9:06 a.m. February 12, 2012
Analysis on arithmetic quotients: SL(2)
Classical and adelic automorphic forms
Bill Casselman
University of British Columbia
cass@math.ubc.ca
This essay will exhibit the realization of discrete series representations of SL2 (R) on spaces of holomorphic functions and relate them to automorphic forms for certain arithmetic groups. In a second essay I
shall relate these in turn to representations of adèle groups, and more generally make remarks about the
connection between arithmetic quotients and adelic quotients.
By now the realization of discrete series in terms of holomorphic functions on the upper half plane H is
so well known that explaining how this remarkable construction arises naturally is rarely done. I’ll not
explore the history of this construction, but short circuit the development by following a suggestion to
be found in [Serre:1970].
The classical theory of automorphic forms, in spite of initial appearances, is about the group GL2 , not
SL2 . The classical theory is concerned with functions on the upper half plane, which is acted on by
pos
pos
fractional linear transformations in GL2 (R), and it happens that the intersection of GL2 (R) with
GL2 (Z) is SL2 (Z). This accident is responsible for some mild confusion.
Contents
1. Representations on holomorphic functions
2. Identification with discrete series
3. Automorphic forms
I. References
1. Representations on holomorphic functions
The space P1 (C) is by definition the quotient of C2 − {0} by scalars. This may be identified with C ∪ {∞}
via the map (z, w) 7→ z/w. The copy of C in P1 (C) is the image of the pairs (z, w) with w 6= 0, and there
is a splitting over it taking z to (z, 1).
The group GL2 (C) acts on C2 in the standard way:
a b
z
az + bw
g=
: ω=
7−→ gω =
.
c d
w
cz + dw
and the corresponding action on P1 (C) is by fractional linear transformations:
g
z
(az + b)/(cz + d)
g(z)
a b
= (cz + d)
= j(g, z)
if g =
.
1
1
1
c d
Let Z be the space of complex pairs (z, w) with w 6= 0 and IM (z/w) > 0 (that is to say, z/w is real, with
positive imaginary part). The map (z, w) 7→ z/w specifies it as a fibre space over the upper half plane H
with fibre equal to C× . As I have already mentioned, this fibring is split by the inverse map z 7→ (z, 1),
and H may be identified with its image in Z .
Classical automorphic forms and representations of SL(2) (9:06 a.m. February 12, 2012)
2
If g is real, we have
az + bw
a(z/w) + b
=
cz + dw
c(z/w) + d
aζ + b
(ζ = z/w)
=
cζ + d
1 aζ + b aζ + b
aζ + b
=
−
IM
cζ + d
2i cζ + d cζ + d
ad − bc
IM (ζ) .
=
|cζ + d|2
pos
Therefore the group G = GL2 (R) takes Z into itself. Since it commutes with scalar multiplication it
preserves the fibring over H. The traditional action of G on H is derived from this, and the automorphy
factor j(g, z) measures the extent to which the splitting fails to be G-covariant. That j(g, z) measures an
obstruction explains why the equation
j(gh, z) = j g, h(z) j(h, z)
is reminiscent of a cocycle relation.
The group G acts also on the space of smooth functions on Z :
F 7−→ Lg F,
[Lg F ](ω) = F (g −1 ω)
is the associated right action. The space Z is an open subset of C × C, hence possesses an inherited
complex structure. Since G preserves this complex structure, it takes the subspace of holomorphic
∞
functions into itself. Since G commutes with scalar multiplication it leaves stable the subspace Cm
(Z)
of smooth functions of weight m—those F such that
F (λz, λw) = λ−m F (z, w)
hol
∞
as well as its subspace Cm
(Z) of holomorphic functions. In effect, the functions in Cm
(Z) are smooth
sections of a holomorphic bundle over H.
The group G acts transitively on Z , and the isotropy subgroup of (i, 1) is just I , so Z is in fact a principal
homogeneous space over G. That is to say, it is G-isomorphic to G given a choice of base point. The
stabilizer of any fibre of the projection onto H is a copy of C× in G. In particular, the fibre over i in H
may be identified with the copy of C× given by the embedding of C into M2 (R)
ι: a + ib 7−→
since if λ = a + bi then
a −b
b
a
,
i
a −b
i
i(λ)
=
1
b
a
1
ai − b
=
bi + a
(ai − b)/(a + bi)
= (a + bi)
1
i
= (a + bi)
1
i
=λ
.
