THE WEIL REPRESENTATION AND HECKE OPERATORS FOR
VECTOR VALUED MODULAR FORMS
JAN HENDRIK BRUINIER AND OLIVER STEIN
Abstract. We define Hecke operators on vector valued modular forms transforming with
the Weil representation associated to a discriminant form. We describe the properties of
the corresponding algebra of Hecke operators and study the action on modular forms.
1. Introduction
Hecke operators are a fundamental tool in the study of modular forms. They can be
used to obtain information on the arithmetic nature of the Fourier coefficients. They are
vital for the definition of L-functions associated to modular forms and for understanding
their properties. The theory of Hecke operators is well developed for scalar valued modular
forms [Sh1].
In many recent works, vector valued modular forms associated to the Weil representation
play an important role, see e.g. [Bo1], [Bo2], [Br], [McG], [Sch]. For instance, Borcherds
uses them to provide a elegant description of the Fourier expansion of various theta liftings.
The purpose of the present paper is to develop the foundations of a Hecke theory for such
vector valued modular forms. The results can be used to associate an L-function to a
vector valued modular form, essentially the standard L-function.
We now describe the content of this paper in more detail. Let L be a non-degenerate
even lattice of type (b+, b−) and level N . The modulo 1 reduction of the quadratic form
on the dual lattice L′ defines a Q/Z-valued quadratic form on the discriminant group
A = L′/L. To simplify the exposition, we assume throughout the introduction that the
signature sig(L) = b+ − b− of L is even. In the body of the paper, both, odd and even
signature is treated.
The Weil representation associated to the discriminant form A is a unitary representation
of Γ = SL2(Z) on the group ring C[A],
ρA : Γ −→ U(C([A])),
defined by (2.3), (2.4). It factors through the finite quotient
S(N) := SL2(Z/NZ) ∼= Γ/Γ(N).
Date: October 27, 2018.
2000 Mathematics Subject Classification. 11F27, 11F25.
http://arxiv.org/abs/0704.1868v1
2 JAN H. BRUINIER AND OLIVER STEIN
Let k ∈ Z. A holomorphic function f : H → C[A] is called a modular form of weight k
and type ρA for the group Γ, if
f(Mτ) = (cτ + d)kρA(M)f(τ)
for all M = ( a bc d ) ∈ Γ, and f is holomorphic at the cusp ∞. We denote the vector space of
such holomorphic modular forms by Mk,A, and write Sk,A for the subspace of cusp forms.
In order to define Hecke operators on vector valued modular forms of type ρA, we need to
extend the representation ρA to a sufficiently large subgroup of GL
2 (Q). A natural starting
point is to try to extend ρA, viewed as a representation of S(N), to a representation of
G(N) := GL2(Z/NZ). However, it was observed by E. Freitag that such an extension does
not exist in general, see Example 3.1.
Here we consider the subgroup
{M ∈ G(N); det(M) ≡ � (mod N)}
of matrices whose determinant is a square modulo N . It has the extension
Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)},
where U(N) denotes the unit group of Z/NZ. The group Q(N) is isomorphic to S(N) ×
U(N). Consequently, we may extend the Weil representation to Q(N) by taking the tensor
product of ρA on S(N) and a suitable character on U(N), see Proposition 3.3.
If M is an element of G(N) whose determinant is a square modulo N , and r, r′ ∈ U(N)
with det(M) ≡ r2 ≡ r′2 (mod N), then (M, r) and (M, r′) both belong to Q(N). We prove
that the action of ρA(M, r) and ρA(M, r
′) on C[A] differ only by the action of an element
of the orthogonal group O(A), see Proposition 3.5.
This extension of the Weil representation can be used to define a Hecke operator T (M, r)
on Mk,A for every pair (M, r), where M ∈ M2(Z) and r ∈ U(N) with det(M) ≡ r2
(mod N). We compute the action of these operators on the Fourier expansion of a modular
form (see Section 4.1). They generalize the classical Hecke operators on scalar valued
modular forms and Jacobi forms (see e.g. Remark 4.4 and Remark 4.11).
In particular, for every positive integer m coprime to N we obtain a Hecke operator
T (m2)∗ := T
on Mk,A. These operators generate a commutative subalgebra of End(Mk,A), which is actu-
ally already generated by the T (p2)∗ for p prime and coprime to N . The operators T (m2)∗
take cusp forms to cusp forms and are self-adjoint with respect to the Petersson scalar
product (see Theorem 4.12). In particular, Sk,A has a basis of simultaneous eigenforms of
all T (m2)∗ with (m,N) = 1.
In Section 5 we extend the definition of the Hecke operators T (m2)∗ to all positive
integers m, not necessarily coprime to N . This is done by defining the right-action on
C[A] of a matrix α =
by the same formula as in the case where m is coprime to N .
Notice, that the corresponding linear map
C[A] −→ C[A], eλ 7→ eλ |A α = emλ
THE WEIL REPRESENTATION AND HECKE OPERATORS 3
is neither surjective nor injective in general. However, it still can be used to obtain an
“action” of the double coset ΓαΓ, see Proposition 5.1 and Lemma 5.2. This suffices to
define a corresponding Hecke operator T (m2)∗ on Mk,A, which is consistent with our earlier
definition when m is coprime to N .
For any positive integer m, the Hecke operator T (m2)∗ is self adjoint with respect to the
Petersson scalar product. Moreover, if m and n are coprime, then
T (m2)∗T (n2)∗ = T (m2n2)∗,
see Theorem 5.6. Observe that for a prime p dividing N the local Hecke algebra, that is,
the subalgebra of End(Mk,A) generated by the T (p
2ν)∗, is considerably more complicated
than in the case where p is coprime to N . For instance, it is commutative if p is coprime
to N , but in general non-commutative if p divides N .
Let S be a finite set of primes and let NS be the product of the primes in S. Let f ∈ Sk,A
be a common eigenform of all T (m2)∗ with (m,NS) = 1, so
f |k,A T (m2)∗ = λm(f)f.
We can use the above results to define an L-function associated to f by putting
LS(s, f) =
(m,NS)=1
λm(f)m
It is easily seen that LS(s, f) converges for ℜ(s) sufficiently large. By Theorem 5.6, this
L-function has an Euler product expansion. According to [Bö], it should be viewed as
the standard L-function of f . It would be interesting to study the analytic properties of
LS(s, f) in more detail. This could possibly be done by using a variant of the doubling
method (see [Bö], [Ga], [PSR]) involving a Siegel Eisenstein series of genus 2 associated to
the Weil representation of Sp(2,Z) on C[A2].
Moreover, it would be very interesting to develop a theory of new forms for the space
Mk,A. One could try to associate an irreducible automorphic representation to a vector
valued new form and study the properties of the resulting map.
If the signature of L is odd, one can carry over the above results. However, one has to
work with the metaplectic cover of Γ and with similar {±1}-extensions of S(N), G(N),
and Q(N). In this case Mk,A vanishes unless k is half-integral. Following the argument
of Shimura [Sh2], we show that the Hecke operator T (M, r) vanishes identically unless
det(M) is the square of a rational number, see Proposition 4.9. The computation of the
action of the Hecke operators on modular forms is more involved than in the even signature
(i.e. integral weight) case, see Theorem 4.10.
We thank E. Freitag for many valuable discussions on this paper. Moreover, we thank
J. Funke for several useful comments.
2. Discriminant forms and the Weil representation
Here we briefly summarize some facts on lattices, discriminant forms, and the Weil
representation. See also [Bo1], [Bo2], [Br].
4 JAN H. BRUINIER AND OLIVER STEIN
Let L be a non-degenerate even lattice of type (b+, b−). We denote the bilinear form on
L by (·, ·) and the associated quadratic form by x 7→ 1
x2 = 1
(x, x). We let sig(L) = b+−b−
be the signature of L. We write L′ for the dual lattice of L, and denote by N the level of
L, that is, the smallest positive integer such that N
x2 ∈ Z for all x ∈ L′. The finite abelian
group L′/L is called the discriminant group of L. Its order is equal to the absolute value
of the Gram determinant of L.
Recall that a discriminant form is a finite abelian group A together with a Q/Z-valued
non-degenerate quadratic form x 7→ 1
x2, for x ∈ A (see [Ni]). If L is a non-degenerate
even lattice then L′/L is a discriminant form where the quadratic form is given by the
mod 1 reduction of the quadratic form on L′. Conversely, every discriminant form can be
obtained in this way. The quadratic form on L′/L determines the signature of L modulo
8 by Milgram’s formula (see [MH] Appendix 4):
(2.1)
λ∈L′/L
e(λ2/2) =
|L′/L|e(sig(L)/8).
Here and throughout we abbreviate e(z) = e2πiz for z ∈ C. We define the signature
sig(A) ∈ Z/8Z of a discriminant form A to be the signature of any even lattice with that
discriminant form.
If A is a discriminant form, then we write An for the subgroup of elements that are n-th
powers of elements of A. Moreover, we write An for the subgroup of elements of A whose
order divides n. We have an exact sequence
0 −→ An −→ A −→ An −→ 0,(2.2)
and An is the orthogonal complement of An.
Let H = {τ ∈ C; ℑ(τ) > 0} by the complex upper half plane. We write G̃L
2 (R) for
the metaplectic two-fold cover of GL+2 (R). The elements of this group are pairs (M,φ(τ))
where M = ( a bc d ) ∈ GL
2 (R) and φ : H → C is a holomorphic function with φ(τ)2 = cτ +d.
The multiplication is defined by
(M,φ(τ))(M ′, φ′(τ)) = (MM ′, φ(M ′τ)φ′(τ)).
For g = (M,φ) ∈ G̃L
2 (R), we put det(g) = det(M). Moreover, if G is a subset of
GL+2 (R), we write G̃ for its inverse image under the covering map. Throughout we write
Γ = SL2(Z) for the full modular group. It is well known that the integral metaplectic
group Γ̃ is generated by T = (( 1 10 1 ) , 1), and S = ((
1 0 ) ,
τ ). One has the relations
S2 = (ST )3 = Z, where Z =
((−1 0
is the standard generator of the center of Γ̃.
We now recall the Weil representation associated with a discriminant form A (see also
[Bo1], [Bo2]). It is a representation of Γ̃ on the group algebra C[A]. We denote the
standard basis elements of C[A] by eλ, λ ∈ A, and write 〈·, ·〉 for the standard scalar
product (antilinear in the second entry) such that 〈eλ, eµ〉 = δλ,µ. The Weil representation
ρA associated with the discriminant form A is the unitary representation of Γ̃ on the group
THE WEIL REPRESENTATION AND HECKE OPERATORS 5
algebra C[A] defined by
ρA(T )(eλ) = e(λ
2/2)eλ,(2.3)
ρA(S)(eλ) =
e(− sig(A)/8)√
e(−(λ, µ))eµ.(2.4)
Note that
ρA(Z)(eλ) = e(− sig(A)/4)e−λ.(2.5)
The orthogonal group O(A) also acts on C[A] by
ρA(h)(eλ) = eh−1λ(2.6)
for h ∈ O(A), and the actions of Γ̃ and O(A) commute.
If the signature of A is even, then (2.5) implies that Z2 acts trivially. Hence, the
Weil representation factors through Γ. Moreover, it is trivial on the principal congruence
subgroup Γ(N), where N is the level of A, i.e., the level of any even lattice L with L′/L = A
(see e.g. [Eb], Chapter 3, Theorem 3.2). Therefore, ρA factors through the finite group
S(N) := SL2(Z/NZ) ∼= Γ/Γ(N).(2.7)
If the signature of A is odd, we notice that the level of A must be divisible by 4. This
follows from the oddity formula ([CS] p. 383 (30)) which implies that A contains odd 2-adic
Jordan components. On Γ(4) the metaplectic cover has the section
s : Γ(4) −→ Γ̃(4),
cτ + d
given by the theta multiplier system. Here
· denotes the principal branch of the holomor-
phic square root. The same argument as at the end of the proof of Theorem 5.4 in [Bo2]
implies that ρA is trivial on s(Γ(N)) and factors through the central extension of S(N) by
{±1} given by
S1(N) := Γ̃/s(Γ(N)).(2.8)
We will also need the action of ρA on diagonal matrices in S(N). Following [McG], for
integers a, d coprime to N such that ad ≡ 1 (mod N), we put
Rd := ST
dS−1T aST d.(2.9)
It is easily checked that Rd = (M,φ) where M ≡ ( a 00 d ) (mod N).
Lemma 2.1. (See [McG] Lemma 4.6.) For a, d as above we have
ρA(Rd)eλ =
gd(A)
edλ.(2.10)
Here gd(A) denotes the Gauss sum
gd(A) =
e(dλ2/2)(2.11)
6 JAN H. BRUINIER AND OLIVER STEIN
and g(A) = g1(A). �
Notice that by Milgram’s formula we have g(A) =
|A|e(sig(A)/8). Moreover, one easily
checks that |gd(A)| =
|A|. If r ∈ Z is coprime to N , then we have gdr2(A) = gd(A). In
particular, gd(A) = ga(A). Finally, Lemma 2.1 and the fact that ρA is a representation
imply the relation
gdr(A)g(A)
gd(A)gr(A)
1, if sig(A) is even,
±1, if sig(A) is odd.
(2.12)
The following more general formula was given by Borcherds.
