Last revised 10:24 a.m. February 10, 2016
Continuous representations
Bill Casselman
University of British Columbia
cass@math.ubc.ca
This essay contains somewhat dry material most useful in motivating eventually a certain crucial but at first
sight somewhat technical transition from representations of groups to representations of Lie algebras. Parts
of it will also be used in the theory of automorphic forms. I have made some effort to reduce everything to
well known facts in measure theory and topology. The standard reference for the material here is [Borel:1972].
I have also used [Weil:1965].
This topic necessarily involves rather general topological vector spaces. In this paper, a TVS will always be
assumed to be locally convex and Hausdorff. But as a rule, all topological vector spaces in representation
theory are also assumed to be quasi-complete. The theory of such spaces is not particularly well known, but
I shall not include here a detailed account. In practice a quasi-complete TVS V is one for which integrals of
V -valued functions are well defined, and in which derivatives can be characterized in a particularly useful
way. Nearly all TVS encountered in the real world are quasi-complete, and it is rare that one has to think
much about it. The standard references on this material are [Treves:1967], §VI.5 of [Bourbaki:Integration],
and §III.8 of [Bourbaki:TVS].
Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
Continuous representations
Representation of measures
Interlude: Fourier series
Representations of a compact group I. Finite-dimensional
Representations of a compact group II. Infinite-dimensional
Smooth representations
Representations of G and of (g, K)
Realization
References
I wish to thank Murat Güngör for pointing out to me some gaps in the exposition on smooth representations.
Note: I decline ‘TVS’ as I do ‘sheep’: ‘one TVS’, ‘two TVS’, etc.
1. Continuous representations
Let G in this section be any locally compact topological group with a countable basis of neighbourhoods of
1, and which is a countable union of compact subsets. Fix on G a right invariant Haar measure.
A representation of G on a finite-dimensional space V is simply a continuous homomorphism from G to
GL(V ). But infinite-dimensional representations require more care. Suppose V to be a quasi-complete
Hausdorff topological vector space. A continuous representation of G on V is a map π from G to the group
of linear translations of V such that
G × V → V : (g, v) 7−→ π(g)v
is continuous. This means, by definition, that whenever we are given g0 in G, v0 in V , and a semi-norm ρ of
V then we can find a neighbourhood X of 1 in G and a semi-norm σ such that
π(xg0 )(v0 + u) − π(g0 )v0 < ε
ρ
Continuous representations
2
whenever x lies in X and kukσ < δ .
It is often annoying to check this condition directly, but verification can be reduced to two simpler steps.
1.1. Proposition. The representation (π, V ) is continuous if and only if these two conditions are satisfied:
(a) for a fixed v in V the map g 7→ π(g)v is continuous;
(b) if X is a compact subset of G and ρ a semi-norm of V , there exists a semi-norm σ such that
π(x)v ≤ kvkσ
ρ
(x ∈ X) .
The first condition, in our circumstances, means
(a’) Suppose v0 in V . For every continuous semi-norm ρ and ε > 0 there exists a neighbourhood X of 1
in G such that kπ(g)v0 − v0 kρ < ε whenever x lies in X .
The second condition (b) here is usually the easier to verify, since it says that the family of norms on V is in
some sense invariant under G, and this is often transparently true. In fact, under a mild restriction on V (that
it be barreled) (b) follows from (a). This is to be found as Proposition 1 of VIII.§1 in Bourbaki’s Integration.
Proof. The necessity of (a’) and (b) is immediate from the definition of continuity. As for sufficiency, suppose
g0 , v0 , and ρ given. Then
π(xg0 )(v0 + u) − π(g0 )v0 = π(x)g0 )v0 − π(g0 )v0 + π(x)π(g0 )u
ρ
ρ
≤ π(x)π(g0 )v0 − π(g0 )v0 ρ + π(x)π(g0 )uρ .
According to (a’) we can find a neighbourhood X of 1 in G such atht
kπ(x)π(g0 )v0 − π(g0 )v0 kρ < ε/2 ,
and then by (b) we can find σ such that
k + π(x)π(g0 )ukρ ≤ kukσ < ε/2
for x in X , if kuk < ε/2.
1.2. Corollary. The representation (π, V ) is a continuous representation of G if and only if the embedding of
V into the space of functions from G to V , taking v to the function Fv (g) = π(g)v , is a continuous map from
V to C(G, V ).
The space C(G, V ) is that of all continuous functions on G with values in V . Its topology is defined by
semi-norms
kF kΩ,ρ = sup F (g)ρ
g∈Ω
where Ω is a compact subset of G and ρ a continuous semi-norm on V . The image of V is in fact the closed
subspace of all functions F such that F (gx) = π(g)F (x) for all g , x in G.
Proof. Condition (a) means that the image of V lies in C(G, V ). Condition (b) means that the map from V
to C(G, V ) is continuous.
If (π, V ) is a continuous representation of G, then a priori we have two topologies on V , the original one and
that induced from C(G, V ). It is easy to see that these are the same.
The simplest general class of continuous representations is that of representations of G on various spaces of
functions on itself and on quotient spaces H\G, as well as on certain spaces of induced representations. The
group G acts on these by means of the right regular, and in some cases left regular, representations:
Rg F (x) := F (xg),
Lg F (x) := F (g −1 x) .
Continuous representations
3
On G itself these commute, hence define an action of G × G on various spaces of functions on G.
Suppose given on G a right-invariant Haar measure. Recall L2 (G) to be the Hilbert space of all measurable
functions F on G such that
Z
1/2
|f (g)|2 dg
kF k2 =
< ∞.
G
It is the closure in this norm of the continuous functions on G of compact support. This L2 norm is invariant
under the right-regular representation of G.
Define C(G) to be the vector space of continuous C-valued functions on G. It becomes a Fréchet space if
assigned the norms
kf k∞,Ω = sup f (g)g∈Ω
for compact subsets Ω. The notation is standard and justified because, for example, the ‘disk’
|x|p + |y|p ≤ 1
has as limit the square max |x|, |y| ≤ 1 as p → ∞.
In accord with this definition, the space L∞ (G) is that of all bounded continuous functions on G—i.e.
functions F for which kF k∞ is finite.
