REPRESENTATION THEORY FOR GL(2, R)
REPRESENTATION THEORY FOR GL(2, R)
MARTIN ANDLER
Notes de cours : version provisoire.
Cours de formation doctorale sur GL(2), CIRM, Luminy, 17-21 mars 2008
1. Introduction
Why would one want to study representations of a group ? A good motivation comes from Fourier theory. Let G = R/Z or R. Indeed, what is
Fourier series about ? For a sufficiently regular function (say a test function
in S(R)), put
Z
f (x)e−2iπnx dx;
an (f ) =
R/Z
then, as is well known
T
L2 (R) −−−x−→ L2 (R)
Fy
Fy
M
2iπx·
e
`2 −−−
−−→ `2
where Tx be the translation operator f (·) 7→ f (· + x),
X
f (x) =
an(f ) e2iπnx ,
n∈Z
and Mu (v)(n) = un vn . Let F : L2 (R) → `2 the map f 7→ (an (f ))n∈Z . It
is of course an isometry, and more significantly from our point of view, one
has the following diagram, for any φ ∈ S
Cφ
L2 (R) −−−−→ L2 (R)
Fy
Fy
MF (φ)
`2 −−−−→ `2
where Cφ is convolution by φ : f 7→ φ ∗ f , and Mφ0 is the diagonal operator given by multiplication by φ = F(φ). In other words, Fourier series
transforms translation operators and convolution operators into multiplication operators : it is a simultaneous diagonalization for such operators. It
should come as no surprise that a commuting set of operators should be
simultaneously diagonalizable.
Date: March 17, 2008.
1
2
MARTIN ANDLER
Do the same thing for G = R. The formulas are the same, where now F is
Fourier transform
Z
f (x)e−2iπνx dx.
F(f )(ν) =
R
The difference is that the operators Tx do not have eigenvalues or eigenvectors; nonetheless, spectral theory replaces diagonalization. But if we accept
to see things formally, we do have a diagonalization the Dirac function µν
is an eigenfuction for Tx :
FTx F −1 (µnu ) = e2iπνx µν .
Actually, Fourier theory does something else : not only do we have a Hilbert
isomorphism transforming translation operators (or rather the integrated
form : convolution operators) into diagonal (= multiplication) operators,
but we also have an inversion formula expressing the original function in
terms of its Fourier transform
Note that, in both cases, performing spectral theory involves the unitary
characters of the group. But the situation that arises in these examples is
far from being generic. Spectral theory for one operator (or for the group
generated by that operator, Z or a quotient) can involve both continuous
and discrete spectrum, so we should prepared for much more complicated
situations.
If we want to do the same thing for a non abelian group G (simultaneous
spectral theory for translation operators), we are confronted with a number
of problems
• even formally, there are no one dimensional invariant subspaces under all translation operators
• left and right translation operators are different
• clearly, if there is a minimal invariant subspace, then it affords an
irreducible representation of G
• one “sees” that translation operators are going to be “diagonalized”
into representations of G
• but then, how does one get an inversion formula for functions ?
The next example is that of a compact group K. The Peter-Weyl theory
(see below) gives an indication of what should be done in general.
•
•
•
•
determine (almost) all unitary representations of G
define a measure on their set
show that the regular representation
R can be decomposed
have an inversion formula φ(e) = Gb tr(π(φ))dµ(π).
1.1. Sketchy historical notes. It is impossible to give an overview of the
history of this subject without overlooking some major contributions. Let us
at least give some landmarks and some names associated to these landmarks.
The notes in [Knapp] provide a much more detailed view.
GL(2, R)
3
The idea of looking at infinite dimensional representations of non compact
groups arose in the 1930’s in connection with physics, and more precisely
quantum mechanics (see [Wign 1], [Wign 2]). The first example which was
studied in a completely thorough and mathematical way is that of the
Lorentz group SU(1, 1) in a remarkable paper by a Princeton professor,
V. Bargmann published in 1947 ([Barg]). It is interesting to note that
Bargmann was also a mathematical physicist. Also in 1947, in Moscow,
Gelfand and Naimark, published similar results ([Gel-Nai]); around the
same time, Gelfand, in Moscow and Godement, in Paris and Nancy worked
on spherical functions for general semi-simple groups. In the early fifties,
Gelfand and his collaborators, on the one hand, Harish-Chandra (a student
of Dirac who drifted to mathematics) obtained a Plancherel formula for
SL(2, R) ([Gel-Gra], [Hari 1]).
It is fair to say that the subject developed in the 50’s and 60’s under the very
strong influence of Harish-Chandra, who obtained a completely general view
of the representation theory for semi-simple groups, including the general
Plancherel formula obtained in the early seventies. A fair number of US
mathematicians were also involved : Kostant; starting in the late 60’s and
70’s : Wallach, Knapp, Schmid (actually German, but in the US). Other
players in the game were notably the Russian school around Gelfand, and
the French (Godement, Bruhat...)
The next major influence is the connection with number theory, appearing
with the spectral decomposition of L2 (G/Γ and the connection to automorphic forms which appeared in the work of Godement, Jacquet, Langlands in
the sixties (see [God 1], [Langl]).
If representation theory for real groups had remained somewhat immune to
“modernity” (for instance, there was little homological algebra) until the
mid seventies, this changed around 76 because of the work of Zuckermann
and Vogan ([Vog 1]); some deep connections with differential geometry (index theorem) also became apparent. Also, the study of composition series of
representation showed some deep connections to algebraic geometry, specifically to the geometry of Schubert varieties. This led to the localization
theory of Beilinson-Bernstein and Brylinski-Kashiwara establishing the connection between representations and D-modules.
Another link to geometry (symplectic geometry) appeared in the sixties
through the “orbit method”, first imagined by Kirillov to study representation theory for nilpotent Lie groups, and which became a general “philosophy” for studying the dual of a Lie group, as developed by Kostant and a
French school around Duflo and Vergne.
2. The group GL(2, R)) and its siblings
There are five closely related groups :
4
MARTIN ANDLER
a b
GL(2, R) = {g =
, a, b, c, d ∈ R, ad − bc = 1}
c d
GL + (2, R) = {g ∈ GL(2, R), det g > 0}
SL(2, R) = {g ∈ GL(2, R), det g = 1}
SL ± (2, R) = {g ∈ GL(2, R), det g = ±1}
PSL(2, R) = SL(2, R)/{±1} = PGL(2, R) = GL(2, R)/Z,
where Z is the center of G. There is a companion group to SL(2, R), with its
own variants, corresponding to SL ± (2, R), GL(2, R), GL + (2, R), PSL(2, R) :
Exercise 1. Show that
SU(1, 1) = SU(1, 1) = {M ∈ SL(2, C), t M M = Id}
and SL(2, R) are isomorphic
Until the mid-seventies, this class would have been on SL(2, R). And indeed
the main reference, Lang’s book [Lang], is called SL(2, R), which is of course
semi-simple. For reasons that will hopefully become apparent as we go on,
reductive groups are more interesting than semi-simple groups, so we study
GL(2, R) instead. But the representation theory of these groups is almost
the same.
2.1. Generalities about the Lie algebra of a Lie group. Let G be a
Lie group. Its Lie algebra g is at least two things at the same time :
g = Te G = left invariant vector fields on G.
Recall that a left invariant vector field on G is a vector field L such that
(`g )∗ L = L where `g is the left translation by g : x 7→ gx. For a C ∞ -function
φ on G, this means
L(φ(g·))(h) = L(φ)(gh)
for all g, h in G. We can make the identification X ∈ g 7→ LX vector field
concrete using the exponential on G (in the case of GL(2, R), SL(2, R) it is
the usual exponential of matrices) :
(1)
d
φ(h exp tX).
dt |t=0
is a Lie algebra map from g to vector
∀X ∈ g, LX φ(h) =
One shows that the map X 7→ LX
fields on G.
We will see below how to get expressions for the LX in coordinates.
Let ψ : G → H be a homomorphism of Lie groups. By derivation, one gets a
map, also written ψ : g → h given as ψ = dψe . Any book (resp. good book,
like [God 2]) on Lie groups explains (resp. explains in a clear way) that this
ψ is a Lie algebra homomorphism g → h.
GL(2, R)
5
2.2. The Lie algebras. The Lie algebra g0 = gl(2, R) of GL(2, R) is the set
of all 2 × 2 matrices; the Lie bracket is the one coming from the associative
algebra structure.
gl(2, R) has a basis
1 0
1 0
0 1
0 0
i=
,h =
,x =
,y =
,
0 1
0 −1
0 0
1 0
with the usual relations [h, x] = 2x, [h, y] = −2y, [x, y] = h. The Lie algebra
sl(2, R) = {m ∈ gl(2, R), tr(m) = 0} has generators h, x, y. The root system
is {α, −α}, with α the linear form on h such that α(h) = 2. The “half-sum
of positive roots” is ρ = α/2.