Proposition 2.2. (See [Bo2] Theorem 5.4.) Let g = (( a bc d ) ,
cτ + d) ∈ Γ̃, and suppose
that b and c are divisible by N . Then
ρA(g)eλ = χA(g)edλ.(2.13)
Here χA denotes the character of Γ̃0(N) defined in [Bo2] Theorem 5.4. �
Lemma 2.3. Let U =
( 1 01 1 ) ,
τ + 1
∈ Γ̃. The action of Um is given by
m)eλ =
µ,ν∈A
−mµ2/2 + (µ, λ− ν)
Proof. Since Um = ST−mS−1, this follows immediately from (2.3) and (2.4). �
In many recent works vector valued modular forms associated to the Weil representation
are considered (see e.g. [Bo1], [Bo2], [Br], [McG], [Sch]). Let k ∈ 1
Z, and let Γ′ ⊂ Γ̃ be a
subgroup of finite index. A holomorphic function f : H → C[A] is called a modular form
of weight k and type ρA for the group Γ
′, if
f(Mτ) = φ(τ)2kρA(M,φ)f(τ)
for all (M,φ) ∈ Γ′, and f is holomorphic at the cusps of Γ′. We denote the C-vector space
of such holomorphic modular forms by Mk,A(Γ
′). Moreover, for the full modular group we
put Mk,A = Mk,A(Γ̃). Formula (2.5) implies that Mk,A = {0} unless
2k ≡ sig(A) mod 2.(2.14)
Recall that for f, g ∈ Mk,A(Γ′) the Petersson scalar product is defined by
(f, g) =
[Γ̃ : Γ′]
〈f(τ), g(τ)〉 yk dx dy
.(2.15)
Here x denotes the real part and y the imaginary part of τ ∈ H. The Petersson scalar
product converges when f ⊗ g is a cusp form.
THE WEIL REPRESENTATION AND HECKE OPERATORS 7
3. Extending the Weil representation
In the classical theory of scalar valued modular forms Hecke operators play an impor-
tant role (see e.g. [Sh1]). It is natural to try to define Hecke operators on vector valued
modular forms of type ρA as well. This requires the extension of the representation ρA to
a representation (of a sufficiently large subgroup) of G̃L
2 (Q). However, it is not obvious
how this can be done.
A natural starting point is to try to extend ρA, viewed as a representation of S(N)
(respectively S1(N)), to a representation of (a double cover of) GL2(Z/NZ). However, it
was observed by E. Freitag that such an extension does not exist in general. This follows
from the following example.
Example 3.1. Let d ≡ 1 (mod 4) be an integer such that p := |d| is a prime. We consider
the ring of integers O in the quadratic field Q(
d) of discriminant d. Together with the
norm form, it is an even lattice of type (1, 1) if d > 0, and of type (2, 0) if d < 0. The
dual lattice is 1√
O, the inverse of the different, and the corresponding discriminant form A
can be identified with the finite field Fp together with the quadratic form x 7→ −1dx
2. The
associated Weil representation ρA is a p-dimensional representation of S(p) = SL2(Fp) on
C[A]. The action of the orthogonal group O(A) = {±1} splits C[A] into two S(p)-invariant
subspaces
C[A]+ = span{eλ + e−λ; λ ∈ A},
C[A]− = span{eλ − e−λ; λ ∈ A}.
They have dimension p+1
, and p−1
, respectively. It follows from [NW], Theorem 4, that
the corresponding representations of S(p) are irreducible.
On the other hand, the character table of GL2(Fp) is well known, see e.g. [FH] §5.2. It
has p− 1 one-dimensional representations, p− 1 irreducible p-dimensional representations,
(p− 1)(p− 2)/2 irreducible (p+1)-dimensional representations, and (p2 − p)/2 irreducible
(p− 1)-dimsional representations.
Now assume that p ≥ 5 and that ρA has an extension ρ̃A to a representation of GL2(Fp).
Because of the irreducibility of C[A]±, such an extension would have to be a p-dimensional
irreducible representation of GL2(Fp). But these representations remain irreducible under
restriction to SL2(Fp), see [FH], p.72 (2). We obtain a contradiction.
Remark 3.2. In [McG], McGraw continues ρA to an action of GL2(Z/NZ). However, this
action is not C-linear, causing serious difficulties when one tries to define Hecke operators.
Here we consider a different group extension Q(N) of S(N) and show that ρA can be
continued to a representation of Q(N). Together with the considerations of Section 5 this
will suffice for many applications of Hecke operators; for instance, to define the standard
L-function of a modular form of type ρA.
Let A be a discriminant form as in the previous section, and let N be the level of A. We
denote by U(N) the unit group of Z/NZ. We briefly write G(N) = GL2(Z/NZ) for the
general linear group modulo N . The determinant homomorphism G(N) → U(N) gives
8 JAN H. BRUINIER AND OLIVER STEIN
rise to the exact sequence
1 −→ S(N) −→ G(N) −→ U(N) −→ 1.(3.1)
This sequence splits and G(N) can be viewed as a semidirect product of S(N) and U(N).
3.1. The case of even signature. Throughout this subsection we assume that sig(A) is
even. Let Q(N) be the group
Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)}(3.2)
with the product defined component-wise. We have an exact sequence
1 −→ S(N) −→ Q(N) −→ U(N) −→ 1,(3.3)
where S(N) → Q(N) is given by M 7→ (M, 1), and Q(N) → U(N) is given by (M, r) 7→ r.
The latter homomorphism has the section
U(N) −→ Q(N), r 7→
.(3.4)
For (M, r) ∈ Q(N) the assignment (M, r) 7→ (M ( r 00 r )
, r) defines an isomorphism
Q(N) ∼= S(N)× U(N).
We consider the action of S(N) on C[A] by the Weil representation ρA. In view of (2.12),
the assignment r 7→ g(A)
gr(A)
defines a character of U(N). We define a unitary representation
of U(N) on C[A] by putting
ρA(r)eλ = ρA
gr(A)
eλ(3.5)
for r ∈ U(N).
Proposition 3.3. The Weil representation ρA of the group S(N) extends to a unitary
representation of Q(N) by (3.5).
Proof. Since Q(N) = S(N) × U(N) we only have to check that the actions of S(N) and
U(N) commute. This is obvious. �
Remark 3.4. Clearly we could take any character of U(N) to define the action of the Weil
representation on U(N). The above choice is compatible with the definition of the Weil
representation on double cosets in Section 5. Moreover, it is compatible with the usual
Hecke operators on scalar valued modular forms, see Remark 4.4 and Remark 4.11.
The following proposition shows that the first entry of an element (M, r) ∈ Q(N) gives
the “essential contribution” to the Weil representation. If we fix M then for the different
choices of r ∈ U(N) the Weil representation ρA(M, r) differs by the action of an element
of the orthogonal group O(A).
Proposition 3.5. Let (M, r1), (M, r2) ∈ Q(N). Then h : A → A, λ 7→ r1r−12 λ is an
orthogonal transformation in O(A), and
ρA(M, r1)eλ = ρA(M, r2)ρA(h)eλ = ρA(M, r2)eh−1λ.
THE WEIL REPRESENTATION AND HECKE OPERATORS 9
Proof. We have (M, r2)
−1(M, r1) = (1, r1r
2 ), and t = r1r
2 ∈ U(N) has the property that
t2 ≡ 1 (mod N). According to (2.10) and (3.5), the action of (1, t) is given by
ρA(1, t)eλ = et−1λ.
Since (tλ, tµ) = (t2λ, µ) = (λ, µ) for all λ, µ ∈ A, multiplication by t−1 is an orthogonal
transformation. �
Lemma 3.6. We have
eλ = er−1λ,(3.6)
eλ = erλ.(3.7)
Proof. In Q(N) we have
= (Rr−1 , 1)
Therefore the fist formula follows from Lemma 2.1 and (3.5). The second formula follows
similarly. �
3.2. The case of odd signature. Throughout this subsection we assume that sig(A) is
odd. In this case the argument of the previous section only shows that the Weil repre-
sentation extends to a projective representation of Q(N). More precisely, it extends to a
group homomorphism
Q(N) −→ U(C[A])/{±1}.
This projective representation gives rise to a 2-cocycle of Q(N) with values in {±1}. The
cocycle defines a central extension Q1(N) of Q(N) by {±1}. The group S1(N), see (2.8),
can be identified with a subgroup of Q1(N) (see Section 4.2 for more details).
Proposition 3.7. The assignment (3.5) defines an extension of the Weil representation
to a projective representation of Q(N). It lifts to a unitary representation of Q1(N).
4. Hecke operators on vector valued modular forms
We now use the results of Section 3 to define Hecke operators on vector valued modular
forms of type ρA. We consider the groups
G(N) = {M ∈ GL+2 (Q); ∃n ∈ Z with (n,N) = 1 such that nM ∈ M2(Z)(4.1)
and (det(nM), N) = 1},
Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)}.(4.2)
We view Γ as a subgroup of Q(N) by the embedding M 7→ (M, 1). We are interested in
the action of the Hecke algebra of the pair (Q(N),Γ) on modular forms of type ρA and
weight k. We have to distinguish the cases whether sig(A) is even or odd.
10 JAN H. BRUINIER AND OLIVER STEIN
4.1. The case of even signature. The composition of the reduction map Q(N) → Q(N)
with the Weil representation ρA : Q(N) → U(C[A]) induces a unitary representation of
Q(N) on C[A], which we will denote by ρA as well. This left action induces a corresponding
right action by
a |A (M, r) = ρA(M, r)−1a.(4.3)
In view of (2.14) we only have to consider modular forms of integral weight k ∈ Z. Let
f be a complex valued function on H. For M ∈ GL+2 (R) the Petersson slash operator is
defined by
(f |k M)(τ) = det(M)k/2j(M, τ)−kf(Mτ).(4.4)
This defines a right action of G(N) on functions H → C. The center acts by multiplication
with ±1.
If f : H → C[A] is a function we write f =
λ∈A fλ ⊗ eλ for its decomposition in
components with respect to the standard basis of C[A]. The tensor product of the above
two actions yields a right action of Q(N) on such functions, denoted by
f |k,A (M, r) =
fλ |k M
eλ |A (M, r)
.(4.5)
Notice that a holomorphic function f : H → C[A] belongs to Mk,A, if and only if
f |k,A M = f
for all M ∈ Γ, and f is holomorphic at the cusp ∞.
We consider the Hecke algebra of the pair (Q(N),Γ) in the sense of Shimura [Sh1]. If
(M, r) ∈ Q(N), the corresponding double coset decomposes into a finite union of left cosets
Γ · (M, r) · Γ =
γ∈Γ\ΓMΓ
Γ · (γ, r).
Definition 4.1. For (M, r) ∈ Q(N) we define the corresponding Hecke operator T (M, r)
on Mk,A by
f |k,A T (M, r) = det(M)k/2−1
γ∈Γ\ΓMΓ
f |k,A (γ, r), f ∈ Mk,A.(4.6)
The usual argument now shows that f |k,A T (M, r) ∈ Mk,A. Hence T (M, r) defines an
endomorphism of Mk,A. A modular form f ∈ Mk,A has a Fourier expansion of the form
f(τ) =
n∈Z+λ2/2
c(λ, n)e(nτ)⊗ eλ.(4.7)
Theorem 4.2. Let p be a prime which is a square modulo N , and assume that r ∈ U(N)
with p ≡ r2 (mod N). Let f ∈ Mk,A and denote the Fourier expansion as in (4.7). Then
f |k,A T
n∈Z+λ2/2
b(λ, n)e(nτ)⊗ eλ,
THE WEIL REPRESENTATION AND HECKE OPERATORS 11
where
b(λ, n) = c(rλ, pn) + pk−1c(λ/r, n/p).
Here we understand that c(λ/r, n/p) = 0 if p ∤ n, i.e., if ordp(n) = 0.
Proof. Using Lemma 3.6, the formula follows in the same way as in the scalar valued
case. �
Proposition 4.3. Let p be a prime coprime to N . Let f ∈ Mk,A and denote the Fourier
expansion as in (4.7). Then
f |k,A T
n∈Z+λ2/2
b(λ, n)e(nτ)⊗ eλ,
where
b(λ, n) = c(pλ, p2n) +
gp(A)
pk−2(δp(n)− 1)c(λ, n) + p2k−2c(λ/p, n/p2),
δp(n) =
p, if p | n,
0, if p ∤ n.
Moreover, we understand that c(λ/p, n/p2) = 0 if p2 ∤ n.
Proof. We omit the proof, which is similar (but easier) to the proof of Theorem 4.10. �
Remark 4.4. If |A| = ℓ is a prime, then Mk,A can be identified with the plus or minus
subspace of the space Mk(ℓ, χℓ) of scalar valued modular forms for Γ0(ℓ) with nebentypus,
see [BB]. Under this identification, the above Hecke operators correspond to the usual
Hecke operators onMk(ℓ, χℓ). This follows by comparing the actions on Fourier expansions.
For Hecke operators on Mk(ℓ, χℓ) see e.g. [Mi], Lemma 4.5.14.
4.2. The case of odd signature. We now assume that sig(A) is odd. In view of (2.14)
we only have to consider modular forms of half integral weight k ∈ Z + 1/2. In this case,
the problem arises that both, (4.3) and (4.4), only define projective actions of Q(N). To
obtain honest actions, one has to consider appropriate central extensions.
We begin by considering the action on C[A]. The composition of the natural reduction
Q(N) → Q(N) with the projective Weil representation ρA : Q(N) → U(C[A])/{±1}
induces a projective representation
ρA : Q(N) −→ U(C[A])/{±1}, g 7→ ρA(g).