1
Classical automorphic forms and representations of SL(2) (9:06 a.m. February 12, 2012)
3
Choosing a base point in Z allows us to identify functions on Z with functions on G. For each smooth
F on Z , therefore, let ΦF be the corresponding function on G:
i
ΦF (g) = F g
.
1
This follows immediately from the equation above:
[lifting-K] Lemma 1.1.
∞
The map F 7→ ΦF identifies Cm
(Z) with the space of smooth Φ satisfying the condition
for λ in C× .
Φ g · ι(λ) = λ−m Φ(g)
The group G acts on this by means of the left regular representation, compatibly with its action on
∞
Cm
(Z).
hol
Next, I’ll show how to characterize the image of Cm
(Z). with respect to Φ. Before I state the main result,
I’ll recall some elementary facts of complex analysis. A smooth C-valued function f = u(x, y) + iv(x, y)
on an open subset of C is holomorphic if and only if the real Jacobian matrix of f

∂u
 ∂x
 ∂v
∂x

∂u
∂y 
∂v 
∂y
considered as a map from R2 to itself lies in the image of C in M2 (R). (Since this image generically
coincides with the group of orientation-preserving similitudes, this means precisely that it is conformal.)
This condition is equivalent to the Cauchy-Riemann equations
∂u
∂v
=
,
∂x
∂y
∂u
∂v
=−
.
∂y
∂x
Holomorphicity may also be expressed by the single equation
∂f
=0
∂z
where
1
∂
=
∂z
2
∂
∂
+i
∂x
∂y
.
∂
∂
−i
∂x
∂y
.
When f is holomorphic, its complex derivative is
∂f
1
=
∂z
2
The notation is designed so that for an arbitrary smooth function
df =
where dz = dx + i dy .
∂f
∂f
dz +
dz
∂z
∂z
Classical automorphic forms and representations of SL(2) (9:06 a.m. February 12, 2012)
4
The Lie algebra of G has as basis:
1 0
ζ=
0 1
1
0
α=
0 −1
= (1/2)(α + ζ)
0 −1
κ=
1
0
0 1
ν+ =
0 0
In addition, we shall find useful these:
η=
x+ =
1 0
0 0
1 −i
1
0
0 1
=
−i
−i −1
0 −1
1 0
= α − i(2ν+ + κ)
= (α + ζ) − ζ − i(2ν+ + κ)
= (2η − ζ) − i(2ν+ + κ) .
The significance of x+ and its conjugate x− is that
[κ, x± ] = ±2ix± .
Now for F in C ∞ (Z), define a function f on the upper half-plane H to be its restriction to its image in
Z:
f (z) = F (z, 1) .
[lifting-hol] Lemma 1.2.
∞
For F in Cm
(Z) we have
Rx+ ΦF (p) = −4iy
if
p=
y x
,
0 1
∂f (z)
∂z
p(i) = z = x + iy .
Proof. Since κ and ζ are in the Lie algebra of ι(C× ):
Rκ F = −miF,
But then
Rζ F = −mF .
Rx+ F (p) = (Rα − 2iRν+ − iRκ )F (p)
= (2Rη − Rζ − 2iRν+ − iRκ )F (p)
= (2Rη − 2iRν+ + m − i(−mi))F (p)
= (2Rη − 2iRν+ )F (p)
Classical automorphic forms and representations of SL(2) (9:06 a.m. February 12, 2012)
5
Now I apply the basic formula RX f (g) = [LgXg−1 f ](g) to get
(2Rη − 2iRν+ )F (p) = (2Lpηp−1 − 2iLpν+ p−1 )F (p) .
But
Lη = x
∂
∂
+y
∂x
∂y
and
pηp−1 = yη − xν+
so
∂f
∂f
− 2iy
∂y
∂x
∂f
∂f
= −2iy
+i
∂x
∂y
∂f
= −4iy
.
∂z
(2Lpηp−1 − 2iLpν+ p−1 )F (p) = 2y
This has as consequence:
hol
hol
The image of Cm
(Z) under the map F 7→ ΦF is the space Cm
(G) of all smooth
functions Φ in C (G) such that
(a) Φ g · ι(λ) = λ−m Φ(g) for all λ in C× ;
(b) Rx+ Φ = 0.
[lift-hola] Proposition 1.3.