Subalgebras of g0 = sl(2, R)
• the standard “split” Cartan subalgebra is a0 = Rh;
−
• the two opposed nilpotent algebras n+
0 = n0 = Rx, n0 = Ry
+
+
• the positive Borel subalgebra is b0 = b0 = a0 ⊕ n0 ;
• the “compact” Cartan subalgebra k0 = Rz = R(x − y) is also the Lie
algebra of a maximal compact subgroup.
Any Cartan subalgebra is conjugate to a0 or k0 .
The Cartan involution on g0 is the map θ : m 7→ −t m. The fixed points is
k0 , the −1-eigenspace is the set s0 of symmetric matrices. Note that a0 ⊂ s.
One has various decompositions:
• Cartan decomposition g0 = k0 + s0
• Iwasawa decomposition g0 = k0 ⊕ a0 ⊕ n0
• g0 = n−
0 ⊕ a0 ⊕ n0 .
The complexification of sl(2, R) is sl(2, C) = sl(2, R) + i sl(2, R). There is
another real form : sl(2, C) = su(2) + i su(2).
For gl(2, R), nothing changes except that a0 has to be replaced by a0 ⊕ z0 ,
where z0 is the center.
We observe that
1 1 −i
1 1 i
1 0
0 i
=
,
0 −1
−i 0
2 −i 1
2 i 1
in other words h and iz are conjugate in SL(2, C).
Recall that GL(2, R) acts on itself by conjugation ig (m) = gmg −1 . ig ( SL(2, R)) =
SL(2, R), and the map g 7→ ig descends to a map from PSL(2, R) to the set
of automorphisms of GL(2, R). For a fixed g, the differential of ig at e = Id
is the map
h 7→ Ad g (n) = gng −1 .
For m ∈ sl(2, R), Ad g (m) ∈ sl(2, R) The exponential for G is the usual expoP mn
nential of matrices : exp m = ∞
0 n! . One checks easily that ig (exp m) =
exp(Ad g (m)). Now differentiate at e the map g ∈ GL(2, R) 7→ Ad g ∈
GL(gl(2, R)) : ad = (d Ad )e . One has
ad m (m0 ) = [m, m0 ].
6
MARTIN ANDLER
There is an interesting bilinear symmetric form on gl(2, R), the Killing form,
defined :
B(m, m0 ) = tr(ad m ad m0 ).
Relative to the basis I, X, Y, H, the matrix of B is
0 0 0 0
0 0 4 0
0 4 0 0 .
0 0 0 8
The restiction to sl(2, R) is non degenerate. As a consequence of the definition, for all g ∈ GL(2, R), Ad g ∈ O(gl(2, R), B).
2.3. Subgroups of SL(2, R).
• the connected component of the spilt Cartan subgroup A =
def
diag(a) =
(it is the centralizer of a0 ), and has
Lie algebraa0 .
0
def a
• H the full Cartan subgroup H = diag(a) =
, a 6= 0
0a−1 1 x
• the positive maximal nilpotent subgroup N =
,x ∈ R ,
0 1
with Lie algebra n0 .
1 0
−
• the negative maximal nilpotent subgroup N =
,y ∈ R ,
y 1
with Lie algebra n−
0.
a x
• the Borel subgroup B =
, a > 0, x ∈ R , with Lie alge0 a−1
bra b0 .
cos θ sin θ
• the maximal compact K = k(θ) =
, θ ∈ R . It is
− sin θ cos θ
also the compact Cartan subgroup. It has Lie algebra k0 .
• the centralizer M = {±Id} of A in K
• the minimal parabolic P = M AN
Any Cartan subgroup is conjugate to H or K
The Weyl group is the quotient of the normalizer of a Cartan subgroup by
its centralizer.
the split Cartan H, Z(H) = H, and N (H) = H ∪ wH,
For 0 1
where w =
, so that W = Z/2Z. For the compact Cartan K, the
−1 0
normalizer is K.
One has the following decompositions
+
• Polar decomposition G = K exp s+
0 (where s0 ⊂ s0 is the set of
symmetric positive definite matrices)
• Cartan decomposition G = KAK
• Iwasawa decomposition G = KAN
• Bruhat decomposition G = P W P .
a 0
,a > 0
0 a−1
GL(2, R)
7
For GL + (2, R), the only difference comes from the center Z ∼ R∗+ . The
groups
K,
N are unchanged, A is defined as the set of all diagonal matrices
a1 0
with a1 a2 > 0. The centralizer M of A in K in unchanged, P is
0 a2
defined again as M AN .
2.4. Subgroups of SL ± (2, R), GL(2, R). In both cases SL ± (2, R), GL(2, R)
the maximal compact is O(2). It is not connected or abelian. The group A
is
a
0
A={
, a ∈ R∗+ , = ±1} for SL ± (2, R)
0 a−1
a1 0
A={
a1 , a2 ∈ R∗ } for GL(2, R)
0 a2
2.5. Algebraic or not. Only five of these groups ( SL(2, R), PSL(2, R),
SL ± (2, R), GL + (2, R), GL(2, R)) are connected. The question is : should
one restrict oneself to studying connected groups ? The answer is no, although it is not completely clear in the cases studied here. As we will
mention at various points in these notes, representation theory for reductive
groups is an inductive procedure going from smaller groups to larger groups
– the starting point being groups of dimension 1 : R/Z, R or R∗ and groups
of dimension 3 : SL(2, R), PSL(2, R), SL ± (2, R), GL + (2, R), GL(2, R) (in
the non-compact case), SU(2) and its variants in the compact case.
Just to give a hint of this inductive procedure, representations of SL(n, R)
are obtained from representations of some “parabolic subgroups” P which
are the stabilizers of generalized flag manifolds, i.e. groups which are blockupper triangular :
∗
∗
0
0
0
0
0
0
0
∗
∗
0
0
0
0
0
0
0
∗
∗
∗
∗
∗
0
0
0
0
∗
∗
∗
∗
∗
0
0
0
0
∗
∗
∗
∗
∗
0
0
0
0
∗
∗
∗
∗
∗
∗
0
0
0
∗
∗
∗
∗
∗
∗
∗
0
0
∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗.
8
Such a group has a Levi
∗ ∗
∗ ∗
0 0
0 0
L is of the form
0 0
0 0
0 0
0 0
0 0
MARTIN ANDLER
decomposition P = LU , where
0 0 0 0 0 0 0
1
0 0 0 0 0 0 0
0
0
∗ ∗ ∗ 0 0 0 0
0
∗ ∗ ∗ 0 0 0 0
0
∗ ∗ ∗ 0 0 0 0
and
U
0
0 0 0 ∗ 0 0 0
0
0 0 0 0 ∗ 0 0
0
0 0 0 0 0 ∗ ∗
0 0 0 0 0 ∗ ∗.
0
0
1
0
0
0
0
0
0
0
∗
∗
1
0
0
0
0
0
0
∗
∗
0
1
0
0
0
0
0
∗
∗
0
0
1
0
0
0
0
∗
∗
∗
∗
∗
1
0
0
0
The point here is that even if the large group SL(n, R) is connected and
semi-simple, the Levi component is neither connected nor semi-simple. This
explains why we should study reductive groups (and the galoisian point of
view in Langlands’ theory will make the argument stronger a bit later).
The class of reductive groups G that are often considered in the litterature
([Vog 1], Springer’s article in [Bo-Cas]) should verify the following axioms :
(1) G is a real Lie group, K is maximal compact subgroup, θ is an
involution of g0
(2) g0 is a real reductive Lie algebra
(3) for any g ∈ G, Ad g is an inner automorphism of g
(4) k0 is the fixed point set for θ
(5) if p0 is the −1 eigensapce for θ, the map (k, X) ∈ K × p0 7→ k exp X
is a diffeomorphism
(6) G has a faithful finite dimensional representation
(7) (optional) for any h0 Cartan subalgebra of g0 , the centralizer H is
abelian.
The various groups concerned here verify all of these conditions.
Occasionally, we will consider non algebraic groups. Since SL(2, C) is simply
connected (its maximal compact subgroup is SU(2) whih is diffeomorphic
to a 3-sphere in R4 ), SL(2, R) is simply connected as an algebraic group –
there is no algebraic covering. But of course, as a Lie group it is not simply
connected. There are two coverings which are of particular interest : the
f
universal simply connected covering SL(2,
R), whose center is isomorphic to
Z, and the metaplectic covering (of order 2), which plays a very important
role in number theory and representation theory.
2.6. Haar measures. On a locally compact group G, there is, up to a
positive constant, a unique left invariant measure dg (it is easy to show for
a Lie group). SL(2, R), GL(2, R) are unimodular, meaning that the Haar
measure is also right invariant. We want to give formulas for the Haar
measure on GL(2, R), SL(2, R), particularly in terms of the decompositions
of those groups. The Haar measure on GL(2, R) is easily seen to be
1
d+ g,
d× g =
| det g|2
∗
∗
∗
∗
∗
∗
1
0
0
∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗
∗ ∗
1 0
0 1.
GL(2, R)
9
where dL g is the Lebesgue measure on gl(2, R) (the Haar measure on (gl(2, R), +).
This measure can be restricted to GL + (2, R), which is open. We drop the ×
from d× g. Since GL + (2, R) = SL(2, R)×R∗+ , the measure d× g decomposes
accordingly : there exists a measure dS γ on SL(2, R) such that
ZZ
Z
da
a 0
φ γ
φ(g)dg =
dS γ .