If we choose for every g ∈ Q(N) a ρ̃A(g) ∈ U(C[A]) such that ρ̃A(g) 7→ ρA(g) under the
projection to U(C[A])/{±1}, we obtain a 2-cocycle c with values in {±1} defined by
ρ̃A(g1g2) = c(g1, g2)ρ̃A(g1)ρ̃A(g2)
for g1, g2 ∈ Q(N). This cocycle gives rise to a central extension
Q1(N) = Q(N)× {±1},(4.8)
12 JAN H. BRUINIER AND OLIVER STEIN
where the multiplication is defined by
(g1, t1)(g2, t2) = (g1g2, t1t2c(g1, g2)
Here the cocycle condition
c(g1g2, g3)c(g1, g2) = c(g1, g2g3)c(g2, g3)
is equivalent to the associativity law for the above multiplication. We obtain a unitary
representation
Q1(N) −→ U(C[A])(4.9)
by putting ρA(g, t) = tρ̃A(g). This left action induces a corresponding right action
a |A (g, t) = ρA(g, t)−1a,(4.10)
for (g, t) ∈ Q1(N) and a ∈ C[A].
Without loss of generality, for (M, 1) ∈ Γ× {1} ⊂ Q(N) we choose
ρ̃A(M, 1) = ρA(M,
j(M, τ)).(4.11)
Then we have an injective homomorphism
Γ̃ −→ Q1(N), (M,±
j(M, τ)) 7→ (M, 1,±1).(4.12)
Moreover, for a positive integer m coprime to N , we put
eλ = em−1λ.(4.13)
To define an action on functions we consider the metaplectic group G̃L
2 (R). It acts on
functions f : H → C by
(f |k (M,φ))(τ) = det(M)k/2φ(τ)−2kf(Mτ)(4.14)
for (M,φ) ∈ G̃L
2 (R). In particular, we have an action of H̃(N) ⊂ G̃L
2 (R), where
H(N) = {M ∈ G(N); det(M) is a square mod N} ⊂ GL+2 (R)(4.15)
is the image of the projection of Q(N) to the first component. Notice that the cocycle
of Q(N) given by the choice of ±
j(M, τ) for (M, r) ∈ Q(N) and the cocycle c are not
isomorphic (this follows for instance from Lemma 4.8 below). However, their restrictions
to Γ̃ are isomorphic.
To define an action on C[A]-valued functions in weight k we have to consider a combi-
nation of the above extensions. We let Q2(N) be the group of tuples (M,φ, r, t), where
g = (M,φ) ∈ H̃(N), and r ∈ U(N) with det(M) ≡ r2 (mod N), and t ∈ {±1}. The
composition law is defined by
(g1, r1, t1)(g2, r2, t2) = (g1g2, r1r2, t1t2c((M1, r1), (M2, r2))
−1)(4.16)
for (gi, ri, ti) ∈ Q2(N) and gi = (Mi, φi). We denote by P : Q2 → H̃(N) the natural
projection. It has the kernel
{(1, 1, r, t) ∈ Q2; r2 ≡ 1 (N)}.
THE WEIL REPRESENTATION AND HECKE OPERATORS 13
Over Γ̃ the projection P has the section
L : Γ̃ −→ Q2, (M,±
j(M, τ)) 7→ (M,±
j(M, τ), 1,±1).(4.17)
We write
∆ = L(Γ̃).(4.18)
We define the Weil representation ρA on Q2(N) by composing the natural map to Q1(N)
with the Weil representation on that group. For γ ∈ Γ̃ we have
ρA(γ) = ρA(L(γ)).(4.19)
The tensor product of the above two actions yields a right action of Q2(N) on functions
f : H → C[A]. If we write f =
λ∈A fλ ⊗ eλ for the decomposition in components with
respect to the standard basis of C[A], the action is given by
f |k,A (M,φ, r, t) =
fλ |k (M,φ)
eλ |A (M, r, t)
(4.20)
for (M,φ, r, t) ∈ Q2(N). A holomorphic function f : H → C[A] belongs to Mk,A, if and
only if f |k,A (M,φ) = f for all (M,φ) ∈ Γ̃, and f is holomorphic at the cusp ∞.
We may now define Hecke operators on modular forms of type ρA following Shimura
[Sh2]. For α = (M,φ) ∈ H̃(N) and ξ = (α, r, t) ∈ Q2(N) we consider the double coset
∆ξ∆. If
∆ξ∆ =
is a left coset decomposition, we define the Hecke operator T (ξ) by
f |k,A T (ξ) = det(α)k/2−1
f |k,A ξi(4.21)
for f ∈ Mk,A. It is easily seen that T (ξ) is independent of the choice of the coset represen-
tatives and defines an endomorphism of Mk,A. We recall the following standard lemma.
Lemma 4.5. Let the notation be as above. Then
(∆ ∩ ξ−1∆ξ) · γi
(γi ∈ ∆) is a disjoint left coset decomposition if and only if
∆ξ∆ =
∆ · ξγi
is a disjoint left coset decomposition. �
To compute the action of Hecke operators we have to compare the groups ∆ ∩ ξ−1∆ξ
and L(Γ̃ ∩ α−1Γ̃α). Let α ∈ H̃(N) and ξ ∈ Q2(N) with P (ξ) = α. For γ ∈ Γ̃ ∩ α−1Γ̃α we
L(αγα−1) = ξL(γ)ξ−1 · (1, 1, t(γ))(4.22)
14 JAN H. BRUINIER AND OLIVER STEIN
with t(γ) ∈ {±1}. Here t(γ) is independent of the choice of ξ and is determined by the
condition
ρA(αγα
−1) = t(γ)ρA(ξ)ρA(γ)ρA(ξ)
−1.(4.23)
Hence t defines a group homomorphism
t : Γ̃ ∩ α−1Γ̃α −→ {±1}.(4.24)
Lemma 4.6. Let the notation be as above. We have L(ker(t)) = ∆∩ ξ−1∆ξ. Moreover, if
t is non-trivial, then f |k,A T (ξ) = 0 for all f ∈ Mk,A.
Proof. The proof is analogous to Proposition 1.0 in [Sh2] �
Lemma 4.7. Let the notation be as above. The homomorphism t is trivial if and only if P
gives a bijective map of ∆ξ∆ onto Γ̃αΓ̃. Moreover, when this is the case then ∆ξ∆ =
i ∆ξi
(where ξi ∈ ∆ξ∆) is a disjoint union if and only if Γ̃αΓ̃ =
i Γ̃P (ξi) is a disjoint union.
Proof. The proof is analogous to Proposition 1.1 in [Sh2] �
Lemma 4.8. Let m,n be positive integers coprime to N , α = ((m 00 n ) ,
n) ∈ H̃(N), and
ξ = (α, r, t) ∈ Q2(N). Define t : Γ̃ ∩ α−1Γ̃α −→ {±1} as in (4.24). Then
cτ + d
Here the quadratic residue symbol is defined as in [Sh2].
Proof. Define Γ′ = Γ0(m) ∩ Γ0(n) ⊂ Γ. We first notice that
Γ̃′ = Γ̃ ∩ α−1Γ̃α.
Since t is a homomorphism it suffices to prove the assertion for a set of generators of Γ̃′.
It is easily verified that Γ̃′ is generated by T n, Um, and Γ̃′ ∩ Γ̃00(N).
For γ = T n ∈ Γ̃′ we have αγα−1 = Tm. We compute t(γ) using (4.23). There is a
constant Cξ of modulus 1 such that ρA(ξ)eλ = Cξenr−1λ. Hence
ρA(ξ)ρA(T
n)ρA(ξ)
eλ = C
ξ ρA(ξ)ρA(T
n)en−1rλ
= C−1ξ e(n
−1r2λ2/2)ρA(ξ)en−1rλ
= e(n−1r2λ2/2)eλ
= e(mλ2/2)eλ.
Here n−1 in the exponentials means the inverse of n in U(N). On the other hand, we have
m)eλ = e(mλ
2/2)eλ. Hence t(T
n) = 1 in accordance with the formula.
THE WEIL REPRESENTATION AND HECKE OPERATORS 15
For γ = Um ∈ Γ̃′ we have αγα−1 = Un. We find
ρA(ξ)ρA(U
m)ρA(ξ)
eλ = C
ξ ρA(ξ)ρA(U
m)en−1rλ
ρA(ξ)
µ,ν∈A
−mµ2/2 + (µ, n−1rλ− ν)
µ,ν∈A
−mµ2/2 + (n−1rµ, λ− ν)
µ,ν∈A
−nµ2/2 + (µ, λ− ν)
This is equal to ρA(U
n). Hence t(Um) = 1 in accordance with the formula.
To compute t(γ) for γ ∈ Γ̃′ ∩ Γ̃00(N), we use the formula for the Weil representation of
Proposition 2.2. Using the definition one easily checks that if γ ∈ Γ̃′ ∩ Γ̃00(N), we have
χA(γ) = χA(αγα
Therefore
ρA(ξ)ρA(γ)ρA(ξ)
eλ = C
ξ ρA(ξ)ρA(γ)en−1rλ
= χA(γ)edλ
ρA(αγα
−1)eλ.
So t(γ) =
as claimed. �
Proposition 4.9. Let α = (M,φ) ∈ H̃(N) and ξ = (α, r, t) ∈ Q2(N). Then the Hecke
operator T (ξ) on Mk,A vanishes identically unless det(M) is a square in Q.
Proof. By multiplying with a positive integer we may assume without loss of generality
that M has entries in Z. According to the elementary divisor theorem for Γ we may further
assume that M = (m 00 n ) with positive integers m,n. So we may assume that
Hence the assertion follows from Lemma 4.6 and Lemma 4.8. �
We now study the relation between Hecke operators and Fourier coefficients. The fol-
lowing theorem is the analogue of Proposition 4.3 in the odd signature case. It can be
viewed as a generalization of Theorem 1.7 in [Sh2].
Theorem 4.10. Let p be a prime coprime to N , and put
∈ H̃(N),
ξ = (α, p, 1) ∈ Q2(N).
16 JAN H. BRUINIER AND OLIVER STEIN
Let f ∈ Mk,A and write
f(τ) =
n∈Z+λ2/2
c(λ, n)e(nτ)⊗ eλ,
f |k,AT (ξ) =
n∈Z+λ2/2
b(λ, n)e(nτ)⊗ eλ.
b(λ, n) = c(pλ, p2n) + ǫ
sig(A)+(−1|A|)
|A|2sig(A)
pk−3/2
c(λ, n) + p2k−2c(λ/p, n/p2).
Here, for an odd integer d we put
1, if d ≡ 1 (mod 4),
i, if d ≡ −1 (mod 4).
Moreover, we understand that c(λ/p, n/p2) = 0 if p2 ∤ n.
Proof. To compute f |k,AT (ξ), we need a set of representatives for ∆\∆ξ∆. In view of
Lemma 4.7 and Lemma 4.8, the map
Lξ : Γ̃αΓ̃ −→ ∆ξ∆, δ = γαγ′ 7→ Lξ(δ) := L(γ)ξL(γ′)
is a bijection (where γ, γ′ ∈ Γ̃). Here we have Lξ(δ) = (δ, p, t), and t = t(δ) is uniquely
determined by the condition
ρA(δ, p, t) = ρA(γ)ρA(ξ)ρA(γ
′).(4.25)
We have the disjoint left coset decomposition
Γ̃αΓ̃ = Γ̃α ∪
h (p)∗
Γ̃βh ∪
b (p2)
Γ̃γb,
where
Nsτ + p
−Nps 1
−Npsτ + 1
−N p2
−Nτ + p2
1 −Nt
N p2d
Nτ + p2d
and r, s ∈ Z are chosen such that pr − N2hs = 1, and d, t ∈ Z are chosen such that
p2d+N2t = 1. Consequently, we obtain the disjoint left coset decomposition
∆ξ∆ = ∆ξ ∪
h (p)∗
∆Lξ(βh) ∪
b (p2)
∆Lξ(γb).(4.26)
THE WEIL REPRESENTATION AND HECKE OPERATORS 17
The action of Lξ(βh) and Lξ(γb) in the Weil representation can be computed by means
of (4.25) and the above decompositions. Using Proposition 2.2 and the fact that ρA(ξ)eλ =
ep−1λ by (4.13), we find that
ρA(Lξ(γb))eλ = χA
−N p2
−Nτ + p2
1 −Nt
N p2d
Nτ + p2d
e(bλ2/2)epλ.
Since d is a square modulo N2, it is a square modulo the square-free part of |A| and modulo
8. Therefore, a quick calculation shows that the character values are 1. Consequently,
ρA(Lξ(γb))eλ = e(bλ
2/2)epλ.(4.27)
In the same way we obtain
ρA(Lξ(βh))eλ = ρA
Nsτ + p
ρA(ξ)eλ
Nsτ + p
ep−1λ
Nsτ + p
1−(−1|A|)−sig(A)
|A|2sig(A)
Here in the last line we have used the explicit formula for χA. Since −N2hs ≡ 1 (mod p),
we find
ρA(Lξ(βh))eλ = ǫ
1−(−1|A|)−sig(A)
|A|2sig(A)
eλ.(4.28)
Now we can compute the Fourier expansion of f |k,AT (ξ). We have
f |k,AT (ξ) = pk−2f |k,Aξ + pk−2
h (p)∗
f |k,ALξ(βh) + pk−2
b (p2)
f |k,ALξ(γb).(4.29)
For the first summand we find
pk−2f |k,Aξ = pk−2
fλ |k α
eλ |A ξ
= p2k−2
2τ)⊗ epλ.