∞
∞
A function F in Cm
(Z) is determined by its restriction f to H:
F (z, w) = w−m F (z/w, 1) = w−m f (z/w) .
hol
How does the action of G on F translate to an action of G on f ? If F in Cm
(Z) restricts to f then
[Lg F ](z, 1) = F (az + b, cz + d) where g −1 =
a b
c d
= (cz + d)−m F (az + b)/(cz + d), 1
az + b
−m
= j(g −1 , z)−m f g −1 (z) .
= (cz + d) f
cz + d
For z in H and g in G, let
f (g)m (z) = j(g, z)−m f g(z) .
This defines a right action of G, whereas f 7→ f (g −1 )m defines a left action.
[compatibility] Proposition 1.4.
The following diagram is commutative:
hol
Cm
(Z)
images/cm-diagram.eps
hol
Cm
(G)
C hol (H)
The arrows are linear isomorphisms of G-spaces, if the action of G on the lower space is
f 7−→ f (g)m .
Classical automorphic forms and representations of SL(2) (9:06 a.m. February 12, 2012)
6
hol
The map from C hol (H) to Cm
(G) takes f to
f g(i) j(g, i)−m .
It will be convenient in the rest of this lecture to modify things slightly. Let
ΦF = ΦF (g) · det−m/2 (g) .
The functions ΦF are now characterized by the conditions (a) Φ g · ι(λ) = (λ/|λ|)−m Φ(g) and (b)
Rx+ Φ = 0. In particular, Φ is fixed by the connected component of the center of G. To go with this
modification, redefine
f [g]m (z) = detm/2 (g)(cz + d)−m f g(z) .
Here, the connected component of the scalars also acts trivially. The left regular action of G on the
functions Φ and this new one on functions on H are compatible. This normalization is just one of several
in the literature. Each has its own charms, but for us invariance under the positive real scalars turns out to
be most convenient. Keep in mind that the embedding of SL2 (R) into GL2 (R) induces an isomorphism
pos
of it with the quotient GL2 (R)/Rpos .
The explicit relation between f and ΦF is
ΦF (g) = f g(i) detm/2 (g) j(g, i)−m .
pos
Instead of working with GL2 (R) we can work with SL2 (R).
[sl2-lift] Proposition 1.5.
The map from C hol (H) to C ∞ (SL2 (R) taking
f 7−→ f g(i) j(g, i)−m
is an SL2 (R) isomorphism of C hol (H) with the space of smooth functions Φ on SL2 (R) such that
Φ g · ι(λ) = λ−m Φ(g) for λ such that |λ| = 1
Rx+ Φ = 0 .
The image of λ |λ| = 1 is the maximal compact subgroup
SO(2) =
in SL2 (R).
cos θ
sin θ
− sin θ
cos θ
Classical automorphic forms and representations of SL(2) (9:06 a.m. February 12, 2012)
7
2. Identification with discrete series
pos
Let πm be the left representation of G = GL2 (R) on the space of holomorphic functions on H taking
hol
f to f [g −1 ]m . This is equivalent to the left representation on Cm
(Z). This is trivial on the connected
component of the scalar matrices, hence may be identified with a representation of SL2 (R). In this
+
section we’ll see that for m > 1 it is an incarnation of the discrete series Dm−1
and for m = 1 one of the
so-called ‘limit’ of discrete series.
Let K be the maximal compact subgroup SO2 of G. Recall the character
ε:
c −s
7 → c + is .
−
s
c
The restriction of πm to K will decompose into a sum of one-diemnsional eigenspaces with respect to K .
[k-finite-holomorphic] Proposition 2.1.
The eigenfunctions of K in πm are the functions
fp (z) =
on H with p ≥ 0. For k in K
z−i
z+i
p 1
z+i
m
πm (k)fp = εm+2p (k)fp .
Proof. The action of G on Z is inherited from a linear one. It maps linear functions to linear functions,
polynomials to polynomials. Linear functions correspond to row vectors, since
pz + qw = [ p q ]
z
w
.
So Lg takes the linear function pz + qw to Lg ([p q]), with
[Lg ([ p q ])
z
w
[p q]g
−1
z
w
so
[ p q ] 7→ [ p q ] g −1 .
The linear functions corresponding to [1, ±i] are eigenfunctions of K in this representation, since
[1, ±i]
c −s
= (c ± is)[1, ±i]
s
c
Lk (z ± iw) = ε∓2 (z ± iw)
Lk (z − iw)p (z + iw)q = εp−q (k)(z − iw)p (z + iw)q .
