)(z) =f(g^{−1}z), one has

Tg·f =π_{m}(g)T_{f}π_{m}(g)^{∗}.

If f ∈ L^{∞}(H) is Γ-invariant, then T_{f} commutes with π_{m}(Γ). Therefore, T can be
regarded as an operator fromL^{∞}(F) intoA_{m}. We freely viewL^{∞}(F) as Γ-invariant
functions in L^{∞}(H). Let D be the positive function on F × F defined by

D(z, ζ) = c_{m}X

g∈Γ

δ(z, gζ).

This function D is called an automorphic kernel. Note that Z

F

D(z, ζ)dµ_{0}(ζ) = c_{m}
Z

H

δ(z, ζ)dµ_{0}(ζ) = 1.

In particular, the infinite sum in the definition ofDis convergent almost everywhere.

Since δis symmetric and Γ-invariant, the kernel Dis symmetric. It follows that the
integral operator associated with D is contractive on L^{p}(F, µ_{0}) for all 1 ≤ p ≤ ∞.

(Indeed, it is easy to check this for p = 1,∞. For the rest of p ∈ (1,∞), use interpolation or H¨older’s inequality.) We write the operator by B∆:

(B_{∆}f)(z) = c_{m}
Z

H

δ(z, ζ)f(ζ)dµ_{0}(ζ) =
Z

F

D(z, ζ)f(ζ)dµ_{0}(ζ).

One has ˆT_{f} =B_{∆}f for f ∈L^{∞}(F).

The operatorB_{∆}onL^{2}(F, µ_{0}) is a function of ∆ in the spectral sense (Theorem 7.4
in [Iw]). Indeed, by [Be, (4.17)] (see also Section 2.2 in [HKZ]), one has

B_{∆} =

∞

Y

k=m

1 + ∆

(k+ 1)(k+ 2) −1

.

It follows that B_{∆} is positive and injective on L^{2}(F, µ_{0}). Injectivity of B_{∆} also
follows from Proposition 2.6 in [HKZ].

Theorem 8.1. For A ∈ Am and f ∈L^{∞}(F), one has
τ(AT_{f}) = 1

µ0(F) Z

F

A(z)fˆ (z)dµ_{0}(z).

Moreover, the operator L^{∞}(F)3f 7→T_{f} ∈ A_{m} extends to a bounded linear operator
S: L^{2}(F, µ_{0})→L^{2}(A_{m}, τ) which satisfies S^{∗}S =µ_{0}(F)^{−1}B_{∆} and S^{∗}A=µ_{0}(F)^{−1}Aˆ
for A∈ A_{m} ⊂L^{2}(A_{m}, τ).

Proof. Let f ∈L^{∞}(F), which is regarded as Γ-invariant function in L^{∞}(H). Then,
one calculates (all integrals below are absolutely convergent)

τ(ATf)= _{µ}^{c}^{m}

0(F)

R

Γ\(H×H)δ(z, ζ) ˆA(z, ζ) ˆTf(z, ζ)dµ^{2}_{0}(z, ζ)

= _{µ}^{c}^{2}^{m}

0(F)

R

Γ\(H×H×H)δ(z, ζ) ˆA(z, ζ)f(w)K(w,z)K(w,ζ)

K(ζ,z)K(w,w)dµ^{3}_{0}(z, ζ, w)

= _{µ}^{c}^{2}^{m}

0(F)

R

Γ\(H×H×H)f(w) ˆA(z, ζ)K(ζ,z)K(w,z)K(w,ζ)

K(z,z)K(ζ,ζ)K(w,w)dµ^{3}_{0}(z, ζ, w)

= _{µ} ^{1}

0(F)

R

Ff(w)R

H×HhAeζ, ezi^{e}^{w}_{K(w,w)}^{(z)e}^{w}^{(ζ)}dµ^{2}_{m}(z, ζ)dµ0(w)

= _{µ} ^{1}

0(F)

R

Ff(w)^{hAe}_{K(w,w)}^{w}^{,e}^{w}^{i}dµ_{0}(w)

= _{µ} ^{1}

0(F)

R

Ff(z) ˆA(z)dµ0(z).
Hence by letting A =T_{f}^{∗} =Tf¯, one obtains

τ(T_{f}^{∗}T_{f}) = 1
µ0(F)

Z

F

f(z)(B_{∆}f)(z)dµ_{0}(z) = 1

µ0(F)hf, B_{∆}fi_{L}2(F,µ0).