18 JAN H. BRUINIER AND OLIVER STEIN
For the second summand in (4.29) we get
h (p)∗
f |k,ALξ(βh) = pk−2
h (p)∗
fλ |k βh
eλ |A Lξ(βh)
sig(A)+(−1|A|)−1
|A|2sig(A)
h (p)∗
fλ(τ +Nh/p)⊗ eλ.
By means of the formula for the Gauss sum
h(p)∗
e(kh/p) =
p we obtain
h (p)∗
fλ(τ +Nh/p) =
n∈Z+λ2/2
c(λ, n)e(nτ),
and therefore
h (p)∗
f |k,ALξ(βh)
sig(A)+(−1|A|)
|A|2sig(A)
pk−3/2
n∈Z+λ2/2
c(λ, n)e(nτ)⊗ eλ.
Finally, for the third summand in (4.29) we get
b (p2)
f |k,ALξ(γb) = pk−2
b (p2)
fλ |k γb
eλ |A Lξ(γb)
= p−2
b (p2)
e(−b(p−1λ)2/2)fλ(τ/p2 + b/p2)⊗ ep−1λ
n∈Z+λ2/2
c(pλ, p2n)e(nτ)⊗ eλ.
This concludes the proof of the theorem. �
Remark 4.11. Let m be a positive integer. Let L be the lattice Z with the quadratic form
x 7→ −mx2. Then L′ = 1
Z, andMk,L′/L is isomorphic to the space Jk+1/2,m of Jacobi forms
of weight k+1/2 and index m (cf. [EZ], Theorem 5.1). Under this isomorphism the Hecke
operator T (ξ) of Theorem 4.10 corresponds to the Hecke operator Tp on Jk+1/2,m defined
in [EZ] §4 (3). This follows from Theorem 4.10 and [EZ], Theorem 4.5, by comparing the
actions on Fourier expansions. Notice that we have in this particular case
sig(A)+(−1|A|)
|A|2sig(A)
For the lattice L = Z with the quadratic form x 7→ mx2 the space Mk,L′/L is isomorphic
to the space Jskewk+1/2,m of skew holomorphic Jacobi forms of weight k + 1/2 and index m
defined in [Sk]. Again, the Hecke operators of Theorem 4.10 correspond to the usual Hecke
operators on skew-holomorphic Jacobi forms.
THE WEIL REPRESENTATION AND HECKE OPERATORS 19
4.3. A Hecke algebra on vector valued modular forms. Using the double coset
actions of the previous section, we may define for every positive integer m coprime to N a
Hecke operator T (m2)∗ : Mk,A → Mk,A, by
f 7→ f |k,A T (m2)∗ =
f |k,A T
, if sig(A) is even,
f |k,A T
, 1, m, 1
, if sig(A) is odd.
(4.30)
In the case of even signature (that is integral weight), the operator T (m2)∗ differs from
the usual Hecke operator T (m2) which is given by the sum of double cosets consisting of
all integral matrices of determinant m2. This is the reason for our notation. In the case of
odd signature (that is half-integral weight), the operator T (m2)∗ is analogous to the Hecke
operator in [Sh2] on scalar valued modular forms.
Theorem 4.12. The Hecke operators T (m2)∗ (for m coprime to N) generate a commu-
tative subalgebra of End(Mk,A), which is actually already generated by the T (p
2)∗ for p
prime and coprime to N . The operators T (m2)∗ take cusp forms to cusp forms and are
self-adjoint with respect to the Petersson scalar product.
Proof. Using the actions (4.5) and (4.20), this follows in the usual way from the properties
of the abstract Hecke algebra of the pair (Q(N),Γ), respectively (Q2(N),∆). �
5. The Weil representation on double cosets
We now want to define Hecke operators T (m2)∗ as in Section 4.3 for all positive integers
m, not necessarily coprime to N . If m and N are not coprime, then the reduction of
does not belong to GL2(Z/NZ). So we cannot use the results of the previous sections.
However, it is still possible to extend the Weil representation ρA to the corresponding
double coset in a compatible way, as we will see.
Let m be a positive integer and α =
∈ G̃L
2 (R). We define a right action on
C[A] by
eλ |A α = emλ.(5.1)
To lighten the notation, we will frequenty drop the subscript from the slash operator.
Comparing with (3.6) and (4.13), we see that (5.1) is compatible with our earlier definition
in the case that (m,N) = 1. Moreover, if δ = γαγ′ ∈ Γ̃αΓ̃, we put
eλ | δ = eλ | γ | α | γ′.(5.2)
We now show that this right action is well defined, that is, independent of the decomposition
of δ.
Proposition 5.1. Let δ = γαγ′ = γ1αγ
1 ∈ Γ̃αΓ̃ (where γ, γ′, γ1, γ′1 ∈ Γ̃). Then
eλ | γ | α | γ′ = eλ | γ1 | α | γ′1.
Proof. First, one easily shows that it suffices to prove the proposition in the case that
γ′ = γ1 = 1. So we have δ = γα = αγ
1 and need to show that
eλ | γ | α = eλ | α | γ′1.(5.3)
20 JAN H. BRUINIER AND OLIVER STEIN
If we write γ =
( a bc d ) ,±
cτ + d
, then γ′1 =
a b/m2
m2c d
m2cτ + d
. In particular,
γ ∈ Γ̃0(m2) and γ′1 ∈ Γ̃0(m2). It suffices to prove (5.3) for γ in a set of generators of
Γ̃0(m2).
It is easily seen that Γ̃0(m2) is generated by Γ̃0(m2) ∩ Γ̃00(N), Tm
, and U . For γ ∈
Γ̃0(m2) ∩ Γ̃00(N) the identity (5.3) immediately follows from Proposition 2.2. For γ = Tm
it is easily verified as well.
We now consider (5.3) for γ = U . Using Lemma 2.3, we see that the left hand side of
(5.3) is equal to
eλ | U | α =
µ,ν∈A
µ2/2 + (µ, λ− ν)
emν .
Using (2.2), in the sum over ν we write ν = ν ′/m + ν ′′ where ν ′ ∈ Am and ν ′′ ∈ Am. We
obtain
eλ | U | α =
ν′∈Am
ν′′∈Am
µ2/2 + (µ, λ− ν ′/m− ν ′′)
The sum over ν ′′ is equal to |Am| if µ ∈ Am and 0 otherwise. Hence
eλ | U | α =
µ2/2 + (µ, λ− ν/m)
(mµ)2/2 + (mµ, λ− ν/m)
On the other hand, the right hand side of (5.3) is equal to
eλ | α | Um
µ,ν∈A
m2µ2/2 + (µ,mλ− ν)
|A||Am|
µ,ν∈A
µ′∈Am
m2µ2/2 + (µ+ µ′, mλ− ν)
The sum over µ′ is equal to |Am| if ν ∈ Am and 0 otherwise. Hence
eλ | α | Um
m2µ2/2 + (µ,mλ− ν)
(mµ)2/2 + (mµ, λ− ν/m)
This concludes the proof of the proposition. �
Lemma 5.2. Let δ ∈ Γ̃αΓ̃, and let γ, γ′ ∈ Γ̃. Then
eλ | (γδγ′) = eλ | γ | δ | γ′.
Proof. This follows immediately from the definition and Proposition 5.1. �
THE WEIL REPRESENTATION AND HECKE OPERATORS 21
The element β =
∈ G̃L
2 (R) belongs to the double coset Γ̃αΓ̃. The following
Proposition gives its action on C[A].
Proposition 5.3. We have
eλ | β =
eµ.(5.4)
Moreover, for the standard scalar product on C[A] we have
〈a | α, b〉 = 〈a, b | β〉, a, b ∈ C[A].(5.5)
Proof. The first assertion follows from the fact that β = SαS−1 and Lemma 5.2. The
second assertion is verified by a straightforward computation. �
Proposition 5.4. Let m,n be coprime positive integers, and put α =
. Then for g ∈ Γ̃αΓ̃ and h ∈ Γ̃βΓ̃ we have
eλ | g | h = eλ | (gh).(5.6)
Proof. We write g = γαγ′ and h = δβδ′ with γ, γ′, δ, δ′ ∈ Γ̃. Since (m,n) = 1, a simple
argument using the elementary divisor theorem shows that gh = ǫαβǫ′ for suitable ǫ, ǫ′ ∈ Γ̃.
In view of Lemma 5.2 it suffices to prove the assertion in the case that γ = δ′ = 1.
As a second reduction step, we now show that we may in addition assume that ǫ′ = 1.
In fact, the identity gh = αγ′δβ = ǫαβǫ′ implies that
δβǫ′−1 = γ′−1α−1ǫαβ.
The matrix component of the left hand side has integral entries, hence the same is true for
the right had side. Using the coprimality of m and n we may infer that
δ̃ := γ′−1α−1ǫα
belongs to Γ̃. We obtain
δβ = δ̃βǫ′,
(αγ′)(δ̃β) = ǫαβ.
From the assertion in the ǫ′ = 1 case we get
eλ | (αγ′) | (δ̃β) = eλ | (ǫαβ),
which implies the assertion for arbitrary ǫ′.
Finally we need to prove the claim in the case that γ = δ′ = ǫ′ = 1. So g = αγ′, h = δβ
gh = αγ′δβ = ǫαβ.
Since αγ′δ = ǫα, Lemma 5.2 implies that
eλ | (αγ′δ) = eλ | (ǫα),
eλ | (αγ′δ) | β = eλ | (ǫα) | β.
Now the claim follows from Lemma 5.2 and the fact that eλ | α | β = eλ | (αβ). �
22 JAN H. BRUINIER AND OLIVER STEIN
Definition 5.5. Let m be a positive integer and α =
∈ G̃L
2 (R). Let
Γ̃ · α · Γ̃ =
Γ̃ · δi
be a disjoint left coset decomposition. We define the Hecke operator T (m2)∗ on modular
forms f ∈ Mk,A by
f 7→ f |k,A T (m2)∗ = mk−2
f |k,A δi = mk−2
fλ |k δi
eλ |A δi
Lemma 5.2 implies that the definition does not depend on the choice of the coset repre-
sentatives. Notice that for m coprime to N , this definition agrees with the earlier definition
in Section 4.3.
Theorem 5.6. For any positive integer m, the Hecke operator T (m2)∗ is a linear operator
on Mk,A taking cusp forms to cusp forms. It is self adjoint with respect to the Petersson
scalar product. Moreover, if m,n are coprime, then
T (m2)∗T (n2)∗ = T (m2n2)∗.
Proof. The first assertion is a consequence of Lemma 5.2. The self adjointness follows from
Proposition 5.3 along the standard argument for scalar valued modular forms (cf. [Bu],
Theorem 1.4.3). The last assertion is a consequence of Proposition 5.4 and the correspond-
ing property of the abstract Hecke algebra. �
Remark 5.7. For a prime p dividing N the local Hecke algebra, that is, the subalgebra
of End(Mk,A) generated by the T (p
2ν)∗, is considerably more complicated than in the case
where p is coprime to N . For instance, it is commutative if p is coprime to N , but in
general non-commutative if p divides N .
References
[Bö] S. Böcherer, Über die Funktionalgleichung automorpher L-Funktionen zur Siegelschen Modul-
gruppe, J. Reine Angew. Math. 362 (1985), 146–168.
[Bo1] R. Borcherds, Automorphic forms with singularities on Grassmannians, Inv. Math. 132 (1998),
491-562.
[Bo2] R. Borcherds, Reflection groups of Lorentian lattices, Duke Math. J. 104 (2000), 319-366.
[Br] J. H. Bruinier, Borcherds products on O(2, l) and Chern classes of Heegner divisors, Springer
Lecture Notes in Mathematics 1780, Springer-Verlag (2002).
[BB] J. H. Bruinier and M. Bundschuh, On Borcherds products associated with lattices of prime dis-
criminant, Ramanujan J. 7 (2003), 49–61.
[Bu] D. Bump, Automorphic Forms and Representations, Cambridge University Press (1998).
[CS] J. H. Conway and H. J. Sloane, Sphere packings, lattices and groups. Third edition. Grundlehren
der Mathematischen Wissenschaften, 290, Springer-Verlag, New York (1999).
[Eb] W. Ebeling, Lattices and codes. A course partially based on lectures by F. Hirzebruch. Second
revised edition. Advanced Lectures in Mathematics, Vieweg, Braunschweig (2002).
[EZ] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in Math. 55, Birkhäuser (1985).
[FH] W. Fulton and J. Harris, Representation Theory, Springer GTM 129, Springer Verlag (1991).
THE WEIL REPRESENTATION AND HECKE OPERATORS 23
[Ga] P. Garrett, Pullbacks of Eisenstein series; Applications. In: Automorphic forms of several variables,
Taniguchi Symposium, Katata, 1983, Birhäuser (1984).
[McG] W. J. McGraw, The rationality of vector valued modular forms associated with the Weil represen-
tation, Math. Ann. 326 (2003), 105–122.
[MH] J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer
Grenzgebiete 73, Springer-Verlag, New York-Heidelberg (1973).
[Mi] T. Miyake, Modular forms, Springer, 1989.
[Ni] V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications. (Russian)
Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177, 238. English translation in Mathematics of
the U.S.S.R., Izvestia 14 (1980), 103–167.