1.3. Proposition. The right regular representations of G on the spaces C(G) and L2 (G) are continuous.
These two spaces are typical. I’ll not give details in other similar cases encountered.
Proof. First to verify (a’) for C(G). We are given a compact subset Ω ⊆ G and ε > 0, and must find a
neighbourhood X of 1 in G such that f (gx) − f (g) < ε for g in Ω, x in X .
Because f is continuous, we can find for every g in Ω a neighbourhood Xg of 1 such that |f (gx) − f (g)| < ε
for x in Xg . Because Ω is compact, it can be covered by a finite number of the sets gXg . Let X be the
intersection of these Xg .
I leave condition (b) for C(G) as an exercise.
As for L2 (G), condition (b) is trivial, whereas (a) requires a bit of work. I leave this, too, as an exercise. (Hint:
use the density of Cc (G) in L2 (G).)
If (π, V ) is a continuous representation of G then its dual (b
π , Vb ) on the continuous linear dual of V is that
defined by the condition
In other words,
π
b(g)b
v , π(g)v = hb
v , vi .
π
b(g)b
v , v = vb, π(g)−1 v .
The continuous linear dual of a TVS may be assigned the weak topology, with norms kb
v kv = |hb
v , vi| for v in
V . It is straightforward to prove:
Continuous representations
4
1.4. Proposition. If (π, V ) is a continuous representation of G, the dual representation π
b is continuous in the
weak topology on Vb .
The matrix coefficient corresponding to the pair b
v and v is the continuous function
g 7−→ hb
v , π(g)vi
on G. If π is finite-dimensional and (ei ) is a basis of V with dual basis ebi , then hb
ej , ei i is in fact a matrix entry.
2. Representation of measures
Let Mc (G) be the space of bounded measures on G of compact support. It can be identified with the space
of continuous linear functionals on the space C(C, C) of all continuous functions on G. If µ1 and µ2 are two
measures in Mc (G) their convolution is defined by the formula
hµ1 ∗ µ2 , f i =
Z
f (xy) dµ1 dµ2 .
G×G
The identity in Mc (G) is the Dirac δ1 taking f to f (1).
If G is assigned a Haar measure dx, the space Cc (G) of continuous functions with compact support may be
embedded in it: f 7→ f dx. The definition of convolution of measures then agrees with the formula for the
convolution of two functions in Cc (G):
f1 ∗ f2 (y) =
Z
f1 (x)f2 (x−1 y) dx
G
I have assumed that the TVS of a continuous representation is quasi-complete. Because of this assumption,
the space of a continuous representation π becomes a module over Mc (G) in accordance with the formula
π(f )v =
Z
π(g)v dµ .
G
This integral is characterized uniquely by the condition that
E Z D Z
vb, π(g)v dµ
π(g)v dµ =
v,
b
G
G
for every continuous linear function vb on V .
Suppose Ω to be the support of µ. Condition (b) of Proposition 1.1 implies that for every semi-norm ρ of V
there exists a semi-norm σ such that
kπ(g)vkρ ≤ kvkσ
for all g in Ω. Then
π(µ)v ≤
ρ
Z
π(g)v |dµ|
ρ
Ω
Z
≤
|dµ| kvkσ .
Ω
Hence:
2.1. Proposition. For every µ in Mc (G) the operator π(µ) is continuous.
Continuous representations
5
For a given v in V the map µ 7→ π(µ)v from Mc (G) to V is covariant with respect to the left regular
representation, since
u • π(Lg µ)v = hLg µ, u • π(•)vi
= hµ, u • π(g −1 π(•)vi
= hµ, π(g)u • π(•)vi
= π(g)u • π(µ)v
= u • π(g)π(µ)v .
The map µ 7−→ π(µ) is a ring homomorphims, since the composition of two operators π(µ1 )π(µ2 ) can be
calculated easily to agree with π(µ1 ∗ µ2 ). For the left regular representation, Lµ f = µ ∗ f .
I define a Dirac function on G to be a function f in Cc∞ (G, R) satisfying the conditions
(a) f (g) = f (g −1 );
(b) f (g) ≥ 0 forRall g ;
(c) the integral G f (g) dg is equal to 1.
A Dirac sequence is a sequence of Dirac functions fn with support tending to 1.
2.2. Corollary. If {vi } is a finite set in V , ρ a semi-norm on V , ε > 0, and {fn } a Dirac sequence on G, and ρ
a semi-norm on V then for some N
π(fn )vi − vi < ε
ρ
for all i, all n > N .
3. Interlude: Fourier series
In the next section I’ll take up representation theory for an arbitrary compact group, but in this one I’ll look
at a familiar case. It is a simple example, but in some sense prototypical. Let S be the multiplicative group of
complex numbers z with |z| = 1.
For each m in Z, the map taking z to z m is a differentiable one-dimensional representation of S. The classical
theory of Fourier series asserts that every f in L2 (S) can be expressed as a sum
f=
X
cn z n
n
in the sense that the associated finite sums
fN =
X
cn z n
|n|≤N
converge to f in L2 (S). The sum is orthogonal, and consequently the coefficients are given by the formula
Z
cm =
z −m f (z)
S
dz
.
z
In other words, the space L2 (S) is the Hilbert direct sum of the spaces spanned by the characters z n . This
induces an isomorphism of L2 (Z) with L2 (S). The smooth functions in the representation of S on L2 (S)
correspond to the sequences (cm ) with cm rapidly decreasing as a function of m—for each k the sequence
|m|k cm is bounded. As we’ll see later, these are the smooth vectors of the representation.
If (π, V ) is any smooth representation of S, let Πm be the projection operator
Πm v =
Z
|z|=1
z −m π(z)v
dz
.
z
Continuous representations
6
The image is the subspace Vm of v in V such that π(z)v = z m v for all z . Of course it may happen that
Vm = 0. Let Vfin be the algebraic direct sum of the spaces Vm . This may also be characterized as the S-finite
vectors in V , those contained in some finite-dimensional S-stable subspace.
3.1. Lemma. The subspace Vfin is dense in V .
Proof. According to the Hahn-Banach theorem, it suffices to show that any continuous linear functional F
on V that vanishes on all Vm vanishes everywhere. Using a Dirac sequence to approximate δ1 , it must be
shown that
ϕ in C ∞ (S) and v in V we have hF, π(ϕ)vi = 0. But ϕ will be the limit of finite sums of
Pfor any
functions
cm z m , and the assumption on f therefore implies that hF, π(ϕ)vi = 0.