−1
0
a
a
SL(2,R)×R∗+
GL + (2,R)
This dS γ is the Haar measure on SL(2, R). When there is no ambiguity
about the group, dg will be its Haar measure.
Now, it is fairly easy to check that the Haar measure on SL(2, R) decomposes as
Z
Z
φ(g)dg =
SL(2,R)
R∗+
da
a 0
1 x
cos θ sin θ
dxdθ.
φ
−1
0 1
− sin θ cos θ
0 a
a
[0,2π]
Z Z
R
It is sometimes more convenient to have the decomposition in a different
order. Let
Z
Z Z
Z
da
a 0
1 x
cos θ sin θ
φ(g)dg =
φ
dx 3 dθ.
−1
0 1
− sin θ cos θ
0 a
a
SL(2,R)
R R∗+ [0,2π]
We will often let SL(2, R) act on the Poincar´e upper half plane P + = {z ∈
C, im z > 0} by
az + b
a b
·z =
.
c d
cz + d
The action is transitive, and the stabilizer of i is K. Therefore G/K ∼ P + .
Since G/K ∼ N A, we have a bijection N A → P + ; explicitly
a 0
1 x
7→ z = x + ia2 .
0 1
0 a−1
is identified to dx dy
on P + .
The measure dx da
a3
y2
2.7. Left invariant vector fields. Using local coordinates associated with
these decompositions, one can express the left invariant vector fields LX
defined above (2). Let us do it in one case which will be useful later. From
the Iwasawa decomposition, any g ∈ GL + (2, R) can be expressed as g = ank
with a ∈ A, n ∈ N, k ∈ K, which gives in turn a unique decomposition
a b
u 0
y x 0 1 k(θ).
=
c d
0 u
The explicit formulas are
ueiθ = d − ic
eiθ
(a + ib)
u
Any function F on GL(2, R) has an expression in coordinates F (u, x, y, θ).
A function F on SL(2, R) is identified to a function F (x, y, θ). (Using these
x + iy =
10
MARTIN ANDLER
coordinates rather than those coming from the Iwasawa decomposition of
SL(2, R) is better suited to the identification G/K ∼ P + , since z = x + iy
is then interpreted as the element of P + corresponding to the class of g in
G/K. For any element X ∈ gl(2, R), g ∈ G, decompose
u(t)
0
y(t) x(t) 0 1 k(θ(t)).
g exp(tX) =
0
u(t)
Then
LX =
∂F
d
∂F
∂F
∂F
0
+ x0 (0)
+ y 0 (0)
+ θ0 (0)
|t=0 F (g exp tX) = u (0)
dt
∂u
∂x
∂y
∂θ
where all values of the functions are taken at (u, x, y, θ). We do the computation for the basis of gl(2, R) : x, h, I, w = x − y. This gives the following
formulas
∂
∂
∂
Lx = y cos 2θ
(2)
+ y sin 2θ
+ sin2 θ
∂x
∂y
∂θ
∂
∂
∂
(3)
+ 2y cos θ
+ sin 2θ
Lh = −2y sin 2θ
∂x
∂y
∂θ
∂
LI =
(4)
∂u
∂
(5)
.
Lw =
∂θ
2.8. The enveloping algebra. The enveloping algebra of a Lie algebra is
in general an associative algebra (over R) which is universal for Lie algebra
homomorphisms of the Lie algebra into the Lie algebra underlying an associative algebra. We are interested in the enveloping algebras U (g0 ), U (g)
of g0 and g = (g0 )C = sl(2, C) the complexification of g0 . The Poincar´eBirkhoff-Witt asserts that U (g) is the set of (noncommutative) polynomials
of the form
X
αk,l,m xk y l hm .
k,l,m
We will view U (g) in various different ways
• g0 being identified to left invariant vector fields on G by
∀a ∈ g0 , a · φ(g) =
d
φ(g exp(ta),
dt |t=0
U (g0 is the set of left invariant differential operators on G;
• for u ∈ U (g0 ) seen as a left invariant differential operator, define u
e
as the distribution with support e
<u
e, φ >= (u · φ)(e)
for φ a test function on G. For a ∈ g, < e
a, φ >=
d
dt |t=0 φ(exp(ta)).
GL(2, R)
11
2.9. The center of the enveloping algebra Z(g). It plays an important
role in representation theory. For a general complex semi-simple algebra
over C, Z(g is a polynomial algebra with ` generators, where ` is the rank
(i.e. the dimension of a Cartan subalgebra). Again, if g is semi-simple with
Killing
P ij form B, take aijbasis Ai of g, and let αij = B(Ai , Aj ). The element
ij α Ai Aj (where α is the inverse matrix) is
(1) independent of the choice of a basis
(2) in Z(g).
It is called the Casimir element. In the case of sl(2, C), the Casimir element
Ω0 is 4(2H 2 + 4XY + 4Y X). It generates Z(g).
Since we have the decomposition g = h ⊕ n+ ⊕ n− , we get a decomposition
U (g) = U (h) ⊕ (n− U (g) + U (g)n).
Let ξ be the projection from Z(g) to U h) parallel to (n− U (g) + U (g)n) : for
all z ∈ Z(g), z − ξ(z) ∈ yU (g) + U (g)x. Since h is abelian, U (h) = S(h),
which is identified with polynomial functions on h∗ . Consider the shift :
P ∈ S(h) 7→ τρ (P ) ∈ S(h) such that τρ (P )(λ) = P (λ − ρ).
If h) is identified to C using the basis h, S(h) is identified to polynomials in
one variable C[y], and the map τρ is the map P (y) 7→ P (y − 1).
Theorem 1 (Harish Chandra). The map τρ ◦ξ : Z(g] → S(h) induces an isomorphism of algebras from Z(g) to the set of even elements (W -invariants)
S(h)W in S(h).
3. Generalities on representations of GL(2, R)
3.1. Abstract theory. Let for now G be a locally compact group, and H
be a Hilbert space. On H, there are the unitary operators U (H) and the
bounded operators B(H). A representation of G in H is a homomorphism
π of G into B(H) which is continuous with respect to the weak topology on
B(H) – i.e. :
(g, v) ∈ G × H 7→ π(g)(v) ∈ H
is continuous. It is enough to require
(1) continuity of g → π(g)v at g = 1 for all v
(2) a bound for kπ(g)k in a neighborhood of e.
The representation is unitary if π(G) ⊂ U (H); it is irreducible if there
are no closed spaces other than 0, H which are invariant under all π(g).
Two representations (π, H), (π 0 , H0 ) are equivalent if there exists a bounded
operator J (interwining operator) H → H0 such that
∀g, π 0 (g)J = Jπ(g).
Most of the time we will not really make the distinction beteween a representation and its class. But as we will see, it is often interesting and sometimes
crucial to consider equivalent representations as different objects. Unitary
representations are said do be unitarily equivalent if there exists a unitary
12
MARTIN ANDLER
intertwining operator. The set of equivalence classes of irreducible unitary
b of G.
representations is the unitary dual G
Theorem 2 (Schur’s lemma). Let S a set of operators in a Hilbert space
H leaving no closed subspace other than 0, H invariant. Let A a hermitian
operator commuting with every element of S. Then A = cId for some c ∈
R. If A is not hermitian, but normal, and both A, A∗ commute with every
element of S, then A = cId for some c ∈ C.
The proof involves the spectral theorem.
Corollary 3. A unitary representation is irreducible
twining operators with itself is λId
iff the only inter-
It follows from Schur’s lemma and the structure of GL(2, R), GL + (2, R)
that representation theory for GL(2, R) very close to representation theory
for SL ± (2, R) and representation theory for GL + (2, R) is very close to
representation theory for SL(2, R).
Let π be a representation of G in H. Let f be in L1 (G); define
Z
π(f ) =
f (g)π(g)dg
G
where dg is a left invariant (Haar) measure. The map f 7→ π(f ) is a representation on the convolution algebra L1 (G) in H (recall that the convolution
on G is defined by
Z
φ ∗ ψ(g) =
φ(gh−1 )ψ(h)dh
G
when it makes sense (for instance φ, ψ ∈ Cc (G)).
3.2. Compact groups. Recall that for G compact, the following hold
• any representation is unitary (there exists a (second) hermitian form
on H defining a norm equivalent to the first one such that π is unitary
with respect to it
• any irreducible representation is finite dimensional
• the coefficients (matrix coefficients) cπv,w =< π(·)v, w > verify
< cπv,w , cπv0 ,w0 > = 0 if π, π 0 are inequivalent
1
< v, v 0 >< w, w0 > if π = π 0
=
dim(π)
P
• the left regular representation of G on L2 (G) is equivalent to π∈Gb dim(π)π
• any unitary representation of G on a Hilbert space is the (orthogonal)
sum of irreducible representations of G.
GL(2, R)
13
3.3. Finite dimensional representations. Let G be a Lie group. A finite
dimensional representation π of G in V is a continuous homomorphism in
GL(V ), therefore it is a Lie group homomorphism G → GL(V ), therefore
there is a derived (Lie algebra) homomorphism
dπe : g 7→ gl(V ).