[NW] A. Nobs and J. Wolfart, Die irreduziblen Darstellungen der Gruppen SL2(Zp), insbesondere
SL2(Z2). I. Teil, Comment Math. Helvetici 51 (1976), 491–526.
[PSR] I. Piatetski-Shapiro, S. Rallis, L-functions for classical groups, Lecture Notes in Mathematics
1254, Springer-Verlag, Berlin (1987).
[Sch] N. R. Scheithauer, On the classification of automorphic products and generalized Kac-Moody alge-
bras, Invent. Math. 164 (2006), 641–678.
[Sh1] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univer-
sity Press, Princeton (1971).
[Sh2] G. Shimura, On modular forms of half integral weight, Annals of Math. 97 (1973), 440-481.
[Shin] T. Shintani, On the construction of holomorphic cusp forms of half integral weight, Nagoya Math.
J. 58 (1975), 83-126.
[Sk] N.-P. Skoruppa, Developments in the theory of Jacobi forms. In: Proceedings of the conference on
automorphic funtions and their applications, Chabarovsk (eds.: N. Kuznetsov and V. Bykovsky),
The USSR Academy of Science (1990), 167–185. (see also MPI-preprint 89-40, Bonn (1989).)
Mathematisches Institut, Universität zu Köln, Weyertal 86–90, D-50931 Köln, Germany
E-mail address : bruinier@math.uni-koeln.de
E-mail address : ostein@math.uni-koeln.de
	1. Introduction
	2. Discriminant forms and the Weil representation
	3. Extending the Weil representation
	3.1. The case of even signature
	3.2. The case of odd signature
	4. Hecke operators on vector valued modular forms
	4.1. The case of even signature
	4.2. The case of odd signature
	4.3. A Hecke algebra on vector valued modular forms
	5. The Weil representation on double cosets
	References

  We define Hecke operators on vector valued modular forms transforming with
the Weil representation associated to a discriminant form. We describe the
properties of the corresponding algebra of Hecke operators and study the action
on modular forms.

Introduction
Hecke operators are a fundamental tool in the study of modular forms. They can be
used to obtain information on the arithmetic nature of the Fourier coefficients. They are
vital for the definition of L-functions associated to modular forms and for understanding
their properties. The theory of Hecke operators is well developed for scalar valued modular
forms [Sh1].
In many recent works, vector valued modular forms associated to the Weil representation
play an important role, see e.g. [Bo1], [Bo2], [Br], [McG], [Sch]. For instance, Borcherds
uses them to provide a elegant description of the Fourier expansion of various theta liftings.
The purpose of the present paper is to develop the foundations of a Hecke theory for such
vector valued modular forms. The results can be used to associate an L-function to a
vector valued modular form, essentially the standard L-function.
We now describe the content of this paper in more detail. Let L be a non-degenerate
even lattice of type (b+, b−) and level N . The modulo 1 reduction of the quadratic form
on the dual lattice L′ defines a Q/Z-valued quadratic form on the discriminant group
A = L′/L. To simplify the exposition, we assume throughout the introduction that the
signature sig(L) = b+ − b− of L is even. In the body of the paper, both, odd and even
signature is treated.
The Weil representation associated to the discriminant form A is a unitary representation
of Γ = SL2(Z) on the group ring C[A],
ρA : Γ −→ U(C([A])),
defined by (2.3), (2.4). It factors through the finite quotient
S(N) := SL2(Z/NZ) ∼= Γ/Γ(N).
Date: October 27, 2018.
2000 Mathematics Subject Classification. 11F27, 11F25.
http://arxiv.org/abs/0704.1868v1
2 JAN H. BRUINIER AND OLIVER STEIN
Let k ∈ Z. A holomorphic function f : H → C[A] is called a modular form of weight k
and type ρA for the group Γ, if
f(Mτ) = (cτ + d)kρA(M)f(τ)
for all M = ( a bc d ) ∈ Γ, and f is holomorphic at the cusp ∞. We denote the vector space of
such holomorphic modular forms by Mk,A, and write Sk,A for the subspace of cusp forms.
In order to define Hecke operators on vector valued modular forms of type ρA, we need to
extend the representation ρA to a sufficiently large subgroup of GL
2 (Q). A natural starting
point is to try to extend ρA, viewed as a representation of S(N), to a representation of
G(N) := GL2(Z/NZ). However, it was observed by E. Freitag that such an extension does
not exist in general, see Example 3.1.
Here we consider the subgroup
{M ∈ G(N); det(M) ≡ � (mod N)}
of matrices whose determinant is a square modulo N . It has the extension
Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)},
where U(N) denotes the unit group of Z/NZ. The group Q(N) is isomorphic to S(N) ×
U(N). Consequently, we may extend the Weil representation to Q(N) by taking the tensor
product of ρA on S(N) and a suitable character on U(N), see Proposition 3.3.
If M is an element of G(N) whose determinant is a square modulo N , and r, r′ ∈ U(N)
with det(M) ≡ r2 ≡ r′2 (mod N), then (M, r) and (M, r′) both belong to Q(N). We prove
that the action of ρA(M, r) and ρA(M, r
′) on C[A] differ only by the action of an element
of the orthogonal group O(A), see Proposition 3.5.
This extension of the Weil representation can be used to define a Hecke operator T (M, r)
on Mk,A for every pair (M, r), where M ∈ M2(Z) and r ∈ U(N) with det(M) ≡ r2
(mod N). We compute the action of these operators on the Fourier expansion of a modular
form (see Section 4.1). They generalize the classical Hecke operators on scalar valued
modular forms and Jacobi forms (see e.g. Remark 4.4 and Remark 4.11).
In particular, for every positive integer m coprime to N we obtain a Hecke operator
T (m2)∗ := T
on Mk,A. These operators generate a commutative subalgebra of End(Mk,A), which is actu-
ally already generated by the T (p2)∗ for p prime and coprime to N . The operators T (m2)∗
take cusp forms to cusp forms and are self-adjoint with respect to the Petersson scalar
product (see Theorem 4.12). In particular, Sk,A has a basis of simultaneous eigenforms of
all T (m2)∗ with (m,N) = 1.
In Section 5 we extend the definition of the Hecke operators T (m2)∗ to all positive
integers m, not necessarily coprime to N . This is done by defining the right-action on
C[A] of a matrix α =
by the same formula as in the case where m is coprime to N .
Notice, that the corresponding linear map
C[A] −→ C[A], eλ 7→ eλ |A α = emλ
THE WEIL REPRESENTATION AND HECKE OPERATORS 3
is neither surjective nor injective in general. However, it still can be used to obtain an
“action” of the double coset ΓαΓ, see Proposition 5.1 and Lemma 5.2. This suffices to
define a corresponding Hecke operator T (m2)∗ on Mk,A, which is consistent with our earlier
definition when m is coprime to N .
For any positive integer m, the Hecke operator T (m2)∗ is self adjoint with respect to the
Petersson scalar product. Moreover, if m and n are coprime, then
T (m2)∗T (n2)∗ = T (m2n2)∗,
see Theorem 5.6. Observe that for a prime p dividing N the local Hecke algebra, that is,
the subalgebra of End(Mk,A) generated by the T (p
2ν)∗, is considerably more complicated
than in the case where p is coprime to N . For instance, it is commutative if p is coprime
to N , but in general non-commutative if p divides N .
Let S be a finite set of primes and let NS be the product of the primes in S. Let f ∈ Sk,A
be a common eigenform of all T (m2)∗ with (m,NS) = 1, so
f |k,A T (m2)∗ = λm(f)f.
We can use the above results to define an L-function associated to f by putting
LS(s, f) =
(m,NS)=1
λm(f)m
It is easily seen that LS(s, f) converges for ℜ(s) sufficiently large. By Theorem 5.6, this
L-function has an Euler product expansion. According to [Bö], it should be viewed as
the standard L-function of f . It would be interesting to study the analytic properties of
LS(s, f) in more detail. This could possibly be done by using a variant of the doubling
method (see [Bö], [Ga], [PSR]) involving a Siegel Eisenstein series of genus 2 associated to
the Weil representation of Sp(2,Z) on C[A2].
Moreover, it would be very interesting to develop a theory of new forms for the space
Mk,A. One could try to associate an irreducible automorphic representation to a vector
valued new form and study the properties of the resulting map.
If the signature of L is odd, one can carry over the above results. However, one has to
work with the metaplectic cover of Γ and with similar {±1}-extensions of S(N), G(N),
and Q(N). In this case Mk,A vanishes unless k is half-integral. Following the argument
of Shimura [Sh2], we show that the Hecke operator T (M, r) vanishes identically unless
det(M) is the square of a rational number, see Proposition 4.9. The computation of the
action of the Hecke operators on modular forms is more involved than in the even signature
(i.e. integral weight) case, see Theorem 4.10.
We thank E. Freitag for many valuable discussions on this paper. Moreover, we thank
J. Funke for several useful comments.
2. Discriminant forms and the Weil representation
Here we briefly summarize some facts on lattices, discriminant forms, and the Weil
representation. See also [Bo1], [Bo2], [Br].
4 JAN H. BRUINIER AND OLIVER STEIN
Let L be a non-degenerate even lattice of type (b+, b−). We denote the bilinear form on
L by (·, ·) and the associated quadratic form by x 7→ 1
x2 = 1
(x, x). We let sig(L) = b+−b−
be the signature of L. We write L′ for the dual lattice of L, and denote by N the level of
L, that is, the smallest positive integer such that N
x2 ∈ Z for all x ∈ L′. The finite abelian
group L′/L is called the discriminant group of L. Its order is equal to the absolute value
of the Gram determinant of L.
Recall that a discriminant form is a finite abelian group A together with a Q/Z-valued
non-degenerate quadratic form x 7→ 1
x2, for x ∈ A (see [Ni]). If L is a non-degenerate
even lattice then L′/L is a discriminant form where the quadratic form is given by the
mod 1 reduction of the quadratic form on L′. Conversely, every discriminant form can be
obtained in this way. The quadratic form on L′/L determines the signature of L modulo
8 by Milgram’s formula (see [MH] Appendix 4):
(2.1)
λ∈L′/L
e(λ2/2) =
|L′/L|e(sig(L)/8).
Here and throughout we abbreviate e(z) = e2πiz for z ∈ C. We define the signature
sig(A) ∈ Z/8Z of a discriminant form A to be the signature of any even lattice with that
discriminant form.
If A is a discriminant form, then we write An for the subgroup of elements that are n-th
powers of elements of A. Moreover, we write An for the subgroup of elements of A whose
order divides n. We have an exact sequence
0 −→ An −→ A −→ An −→ 0,(2.2)
and An is the orthogonal complement of An.
Let H = {τ ∈ C; ℑ(τ) > 0} by the complex upper half plane. We write G̃L
2 (R) for
the metaplectic two-fold cover of GL+2 (R). The elements of this group are pairs (M,φ(τ))
where M = ( a bc d ) ∈ GL
2 (R) and φ : H → C is a holomorphic function with φ(τ)2 = cτ +d.
The multiplication is defined by
(M,φ(τ))(M ′, φ′(τ)) = (MM ′, φ(M ′τ)φ′(τ)).
For g = (M,φ) ∈ G̃L
2 (R), we put det(g) = det(M). Moreover, if G is a subset of
GL+2 (R), we write G̃ for its inverse image under the covering map. Throughout we write
Γ = SL2(Z) for the full modular group. It is well known that the integral metaplectic
group Γ̃ is generated by T = (( 1 10 1 ) , 1), and S = ((
1 0 ) ,
τ ). One has the relations
S2 = (ST )3 = Z, where Z =
((−1 0
is the standard generator of the center of Γ̃.
We now recall the Weil representation associated with a discriminant form A (see also
[Bo1], [Bo2]). It is a representation of Γ̃ on the group algebra C[A]. We denote the
standard basis elements of C[A] by eλ, λ ∈ A, and write 〈·, ·〉 for the standard scalar
product (antilinear in the second entry) such that 〈eλ, eµ〉 = δλ,µ. The Weil representation
ρA associated with the discriminant form A is the unitary representation of Γ̃ on the group
THE WEIL REPRESENTATION AND HECKE OPERATORS 5
algebra C[A] defined by
ρA(T )(eλ) = e(λ
2/2)eλ,(2.3)
ρA(S)(eλ) =
e(− sig(A)/8)√
e(−(λ, µ))eµ.(2.4)
Note that
ρA(Z)(eλ) = e(− sig(A)/4)e−λ.(2.5)
The orthogonal group O(A) also acts on C[A] by
ρA(h)(eλ) = eh−1λ(2.6)
for h ∈ O(A), and the actions of Γ̃ and O(A) commute.
If the signature of A is even, then (2.5) implies that Z2 acts trivially. Hence, the
Weil representation factors through Γ. Moreover, it is trivial on the principal congruence
subgroup Γ(N), where N is the level of A, i.e., the level of any even lattice L with L′/L = A
(see e.g. [Eb], Chapter 3, Theorem 3.2). Therefore, ρA factors through the finite group
S(N) := SL2(Z/NZ) ∼= Γ/Γ(N).(2.7)
If the signature of A is odd, we notice that the level of A must be divisible by 4. This
follows from the oddity formula ([CS] p. 383 (30)) which implies that A contains odd 2-adic
Jordan components. On Γ(4) the metaplectic cover has the section
s : Γ(4) −→ Γ̃(4),
cτ + d
given by the theta multiplier system. Here
· denotes the principal branch of the holomor-
phic square root. The same argument as at the end of the proof of Theorem 5.4 in [Bo2]
implies that ρA is trivial on s(Γ(N)) and factors through the central extension of S(N) by
{±1} given by
S1(N) := Γ̃/s(Γ(N)).(2.8)
We will also need the action of ρA on diagonal matrices in S(N). Following [McG], for
integers a, d coprime to N such that ad ≡ 1 (mod N), we put
Rd := ST
dS−1T aST d.(2.9)
It is easily checked that Rd = (M,φ) where M ≡ ( a 00 d ) (mod N).