In the subsequent sections, I am going to follow a slightly different convention from the classical one sketched
here. If π is any continuous representation of the locally compact group G and f in Cc (G), I will take its
Fourier transform at π to be the endomorphism π(f ). This turns out to make notation somewhat simpler. In
this scheme, if G = S and f lies in C(G) its Fourier transform will be
πm (f ) =
Z
f (z)z m
S
dz
.
z
Of course, adopting this new convention does not change the underlying mathematics.
4. Representations of a compact group I. Finite-dimensional
Representations of compact groups are a model for all of representation theory as well as a necessary
preliminary to much of it. In this section, let K be an arbitrary compact group with a countable basis of
neighbourhoods of 1. I do not assume it, at first, to be a Lie group. For example, the additive group of p-adic
integers Zp = lim Z/(pn ) is allowable.
←−
The representations of compact groups is much better known than most of the material in this essay, and I
make no claim to originality. The exposition pretty much follows a straightforward line, but because it is
rather long I have divided the exposition into two major pieces, mostly separating finite dimensions from
infinite dimensions, and each of these in turn into smaller pieces. The theory of continuous finite dimensional
representations of compact groups differs from the theory of representations of finite groups primarily in so
far as integration replaces finite sums, so this first part is especially straightforward.
Assign K a left-invariant measure dk of total measure 1.
A continuous finite-dimensional (cfd) representation of K on the space V is simply a
continuous map from K to GL(V ). I start off with:
SEMI-SIMPLICITY .
4.1. Proposition. The group K is unimodular.
This means that every right-invariant Haar measure is also left-invariant.
Proof. A left-invariant Haar measure dℓ x is unique up to constants. For k in K , dℓ xk is also left-invariant,
hence dℓ xk = δ(k) dℓ x for some positive constant δ(k). The map taking k to δ(k) is a continuous homomorphism from K to the multiplicative group of positive real numbers, hence trivial.
Define 1 to be the constant function equal to 1 on all of K . If (π, V ) is a continuous representation of K then
π(1)v =
Z
π(k)v dk .
K
The vector π(1)v is fixed by all of K , and π(1) is idempotent.
4.2. Proposition. If (π, V ) is a continuous representation of K then v is fixed by all of K if and only if
π(1)v = v . The kernel of π(1) is a closed K -stable summand of V and
V = V K ⊕ Ker π(1) .
Continuous representations
7
In other words, π(1) is the unique K -equivariant projection onto the subspace V K of K -fixed vectors, which
is a closed subspace of V .
Proof. We have
v=
Z
K
I − π(k) v dk +
Z
π(k)v dk .
K
The second term is π(1)v and is fixed by K . The first is in the kernel of π(1).
The proof shows that Ker π(1) is the closure of the spans of the π(k)v − v .
4.3. Corollary. If
0 −→ U −→ V −→ W −→ 0
is an exact sequence of continuous representations of K then
0 −→ U K −→ V K −→ W K −→ 0
is also exact.
If W has finite dimension the map from V K to W K splits continuously, as does any continuous map from V
onto a finite-dimensional vector space.
This decomposition is a special case of a more general result. One major point of this section is to find a
generalization of the projection π(1) associated to representations of K other than the trivial one.
Suppose (π1 , V1 ) and (π2 , V2 ) to be two cfd representations. There are two natural ways to define a representation of K × K on the vector space HomC (V1 , V2 ), the difference being a matter of order:
(a)
f 7−→ π2 (k2 ) f π1 (k1−1 )
(b) f 7−→ π2 (k1 ) f π1 (k2−1 )
I’ll generally choose (a). This is just one of a number of somewhat similar choices to be made in this subject
For either choice here, the space HomK (V1 , V2 ) of K -equivariant linear maps from V1 to V2 is the subspace
of invariants with respect to the diagonal copy of K in K × K . A special case of the Hom representation is
the representation of K on Vb dual to V , with
hb
π (k)b
v , vi = hb
v , π(k −1 )vi .
4.4. Proposition. Any short exact sequence
0 −→ U −→ V −→ W −→ 0
of continuous K -representations, where W has finite dimension, splits continuously.
Thus V ∼
= W ⊕ U.
Proof. From the exact sequence in the Corollary we obtain an exact sequence
0 −→ HomC (W, U ) −→ HomC (W, V ) −→ HomC (W, W ) −→ 0
of K × K representations. By the Proposition, this gives rise to an exact sequence
0 −→ HomK (W, U ) −→ HomK (W, V ) −→ HomK (W, W ) −→ 0
Any lifting of the identity map in the last term is a splitting.
Continuous representations
8
For a more explicit proof, choose a linear splitting f : W → V , which is necessarily continuous. Then
f : w 7−→
Z
K
π(k) f π(k −1 )w dk
is a K -equivariant splitting.
4.5. Corollary. Every cfd representation of K is a direct sum of irreducible representations.
This follows by induction from Proposition 4.4. It ceases to be true if π is not finite-dimensional, but it has a
useful generalization, as we shall see later.
4.6. Corollary. Suppose (π, V ) to be a cfd representation. It is irreducible if and only if EndK (V ) = C.
4.7. Corollary. Suppose π1 , π2 to be two irreducible cfd representations of K . Then
HomK (V1 , V2 ) =
0 unless π1 is isomorphic to π2
C if π1 = π2 .
Proof. Should be clear, since any kernel or cokernel must be trivial.
TENSOR PRODUCTS . If (π1 , V1 ) and (π2 , V2 ) are two cfd representations, the tensor product representation
of K × K is defined by the formula
[π1 ⊗ π2 ](k1 , k2 ): v1 ⊗ v2 7−→ π1 (k1 )v1 ⊗ π2 (k2 )v2 .
4.8. Proposition. If K1 , K2 are compact groups and (πi , Vi are cfd representations of Ki , then π1 ⊗ π2 is
irreducible, and every irreducible representation of K1 × K2 is of this form.