Corollary 8.2. The operator S is injective and has dense range.

Proof. Since S^{∗}S = B_{∆} is injective, so is S. We prove the injectivity of S^{∗}. We
essentially prove that the formula S^{∗}A =µ_{0}(F)^{−1}Aˆ is valid on L^{2}(A_{m}, τ). By the
proof of Theorem 7.4, the map A_{m} 3 A 7→ A(z, ζˆ ) extends to a scalar multiple of
an isometry R from L^{2}(A_{m}) into L^{2}(Γ\(H×H), δ·(µ_{0}×µ_{0})). We note that ranR
consists of Γ-invariant sesqui-holomorphic functions onH×H. By Theorem 8.1, the
operatorS^{∗}R^{∗} onR(A_{m}) is the map ˆA(z, ζ)7→A(z)ˆ ∈L^{2}(F, µ_{0}), and a fortiori, the
latter map is bounded and well-defined on ranR. By sesqui-holomorphic property,

the operator S^{∗}R^{∗} is injective on ranR.

9. Hecke operators acting on von Neumann algebras

(Ref: [Ra2].) Sticking to the notation of previous section, let Γ = SL(2,Z) (or
any other lattice in SL(2,R)). The group SL(2,R) acts on B(H_{m}) by conjugation
Ad_{π}_{m} and one hasA_{m} =π_{m}(Γ)^{0} =B(H_{m})^{Γ}. Since the action Ad_{π}_{m} is trivial on the
center, it can be regarded as an action of PSL(2,R). We still denote by Γ the image

of Γ in PSL(2,R). Let G = PGL^{+}(2,Q) ≤ PSL(2,R) (or any other intermediate
subgroup Γ ≤G≤ PSL(2,R) such that Γ≤G is a Hecke pair). By the arguments
in Section 2, the Hecke algebra of Γ≤G acts onA_{m} by completely positive maps:

Ψ_{α}(x) =X

k

π_{m}(g_{k})xπ_{m}(g_{k})^{∗},
where α=F

g_{k}Γ∈Γ\G/Γ.

Theorem 9.1. Let α ∈ Γ\G/Γ. Then, the completely positive map Ψ_{α}, viewed as
an operator on L^{2}(A_{m}, τ), is unitarily equivalent to the classical Hecke operator T_{α}
on L^{2}(F, µ_{0}).

Proof. Let T: L^{∞}(H) → B(H_{m}) and S: L^{2}(F, µ_{0}) → L^{2}(A_{m}, τ) be the operators
defined in Section 8 and recall that Tg·f = π_{m}(g)T_{f}π_{m}(g)^{∗}. Hence, for every f ∈
L^{∞}(F)⊂L^{2}(F, µ_{0}) and α=F

g_{k}Γ∈Γ\G/Γ, one has
ST_{α}f =SX

g_{k}·f =T^{P}_{g}_{k}·f =X

π_{m}(g_{k})T_{f}π_{m}(g_{k})^{∗} = Ψ_{α}(T_{f}).

This implies ST_{α} = Ψ_{α}S on L^{2}(F, µ_{0}). Let S =U|S| be the polar decomposition.

By Corollary 8.2, U is a unitary operator. Moreover, since |S| = B^{1/2}_{∆} commutes
with T_{α} and is a positive injective operator, one has U T_{α} = Ψ_{α}U.
We are inclined to conjecture that the operator U∆U^{∗} onL^{2}(A_{m}, τ) gives rise to
a quantum Dirichlet form in the sense of [Sa].

Part 3. Von Neumann representations for Hecke operators (Ref: [Ra2]).