Lemma 2.1. (See [McG] Lemma 4.6.) For a, d as above we have
ρA(Rd)eλ =
gd(A)
edλ.(2.10)
Here gd(A) denotes the Gauss sum
gd(A) =
e(dλ2/2)(2.11)
6 JAN H. BRUINIER AND OLIVER STEIN
and g(A) = g1(A). �
Notice that by Milgram’s formula we have g(A) =
|A|e(sig(A)/8). Moreover, one easily
checks that |gd(A)| =
|A|. If r ∈ Z is coprime to N , then we have gdr2(A) = gd(A). In
particular, gd(A) = ga(A). Finally, Lemma 2.1 and the fact that ρA is a representation
imply the relation
gdr(A)g(A)
gd(A)gr(A)
1, if sig(A) is even,
±1, if sig(A) is odd.
(2.12)
The following more general formula was given by Borcherds.
Proposition 2.2. (See [Bo2] Theorem 5.4.) Let g = (( a bc d ) ,
cτ + d) ∈ Γ̃, and suppose
that b and c are divisible by N . Then
ρA(g)eλ = χA(g)edλ.(2.13)
Here χA denotes the character of Γ̃0(N) defined in [Bo2] Theorem 5.4. �
Lemma 2.3. Let U =
( 1 01 1 ) ,
τ + 1
∈ Γ̃. The action of Um is given by
m)eλ =
µ,ν∈A
−mµ2/2 + (µ, λ− ν)
Proof. Since Um = ST−mS−1, this follows immediately from (2.3) and (2.4). �
In many recent works vector valued modular forms associated to the Weil representation
are considered (see e.g. [Bo1], [Bo2], [Br], [McG], [Sch]). Let k ∈ 1
Z, and let Γ′ ⊂ Γ̃ be a
subgroup of finite index. A holomorphic function f : H → C[A] is called a modular form
of weight k and type ρA for the group Γ
′, if
f(Mτ) = φ(τ)2kρA(M,φ)f(τ)
for all (M,φ) ∈ Γ′, and f is holomorphic at the cusps of Γ′. We denote the C-vector space
of such holomorphic modular forms by Mk,A(Γ
′). Moreover, for the full modular group we
put Mk,A = Mk,A(Γ̃). Formula (2.5) implies that Mk,A = {0} unless
2k ≡ sig(A) mod 2.(2.14)
Recall that for f, g ∈ Mk,A(Γ′) the Petersson scalar product is defined by
(f, g) =
[Γ̃ : Γ′]
〈f(τ), g(τ)〉 yk dx dy
.(2.15)
Here x denotes the real part and y the imaginary part of τ ∈ H. The Petersson scalar
product converges when f ⊗ g is a cusp form.
THE WEIL REPRESENTATION AND HECKE OPERATORS 7
3. Extending the Weil representation
In the classical theory of scalar valued modular forms Hecke operators play an impor-
tant role (see e.g. [Sh1]). It is natural to try to define Hecke operators on vector valued
modular forms of type ρA as well. This requires the extension of the representation ρA to
a representation (of a sufficiently large subgroup) of G̃L
2 (Q). However, it is not obvious
how this can be done.
A natural starting point is to try to extend ρA, viewed as a representation of S(N)
(respectively S1(N)), to a representation of (a double cover of) GL2(Z/NZ). However, it
was observed by E. Freitag that such an extension does not exist in general. This follows
from the following example.
Example 3.1. Let d ≡ 1 (mod 4) be an integer such that p := |d| is a prime. We consider
the ring of integers O in the quadratic field Q(
d) of discriminant d. Together with the
norm form, it is an even lattice of type (1, 1) if d > 0, and of type (2, 0) if d < 0. The
dual lattice is 1√
O, the inverse of the different, and the corresponding discriminant form A
can be identified with the finite field Fp together with the quadratic form x 7→ −1dx
2. The
associated Weil representation ρA is a p-dimensional representation of S(p) = SL2(Fp) on
C[A]. The action of the orthogonal group O(A) = {±1} splits C[A] into two S(p)-invariant
subspaces
C[A]+ = span{eλ + e−λ; λ ∈ A},
C[A]− = span{eλ − e−λ; λ ∈ A}.
They have dimension p+1
, and p−1
, respectively. It follows from [NW], Theorem 4, that
the corresponding representations of S(p) are irreducible.
On the other hand, the character table of GL2(Fp) is well known, see e.g. [FH] §5.2. It
has p− 1 one-dimensional representations, p− 1 irreducible p-dimensional representations,
(p− 1)(p− 2)/2 irreducible (p+1)-dimensional representations, and (p2 − p)/2 irreducible
(p− 1)-dimsional representations.
Now assume that p ≥ 5 and that ρA has an extension ρ̃A to a representation of GL2(Fp).
Because of the irreducibility of C[A]±, such an extension would have to be a p-dimensional
irreducible representation of GL2(Fp). But these representations remain irreducible under
restriction to SL2(Fp), see [FH], p.72 (2). We obtain a contradiction.
Remark 3.2. In [McG], McGraw continues ρA to an action of GL2(Z/NZ). However, this
action is not C-linear, causing serious difficulties when one tries to define Hecke operators.
Here we consider a different group extension Q(N) of S(N) and show that ρA can be
continued to a representation of Q(N). Together with the considerations of Section 5 this
will suffice for many applications of Hecke operators; for instance, to define the standard
L-function of a modular form of type ρA.
Let A be a discriminant form as in the previous section, and let N be the level of A. We
denote by U(N) the unit group of Z/NZ. We briefly write G(N) = GL2(Z/NZ) for the
general linear group modulo N . The determinant homomorphism G(N) → U(N) gives
8 JAN H. BRUINIER AND OLIVER STEIN
rise to the exact sequence
1 −→ S(N) −→ G(N) −→ U(N) −→ 1.(3.1)
This sequence splits and G(N) can be viewed as a semidirect product of S(N) and U(N).
3.1. The case of even signature. Throughout this subsection we assume that sig(A) is
even. Let Q(N) be the group
Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)}(3.2)
with the product defined component-wise. We have an exact sequence
1 −→ S(N) −→ Q(N) −→ U(N) −→ 1,(3.3)
where S(N) → Q(N) is given by M 7→ (M, 1), and Q(N) → U(N) is given by (M, r) 7→ r.
The latter homomorphism has the section
U(N) −→ Q(N), r 7→
.(3.4)
For (M, r) ∈ Q(N) the assignment (M, r) 7→ (M ( r 00 r )
, r) defines an isomorphism
Q(N) ∼= S(N)× U(N).
We consider the action of S(N) on C[A] by the Weil representation ρA. In view of (2.12),
the assignment r 7→ g(A)
gr(A)
defines a character of U(N). We define a unitary representation
of U(N) on C[A] by putting
ρA(r)eλ = ρA
gr(A)
eλ(3.5)
for r ∈ U(N).
Proposition 3.3. The Weil representation ρA of the group S(N) extends to a unitary
representation of Q(N) by (3.5).
Proof. Since Q(N) = S(N) × U(N) we only have to check that the actions of S(N) and
U(N) commute. This is obvious. �
Remark 3.4. Clearly we could take any character of U(N) to define the action of the Weil
representation on U(N). The above choice is compatible with the definition of the Weil
representation on double cosets in Section 5. Moreover, it is compatible with the usual
Hecke operators on scalar valued modular forms, see Remark 4.4 and Remark 4.11.
The following proposition shows that the first entry of an element (M, r) ∈ Q(N) gives
the “essential contribution” to the Weil representation. If we fix M then for the different
choices of r ∈ U(N) the Weil representation ρA(M, r) differs by the action of an element
of the orthogonal group O(A).
Proposition 3.5. Let (M, r1), (M, r2) ∈ Q(N). Then h : A → A, λ 7→ r1r−12 λ is an
orthogonal transformation in O(A), and
ρA(M, r1)eλ = ρA(M, r2)ρA(h)eλ = ρA(M, r2)eh−1λ.
THE WEIL REPRESENTATION AND HECKE OPERATORS 9
Proof. We have (M, r2)
−1(M, r1) = (1, r1r
2 ), and t = r1r
2 ∈ U(N) has the property that
t2 ≡ 1 (mod N). According to (2.10) and (3.5), the action of (1, t) is given by
ρA(1, t)eλ = et−1λ.
Since (tλ, tµ) = (t2λ, µ) = (λ, µ) for all λ, µ ∈ A, multiplication by t−1 is an orthogonal
transformation. �
Lemma 3.6. We have
eλ = er−1λ,(3.6)
eλ = erλ.(3.7)
Proof. In Q(N) we have
= (Rr−1 , 1)
Therefore the fist formula follows from Lemma 2.1 and (3.5). The second formula follows
similarly. �
3.2. The case of odd signature. Throughout this subsection we assume that sig(A) is
odd. In this case the argument of the previous section only shows that the Weil repre-
sentation extends to a projective representation of Q(N). More precisely, it extends to a
group homomorphism
Q(N) −→ U(C[A])/{±1}.
This projective representation gives rise to a 2-cocycle of Q(N) with values in {±1}. The
cocycle defines a central extension Q1(N) of Q(N) by {±1}. The group S1(N), see (2.8),
can be identified with a subgroup of Q1(N) (see Section 4.2 for more details).
Proposition 3.7. The assignment (3.5) defines an extension of the Weil representation
to a projective representation of Q(N). It lifts to a unitary representation of Q1(N).
4. Hecke operators on vector valued modular forms
We now use the results of Section 3 to define Hecke operators on vector valued modular
forms of type ρA. We consider the groups
G(N) = {M ∈ GL+2 (Q); ∃n ∈ Z with (n,N) = 1 such that nM ∈ M2(Z)(4.1)
and (det(nM), N) = 1},
Q(N) = {(M, r) ∈ G(N)× U(N); det(M) ≡ r2 (mod N)}.(4.2)
We view Γ as a subgroup of Q(N) by the embedding M 7→ (M, 1). We are interested in
the action of the Hecke algebra of the pair (Q(N),Γ) on modular forms of type ρA and
weight k. We have to distinguish the cases whether sig(A) is even or odd.
10 JAN H. BRUINIER AND OLIVER STEIN
4.1. The case of even signature. The composition of the reduction map Q(N) → Q(N)
with the Weil representation ρA : Q(N) → U(C[A]) induces a unitary representation of
Q(N) on C[A], which we will denote by ρA as well. This left action induces a corresponding
right action by
a |A (M, r) = ρA(M, r)−1a.(4.3)
In view of (2.14) we only have to consider modular forms of integral weight k ∈ Z. Let
f be a complex valued function on H. For M ∈ GL+2 (R) the Petersson slash operator is
defined by
(f |k M)(τ) = det(M)k/2j(M, τ)−kf(Mτ).(4.4)
This defines a right action of G(N) on functions H → C. The center acts by multiplication
with ±1.
If f : H → C[A] is a function we write f =
λ∈A fλ ⊗ eλ for its decomposition in
components with respect to the standard basis of C[A]. The tensor product of the above
two actions yields a right action of Q(N) on such functions, denoted by
f |k,A (M, r) =
fλ |k M
eλ |A (M, r)
.(4.5)
Notice that a holomorphic function f : H → C[A] belongs to Mk,A, if and only if
f |k,A M = f
for all M ∈ Γ, and f is holomorphic at the cusp ∞.
We consider the Hecke algebra of the pair (Q(N),Γ) in the sense of Shimura [Sh1]. If
(M, r) ∈ Q(N), the corresponding double coset decomposes into a finite union of left cosets
Γ · (M, r) · Γ =
γ∈Γ\ΓMΓ
Γ · (γ, r).
Definition 4.1. For (M, r) ∈ Q(N) we define the corresponding Hecke operator T (M, r)
on Mk,A by
f |k,A T (M, r) = det(M)k/2−1
γ∈Γ\ΓMΓ
f |k,A (γ, r), f ∈ Mk,A.(4.6)
The usual argument now shows that f |k,A T (M, r) ∈ Mk,A. Hence T (M, r) defines an
endomorphism of Mk,A. A modular form f ∈ Mk,A has a Fourier expansion of the form
f(τ) =
n∈Z+λ2/2
c(λ, n)e(nτ)⊗ eλ.(4.7)
Theorem 4.2. Let p be a prime which is a square modulo N , and assume that r ∈ U(N)
with p ≡ r2 (mod N). Let f ∈ Mk,A and denote the Fourier expansion as in (4.7). Then
f |k,A T
n∈Z+λ2/2
b(λ, n)e(nτ)⊗ eλ,
THE WEIL REPRESENTATION AND HECKE OPERATORS 11
where
b(λ, n) = c(rλ, pn) + pk−1c(λ/r, n/p).
Here we understand that c(λ/r, n/p) = 0 if p ∤ n, i.e., if ordp(n) = 0.