Proof. Let K = K1 × K2 . Suppose V 6= 0 to be any K -stable subspace of V1 ⊗ V2 , and U 6= 0 an irreducible
K1 -stable subspace of V . As a representation of K1 , V is a direct sum of copies of V1 , and the canonical
map from U ⊗ Hom(U, V ) to V is an isomorphism. If V = V1 ⊗ V2 then Hom(U, V is isomorphic to V2 ,
and since V2 is irreducible, the embedding of V into V1 ⊗ V2 must induce an ismorphism of Hom(U, V ) with
Hom(U, V1 ⊗ V2 ). Therefore V = V1 ⊗ V2 .
Similarly, if V is any irreducible representation of K1 × K2 and U 6= 0 is any irreducible K1 -stable subspace,
then V is isomorphic to U ⊗ Hom(U, V ), and Hom(U, V ) is an irreducible representation of K2 .
The canonical pairing
vb ⊗ v 7−→ hb
v , vi
is a K -equivariant map from Vb ⊗ V to the trivial representation, and indeed this is the point of the definition
of the dual representation. If π is irreducible, any K -invariant linear function on Vb ⊗ V is a multiple of this
one.
The ⊗ and Hom representations are closely related. For each pair vb1 in Vb1 and v2 in V2 define a linear map
from V1 to V2 by the formula
v 7−→ hb
v1 , vi v2 .
The range of this map is the one-dimensional space spanned by v2 , and its kernel is the space of codimension
one annihilated by vb1 . The map taking (b
v1 , v2 ) to Fv̂1 ,v2 is bilinear and hence extends to a linear map
Φ: Vb1 ⊗C V2 −→ HomC (V1 , V2 ), vb1 ⊗ v2 7−→ Fv̂1 ⊗v2 .
The representation of K × K on Hom is designed precisely so that this isomorphism is K × K -equivariant.
If we choose bases (ei ) and (fj ) for U and V , as well as the dual bases (b
ei ) and (fbi ), the matrix Ej,i of Φêi ,fj
Continuous representations
9
has zero entries everywhere except a 1 in column i and row j . What this means is that the map taking the
matrix (Fj,i ) in Hom(U, V ) to
is inverse to the one I have defined.
X
Fj,i ebj ⊗ ei
In the case where the two spaces are the same, the identity map from V to itself is the image of the sum
P
bi ⊗ ei (which is therefore independent of the choice of basis).
ie
That the map one way is straightforward and the map back the other way uses a basis reflects the serious
difficulty that occurs when V has infinite dimension. One has to be especially careful for topological vector
spaces. Only for nuclear vector spaces is there a reasonable identification of a (topological) tensor product
with a space of linear maps.
HERMITIAN FORMS .
A Hermitian pairing between two vector spaces is a map
v1 ⊗ v2 7−→ v1 • v2
from V1 × V2 to C, linear in the first factor, conjugate linear in the second. A Hermitian form on V is a
Hermitian pairing of V with itself. If H is a Hermitian form on V and c is in C, then cH is Hermitian if and
only if c is real. The set of all positive definite Hermitian forms on V is a convex open cone in the space of all
Hermitian forms.
4.9. Proposition. If (π, V ) is an irreducible cfd representation, then the space of K -invariant Hermitian forms
on V has dimension one over R. The subset of positive definite ones is a real ray.
The last assertion means that there exists at least one Hermitian form on V that is K -invariant, and all others
are positive multiples of it. Consequently, every cfd representation is unitary.
Proof. The only non-trivial point to verify is that there always exists a non-trivial invariant Hermitian form
on V . But if one starts with an arbitrary positive definite form on V its K -average will still be positive
definite.
4.10. Proposition. If (π1 , V1 ) and (π2 , V2 ) are two irreducible cfd representations, the space of K -invariant
Hermitian pairings of V1 with V2 is trivial if π1 is not isomorphic to π2 .
Proof. Any Hermitian pairing is a conjugate-linear map from V1 to Vb2 . Because of irreducibility, it must be
an isomorphism. But the previous result tells us that there exist conjugate-linear K -isomorphism of V2 with
Vb2 , hence a linear K -isomorphism of V1 with V2 .
Suppose (π, V ) to be an irreducible cfd representation, u • v an invariant positive definite Hermitian form on
V . Since it is non-degenerate, for each vb in Vb there exists a vector ϕ(b
v ) in V such that
u • ϕ(b
v ) = hb
v , ui
for all u in V . The map vb 7→ ϕ(b
v ) is conjugate-linear. We can assign a positive definite Hermitian form on Vb
by the formula
u
b b• vb = ϕ(b
v ) • ϕ(b
u)
which is conjugate-linear in the second factor. How does this depend on the choice of Hermitian inner
product? If we are given two Hermitian inner products • 1 and • 2 = c · • 1 , then
(4.11)
b• 2 = c−1 · b• 1 .
Given a Hermitian form on V , one can then define one on Vb ⊗ V according to the formula
(b
v1 ⊗ v1 ) • (b
v2 ⊗ v2 ) = (b
v1 b• vb2 ) · (v1 • v2 ) .
Continuous representations
10
Because of (4.11):
b ⊗ V is canonical.
4.12. Lemma. The definition of Hermitian inner product on V
That is to say, it does not depend on the choice of Hermitian inner product on V . We shall see another way
to interpret this in a moment.
MATRIX COEFFICIENTS .
The product K × K acts on C(K), the space of continuous functions on K , by
L(k1 , k2 )f (k) = Lk1 Rk2 f (k) = f (k1−1 kk2 ) .