One can see this by identifying g = Te G = left invariant vector fields . If
a ∈ Te G and e
a is the corresponding vector field, π∗ e
a is the restriction to
π(G) of the left invariant vector field defined by dπe (a). Since π∗ preserves
brackets, we are done. Most of the time, we will write also π for the derived
representation.
There is a converse. If G is simply connected, any finite dimensional Lie
algebra representation of g is the derived representation of a finite dimensional representation of G. This holds also over C if we restrict ourselves
to the correspondance between holomorphic representations of a complex
Lie group G and C-linear representations of g. We apply these remarks to
SL(2, R).
Theorem 4 (Unitary trick). Let V be a complex finite dimensional vector
space. There is a one to one correspondence between the following
(1)
(2)
(3)
(4)
(5)
(6)
representations of SL(2, R) in V
representations of SU(2) in V
holomorphic representations of SL(2, C) in V
representations of sl(2, R) in V
representations of su(2) in V
complex linear representations of sl(2, C) in V .
Proof. We use
sl(2, C) = sl(2, R) + i sl(2, R) = su(2) + i su(2).
From this, we get the correspondence for the three Lie algebras representation. Since SU(2) is simply connected (it is a 3-sphere in R4 ), so is
SL(2, C). So any representation of sl(2, C) (resp su(2)) yields a representation of SL(2, C) (resp SU(2)). The other implications are obvious.
Recall that the finite dimensional irreducible representations of sl(2, C) are
given by
Theorem 5. For any integer n ≥ 0, there is a unique irreducible complex
representation πn of sl(2, C) of dimension n + 1. The action of x, y, h on a
basis v0 , . . . , vn is given by
πn (h)(vi ) = (n − 2i)(vi )
πn (x)(vi ) = i(n − i + 1)vi−1 with the convention that v−1 = 0
πn (y)(vi ) = vi+1 with the convention that vn+1 = 0.
14
MARTIN ANDLER
The corresponding representation of SL(2, C) is given the following action
on homogeneous complex polynomials of degree n in two variables
z1
−1 z1
).
πn (g)(P )(
) = P (g
z2
z2
Now we are ready for the disaster :
Proposition 6. The only finite dimensional unitary representation of SL(2, R)
is the trivial one-dimensional representation.
Proof. Let π be a finite dimensional unitary irreducible representation of
SL(2, R) in V . The corresponding representation π † of SU(2) is unitary :
there is an hermitian product on V such that
∀u ∈ SU(2), π(u) is unitary
therfeore for any a ∈ su(2)(2), π(a) is skew hermitian. A skew hermitian
operator is normal, therefore diagonalizable in an orthonormal basis, and
its eigenvalues are imaginary. Now, su(2)(2) = k ⊕ ip : for any a ∈ ip, π(a)
has imaginary eigenvalues. But sl(2, R) = k ⊕ p, so for any a0 ∈ ip, π(a0 ) has
imaginary eigenvalues. It follows that for a0 ∈ p, π(a0 ) = 0 But [p, p] = k, so
π = 0 on sl(2, R).
4. Zoology of unitary representations of SL(2, R)
In this section, we will give a “full” description of irreducible unitary representations of SL(2, R), without any attempt to show that indeed the list is
complete, nor to understand where these representations come from.
4.1.
Discrete series. Let m ∈ N, m ≥ 2. Let
ZZ
Hm = {f holomorphic on P + , kf k2 =
|f (z)|2 y m−2 dxdy < ∞.}
Three important points here, two of them non trivial :
• the measure y −2 dxdy on P + is G-invariant
• Hm is a Hilbert space (which is a bit surprising)
• Hm is non zero because (z + i)−m ∈ Hn .
+ on
The holomorphic discrete series representation is the representation Dm
Hm given by
az − c
+
).
Dm
f (z) = (−bz + d)−m f (
−bz + d
It is easily unitary. To verify irreducibility, assume that U is a closed invariant subspace of Hm. It contains
a non zero holomorphic function f (z);
a 0
by replacing it by Dm
f (z) = f (a2 z − ca) for well chosen a, c, U
c a−1
contains a function f such that f (i) 6= 0. Now
cos θz + sin θ
+
Dm
(k)f (z) = (− sin θz + cos θ)−m f
∈U
− sin θz + cos θ
GL(2, R)
15
so
1
2π
Z
2π
e−miθ (− sin θz + cos θ)−m f (
0
cos θz + sin θ
)dθ ∈ U.
− sin θz + cos θ
A shrewd application of the Cauchy formula yields (2i)m f (i)(z + i)−m ∈ U.
If U were not everything, applying the same idea to U ⊥ would lead to a
contradiction.
The discrete series has two important related properties
• its matrix coefficients are in L2 (G)
• it appears as a closed subspace of L2 (G), invariant under the left
regular representation.
− ; it is is given by the
It has a sister, the antiholomorphic discrete series Dm
complex conjugate formulas. It is also irreducible unitary.
4.2. Unitary principal series. Fix a µ ∈ {0, 1} and a real number ν. We
define the representation P µ,ν acting on L2 (R) by
(6)
P µ,iν (g)f (x) = (sgn(−bx + d))µ | − bx + d|−1−iν f (
ax − c
).
−bx + d
This is a unitary representation. There are some equivalences :
P µ,iν ∼ P µ,−iν
that are given by some explicit operators that we will explain later; and
these representations are irreducible for ν 6= 0 (see below) and ν = 0, µ = 0.
P 1,0 is a sum of two representations.
An outline of the argument for irreducibility. Let B be an operator commuting with P µ,iν (g). In particular, it commutes with
µ,iν 1 0
P
: f (x) 7→ f (x − c).
c 1
Therefore F(Bf )(ξ) = m(ξ)F(f )(ξ) for somebounded
measurable function
a
0
m. Using the commutation of B with P µ,iν
we get (Bf )(a2 x) =
0 a−1
B(f (a2 ·))(x), and from this m(a2 ξ) = m(ξ), therefore m is constant on
ξ > 0 and ξ < 0. It remains to show that, except for the case ν = 0, µ =
1, the space of functions whose Fourier transform vanishes on R+ is not
stable under P µ,ν . Now, such functions are boundary values of holomorphic
functions F on P + such that
Z
sup |F (x + iy)|2 dx.
y
R
The conclusion follows, after a bit more work, from a theorem of Privalov
(see [Knapp] p. 37).
16
MARTIN ANDLER
4.3. Complementary series. For u ∈]0, 1[, consider the Hilbert space
(
)
ZZ
f (x)f (y)
Hu = f : R → C,
dxdy < ∞
1−u
R2 |x − y|
with the action
ax − c
a b
f (x) = | − bx + d|1−u f (
).
Cu
c d
−bx + d
The representation Cu are irreducible unitary. We will discuss them later.
4.4. Other unitary representations. There is the trivial representation,
and two “limits of discrete series” D1+ , D1− . They are defined like for the discrete series, except that the Hilbert space is the set of holomoprhic functions
on P + such that
Z
kf k2 = sup
|f (x + iy)|2 dx < ∞.
y∈R∗+
R
The only reducible unitary principal series reduces as : P 1,0 = D1+ + D1− .
GL(2, R)
17
5. From representations of the Lie group
to representations of the Lie algebra
It seems natural to want to take the derivative of representations of a Lie
group G to get representations of the Lie algebra g. Unfortunately, in the
infinite dimensional case, functional analysis gets in the way. Take the very
simple example of the regular representation of R in L2 (R). It is defined on
the dense space of continuous functions with compact support by
λx f (y) = f (y + x).
Taking the derivative of this with respect to x requires derivability of f . Let
X = 1 be the generator of the Lie algebra of R, we get
dλ(X)(f )(y) = f 0 (y).
The operator f 7→ f 0 is an unbounded operator on L2 (R) with dense domain,
for instance, the Schwartz space S. But there are a lot of other possibilities
for a domain. There is no way that we can escape these fine points in our
discussion, but we will almost never go into the heart of the matter – just
stating the results that are needed.
5.1. C ∞ -vectors. In what follows, G = SL(2, R) though most of the results
are general. Let (π, H) a representation of G. An element v ∈ H is said to
be a C ∞ vector if the map g 7→ π(g)v is a C ∞ map from G to H. The set
H∞ of C ∞ -vectors is a subspace of H. For v ∈ H∞ , X ∈ g define
(7)
def
def
π(X)v = dπe (X)v =
d
π(exp(tX))v.
dt |t=0
Proposition 7. The formula above defines a representation of g in H∞ .
Proof. From the definition (7), we get
π(g)π(exp(tX))v − π(g)v
t
π(g exp(tX))v − π(g)v
= lim
= LX .cv (g),
t→0
t
π(g)π(X)v = lim
t→0
where LX is the left invariant vector field on G defined by X, and cv (g) =
π(g)v. This proves that g 7→ π(g)π(X)v is C ∞ .
The fact that π([X, Y ])v = [π(X), π(Y )]v is proved basically as it was done
in the finite dimensional case (see [Lang], p. 94).