Proof. Using Lemma 3.6, the formula follows in the same way as in the scalar valued
case. �
Proposition 4.3. Let p be a prime coprime to N . Let f ∈ Mk,A and denote the Fourier
expansion as in (4.7). Then
f |k,A T
n∈Z+λ2/2
b(λ, n)e(nτ)⊗ eλ,
where
b(λ, n) = c(pλ, p2n) +
gp(A)
pk−2(δp(n)− 1)c(λ, n) + p2k−2c(λ/p, n/p2),
δp(n) =
p, if p | n,
0, if p ∤ n.
Moreover, we understand that c(λ/p, n/p2) = 0 if p2 ∤ n.
Proof. We omit the proof, which is similar (but easier) to the proof of Theorem 4.10. �
Remark 4.4. If |A| = ℓ is a prime, then Mk,A can be identified with the plus or minus
subspace of the space Mk(ℓ, χℓ) of scalar valued modular forms for Γ0(ℓ) with nebentypus,
see [BB]. Under this identification, the above Hecke operators correspond to the usual
Hecke operators onMk(ℓ, χℓ). This follows by comparing the actions on Fourier expansions.
For Hecke operators on Mk(ℓ, χℓ) see e.g. [Mi], Lemma 4.5.14.
4.2. The case of odd signature. We now assume that sig(A) is odd. In view of (2.14)
we only have to consider modular forms of half integral weight k ∈ Z + 1/2. In this case,
the problem arises that both, (4.3) and (4.4), only define projective actions of Q(N). To
obtain honest actions, one has to consider appropriate central extensions.
We begin by considering the action on C[A]. The composition of the natural reduction
Q(N) → Q(N) with the projective Weil representation ρA : Q(N) → U(C[A])/{±1}
induces a projective representation
ρA : Q(N) −→ U(C[A])/{±1}, g 7→ ρA(g).
If we choose for every g ∈ Q(N) a ρ̃A(g) ∈ U(C[A]) such that ρ̃A(g) 7→ ρA(g) under the
projection to U(C[A])/{±1}, we obtain a 2-cocycle c with values in {±1} defined by
ρ̃A(g1g2) = c(g1, g2)ρ̃A(g1)ρ̃A(g2)
for g1, g2 ∈ Q(N). This cocycle gives rise to a central extension
Q1(N) = Q(N)× {±1},(4.8)
12 JAN H. BRUINIER AND OLIVER STEIN
where the multiplication is defined by
(g1, t1)(g2, t2) = (g1g2, t1t2c(g1, g2)
Here the cocycle condition
c(g1g2, g3)c(g1, g2) = c(g1, g2g3)c(g2, g3)
is equivalent to the associativity law for the above multiplication. We obtain a unitary
representation
Q1(N) −→ U(C[A])(4.9)
by putting ρA(g, t) = tρ̃A(g). This left action induces a corresponding right action
a |A (g, t) = ρA(g, t)−1a,(4.10)
for (g, t) ∈ Q1(N) and a ∈ C[A].
Without loss of generality, for (M, 1) ∈ Γ× {1} ⊂ Q(N) we choose
ρ̃A(M, 1) = ρA(M,
j(M, τ)).(4.11)
Then we have an injective homomorphism
Γ̃ −→ Q1(N), (M,±
j(M, τ)) 7→ (M, 1,±1).(4.12)
Moreover, for a positive integer m coprime to N , we put
eλ = em−1λ.(4.13)
To define an action on functions we consider the metaplectic group G̃L
2 (R). It acts on
functions f : H → C by
(f |k (M,φ))(τ) = det(M)k/2φ(τ)−2kf(Mτ)(4.14)
for (M,φ) ∈ G̃L
2 (R). In particular, we have an action of H̃(N) ⊂ G̃L
2 (R), where
H(N) = {M ∈ G(N); det(M) is a square mod N} ⊂ GL+2 (R)(4.15)
is the image of the projection of Q(N) to the first component. Notice that the cocycle
of Q(N) given by the choice of ±
j(M, τ) for (M, r) ∈ Q(N) and the cocycle c are not
isomorphic (this follows for instance from Lemma 4.8 below). However, their restrictions
to Γ̃ are isomorphic.
To define an action on C[A]-valued functions in weight k we have to consider a combi-
nation of the above extensions. We let Q2(N) be the group of tuples (M,φ, r, t), where
g = (M,φ) ∈ H̃(N), and r ∈ U(N) with det(M) ≡ r2 (mod N), and t ∈ {±1}. The
composition law is defined by
(g1, r1, t1)(g2, r2, t2) = (g1g2, r1r2, t1t2c((M1, r1), (M2, r2))
−1)(4.16)
for (gi, ri, ti) ∈ Q2(N) and gi = (Mi, φi). We denote by P : Q2 → H̃(N) the natural
projection. It has the kernel
{(1, 1, r, t) ∈ Q2; r2 ≡ 1 (N)}.
THE WEIL REPRESENTATION AND HECKE OPERATORS 13
Over Γ̃ the projection P has the section
L : Γ̃ −→ Q2, (M,±
j(M, τ)) 7→ (M,±
j(M, τ), 1,±1).(4.17)
We write
∆ = L(Γ̃).(4.18)
We define the Weil representation ρA on Q2(N) by composing the natural map to Q1(N)
with the Weil representation on that group. For γ ∈ Γ̃ we have
ρA(γ) = ρA(L(γ)).(4.19)
The tensor product of the above two actions yields a right action of Q2(N) on functions
f : H → C[A]. If we write f =
λ∈A fλ ⊗ eλ for the decomposition in components with
respect to the standard basis of C[A], the action is given by
f |k,A (M,φ, r, t) =
fλ |k (M,φ)
eλ |A (M, r, t)
(4.20)
for (M,φ, r, t) ∈ Q2(N). A holomorphic function f : H → C[A] belongs to Mk,A, if and
only if f |k,A (M,φ) = f for all (M,φ) ∈ Γ̃, and f is holomorphic at the cusp ∞.
We may now define Hecke operators on modular forms of type ρA following Shimura
[Sh2]. For α = (M,φ) ∈ H̃(N) and ξ = (α, r, t) ∈ Q2(N) we consider the double coset
∆ξ∆. If
∆ξ∆ =
is a left coset decomposition, we define the Hecke operator T (ξ) by
f |k,A T (ξ) = det(α)k/2−1
f |k,A ξi(4.21)
for f ∈ Mk,A. It is easily seen that T (ξ) is independent of the choice of the coset represen-
tatives and defines an endomorphism of Mk,A. We recall the following standard lemma.
Lemma 4.5. Let the notation be as above. Then
(∆ ∩ ξ−1∆ξ) · γi
(γi ∈ ∆) is a disjoint left coset decomposition if and only if
∆ξ∆ =
∆ · ξγi
is a disjoint left coset decomposition. �
To compute the action of Hecke operators we have to compare the groups ∆ ∩ ξ−1∆ξ
and L(Γ̃ ∩ α−1Γ̃α). Let α ∈ H̃(N) and ξ ∈ Q2(N) with P (ξ) = α. For γ ∈ Γ̃ ∩ α−1Γ̃α we
L(αγα−1) = ξL(γ)ξ−1 · (1, 1, t(γ))(4.22)
14 JAN H. BRUINIER AND OLIVER STEIN
with t(γ) ∈ {±1}. Here t(γ) is independent of the choice of ξ and is determined by the
condition
ρA(αγα
−1) = t(γ)ρA(ξ)ρA(γ)ρA(ξ)
−1.(4.23)
Hence t defines a group homomorphism
t : Γ̃ ∩ α−1Γ̃α −→ {±1}.(4.24)
Lemma 4.6. Let the notation be as above. We have L(ker(t)) = ∆∩ ξ−1∆ξ. Moreover, if
t is non-trivial, then f |k,A T (ξ) = 0 for all f ∈ Mk,A.
Proof. The proof is analogous to Proposition 1.0 in [Sh2] �
Lemma 4.7. Let the notation be as above. The homomorphism t is trivial if and only if P
gives a bijective map of ∆ξ∆ onto Γ̃αΓ̃. Moreover, when this is the case then ∆ξ∆ =
i ∆ξi
(where ξi ∈ ∆ξ∆) is a disjoint union if and only if Γ̃αΓ̃ =
i Γ̃P (ξi) is a disjoint union.
Proof. The proof is analogous to Proposition 1.1 in [Sh2] �
Lemma 4.8. Let m,n be positive integers coprime to N , α = ((m 00 n ) ,
n) ∈ H̃(N), and
ξ = (α, r, t) ∈ Q2(N). Define t : Γ̃ ∩ α−1Γ̃α −→ {±1} as in (4.24). Then
cτ + d
Here the quadratic residue symbol is defined as in [Sh2].
Proof. Define Γ′ = Γ0(m) ∩ Γ0(n) ⊂ Γ. We first notice that
Γ̃′ = Γ̃ ∩ α−1Γ̃α.
Since t is a homomorphism it suffices to prove the assertion for a set of generators of Γ̃′.
It is easily verified that Γ̃′ is generated by T n, Um, and Γ̃′ ∩ Γ̃00(N).
For γ = T n ∈ Γ̃′ we have αγα−1 = Tm. We compute t(γ) using (4.23). There is a
constant Cξ of modulus 1 such that ρA(ξ)eλ = Cξenr−1λ. Hence
ρA(ξ)ρA(T
n)ρA(ξ)
eλ = C
ξ ρA(ξ)ρA(T
n)en−1rλ
= C−1ξ e(n
−1r2λ2/2)ρA(ξ)en−1rλ
= e(n−1r2λ2/2)eλ
= e(mλ2/2)eλ.
Here n−1 in the exponentials means the inverse of n in U(N). On the other hand, we have
m)eλ = e(mλ
2/2)eλ. Hence t(T
n) = 1 in accordance with the formula.
THE WEIL REPRESENTATION AND HECKE OPERATORS 15
For γ = Um ∈ Γ̃′ we have αγα−1 = Un. We find
ρA(ξ)ρA(U
m)ρA(ξ)
eλ = C
ξ ρA(ξ)ρA(U
m)en−1rλ
ρA(ξ)
µ,ν∈A
−mµ2/2 + (µ, n−1rλ− ν)
µ,ν∈A
−mµ2/2 + (n−1rµ, λ− ν)
µ,ν∈A
−nµ2/2 + (µ, λ− ν)
This is equal to ρA(U
n). Hence t(Um) = 1 in accordance with the formula.
To compute t(γ) for γ ∈ Γ̃′ ∩ Γ̃00(N), we use the formula for the Weil representation of
Proposition 2.2. Using the definition one easily checks that if γ ∈ Γ̃′ ∩ Γ̃00(N), we have
χA(γ) = χA(αγα
Therefore
ρA(ξ)ρA(γ)ρA(ξ)
eλ = C
ξ ρA(ξ)ρA(γ)en−1rλ
= χA(γ)edλ
ρA(αγα
−1)eλ.
So t(γ) =
as claimed. �
Proposition 4.9. Let α = (M,φ) ∈ H̃(N) and ξ = (α, r, t) ∈ Q2(N). Then the Hecke
operator T (ξ) on Mk,A vanishes identically unless det(M) is a square in Q.
Proof. By multiplying with a positive integer we may assume without loss of generality
that M has entries in Z. According to the elementary divisor theorem for Γ we may further
assume that M = (m 00 n ) with positive integers m,n. So we may assume that
Hence the assertion follows from Lemma 4.6 and Lemma 4.8. �
We now study the relation between Hecke operators and Fourier coefficients. The fol-
lowing theorem is the analogue of Proposition 4.3 in the odd signature case. It can be
viewed as a generalization of Theorem 1.7 in [Sh2].
Theorem 4.10. Let p be a prime coprime to N , and put
∈ H̃(N),
ξ = (α, p, 1) ∈ Q2(N).
16 JAN H. BRUINIER AND OLIVER STEIN
Let f ∈ Mk,A and write
f(τ) =
n∈Z+λ2/2
c(λ, n)e(nτ)⊗ eλ,
f |k,AT (ξ) =
n∈Z+λ2/2
b(λ, n)e(nτ)⊗ eλ.
b(λ, n) = c(pλ, p2n) + ǫ
sig(A)+(−1|A|)
|A|2sig(A)
pk−3/2
c(λ, n) + p2k−2c(λ/p, n/p2).
Here, for an odd integer d we put
1, if d ≡ 1 (mod 4),
i, if d ≡ −1 (mod 4).
Moreover, we understand that c(λ/p, n/p2) = 0 if p2 ∤ n.
Proof. To compute f |k,AT (ξ), we need a set of representatives for ∆\∆ξ∆. In view of
Lemma 4.7 and Lemma 4.8, the map
Lξ : Γ̃αΓ̃ −→ ∆ξ∆, δ = γαγ′ 7→ Lξ(δ) := L(γ)ξL(γ′)
is a bijection (where γ, γ′ ∈ Γ̃). Here we have Lξ(δ) = (δ, p, t), and t = t(δ) is uniquely
determined by the condition
ρA(δ, p, t) = ρA(γ)ρA(ξ)ρA(γ
′).(4.25)
We have the disjoint left coset decomposition
Γ̃αΓ̃ = Γ̃α ∪
h (p)∗
Γ̃βh ∪
b (p2)
Γ̃γb,
where
Nsτ + p
−Nps 1
−Npsτ + 1
−N p2
−Nτ + p2
1 −Nt
N p2d
Nτ + p2d
and r, s ∈ Z are chosen such that pr − N2hs = 1, and d, t ∈ Z are chosen such that
p2d+N2t = 1. Consequently, we obtain the disjoint left coset decomposition
∆ξ∆ = ∆ξ ∪
h (p)∗
∆Lξ(βh) ∪
b (p2)
∆Lξ(γb).(4.26)
THE WEIL REPRESENTATION AND HECKE OPERATORS 17
The action of Lξ(βh) and Lξ(γb) in the Weil representation can be computed by means
of (4.25) and the above decompositions. Using Proposition 2.2 and the fact that ρA(ξ)eλ =
ep−1λ by (4.13), we find that
ρA(Lξ(γb))eλ = χA
−N p2
−Nτ + p2
1 −Nt
N p2d
Nτ + p2d
e(bλ2/2)epλ.