This representation on C(K) extends to a continuous representation of K × K on the Hilbert space L2 (K)
with the invariant Hilbert norm
Z
f1 (k)f2 (k) dk .
f1 • f2 =
K
If (π, V ) is a cfd representation, the map from Vb ⊗ V to C(K) taking vb ⊗ v to the matrix coefficient
Mv̂⊗v (k) = hb
v , π(k)vi
is K × K -equivariant. If π is irreducible, then Vb ⊗ V is an irreducible representation of K × K , and according
to Proposition 4.8 this is an embedding. The canonical positive definite Hermitian form on Vb ⊗ V and the
Hilbert norm restricted to the matrix coefficient image must be scalar multiples of each other. Explicitly:
4.13. Proposition. (Schur orthogonality) Suppose π1 , π2 to be two irreducible representations of K . For vi ,
vbi in Vi , Vbi


Z
0
unless π1 is isomorphic to π2
1
v2 , π(k)v2 i dk =
hb
v1 , π(k)v1 ihb
v1 ⊗ v1 ) • (b
v2 ⊗ v2 ) if π = π1 = π2
 dim(π) (b
K
Proof. Only the case π1 = π2 need be examined. For trivial reasons we have
Z
hb
v1 , π(k)v1 ihb
v2 , π(k)v2 i dk = cπ (b
v1 ⊗ v1 ) • (b
v2 ⊗ v2 )
K
The only interesting point is to find the constant cπ . Fix an orthonormal basis ei of V . The dual basis ebi is
also orthonormal. The formula gives us on the one hand
XZ
j
K
eℓ , π(k)ej i dk = dim(π) · cπ (b
ei • ebℓ ) ,
hb
ei , π(k)ej ihb
but on the other hand it is the (i, ℓ)-th entry of the matrix of π(k) tπ(k −1 ) = I (since π(k) is unitary), which
is just ebi • ebℓ . Hence cπ = 1/ dim(π).
If V1 = V2 , the identity transformation I is K -invariant, the linear function f 7→ trace(f ) from
EndC (V ) to C is K -invariant, and the map taking f to trace(f )/ dim(π) times I is the K -projection onto the
one-dimensional space it spans.
THE TRACE .
v , vi.
4.14. Lemma. The trace of Φv̂⊗v is hb
Analogous to the matrix coefficient map is that from EndC (V ) to C(K) taking f to
τ : f 7−→ [k 7−→ trace π(k)f ] .
4.15. Proposition. The diagram
Continuous representations
11
Vb ⊗ V
µ
C(G)
Φ
EndC (V )
τ
is a commutative diagram of K × K -representations.
v , uiv is hb
v , vi.
Proof. Because the trace of u 7→ hb
Suppose (π, V ) to be a cfd representation. I have made a mildly arbitrary choice in defining EndC (V ) as a
representation of K × K , and now I must comment on this. The point is that the vector space EndC (V ) is
canonically isomorphic to its own dual via the pairing
hf1 , f2 i = trace(f1 f2 ) ,
but this is not, as things stand, an invariant K × K -pairing. As I have already observed, are really two
possible definitions of the action of K × K on EndC (V ):
f 7−→ π(k2 )f π(k1−1 ) (the definition I have already made)
(a)
(b) f 7−→ π(k1 )f π(k2−1 ) (the alternative) .
My choice of the first is only a matter of convention, but the second cannot now be ignored. The pairing
trace(f1 f2 ) defines a bilinear map from the tensor product EndC (V ) ⊗ EndC (V ) to C.
4.16. Lemma. This bilinear map is K × K -invariant if K × K acts on the first factor according to (a) but on
the second according to (b).
Proof. It amounts to the identity
trace π(k2 )f1 π(k1−1 )π(k1 )f2 π(k2−1 ) = trace(f1 f2 ) .
This explains the substitution of k −1 for a second k in the following version of Schur orthogonality:
4.17. Proposition. If (π, V ) is an irreducible cfd representation of K then for f1 , f2 in EndC (V )
Z
K
trace π(k)f1 trace π(k −1 )f2 dk =
1
trace π(f1 )π(f2 ) .
dim(π)
Let TR π be the character of π , the function
TR π (k)
= trace π(k) .
If we set f1 = f2 = I in Proposition 4.17, we see that
Z
TR σ (k)TR π (k
−1
) dk =
K
n
0 if σ is not isomorphic to π
1 if π ∼
= σ.
Recall the involution on C(K): f ∨ (k) = f (k −1 ). This is equivariant with respect to the right- and left-regular
representation of K × K on C(G).
4.18. Corollary. If (π, V ) is an irreducible cfd representation, the map taking f in C(K) to dim(π) · π(f ∨ ) is
inverse to that taking f in EndC (V ) to τf in C(K).
v , ui v .
Proof. Apply the Schur orthogonality relations to verify this for the special endomorphism u 7→ hb
Continuous representations
12
That the map takes f to π(f ∨ ) is a consequence of several choices I have made, but seems most natural. It
agrees with the conventions of Fourier series on the circle group S.
For each irreducible cfd representation (π, V ) define the function
ξπ = dim(π) ·
∨
TR π
.
It will replace 1 when dealing with π . It is useful to keep in mind that TR∨π = TRπ̂ .
4.19. Corollary. Suppose (π, V ) an irreducible cfd representation of K . Then
(a)
(b)
(c)
(d)
the function ξπ lies in the centre of C(K);
it is idempotent with respect to convolution;
if π is irreducible then π(ξπ ) = I ;
if ρ, π are two non-isomorphic irreducible representations of K then ρ(ξπ ) = 0.
Proof. The first is true of any conjugation-invariant function in C(K). The second follows from Schur
orthogonality. For the last we start with the formula
π(ξπ )u = dim(π)
and then continue
hb
u, π(ξπ )ui = dim(π)
= dim(π)
Z
K
Z
K
Z
TR π (k
−1
)π(k)u dk
TR π (k
−1
) hb
u, π(k)ui dk
trace π(k −1 )I
K
= trace Φû⊗u
TR (π(k)Φû⊗u ) dk
= hb
u, ui .
FINITE GROUPS . In this section suppose K to be finite. Finite groups are compact groups, so most of the
representation theory of general finite groups has already been seen. But there are some special things worth
noticing.
Integration over K is now defined by
Z
f (k) dk =
K
1 X
f (k) .
|K|
For each irreducible finite dimensional representation (π, V ) of K we have an embedding ιπ of Vb ⊗ V into
the finite-dimensional space C(K).
4.20. Proposition. The space C(K) is the direct sum of the images of the ιπ . For any f in C(K) we have
f=
X
π
ξπ ∗ f =
X
dim(π)(f ∗ TR π ) .
π
Proof. If f lies in C(K) the space generated in C(K) by f will be a direct sum of irreducible representations.
But if (π, V ) is an irreducible representation of K then composition with f 7→ f (1) identifies
HomK V, C(K) ∼
= HomC (V, C) = Vb
so that any copy of π in C(K) lies in the image of ιπ . The images of two distinct representations are orthogonal
since by Schur orthogonality the product of projections for two distinct representations is 0.