We cannot rest here, because we don’t know at this stage whether H∞ is
significantly large; we don’t even know whether it is 6= {0}. In fact
Proposition 8. H∞ is dense in H.
There are several ways to see this. I indicate a couple, before focusing on
one approach more thoroughly.
18
MARTIN ANDLER
Regularising function. We consider a Dirac sequence φn ∈ Cc∞ (G) (compact
support) and consider
Z
π(φn )v =
φn (g)π(g)vdg
G
One shows by the usual arguments that π(φn )v → v when n → ∞. Now we
argue that for any φ ∈ Cc∞ (G), π(φ)v is a C ∞ -vector. Indeed
Z
Z
π(g)π(φ)v = π(g)
φ(h)π(h)vdh =
φ(g −1 h)π(h)vdh,
G
G
so the result follows by taking derivatives under the integral.
Analytic vectors. Juste as there is a notion of C ∞ vectors in a representation,there is a notion of analytic vectors : a vector v is (real) analytic if the
map g → π(g)v is analytic on G with values in H, i.e. it is locally the sum
of its Taylor series. It is an exercise to prove, for φ analytic from G → H
∀g ∈ G, ∃ > 0, ∀X ∈ g, kXk ≤ , ∀t ∈ [0, 1], φ(g exp tX) =
∞ n
X
t
0
n!
(LnX φ)(g).
If v is an analytic vector, then
π(exp X)v =
∞
X
1
(π(X))n v.
n!
0
Now there is a very general theorem of Nelson (citenelshowing that for any
Banach space representation of a Lie group G, the space of analytic vectors
is dense. The special case of semi-simple groups was proved beforehand by
Harish Chandra. Nelson’s proof uses the fundamental
solution of the heat
P
equation associated with an operator ∆ = Xi2 , where Xi is a basis for g.
5.2. K-finite vectors. (In this subsection, we follow mostly [Lang].) For
our purpose, there is another subspace which is very important, the space
HK of K-finite vectors. Their study is made substantially easier in the case
of SL(2, R) since K is commutative. We will see how to adapt definitions
to the cas of SL ± (2, R) later. Consider a representation π of G = SL(2, R)
in a Banach space H. For each integer n, define
Hn = {v ∈ H, π(k(θ))v = einθ v}.
It is clear that Hn is a closed subspace of H.
Theorem 9. Let π be an irreducible representation of G in H. If dim Hn
is finite, then it is 0 or 1. This is always the case when π is unitary.
TheoremP10. Let π be an irreducible representation in a Banach space H.
The sum
Hn is dense in H. If π is unitary on K, this sum is orthogonal.
In order to prove these two theorems, we need to study the K × K representation in Cc (G) :
(k, k 0 ) · f (g) = f (kgk 0 ).
GL(2, R)
19
(Since K is commutative, we dont need to put k −1 instead of k. ) Define
Sn,m = {f ∈ Cc (G), f (k(θ)gk(θ0 )) = e−inθ f (g)e−imθ
for all g, θ, θ0 .
P
Lemma 11. The algebraic sum A = n,m Sn,m is dense for the uniform
norm (and for the L1 -norm) in Cc (G).
This follows from the Fejer theorem for Fourier series.
Lemma 12. We have
(1) Sn,m ∗ Sp,q = 0 if m 6= p and ⊂ Sn,q if m = p
∗
(2) Sn,m
= Sm,n (where φ∗ (g) = φ(g −1 ))
Proof. Let φ ∈ Sn,m , ψ ∈ Sp,q . In the definition of φ ∗ ψ, do the change of
variable y = k(θ)y 0 . We get for all θ
φ ∗ ψ(g) = ei(m−p)θ φ ∗ ψ(g)
which prove the first half of the first statement. The second half follows
from doing the change of variable g = k(θ)g 0 in the formula, and g = g 0 k(θ0 )
in
Z
φ ∗ ψ(g) =
φ(y)ψ(h−1 g)dh.
G
The second statement is straightforward.
Lemma 13. Sn,n is commutative.
Proof. We consider the anti-involution τ : g 7→ t g (t g the transposed), and
the involution
σ : g 7→ ζgζ,
where
1 0
−1
ζ=ζ =
.
0 −1
For k ∈ K, τ (k) = σ(k) = k −1 . On functions, we define
φ† (g) = φ(τ (g)), φ] (g) = φ(σ(g)).
Since G is unimodular, these two maps preserve the Haar measure, so
φ† ∗ ψ † = (ψ ∗ φ)† , φ] ∗ ψ ] = (φ ∗ ψ)] .
Now, for φ ∈ Sn,n , φ† = φ0 . Indeed, for g ∈ G, write g = sk(θ) (polar
decomposition : s symmetric positive, k ∈ K). Then
φ† (g) = φ(k −1 s) = eniθ φ(s)
and
φ] (g) = φ(σ(s)k) = eniθ φ(ζsζ −1 ) = eniθ φ(s).
It follows that φ ∗ ψ = ψ ∗ φ.
Lemma 14. Assume that H is a Hilbert space and the representation π is
unitary on K. Then Hn , Hm are orthogonal.
20
MARTIN ANDLER
Proof. Let v ∈ Hn , w ∈ Hm .
< π(k(θ))v, w >= einθ < v, w >=< v, π(k(−θ))w >= eimθ < v, w > .
Lemma 15. We have
(1) π(Sn,m )H ⊂ Hn
(2) π(Sn,m )Hq = 0 if m 6= q
R
Proof. We compute, for φ ∈ Sn,m , v ∈ H : π(k(θ))[π(φ)v] = G φ(g)π(k(θ)g)vdg.
By a change of variable g 0 = k(θ)g :
Z
π(k(θ))[π(φ)v] =
φ(k(−θ)g 0 )π(g 0 )vdg 0 = eniθ π(φ)v
G
If v ∈ Hq , q =
6 m,
Z
Z
π(φ)v =
φ(g)π(g)vdg =
φ(gk(θ))π(gk(theta))vdg = eiθ(−m+q) π(φ)v
G
G
which implies the second statement.
Lemma 16. Assume that π is irreducible. If Hq 6= 0, it is irreducible under
π(Sq,q ) and π(Sq,q Hq 6= 0.
Proof. Let W be an invariant subspace of Hq and w ∈ W, w 6= 0. We have
P
(1) for f =
fn,m ∈ cA, the Hq -component of π(f )w is π(fq,q )w.
(2) by density of A, and irreducibility of H under Cc (G), there is a
sequence f p ∈ A such that π(f p )w → w0 ∈
/W
The Lemma follows.
Proof of Theorem 9. We know that Sn,n is commutative, and we know that
Hn is finite dimensional and irreducible. Then Hn is one-dimensional. Assume that π is unitary. Since Sn,n is closed under f (g) 7→ f (g −1 ), π(Sn,n )
is closed under taking the adjoint. We can apply Schur’s lemma : the set
π(Sn,n being ∗-closed and commutative, any of its elements is of the form
cId. This forces the dimension to be 0, 1.
Proof of Theorem 10. Let E be the closed subspace generated by the Hn .
By the lemmas above π(A)E ⊂ E; by density of A, E is stable under L1 (G).
By a proper use of a Dirac sequence, E is stable under π(G).
We define a representation π of G in a Banach space H to be admissible
if dim Hn is finite for all n. Theorem 9 shows that any irreducible unitary
representation is admissible, and that for any irreducible admissible representation, dim Hn ≤ 1.
Theorem 17. Let π an irreducible admissible representation of G in H.
Assume Hn 6= {0}. Then Hn is of dimension 1, any of its elements is a C ∞
vector in H, and Hn is the eigenvector of π(z) with eigenvalue n, where
0 −i
z=
.
i 0
GL(2, R)
21
Proof. The only thing that requires a proof is the fact that elements of Hn
∞ = S
∞
are C ∞ . We first argue that Sn,m
n,m ∩ C (G) is dense in Sn,m by
∞ H 6= 0 provided π(S
Fourier theory. Then π(Sn,n
n
n,n )Hn 6= 0. Since Hn is
of dimension 1, it follws that any v ∈ Hn is of the form π(f )v, for some
f ∈ C ∞ (G) :
Z
f (g)π(g)vdg.
v=
It follows that v ∈ H∞ .
G
Remark 18. We will see at some point that actually elements of Hn,n are
analytic.
P
Recall that HK = n Hn is the space of K-finite vectors in H.
Proposition 19. HK is stable under the representation π of g.
We will make a constant use of the basis z, e, f for sl(2, C) :
0 −i
1/2 i/2
1/2 −i/2
z=
,e =
,f = X =
.
i 0
i/2 −1/2
−i/2 −1/2
They verify [z, e] = 2e, [z, f ] = −2f, [e, f ] = z.
Proof. To prove our point, it is enough to show that for v ∈ Hn , π(e)v ∈
HK and π(f )v ∈ HK . The computation goes as for finite dimensional
representations of sl(2, C) :
π(z)π(e)v = π(e)π(z)v + π[z, e]v = nπ(e)v + 2π(e)v = (n + 2)π(e)v
and similarly π(z)π(f )v = (n − 2)π(f )v.