Since d is a square modulo N2, it is a square modulo the square-free part of |A| and modulo
8. Therefore, a quick calculation shows that the character values are 1. Consequently,
ρA(Lξ(γb))eλ = e(bλ
2/2)epλ.(4.27)
In the same way we obtain
ρA(Lξ(βh))eλ = ρA
Nsτ + p
ρA(ξ)eλ
Nsτ + p
ep−1λ
Nsτ + p
1−(−1|A|)−sig(A)
|A|2sig(A)
Here in the last line we have used the explicit formula for χA. Since −N2hs ≡ 1 (mod p),
we find
ρA(Lξ(βh))eλ = ǫ
1−(−1|A|)−sig(A)
|A|2sig(A)
eλ.(4.28)
Now we can compute the Fourier expansion of f |k,AT (ξ). We have
f |k,AT (ξ) = pk−2f |k,Aξ + pk−2
h (p)∗
f |k,ALξ(βh) + pk−2
b (p2)
f |k,ALξ(γb).(4.29)
For the first summand we find
pk−2f |k,Aξ = pk−2
fλ |k α
eλ |A ξ
= p2k−2
2τ)⊗ epλ.
18 JAN H. BRUINIER AND OLIVER STEIN
For the second summand in (4.29) we get
h (p)∗
f |k,ALξ(βh) = pk−2
h (p)∗
fλ |k βh
eλ |A Lξ(βh)
sig(A)+(−1|A|)−1
|A|2sig(A)
h (p)∗
fλ(τ +Nh/p)⊗ eλ.
By means of the formula for the Gauss sum
h(p)∗
e(kh/p) =
p we obtain
h (p)∗
fλ(τ +Nh/p) =
n∈Z+λ2/2
c(λ, n)e(nτ),
and therefore
h (p)∗
f |k,ALξ(βh)
sig(A)+(−1|A|)
|A|2sig(A)
pk−3/2
n∈Z+λ2/2
c(λ, n)e(nτ)⊗ eλ.
Finally, for the third summand in (4.29) we get
b (p2)
f |k,ALξ(γb) = pk−2
b (p2)
fλ |k γb
eλ |A Lξ(γb)
= p−2
b (p2)
e(−b(p−1λ)2/2)fλ(τ/p2 + b/p2)⊗ ep−1λ
n∈Z+λ2/2
c(pλ, p2n)e(nτ)⊗ eλ.
This concludes the proof of the theorem. �
Remark 4.11. Let m be a positive integer. Let L be the lattice Z with the quadratic form
x 7→ −mx2. Then L′ = 1
Z, andMk,L′/L is isomorphic to the space Jk+1/2,m of Jacobi forms
of weight k+1/2 and index m (cf. [EZ], Theorem 5.1). Under this isomorphism the Hecke
operator T (ξ) of Theorem 4.10 corresponds to the Hecke operator Tp on Jk+1/2,m defined
in [EZ] §4 (3). This follows from Theorem 4.10 and [EZ], Theorem 4.5, by comparing the
actions on Fourier expansions. Notice that we have in this particular case
sig(A)+(−1|A|)
|A|2sig(A)
For the lattice L = Z with the quadratic form x 7→ mx2 the space Mk,L′/L is isomorphic
to the space Jskewk+1/2,m of skew holomorphic Jacobi forms of weight k + 1/2 and index m
defined in [Sk]. Again, the Hecke operators of Theorem 4.10 correspond to the usual Hecke
operators on skew-holomorphic Jacobi forms.
THE WEIL REPRESENTATION AND HECKE OPERATORS 19
4.3. A Hecke algebra on vector valued modular forms. Using the double coset
actions of the previous section, we may define for every positive integer m coprime to N a
Hecke operator T (m2)∗ : Mk,A → Mk,A, by
f 7→ f |k,A T (m2)∗ =
f |k,A T
, if sig(A) is even,
f |k,A T
, 1, m, 1
, if sig(A) is odd.
(4.30)
In the case of even signature (that is integral weight), the operator T (m2)∗ differs from
the usual Hecke operator T (m2) which is given by the sum of double cosets consisting of
all integral matrices of determinant m2. This is the reason for our notation. In the case of
odd signature (that is half-integral weight), the operator T (m2)∗ is analogous to the Hecke
operator in [Sh2] on scalar valued modular forms.
Theorem 4.12. The Hecke operators T (m2)∗ (for m coprime to N) generate a commu-
tative subalgebra of End(Mk,A), which is actually already generated by the T (p
2)∗ for p
prime and coprime to N . The operators T (m2)∗ take cusp forms to cusp forms and are
self-adjoint with respect to the Petersson scalar product.
Proof. Using the actions (4.5) and (4.20), this follows in the usual way from the properties
of the abstract Hecke algebra of the pair (Q(N),Γ), respectively (Q2(N),∆). �
5. The Weil representation on double cosets
We now want to define Hecke operators T (m2)∗ as in Section 4.3 for all positive integers
m, not necessarily coprime to N . If m and N are not coprime, then the reduction of
does not belong to GL2(Z/NZ). So we cannot use the results of the previous sections.
However, it is still possible to extend the Weil representation ρA to the corresponding
double coset in a compatible way, as we will see.
Let m be a positive integer and α =
∈ G̃L
2 (R). We define a right action on
C[A] by
eλ |A α = emλ.(5.1)
To lighten the notation, we will frequenty drop the subscript from the slash operator.
Comparing with (3.6) and (4.13), we see that (5.1) is compatible with our earlier definition
in the case that (m,N) = 1. Moreover, if δ = γαγ′ ∈ Γ̃αΓ̃, we put
eλ | δ = eλ | γ | α | γ′.(5.2)
We now show that this right action is well defined, that is, independent of the decomposition
of δ.
Proposition 5.1. Let δ = γαγ′ = γ1αγ
1 ∈ Γ̃αΓ̃ (where γ, γ′, γ1, γ′1 ∈ Γ̃). Then
eλ | γ | α | γ′ = eλ | γ1 | α | γ′1.
Proof. First, one easily shows that it suffices to prove the proposition in the case that
γ′ = γ1 = 1. So we have δ = γα = αγ
1 and need to show that
eλ | γ | α = eλ | α | γ′1.(5.3)
20 JAN H. BRUINIER AND OLIVER STEIN
If we write γ =
( a bc d ) ,±
cτ + d
, then γ′1 =
a b/m2
m2c d
m2cτ + d
. In particular,
γ ∈ Γ̃0(m2) and γ′1 ∈ Γ̃0(m2). It suffices to prove (5.3) for γ in a set of generators of
Γ̃0(m2).
It is easily seen that Γ̃0(m2) is generated by Γ̃0(m2) ∩ Γ̃00(N), Tm
, and U . For γ ∈
Γ̃0(m2) ∩ Γ̃00(N) the identity (5.3) immediately follows from Proposition 2.2. For γ = Tm
it is easily verified as well.
We now consider (5.3) for γ = U . Using Lemma 2.3, we see that the left hand side of
(5.3) is equal to
eλ | U | α =
µ,ν∈A
µ2/2 + (µ, λ− ν)
emν .
Using (2.2), in the sum over ν we write ν = ν ′/m + ν ′′ where ν ′ ∈ Am and ν ′′ ∈ Am. We
obtain
eλ | U | α =
ν′∈Am
ν′′∈Am
µ2/2 + (µ, λ− ν ′/m− ν ′′)
The sum over ν ′′ is equal to |Am| if µ ∈ Am and 0 otherwise. Hence
eλ | U | α =
µ2/2 + (µ, λ− ν/m)
(mµ)2/2 + (mµ, λ− ν/m)
On the other hand, the right hand side of (5.3) is equal to
eλ | α | Um
µ,ν∈A
m2µ2/2 + (µ,mλ− ν)
|A||Am|
µ,ν∈A
µ′∈Am
m2µ2/2 + (µ+ µ′, mλ− ν)
The sum over µ′ is equal to |Am| if ν ∈ Am and 0 otherwise. Hence
eλ | α | Um
m2µ2/2 + (µ,mλ− ν)
(mµ)2/2 + (mµ, λ− ν/m)
This concludes the proof of the proposition. �
Lemma 5.2. Let δ ∈ Γ̃αΓ̃, and let γ, γ′ ∈ Γ̃. Then
eλ | (γδγ′) = eλ | γ | δ | γ′.
Proof. This follows immediately from the definition and Proposition 5.1. �
THE WEIL REPRESENTATION AND HECKE OPERATORS 21
The element β =
∈ G̃L
2 (R) belongs to the double coset Γ̃αΓ̃. The following
Proposition gives its action on C[A].
Proposition 5.3. We have
eλ | β =
eµ.(5.4)
Moreover, for the standard scalar product on C[A] we have
〈a | α, b〉 = 〈a, b | β〉, a, b ∈ C[A].(5.5)
Proof. The first assertion follows from the fact that β = SαS−1 and Lemma 5.2. The
second assertion is verified by a straightforward computation. �
Proposition 5.4. Let m,n be coprime positive integers, and put α =
. Then for g ∈ Γ̃αΓ̃ and h ∈ Γ̃βΓ̃ we have
eλ | g | h = eλ | (gh).(5.6)
Proof. We write g = γαγ′ and h = δβδ′ with γ, γ′, δ, δ′ ∈ Γ̃. Since (m,n) = 1, a simple
argument using the elementary divisor theorem shows that gh = ǫαβǫ′ for suitable ǫ, ǫ′ ∈ Γ̃.
In view of Lemma 5.2 it suffices to prove the assertion in the case that γ = δ′ = 1.
As a second reduction step, we now show that we may in addition assume that ǫ′ = 1.
In fact, the identity gh = αγ′δβ = ǫαβǫ′ implies that
δβǫ′−1 = γ′−1α−1ǫαβ.
The matrix component of the left hand side has integral entries, hence the same is true for
the right had side. Using the coprimality of m and n we may infer that
δ̃ := γ′−1α−1ǫα
belongs to Γ̃. We obtain
δβ = δ̃βǫ′,
(αγ′)(δ̃β) = ǫαβ.
From the assertion in the ǫ′ = 1 case we get
eλ | (αγ′) | (δ̃β) = eλ | (ǫαβ),
which implies the assertion for arbitrary ǫ′.
Finally we need to prove the claim in the case that γ = δ′ = ǫ′ = 1. So g = αγ′, h = δβ
gh = αγ′δβ = ǫαβ.
Since αγ′δ = ǫα, Lemma 5.2 implies that
eλ | (αγ′δ) = eλ | (ǫα),
eλ | (αγ′δ) | β = eλ | (ǫα) | β.
Now the claim follows from Lemma 5.2 and the fact that eλ | α | β = eλ | (αβ). �
22 JAN H. BRUINIER AND OLIVER STEIN
Definition 5.5. Let m be a positive integer and α =
∈ G̃L
2 (R). Let
Γ̃ · α · Γ̃ =
Γ̃ · δi
be a disjoint left coset decomposition. We define the Hecke operator T (m2)∗ on modular
forms f ∈ Mk,A by
f 7→ f |k,A T (m2)∗ = mk−2
f |k,A δi = mk−2
fλ |k δi
eλ |A δi
Lemma 5.2 implies that the definition does not depend on the choice of the coset repre-
sentatives. Notice that for m coprime to N , this definition agrees with the earlier definition
in Section 4.3.
Theorem 5.6. For any positive integer m, the Hecke operator T (m2)∗ is a linear operator
on Mk,A taking cusp forms to cusp forms. It is self adjoint with respect to the Petersson
scalar product. Moreover, if m,n are coprime, then
T (m2)∗T (n2)∗ = T (m2n2)∗.
Proof. The first assertion is a consequence of Lemma 5.2. The self adjointness follows from
Proposition 5.3 along the standard argument for scalar valued modular forms (cf. [Bu],
Theorem 1.4.3). The last assertion is a consequence of Proposition 5.4 and the correspond-
ing property of the abstract Hecke algebra. �
Remark 5.7. For a prime p dividing N the local Hecke algebra, that is, the subalgebra
of End(Mk,A) generated by the T (p
2ν)∗, is considerably more complicated than in the case
where p is coprime to N . For instance, it is commutative if p is coprime to N , but in
general non-commutative if p divides N .
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Mathematisches Institut, Universität zu Köln, Weyertal 86–90, D-50931 Köln, Germany
E-mail address : bruinier@math.uni-koeln.de
E-mail address : ostein@math.uni-koeln.de
	1. Introduction
	2. Discriminant forms and the Weil representation
	3. Extending the Weil representation
	3.1. The case of even signature
	3.2. The case of odd signature
	4. Hecke operators on vector valued modular forms
	4.1. The case of even signature
	4.2. The case of odd signature
	4.3. A Hecke algebra on vector valued modular forms
	5. The Weil representation on double cosets
	References