Continuous representations
13
If δ1 is the characteristic function at 1 multiplied by |K|, then π(δ1 ) is the identity in all representations. It
follows immediately from the previous Proposition that
δ1 =
X
dim(π) TR π .
π
But this is a special case of a more general fact. If f is any conjugation-invariant function on K then it lies in
the center of C(K) and hence π(f ) is scalar multiplication if π is irreducible. If π(f ) = cπ I then
f=
X
cπ ξπ
π
since ππ (ξπ ) = Iπ , and
trace π(f ) = c dim(π),
c = trace π(f )/ dim(π) .
If f = fS is the
function of the conjugacy class S of s then trace π(f ) is also |S| TRπ (s)/|K|.
pcharacteristic p
Its L2 norm is |S|/|K| = 1/ |Ks |. where Ks is the centralizer of s. If
Os =
p
|Ks | fS ,
the OS form an orthonormal basis for conjugate-invariant functions on K . We have proved
4.21. Proposition. If S is the conjugacy class of s then
1 X
−1
OS = p
TR π (s ) TR π .
|Ks | π
Of course we can also write
TR π
=
X
S
1
p
TR π (s)OS .
|Ks |
Both the TRπ and the OS form orthonormal bases of the conjugate-invariant functions on K . That of the
characters is sometimes called the spectral basis, that of the conjugacy classes the geometric basis. One
consequence is that the number of representations is equal to the number of conjugacy classes.
5. Representations of a compact group II. Infinite-dimensional
The principal results of this section are that (a) every irreducible continuous representation of a compact
group is finite-dimensional; (b) an arbitrary continuous representation is a topological direct sum of isotypic
components associated to irreducible representations; and (c) integration against ξπ is a projection onto the
π -component.
The first of these means that the subspace of K -finite vectors is dense in any continuous representation of K .
But one particular case of this is special, and has to be dealt with first.
For each irreducible cfd representation (π, Vπ ), let L2π (K) be the subspace of its matrix coefficients, the
canonical image of Vbπ ⊗ Vπ in C(K) ⊆ L2 (K). The first important result to be proved here is that L2 (K) is
the Hilbert direct sum of the L2π (K). This will take place in several steps.
A K -finite vector in a representation of K is one that is contained in a finite-dimensional K -stable subspace.
5.1. Lemma. The left-K -finite functions in L2 (K) are dense.
Proof. Let V = L2 (K), v in V . Let f be a self-adjoint Dirac function such that Rf v − vk < ε. Since K is
compact, the operator Rf is compact, its eigenspaces Vf,ωi for eigenvalues ωi 6= 0 are finite-dimensional. We
have
X
Rf v − vk2 = kv0 k2 +
(ωi − 1)2 kvi k2 < ε2 .
Continuous representations
14
Each of thespaces
i is finite-dimensional and stable under left multiplication by K . If the set of ωi is
P Vf,ω
finite, then v −
vi = kv0 k < ε. Otherwise, say ωi < 1/2 for i > n, and
X
v −
i≤n
vi < 2ε .
5.2. Lemma. Any left-K -finite function in L2 (K) is continuous.
Proof. Suppose F is contained in the finite-dimensional K -stable subspace V ⊆ L2 (K). The image of C(K)
in EndC (V ) is closed, but by Theorem 2.2 the identity operator I is in its closure. That image therefore
contains I , so for some f in C(K) we have Rf F = F . But it is simple to verify that if f lies in C(K) then
Rf F is continuous.
5.3. Lemma. If π is a cfd representation of K , the image of any K -equivariant map from Vπ to L2 (K) is
contained in L2π (K).
Proof. By the preceding Lemma, the image of Vπ is contained in C(K). A simple version of Frobenius
reciprocity asserts that
HomK V, C(K) = HomC (V, C) = Vb .
But this means precisely that it is in the image of Vb ⊗ V .
5.4. Proposition. If π and σ are not isomorphic the spaces L2π (K) and L2σ (K) are orthogonal.
Proof. This is an immediate consequence of Schur orthogonality.
For F in L2 (K), let Fπ = F ∗ ξπ .
5.5. Theorem. (Peter-Weyl) For any F in L2 (K) and ε > 0 there exists a finite subset Π of irreducible cfd
representations of K such that
X
F −
π∈Π
Fπ < ε .
In other words, L2 (K) is the Hilbert direct sum of the spaces L2π (K).
Immediate from the preceding Lemmas.
5.6. Proposition. Let (π, V ) be any continuous representation of K , (σ, U ) an irreducible cfd representation,
Vσ the image of the operator π(ξσ ). Then:
(a) the image of any K -map from U to V is contained in Vσ ;
(b) the space Vσ is canonically isomorphic to U ⊗ HomK (U, V ).
Proof. The first follows from the previous result, since any K -equivariant map from U to V commutes with
ξσ . The second follows from the identification
b ⊗ U, V )
U ⊗ HomK (U, V ) ∼
= HomK (U
where K acts on the second factor in the right-hand term. This is because of Corollary 4.19 together with the
observation that V ∼
= HomK (Mc (K), V ) with K acting on Mc on the right.
In particular, if Cσ (K) is the image of Vb ⊗ V in C(K), it is the image of Rξπ , which is also the image of
convolution with dim(π) TR π . Keep in mind that TR π itself is in this space.
5.7. Proposition. In any continuous representation of K the K -finite vectors are dense.
Proof. By the Hahn-Banach theorem, it suffices to show that any continuous linear function f on V that
vanishes on each K -stable finite-dimensional subspace vanishes on every v in V . Apply Peter-Weyl to the
continuous function taking k to hf, π(k)vi.
5.8. Corollary. Any irreducible continuous representation of K is finite-dimensional.
Continuous representations
LIE GROUPS .
15
So far, everything I have said is valid for all compact groups. Compact Lie groups are special.
5.9. Corollary. Any compact Lie group K has a faithful, continuous representation of finite dimension.
Proof. The space L2 (K) is the countable direct sum ⊕π (Vbπ ⊗ Vπ ). Let
F0 ⊂ F1 ⊂ F2 ⊂ . . .
be an increasing sequence of finite subsets of the π whose union is all of them.