10. Hecke algebras to group von Neumann algebras

Let Γ ≤ G be a Hecke pair and χ be a character on a central subgroup Z ⊂ Γ∩Z(G). (We have Γ = SL(2,Z), G= SL(2,Q[√

p]) and Z =Z(Γ) in mind.) We
assume that there is a unitary representation π: Gy`^{χ}_{2}Γ such that π|_{Γ} =ρ^{χ}. (See
Section 6.) Let Hbe the Hecke algebra of Γ≤Gandθ: Γ\G/Γ→ L^{χ}Gbe the map
defined by the formal sum

θ(α) = 1

|Z| X

g∈α

hπ(g)ˆ1^{χ},ˆ1^{χ}iλ^{χ}(g),

where ˆ1^{χ} ∈ `^{χ}_{2}Γ is the cyclic trace vector for L^{χ}Γ such that λ^{χ}(s)ˆ1^{χ} = ρ^{χ}(s^{−1})ˆ1^{χ}.
Observe that formally θ(α^{∗}) = θ(α)^{∗}. Moreover, since π(z) = ρ^{χ}(z) = χ(z^{−1}) and
λ^{χ}(z) =χ(z) forz ∈Z, the functiong 7→ hπ(g)ζ, ζiλ^{χ}(g) isZ-invariant and the map
θ factors through Γ^{0}\G^{0}/Γ^{0}, where G^{0} =G/Z and Γ^{0} = Γ/Z. The Zg-th coordinate
of θ(α) is hπ(g)ˆ1^{χ},ˆ1^{χ}iλ^{χ}(g).

Lemma 10.1. For every α ∈ Γ\G/Γ, one has kθ(α)k^{2}_{2} = ind(α). In particular,
ind(α) = ind(α^{∗}).

Proof. Letα =F

Γg_{i}. Then,
kθ(α)k^{2}_{2} =

ind(α)

X

i=1

X

s∈Γ/Z

|hπ(sg_{i})ˆ1^{χ},ˆ1^{χ}i|^{2} =

ind(α)

X

i=1

kπ(g_{i})ˆ1^{χ}k^{2} = ind(α),

since {ρ^{χ}(s)ˆ1^{χ}:s∈Γ/Z} forms a complete orthonormal basis for `^{χ}_{2}Γ.

Because of this lemma, θ is well-defined as a map into L^{2}(L^{χ}G). We consider
L^{2}(L^{χ}G) as the space of closed operators on`^{χ}_{2}Γ which are affiliated withL^{χ}G. The
product L^{2}(L^{χ}G)×L^{2}(L^{χ}G) → L^{1}(L^{χ}G) makes sense and can be computed by a
formal calculation.

Lemma 10.2. View θ(α)’s as vectors in `^{χ}_{2}Γ. Then, one has θ(α)∗θ(β) =θ(α·β).

Proof. Letα =F

g_{i} and x∈G. We will compute the Zx-th coordinate of
θ(α)∗θ(β) = 1

|Z|^{2}
X

g∈α, h∈β

hπ(g)ˆ1^{χ},ˆ1^{χ}ihπ(h)ˆ1^{χ},ˆ1^{χ}iλ^{χ}(gh).

Choose representatives (g_{i}, h_{i}) of Γ-orbits in {(g, h) : gh= x}. Note that there are
c(α, β;x) orbits. Hence theZx-th coordinate is

1

|Z|^{2}

X

1≤i≤c(α,β;x) z∈Z

X

s∈Γ

hπ(zg_{i}s)ˆ1^{χ},ˆ1^{χ}ihπ(s^{−1}h_{i})ˆ1^{χ},ˆ1^{χ}iλ^{χ}(zx)

= 1

|Z|

X

i, z

hπ(hi)ˆ1^{χ}, π(zgi)^{∗}ˆ1^{χ}iλ^{χ}(zx) = c(α, β;x)hπ(x)ˆ1^{χ},ˆ1^{χ}iλ^{χ}(x).

This shows indeed θ(α)∗θ(β) = θ(α·β).

Theorem 10.3. The map θ is well defined and extends to a trace-preserving ∗-
isomorphism from vN(H(Γ^{0}\G^{0}/Γ^{0})) into L^{χ}G.