Proposition 20. Let (π, H) an admissible representation of G, and (π, HK )
the corresponding representation of g0 . There is a bijection between Ginvariant closed subspaces of HK and g0 -invariant subspaces of HK . In
particular, the representation of G is irreducible iff the representation of
g is.
An important point here is that we have not completely lost the fact that
π was originally a representation of G : indeed HK affords a representation
of K.
Definition 21. A (g0 , K)- module (π, V ) is
(1) a representation π of g0 in V
(2) a compatible representation π of K in V (the derived representation
of k is the restriction to k of the representation of g.
P
such that V = n∈Z Vn , the action of K on Vn is through the character χn .
Such a representation is admissible if dim Vn < ∞ for all n.
The representation of g0 in HK associated to an admissible representation
of G is called its Harish-Chandra module.
Proposition 22. Any irreducible (g0 , K)-module is the Harish-Chandra
module of an irreducible representation of G.
22
MARTIN ANDLER
There is an obvious notion of equivalence for (g0 , K)-modules. We say that
two admissible representations of G are “infinitesimally equivalent” if their
Harish-Chandra modules are equivalent. The correspondence is complete
thanks to
Proposition 23. Two unitary representation of G are equivalent
are infinitesimally equivalent.
iff they
6. Structure of (g0 , K)- modules
Let V be an admissible (g0 , K)-module. The representation π of g0 extends
to a representation of g, and then to a representation of the associative
algebra U (g), still denoted π.
We have already seen the basic fact :
If v ∈ Vn , π(e)v ∈ Vn+2 , π(f )v ∈ Vn−2 .
Define an element of U (g) by
Ω = z 2 − 2z + 1 + 4ef = z 2 + 2z + 1 + 4f e = z 2 + 1 + 2ef + 2f e.
Recalling the definition of the Casimir element and the computation of the
Killing form in the basis z, e, f , one sees easily that Ω is in the center Z(g)
of U (g) (Ω − 1 is 8× the Casimir operator). In particular, π(Ω) commutes
with k, so it leaves Vn stable; since Vn is finite dimensional, π(Ω)|Vn has an
eigenvalue. The corresponding eigenspace is stable by π(U (g). Therefore :
Proposition 24. If π is irreducible, π(Ω) is a scalar λ2 (λ ∈ C).
6.1. Classification in terms of K-types. One establishes the following
facts
(1) the subspaces Vn are of dimension 0 or 1
(2) If there is a n such that π(e)v = 0 for 0 6= v ∈ Vn , then V is spanned
by the wn−2m = π(f )m v
(3) if there is a n such that π(f )v = 0 for 0 6= v ∈ Vn , then V is spanned
by the wn+2m = π(e)m v.
It follows that given 0 6= v ∈ Vn one defines (inductively on m) a family of
vectors wj (j ∈ n + 2Z) in the following way
wn = v
2
π(e)wn+2m if λ 6= −n − 2m − 1
λ + (n + 2m + 1)
= 0 if λ = −n − 2m − 1
2
=
π(f )wn−2m if λ 6= n − 2m − 1
λ − (n − 2m − 1
= 0 if λ = n − 2m − 1.
wn+2(m+1) =
wn−2(m+1)
The non-zero wj form a basis of V and
1
1
(8) π(z)wj = jwj , π(e)wj = (λ+(j+1)wj+2 , π(f )wj = (λ−(j−1))wj−2 ,
2
2
GL(2, R)
23
where λ is as before. It follows that
Lemma 25. wn+2(m+1) = 0 iff wn+2m = 0 or λ = ±(n + 2m + 1) and
wn−2(m+1) = 0 iff wn−2m = 0 or λ = ±(n − 2m − 1)
Proposition 26. For all c ∈ C, ∀n ∈ Z, there exists an irreducible (g0 , K)module V such that π(Ω) = c and Vn 6= 0. Two (g0 , K)-modules (π, V ),
(π 0 , V 0 ) are isomorphic if and only if
(1) π(Ω) = π 0 (Ω)
(2) ∃n ∈ Z, Vn 6= {0}, Vn0 6= {0},
Vogan formulates this result in a different way. He defines the set of K-types
of a (g0 , K)-module V as the set of n ∈ Z such that Vn 6= 0 (in general it
would be the set of τ such that Vτ 6= 0), and a lowest K-type as a K-type n
such that |n| is minimal.
Lemma 27. Assume (π, V ) is an irreducible admissible (g, K)-module of
lowest K-type n, and |n| > 1. Then π(Ω) = (|n| − 1)2 . In particular, for
each n with |n| > 1, there is a unique (π, V ) with that property.
Proof. Assume that n > 1 (the case n < −1 is similar), and fix v ∈ Vn . For
n > 1 |n − 2| < |n| so Vn−2 = 0, so π(f )v = 0. Using Ω = (z − 1)2 + 4ef , we
get π(Ω)v = (n − 1)2 .
The next lemma follows from lemma 25 :
Lemma 28. Suppose |n| ≥ 1 and (π, V ) is the irreducible (g0 , K)-module
such that Vn 6= 0 and π(Ω) = (|n|2 − 1)2 . Then the set of K-types is
{n + 2(sgn n)m, m ∈ N}. In particular, n is a lowest Ktype.
We define the discrete series representation (π, Xd (n)) with parameter n 6=
0 to be the unique irreducible admissible (g0 , K)-module with lowest Ktype n + sgn n. For µ = −1, 0, 1, and λ ∈ C, define the continuous series
representation (π, X c (λ)(µ) to be the unique irreducible admissible (g0 , K)module containing the K-type µ with π(Ω) = λ2 . Just as we has a complete
description in terms of K-types for the discrete series,, we can do it for the
continuous series.
Lemma 29. Let µ ∈ {−1, 0, 1} and λ ∈ C.
(1) If λ is not an integer of the same parity as µ + 1, the K-types of
X c (λ)(µ) are {µ + 2m, m ∈ Z}.
(2) If λ is a integer 6= 0 of the same parity as µ + 1, then X c (λ)(µ) is
the finite dimensional representation of highest weight |λ| − 1. Its
K-types are {|λ| − 1, |λ| − 3, . . . , −|λ| + 1}.
(3) If λ = 0 and µ = ±1 the K-types are {µ + 2(sgn µ)m}.
Now we have a complete classification :
Theorem 30. Let π, V be an irreducible admissible (g0 , K)-module with
lowest K-type µ.
24
MARTIN ANDLER
(1) If |µ| > 1, then V ∼ Xd (µ − sgn µ). In this case, µ is the unique
lowest K-type.
(2) If |µ| ≤ 1; let λ be a square root of π(Ω). Then V ∼ X c (λ)(µ).
Furthermore, X c (λ)(µ) ∼ X c (λ0 )(µ) iff λ0 = ±λ. If µ = 0, the
µ is the unique lowest K-type of V . If µ = ±1 and λ 6= 0, then the
lowest K-types of V are ±µ. In this case, X c (λ)(µ) ∼ X c (λ)(−µ).
Finally, if µ = ±1 and λ = 0, then µ is the unique lowest K-type of
V
At this stage, we don’t know which representations are unitary; besides, we
have not made the link between this classification and the (g0 , K)-modules
associated with the representations of G constructed earlier.
6.2. Principal series. We observe that the formulas (25) make sense even
without the complicated condition for wj to be 6= 0 :
Proposition 31. Fix λ ∈ C and = ±1. The principal series representation
Xc ( ⊗ λ) with parameters , λ is defined as follows
(1) It has a basis {wn , n ≡ mod 2Z} of eigenvectors for the action of
k(θ) with eigenvalue einθ
(2) The action of z, e, f on wn is given by
(9)
(10)
(11)
π(z)wn = nwn
1
π(e)wn = (λ + (n + 1))wn+2
2
1
π(f )wn = (λ − (n − 1))wn−2 .
2
Proof. We need to check that the formulas (9) define a representation of g.
An interesting way of doing it is to observe that the relations to be proved
are algebraic in λ, and true for λ ∈ C\Z. In that case, they are the relations
for X c (λ)(µ), so they hold: hence they hold for any λ.
Contrary to the continuous or discrete series of representations, there is no
reason why the Xc ( ⊗ λ) should be irreducible.
Proposition 32. The notations being as before, let µ ∈ {−1, 0, 1} such that
(−1)µ = .
(1) π(Ω) = λ2
(2) If λ is not an integer of parity µ + 1, Xc (, λ) ∼ X c (λ)(µ)
(3) If λ ∈ N∗ , λ ≡ µ + 1, Xc (, λ) contains a submodule isomorphic to
Xd (λ) ⊕ Xd (−λ). The quotient Xc ( ⊗ λ)/Xd (λ) ⊕ Xd (−λ) is the
finite dimensional representation of highest weight λ − 1.
(4) If λ ∈ Z∗− , λ ≡ µ + 1, Xc (, λ) contains a submodule isomorphic
to the finite dimensional representation X c (λ)(µ) of highest weight
−λ − 1. The quotient Xc ( ⊗ λ)/X c (λ)(µ) ∼ Xd (λ) ⊕ Xd (−λ).