For each finite subset F of representations π let
KF =
\
ker(π) .
π∈F
Each of these is a closed subgroup of K , and the intersection, according to the Peter-Weyl Theorem, is 1. If
U is any neighbourhood of 1, then any strictly decreasing sequence of closed sets whose intersection is 1 will
be eventually in U , and since K is a Lie group the subgroups Ki must eventually be just {1}.
If K is a compact Lie group, then the Fourier decomposition of a smooth function f looks much like that of a
smooth function on the unit circle, but I’ll not prove that here. There are two approaches, one concerned with
the asymptotics of the eigenvalues of the Laplacian on an arbitrary Riemannian manifold (see Chapter 12 of
[Taylor:1981]), the other relying on an explicit classification of the irreducible representations for a connected
compact group in terms of characters of a maximal torus, as in [Borel:1972], pages 24–35.
Another thing I’ll not prove here is that every compact subgroup of some GLn is a Lie group. There are
several ways to see this, but one very satisfactory approach is to see that it is in fact an algebraic group, defined
as the zeroes of polynomials (a classic result due, although in a rather different formulation, to Tannaka).
6. Smooth representations
In this section, let G be a Lie group.
Suppose V to be a quasi-complete TVS. There are two ways to define continuously differentiable functions
on G with values in V .
The first just depends on the structure of G as a smooth manifold. The definition in this case just reduces to
the case of an open subset Ω in Rn . The function F in C(Ω, V ) is said to be differentiable at x if
lim
t→0
F (x + tv) − F (x)
= [∂v F ](x)
t
exists for all v in Rn . It is said to be continuously differentiable in Ω if [∂v f ](x) is a continuous function on
Ω × Rn , which case it is a linear function of v and defines a continuous differential dF in C(G, Hom(Rn , V )).
These are standard facts about calculus of functions with values in a quasi-complete space, and often reduce
directly to well known results about the usual calculus by applying the Hahn-Banach theorem.
The second definition depends on the structure of G. It stipulates that
lim
t→0
F (g exp(tX)) − F (g)
= [RX F ](g)
t
exist for all g in G and X in g, and then that [RX F ](g) be a continuous function of g and X .
6.1. Lemma. These two definitions agree.
This is not a very deep result, but I’ll give a proof below, because it brings out where it is required that V
be quasi-complete. The literature usually skips over the question while implicitly assuming it to be true, or
attempts a proof carelessly.
Continuous representations
16
Proof. Verifying one implication is much like verifying the other, so I’ll just do one. The proof reduces to two
steps. The first is a basic fact about the geometry of a neighbourhood of the identity of G: If α(t) and β(t) are
two smooth paths on G with α(0) = β(0) = I and α′ (0) = β ′ (0), then there exists a smooth function X(t)
from some [0, ε) to g such that β(t) = α(t) · exp(t2 X(t)). This is because the exponential map is a smooth
diffeomorphism of a neighbourhood of 0 in g with a neighbourhood of the identity, and β(t) − α(t) = t2 ∆(t)
for some smooth function ∆(t).
The second step takes as model the proof in calculus that directional derivatives depend only on direction,
not on the path that goes off in that direction. We write
f (α(t) · exp(t2 X(t))) − f (0)
f (β(t)) − f (0)
=
t
t
f (α(t) · exp(t2 X(t))) − f (α(t)) f (α(t)) − f (0)
+
.
=
t
t
Therefore it suffices now to show that
f (α(t) · exp(t2 X(t))) − f (α(t))
= 0.
t→0
t
lim
Since
f (α(t) · exp(t2 X(t))) − f (α(t))
f (α(t) · exp(t2 X(t))) − f (α(t))
= t·
t
t2
this will follow from:
6.2. Lemma. Locally on G we have
f (g · exp(sX) − f (g) = sϕ
where ϕ lies in the convex hull of the image of the map
[0, s] −→ V,
u 7−→ [RX f ](g · exp(uX)) .
Proof of the Lemma. Fix g , and consider the map
F (s) = f (g · exp(sX)) .
By assumption, it is in C 1 ([0, ε), V ) for some ε > 0, and
F ′ (s) = [RX f ](g · exp(sX))
since exp((s + h)X) = exp(sX) · exp(hX). According to the Fundamental Lemma for V -valued functions
of one variable
Z
Z
s
F (s) − F (0) =
0
1
F ′ (u) du = s ·
F ′ (θs) dθ .
0
The integral makes sense because V is assumed to be quasi-complete. A basic property of the integral is that
it is equal to some ϕ in V that lies inside the convex hull of the image of [0, s] with respect to θ 7→ F ′ (θs).
If dF is in turn continuously differentiable then the function is said to be continuously differentiable of second
order, in which case RX [RY F ] is a continuous function for all X and Y in g. Proceeding by induction, it is
said to be differentiable of order n + 1 if for each X in g the function RX F is continuously differentiable of
order n. It is smooth if it is differentiable of all orders. Let C m (G, V ) be the space of all functions on G with
values in V that are continuously differentiable of orders ≤ m, C ∞ (G, V ) be the space of smooth functions
from G to V .
The following is immediate, and, because of the Hahn-Banach Theorem, characterizes the derivative RX F :
Continuous representations
17
6.3. Lemma. Suppose F to be a continuously differentiable function on G with values in V and v
b a continuous
linear function on V . Let
ϕ(g) = hb
v , F (g)i .
The function ϕ(g) is a continuously differentiable scalar function on G, and
[RX ϕ](g) = vb, [RX F ](g) .
If (π, V ) is a continuous representation of G, then for each v in V the map taking g to π(g)v is a continuous
function on G with values in V . The vector v is said to be smooth if this map is smooth. The representation
is said to be smooth if all vectors in V are smooth. If (π, V ) is any continuous representation let
V (m) = {v ∈ V | g 7→ π(g)v is in C m (G, V )}
This space, assigned the norms π(X)v ρ for X of order at most m in U (g), is a quasi-complete Hausdorff
TVS, and complete if V is.
6.4. Proposition. If (π, V ) is a continuous representation of G then
(a) the subspace V ∞ is a continuous representation of G;
(b) for each X in U (g) the operator π(X) takes V ∞ to itself, and is continuous;
(c) for every f in Ccm (G) and v in V the vector π(f )v is differentiable of order m, and
π(X)π(f ) = π(LX f ) .
for X in U m (g).