Proof. By above lemmas, the map θ is a ∗-homomorphism from the Hecke algebra
H(Γ^{0}\G^{0}/Γ^{0}) into the∗-algebra of closed operators affiliated withL^{χ}G. Moreover, its
range is in L^{2} and it is trace-preserving. These conditions imply that θ is bounded

and extends to a von Neumann algebra isomorphism.

Recall that since L^{χ}Γ = B(`^{χ}_{2}Γ)^{Γ} where G acts on B(`^{χ}_{2}Γ) by Adπ, the Hecke
algebra of Γ ≤Gacts on L^{χ}Γ by

Ψ_{α}(x) = X

π(g_{k})xπ(g_{k})^{∗}

for α=F

Γgk. This action factors through Γ^{0}\G^{0}/Γ^{0}.

Proposition 10.4. For α∈Γ^{0}\G^{0}/Γ^{0} and x∈ L^{χ}Γ, one has
Ψ_{α}(x) =E_{L}^{L}χ^{χ}Γ^{G}(θ(α)xθ(α)^{∗}).

Proof. Forα =F

g_{k}Γ and x, y ∈Γ, the Zy-th coordinate ofθ(α)λ^{χ}(x)θ(α)^{∗} is
1

|Z|^{2}
X

z∈Z

X

g,h∈α s.t.

gxh^{−1}=zy

hπ(g)ˆ1^{χ},ˆ1^{χ}ihπ(h^{−1})ˆ1^{χ},ˆ1^{χ}iλ^{χ}(gxh^{−1})

= 1

|Z|^{2}
X

z∈Z

X

1≤k≤ind(α^{∗})
s∈Γ

hπ(gks^{−1})ˆ1^{χ},ˆ1^{χ}ihπ(x^{−1}sg_{k}^{−1}zy)ˆ1^{χ},ˆ1^{χ}iλ^{χ}(zy)

= 1

|Z|^{2}
X

z∈Z

X

1≤k≤ind(α^{∗})
s∈Γ

hπ(s^{−1})ˆ1^{χ}, π(g_{k})^{∗}ˆ1^{χ}ihπ(g_{k}^{−1}zy)ˆ1^{χ}, π(s^{−1}x)ˆ1^{χ}iλ^{χ}(zy)

= 1

|Z|^{2}
X

z∈Z

X

1≤k≤ind(α^{∗})
s∈Γ

hπ(s^{−1})ˆ1^{χ}, π(g_{k})^{∗}ˆ1^{χ}ihλ^{χ}(x)π(g^{−1}_{k} zy)ˆ1^{χ}, π(s^{−1})ˆ1^{χ}iλ^{χ}(zy)

= 1

|Z| X

z∈Z

X

1≤k≤ind(α^{∗})

hλ^{χ}(x)π(g_{k}^{−1}zy)ˆ1^{χ}, π(g_{k})^{∗}ˆ1^{χ}iλ^{χ}(zy)

= 1

|Z| X

z∈Z

hΨ_{α}(λ^{χ}(x))λ^{χ}((zy)^{−1})ˆ1^{χ},ˆ1^{χ}iλ^{χ}(zy)

=τ Ψ_{α}(λ^{χ}(x))λ^{χ}(y)^{∗}
λ^{χ}(y),

which coincides with the Zy-th coordinate of Ψ_{α}(λ^{χ}(x)).

Corollary 10.5. Suppose that ξ ∈ L^{2}(L^{χ}Γ) is a unit eigenvector of Ψ_{α} with the
eigenvalue λ(α). Then, one has

E_{ran}^{L}^{χ}^{G}_{θ}(ξ^{∗}θ(α)ξ) = λ(α)
ind(α)θ(α).

Proof. For every β ∈Γ\G/Γ, one has
τL^{χ}G ξ^{∗}θ(α)ξθ(β)^{∗}

=τL^{χ}Γ ξ^{∗}E_{L}^{L}χ^{χ}Γ^{G}(θ(α)ξθ(β)^{∗})

=

τL^{χ}Γ ξ^{∗}Ψ_{α}(ξ)

if β=α

0 if β6=α

by Proposition 10.4, since αΓβ^{∗}∩Γ6=∅ iff α=β.