Since 1/(z − iw) has a singularity on Z , the function (z − iw)p (z + iw)q will be holomorphic on Z only
hol
if p ≥ 0, and it then will lie in Cm
(Z) if and only if −m = p + q . Restricting to the image of H in Z
concludes.
[restriction-to-k] Corollary 2.2.
The restriction of πm to K is the direct sum of characters εm+2p for p = 0, 1, . . . .
When m = 0 the representation πm contains the trivial representation on constants. More generally,
when m ≤ 0 the representation πm contains the unique irreducible representation of dimension |m| + 1.
From now on in this essay, I shall assume that m > 0.
Classical automorphic forms and representations of SL(2) (9:06 a.m. February 12, 2012)
8
We know already that the measure dx dy/y 2 is G-invariant on H. Here is a useful generalization of this:
[H-measure] Proposition 2.3.
The measure y m−2 dx dy is G-invariant for πm .
Proof. What this means is that for f in Cc∞ (H) and f∗ = πm (g)f we have
Z
f∗ (z)2 y m dx dy =
y2
H
Z
f (z)2 y m dx dy .
y2
H
We may as well assume that g is in SL2 (R). Recall that
y g(z) =
y(z)
= y(z)j(g, z)−2 .
|cz + d|2
The first integrand is therefore
f g(z) 2 j(g, z)−2m y m (z) = f g(z) 2 j(g, z)−2m y m g(z) j(g, z)2m
2
= f g(z) y m g(z) ,
so the result follows from the invariance of dx dy/y 2 .
The function (z − i)p /(z + i)p is bounded on H and 1/(z + i)m is square-integrable with respect to the
measure y m−2 dx dy for m > 1. It follows that
the representation πm is square-integrable for m > 1—i.e.
is in fact a discrete summand of L2 SL2 (R) .
3. Automorphic forms
pos
A congruence subgroup Γ of GL2 (R) is a subgroup of
SL2 (Z) = GL2 (Z) ∩ GLpos
2 (R)
containing some principal congruence subgroup
The region
Γ(N ) = γ ∈ SL2 (Z) γ ≡ I (mod N )} .
D = z = x + iy |x| ≤ 1/2, |z| ≥ 1
is a fundamental domain of SL2 (Z) in H, and some finite number of transforms of this region will be a
fundamental domain for Γ. A function on D is said to be of moderate growth if it is O(y M ) for some
M > 0, and rapidly decreasing if O(y −M ) as y → ∞. Similarly for a fundamental domain of Γ.
An automorphic form of weight m for Γ is a holomorphic function f (z) on H satisfying a condition of
moderate growth on every fundamental domain such that:
f [γ]m = f
or
f γ(z) = j(g, z)m f (z)
pos
for all γ in Γ. Following the previous section, define the corresponding function on G = GL2 (R)
Φf (g) = f g(i) detm/2 (g) j(g, i)−m .
It is also invariant under Γ and again in some sense of moderate growth. From the results of the previous
section:
The space of holomorphic forms of weight m for Γ is isomorphic to the space of smooth
(in fact, necessarily real analytic) functions Φ on Γ\G of moderate growth such that
[classical] Theorem 3.1.
Classical automorphic forms and representations of SL(2) (9:06 a.m. February 12, 2012)
9
(a) Φ(gk) = ε−m (k)Φ(g) for k in SO2 ;
(b) Rx+ Φ = 0;
(c) Φ(gz) = Φ(g) for z in the connected component of the center of G.
The existence of an automorphic form of weight m says something about the occurrence of a discrete
series representation of G in C ∞ (Γ\G), but not quite what might be naively expected. Conditions (a)–(c)
mean that the representation of G generated by Φ is a copy of the dual discrete series π
bm . Formally, this
is related to Frobenius reciprocity, since a Γ-invariant vector in πm lies in
HomΓ (b
πm , C)
which according to a kind of Frobenius reciprocity is formally the same as
Hom(g,K) πm , C ∞ (Γ\G) .
This formal analysis is not legal, because the Γ-invariant function is not in πm but in some larger space.
However, this heuristic idea can be justified with some modest amount of trouble.
Part I. References
1. James Milne, Modular functions and modular forms, course notes available at
http://www.jmilne.org/math/CourseNotes/math678.html
2. J-P. Serre, Cours d’Arithmétique, Presses Universitaires de France 1970.
3. G. Shimura, Arithmetic theory of automorphic functions, Princeton University Press (first edition),
1971.