(5) If λ = 0, = −1, Xc ( ⊗ 0) ∼ X c (0)(µ) ⊕ Xc (0)(−µ).
GL(2, R)
25
In the next section, we will make the link between this algebraic description
and the analytic description of section 4.
7. Induced representations
We explain a general method for constructing representations of a group :
unitary induction. Let G be a Lie group with Lie algebra g, P a closed
subgroup with Lie algebra p, and τ a unitary representation of P in some
space V . We assume that G is unimodular (i.e. the Haar measure is also
right invariant) – there is no reason why P should be unimodular. So let δH
be the modular function for P : δP (h) = det ad h (the ad is the one acting
on h). Consider the set of continuous maps φ from G to V , with compact
support mod P , such that
1/2
φ(gh) = δP τ (h)−1 φ(g).
(12)
The map g 7→ kφ(g)k2V has the correct property of covariance under H, so
that there is an “integral”
Z
kφk =
kφ(g)k
˙ 2V dG/P g,
˙
G/P
with adequate properties (one of them being
R
G
=
R
R
G/P
P ).
The space V
IndG
P τ
of π =
is the space of “square integrable” maps from G → V , the
closure of the space of continuous maps verifying (12). The action of G on
V is by
π(g)φ(x) = φ(g −1 x).
This representation π is unitary.
For the geometrically minded, we have a fiber bundle over G/P with generic
fiber the Hilbert space V :
G ×P V
obtained as the quotient of G × V by the relation (g, v) ∼ (gh, τ (h)−1 v).
The space above is then the space of L2 -sections of the fiber bundle.
We apply this construction to
a u
∗
(13)
G = SL(2, R), P = M AN =
,a ∈ R ,u ∈ R .
0 a−1
The modular function is
a u
δP
= |a|2
0 a−1
In this case, thanks to the Iwasawa decomposition G = KAN , the measure
theoretical aspect can be bypassed. We consider λ ∈ C, = ±1, µ ∈ {0, 1}
such that (−1)µ = and the character χ,λ of P defined as
a
t
χ,λ
= (sgn a)µ |a|λ .
0 a−1
26
MARTIN ANDLER
We unitarily induce this character (meaning : if the character is unitary, i.e.
λ imaginary, the resulting representation should be unitary) : the space of
the representation is the Hilbert space H,λ of functions on G such that
(1) φ(gp) = χ,λ+1 (p)−1 φ(g)∀(g, p) ∈ G × P
(2) φ|K ∈ L2 (K).
The principal series representation T,λ is the left regular representation on
H,λ . We need to verify that this it has the required continuity properties,
and perform two comparisons : compare this representation of G with the
representation defined in Section 4, and compare the associated (,K)-module
with Xc (, λ).
Proposition 33. The principal series representation T,λ is admissible. Its
Harish-Chandra module is isomorphic to Xc ( ⊗ λ).
Proof. Using the Iwasawa decomposition, H,λ is easily seen to be Hilbertisomorphic to H = {ψ ∈ L2 (K), φ(−k) = (−1)µ φ(k)}, the map I : H,λ →
H being φ 7→ ψ = φ|K . We transfer the representation to H by I
T,λ
H,λ −−−−→ H,λ
Iy
Iy
0
T,λ
H −−−−→ H
We need to express
−1
y(g, θ) x(g, theta)
a b
k(θ) = k(ζ(g, θ))
c d
0
y(g, θ)−1
An easy calculation gives
p
y(g, θ) = (d cos θ + b sin θ)2 + (c cos θ + a sin θ)2
eiζ(g,θ) = y(g, θ)(d cos theta + b sin θ) + i(c cos θ + a sin θ),
hence the formula
0
T,λ
(g)ψ(k(θ)) = y −λ−1 ψ(k(ζ(g, θ))).
These formulas insures the continuity and the boundedness of the representation. Now, we study the restriction of the representation to K – but this
of course is very easy :
0
T,λ
(k(θ0 ))ψ(k(θ)) = ψ(k(θ − θ0 )).
Not forgetting that imposes a condition on the parity of ψ, Fourier theory
insures that (H,λ )n = Ceinθ if n and have same parity, 0 otherwise. Setting
ψn (k(θ)) = einθ , and φn the corresponding element of H,λ , we want explicit
formulas for T,λ (z), T,λ (e), T,λ (f ) on this basis. We know a priori that we
have formulas T,λ (z)ψn = nψn , T,λ (e)ψn = an ψn+2 , T,λ (f )ψn = bn ψn−2 .
GL(2, R)
27
This implies an = T,λ (e)ψn (1) and similarly for bn . This makes life easier.
We need only to compute the effect of the basis h, x :
−t
d
d
e
0
T,λ (h)ψn (1) = |t=0 φn
= |t=0 e(λ+1)t = (λ + 1)
t
0 e
dt
dt
T,λ (x)ψn (1) = 0.
Therefore
1
i
an = T,λ (e)ψn (1) = T,λ ( h − z + ix)ψn (1)
2
2
n
1
1
= (λ + 1) + = (λ + n + 1), and similarly
2
2
2
1
bn = (λ − (n − 1)).
2
Comparing these formulas with (9), this completes the proof.
Using the algebraic description, we recall the decomposition of the principal
series representation :
(1) T,λ is irreducible if λ is not an integer of parity µ + 1,
(2) If λ ∈ N∗ , λ ≡ µ + 1(2), T,λ contains a submodule isomorphic to
Xd (λ) ⊕ Xd (−λ). The quotient T⊗λ /Xd (λ) ⊕ Xd (−λ) is the finite
dimensional representation of highest weight λ − 1.
(3) If λ ∈ Z∗− , λ ≡ µ + 1(2), T,λ contains a submodule isomorphic to
the finite dimensional representation of highest weight −λ − 1. The
quotient is equivalent to Xd (λ) ⊕ Xd (−λ).
(4) If λ = 0, = −1, T⊗0 ∼ X c (0)(µ) ⊕ Xc (0)(−µ).
We still need to compare this representation to the one defined in section
4. We assume that λ = iν is imaginary. We use a third version of H,λ .
Consider the restriction of φ in H,λ to N − ∼ R. The map
N − × M A × N → G : (n− , a, n) 7→ n− an
is not onto, but is is onto an open everywhere dense set of full measure.
Therefore, the restriction map is one-to-one and onto H,λ → L2 (N − ) for
some measure which turns out to be the Lebesgue measure on N − ∼ R. It
remains to compute the action Te,λ of G in this third “non-compact” picture.
The computation yields
−c + av
T,iν (g)ξ(v) = (sgn(d − bv))µ |d − bv|−iν+1 ξ(
)
d − bv
which is the same formula as (6).
8. Discrete series
8.1. The Harish-Chandra module associated to the (analytic) discrete series. To identify the discrete series as it was defined in paragraph
4.1 to the algebraic construction, i.e. to identify the Harish-Chandra mod± (there m ∈ N, m ≥ 2) with X (n) (n ∈ Z, n 6= 0), we
ule associated with Dm
d
28
MARTIN ANDLER
± . The model we have, of holomorphic
need to compute the K-types of Dm
functions on the Poincar´e half-plane, is not convenient because the action of
K on P + is not simple. The solution consists in making a change of variable
to the unit disk.
Let D = {w, |w| < 1} be the unit disk. The maps
P + → D : z 7→ w =
z−i
w+1
, D → P + : w 7→ z = −i
z+i
w−1
are mutually inverse analytic isomorphisms. Define a measure dνm on D by
dνm = 41−m (1 − |w|2 )m
dudv
= 41−m n(1 − r2 )m−2 rdrdθ.
1 − |w|2 )2
where w = u + iv = reiθ .
The action is actually easier to compute if we replace SL(2, R) by the isomorphic group SU(1, 1). The isomorphism is given explicitly by conjugation:
1 i
−1
u ∈ SU(1, 1) 7→ cuc ∈ SL(2, R) where c =
.
i 1
Proposition 34.
(1) The space Km of holomorphic functions on D which
are square integrable for dνm is a Hilbert space.
(2) The map Tm : Hm → Km defined by
−2i m
w+1
Tm f (w) =
f −i
w−i
w−1
is an isometry.
0 + of SU(1, 1) on L2 (K , dν ) defined as
(3) The representation Dm
m
m
+
−1
−1 is given by
+
0
Dm (g) = Tm Dm (cgc )Tm
αw − β¯
0 + α β
−m
Dm
F
(w)
=
(−βw
+
α
¯
)
F
(
).
β¯ α
¯
−βw + α
¯
iθ
e
0
In the SU(1, 1) picture, the maximal compact subgroup is {
}.
0 e−iθ
It acts by
iθ
0
0 + e
Dm
F (w) = eimθ F (e2iθ w).
0 e−iθ
It follows immediately that the eigenfunctions are the wN , N ∈ N with
the eigenvalue ei(m+2N )θ . In particular, the lowest K-type is m. It is now
+ is isomorphic to
clear that the Harish-Chandra module associated to Dm
Xd (m − 1).
− is
Similarly, one shows that the Harish-Chandra module associated to Dm
isomorphic to Xd (−m + 1).
GL(2, R)
29
8.2. About the construction of the discrete series. We go back to the
original model for the discrete series, where the space of the representation
is Hm , a space of functions on the Poincar´e half plane P + . Recall that the
usual action of GL + (2, R) on P + induces a map
u 0
y x 0 1 k(θ) 7→ g · i = x + iy.
g=
0 u
We identify a function f on P + to a function F on GL + (2, R) by
u 0
y x
F(
k(θ)) = f (x + iy)y m/2 eimθ .
0 1
0 1
0
Clearly F verifies F (gk(θ0 )) = eimθ F (g) and F (gz) = F (g)
for z inthe ceny x
ter. Conversely, such a function F defines f (x + iy) = F (
)y −m/2 .
0 y −1
In this f ↔ F correspondence, C ∞ is preserved.
Consider the representation of G = SL(2, R) unitarily induced by the character e−imθ of K. Its space Vm can be identified to functions on GL + (2, R)
which verify both conditions of F above, plus L2 integrability on G/K. A
simple calculation shows that f ∈ F 7→ F is an isometry for the L2 -norms
and intertwines the two representations.
But elements of Hm verify the additional condition of being holomorphic.
This additional property implies irreducibility of the corresponding representation on Hm . We want to understand what this condition means for
F.
Using the formulas (2), we get
∂
∂
1
+i .
Lf = (Lh − 2iLx + iLw = e−2iθ (−2iy
2
∂ z¯
∂θ
Applying this formula to F (g) = f (x + iy)y m/2 eimθ , we get
∂f
∂ z¯
Therefore, f is holomorphic if and only if LF = 0.
LF (x, y, θ) = ei(m−2)θ y m/2
8.3. An analogy between discrete and principal series. It is very interesting at this stage to compare principal series and discrete series, because
the constructions are much closer than what it appeared at first.
How do we get a principal series representation ? We take a split Cartan
subgroup (split torus) T = M A, take a character of M A, given by a sign
and a complex number λ. We choose a Borel subalgebra b0 containing t0
(since t0 is split, there exists a real Borel subalgebra), extend the character to
a character of the corresponding minimal parabolic subgroup P , induce to G.
With appropriate conditions, the representation is unitary and irreducible.
So the representation space is made of functions φ : SL(2, R) → C such
that φ(gp) = χ(p)−1 f (g)for all p ∈ P plus other conditions; said otherwise,
functions on G such that φ(gt) = χ(t)−1 f (g) for all t ∈ T and φ(gn) = f (g)
30
MARTIN ANDLER
for all n ∈ N . Since N isconnected, this last condition is equivalent to
0 1
Lx φ = 0, where x =
generates n0 .
0 0
Now, how do we get a discrete series representation ? We take a compact
torus T = K, and a character of T given by an integer. We choose a Borel
subalgebra b of sl(2, C) containing t : b = t + Cf , and consider functions on
G satisfying φ(gt) = χ(t)−1 f (g) and Lf φ = 0.
9. Representations of SL ± (2, R)
Since SL(2, R) is a subgroup of index 2 is SL ± (2, R), the representation
theory of the latter can be deduced from that of the former very easily. This is how
works : Let Y be the subgroup of SL ± (2, R) gener it 0 1
ated by y =
, so that SL ± (2, R) = Y SL(2, R), SL ± (2, R) be1 0
ing isomorphic to the semi-direct product Y o SL(2, R). Let τ be an
irreducible representation of SL(2, R) in V . For g ∈ SL ± (2, R), define
τ g : x ∈ SL(2, R) 7→ τ (gxg −1 ); if g ∈ SL(2, R), τ g and τ are automatically
equivalent. There are two possibilities
(1) τ y is equivalent to τ
(2) the stabilizer of τ in G is SL(2, R).
In the first case, let Iy be an intertwining operator : it is a linear isomorphism
V → V such that τ (ygy −1 ) = Iy τ (g)(Iy )−1 . A priori, Iy ie defined up to
multiplication by a constant, but we impose the condition Iy = Iy−1 , which
fixes Iy up to sign. We make a choice of a sign. Define a representation T
of SL ± (2, R) by
T (g) = τ (g) if g ∈ SL(2, R)
T (yg) = Iy τ (g) if g ∈ SL(2, R)
This is an extension of τ .
In the second case, we induce τ from SL(2, R) to SL ± (2, R). The induced
SL ± (2,R)
representation T = Ind SL(2,R) τ is irreducible. The two representations
SL ± (2,R)
SL ± (2,R)
Ind SL(2,R) τ and Ind SL(2,R) τ y are equivalent.
Conversely, all representations of SL ± (2, R) are obtained this way : let T be
a representation of SL ± (2, R), and τ be its restriction to SL(2, R). Either
τ is irreducible, we are in the second cas, and T = Ind τ , or τ is the sum of
two equivalent representations τ1 , τ2 and we are in the first case.
Our goal is to have a description of admissible (g0 , K)-modules for SL ± (2, R)
organised in the same fashion as for SL(2, R). In this section, G = SL ± (2, R)
and G0 = SL(2, R) (indeed it is the connected component). The sub
±1 0
groups A, N don’t change, but K = O(2), K0 = SO(2), M =
,
0 ±1
P = M AN .
GL(2, R)
31
b of irreducible representations of K is {µn , n ∈
Lemma 35. The set K
O(2)
−
±
N}∪{µ+
0 , µ0 }, where µn = Ind SO(2) χn and µ0 are trivial on K0 and restrict
to Y as the trivial or the sign representation.
The set of characters of M is {δ = (δ1 , δ2 ) with δi ∈ {0, 1}} by
0
δ 1
= δ11 δ22 .
0 2
c, ν ∈ C, define the principal series representation
For δ ∈ M
±
(Tδ,ν
, Hδ,ν ) = Ind G
P (δ ⊗ ν),
±
meaning that Tδ,ν
is the left regular representation on the space of functions
f from G to C such that f (gp) = f (g)δ(m−1 )a−i(ν+1) where p = man, m ∈
M, a ∈ A, n ∈ N and f|K ∈ L2 (K).
c, ν ∈ C.
Lemma 36. Let δ ∈ M
Tδ,ν |K = µ+
0 ⊕ µ2 ⊕ µ4 ⊕ . . . for δ = (0, 0)
Tδ,ν |K = µ1 ⊕ µ3 ⊕ µ5 . . . for δ = (1, 0) or (0, 1)
Tδ,ν |K = µ−
0 + ⊕µ2 ⊕ µ4 ⊕ . . . for δ = (1, 1).
Let = δ|M ∩G0 . Then Tδ,ν |
SL(2,R)
= T,ν .
Let Xδ,ν be the Harish-Chandra module of Hδ,ν .
−
Let µδ ∈ {µ1 , µ+
0 , µ0 } such that δ occurs in µM . Fix X δ,ν to be the unique
irreducible subquotient of Xδ,ν containing the K-type µδ . Define the discrete
series representation
g,K
G0
G0
Xd (n) = Ind g,K
g,K0 (Xd (n)) ' Ind g,K0 (Xd (−n)).
One obtains the decomposition of Xδ,ν :
(1) If ν is not a non-zero integer of parity δ1 + δ2 + 1, Xδ,ν is irreducible
isomorphic to X δ,ν
(2) If ν is a positive integer of parity δ1 + δ2 + 1, there is a non-split
exact sequence
0 → Xd (ν) → Xδ,ν → X δ,ν → 0
and the quotient X δ,ν is a finite dimensional module of highest weight
ν−1
(3) If ν is a negative integer of parity δ1 + δ2 + 1, there is a non-split
exact sequence
0 → X δ,ν → Xδ,ν → Xd (ν) → 0.
and the submodule X δ,ν is a finite dimensional module of highest
weight −ν − 1.
32
MARTIN ANDLER
Moreover, the only equivalences between X δ,ν and X δ0 ,ν 0 arise iff (δ, ν) =
(δ 0 , ν 0 ) or (δ, ν) = (sδ, −ν), where s(δ1 , δ2 ) = (δ2 , δ1 ) and the Xd (n), X δ,ν
exhaust the set of representations of SL ± (2, R).
9.1. Representations of GL(2, R). In view of previous remarks, the classification for SL ± (2, R) yields immediately the classification for GL(2, R) :
one simply needs to “add” a character of the connected component of the
center. This gives the following classification. The parameters are δ, as
above, and v¯ = (ν1 , ν2 )inC × C. Set ν = ν1 − ν2 .
(1) Irreducible principal series : Xδ,¯ν for ν1 − ν2 not an integer of parity
δ1 + δ2 + 1
(2) Discrete series Xd (ν1 , ν2 ) with ν1 − ν2 a positive integer
(3) Finite dimensional representation representation X δ,¯ν of highest weight
(ν1 , ν2 ) where ν1 − ν2 is a positive integer of parity δ1 + δ2 + 1.
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´matiques (UMR CNRS 8100), Universite
´ de Versailles
Laboratoire de Mathe
´dex
Saint Quentin, 78035 Versailles Ce
E-mail address: [email protected]
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