Proof. Only the last requires confirmation. We have
π(f ) =
π exp(tX) π(f ) =
=
Z
ZG
ZG
G
f (x)π(x) dx
f (x)π exp(tX)x dx
f (exp(−tX)x π(x) dx .
The map from V to C(G, V ) taking v to the function π(g)v is injective. The image of V ∞ is a closed subspace
of C ∞ (G, V ), and its topology is the inherited one.
In Proposition 6.4, m is allowed to be ∞. Choosing a Dirac sequence of smooth functions, we see:
6.5. Corollary. In any continuous representation of G the smooth vectors are dense.
We have
π [X, Y ] = π(X), π(Y ) ,
on the subspace V (2) . The map from g to continuous endomorphisms of V ∞ associated to a continuous
representation of G is a representation of the Lie algebra g.
6.6. Proposition. Any continuous finite-dimensional representation of a Lie group is smooth.
Proof. If (π, V ) is continuous, the image of Cc∞ (G) in End(V ) is closed and by Theorem 2.2 therefore contains
I . If π(f ) = I , then v = π(f )v for every v , which is therefore smooth.
Continuous representations
18
7. Representations of G and of (g, K)
In this section, let G be the group of R-rational points on a Zariski-connected reductive group defined over
R, K a maximal compact subgroup of G. According to Cartan’s fixed point theorem, all choices of K are
conjugate to each other by an element of the connected component of G. Suppose A to be the group of
R-rational points on a maximal split torus of G. According to §14 of [Borel-Tits:1972], A meets all connected
components of G. This implies the group K meets all of the connected components of G. (This is a special case
of an older result of [Mostow:1955] about arbitrary Lie groups with a finite number of connected components.)
The group G acts trivially by conjugation on Z(g), the center of the enveloping algebra of G.
A continuous representation of G is admissible if the dimension of each K -isotypic component has finite
dimension. Admissible representations are ubiquitous as well as important. All irreducible unitary representations of G are known to be admissible, and admissible representations are those occurring in various
decompositions of natural representations of G, such as that of G × G on Cc∞ (G) or L2 (G). In short, they
are basic objects in the theory.
If (π, V ) is any continuous representation of G, let V(K) be the subspace of K -finite vectors—that is to say,
contained in a K -stable finite-dimensional subspace of V .
7.1. Proposition. If (π, V is an admissible continuous representation of G, then for every v in V(K) there
exists a function f in Cc∞ (G) such that π(f )v = v .
Proof. If ξ is an idempotent in C(K) such that π(ξ) is the identity on the finite dimensional K -isotypic
component U , then the image of π(ξ)π(Cc∞ (G))π(ξ) is a closed subalgebra of the finite-dimensional algebra
EndC (U ). But if we choose a Dirac sequence ϕi in Cc∞ (G), then π(ξ)ϕi π(ξ) will converge to the identity of
U . So the identity operator is in the image, too.
7.2. Corollary. If V is an admissible representation of G, then the subspace V(K) is contained in V ∞ and is
stable with respect to g.
Proof. The first claim follows immediately from the Proposition and Proposition 6.4. If U is a finitedimensional subspace, the map X ⊗ u 7→ π(X)u is a surjection of K -spaces g ⊗ U → π(g)U .
7.3. Corollary. Every K -finite vector in an admissible representation (π, V ) of G is also Z(g)-finite. If π is
irreducible, there exists a homomorphism from Z(g) to C through which elements of Z(g) act.
This homomorphism is traditionally called (for reasons that escape me) the infinitesimal character of π .
Many technical problems are to be found in the theory of infinite-dimensional representations of G that don’t
exist for finite-dimensional ones. The most serious arise because many different continuous representations
of G are all in some sense equivalent, even though analytically of a very different nature. Some are quite
complicated, in fact unnecessarily complicated. The sort of thing I have in mind is illustrated by representations of SL2 (R) associated to its action on P = P1 (R). The spaces of analytic functions, smooth functions,
and locally L2 functions on P are for most purposes best considered as different incarnations of the same
beast. What they all have in common is the same underlying space of K -finite functions. This, happily, is a
representation of the Lie algebra sl2 , if not of the group itself.
Suppose given a representation of g and a continuous representation of K simultaneously on a space V . I’ll
denote noth representations by π . It is called an admissible representation of the pair (g, K) if
(a) as a representation of K it is an algebraic direct sum of irreducible finite-dimensional representations
of K , each with finite multiplicity;
(b) the two representations of k, as Lie algebra of K and as subalgebra of g, are the same.
These are often called Harish-Chandra modules. If V is an admissible representation of G then V(K) is an
admissible representation of (g, K).
In practice, in the theory of representations of the reductive group G, one works with admissible representations of (g, K) rather than continuous representations of G. We shall see in a moment how this can be
justified. The fundamental question is this: to what extent does a representation of the Lie algebra determine
Continuous representations
19
that of the group? One problem that the representation of the Lie algebra alone can’t handle at all is the behaviour of the representation on connected components of G other than that of the identity, but that problem
is addressed by including the action of K , which meets all connected components.
Before taking up the justification of the definition of Harish-Chandra modules, let me point out why replacing
the group G by the pair (g, K) is a good idea. Suppose for the moment that G = SL2 (R), and let P = P1 (R).
The group G acts on the right this space through the action
v=
x
7−→ g −1 v .
y
It therefore acts on the left on various spaces of functions on P. Among these are the space C(P) of continuous
functions, the space C ∞ (P) of smooth functions, the space C ω (P) of real analytic functions, and L2,loc (P) of
locally square-integrable functions. All of them contain exactly the same space of K -finite functions, which
as a representation of K is the direct sum of characters ε2n of K where
ε:
c −s
7 → c + is .
−
s
c
As you can imagine, a great deal of simplification takes place if one looks just at this last space—analysis will
be replaced by algebra.
There are a number of facts that justify the switch from representations of G to those of (g, K). I state them
here briefly:
(a) every Harish-Chandra module is the restriction to K -finite vectors of some continuous representation
of G;
(b) if (π, V ) is an admissible representation of G, the map taking U ⊆ V(K) to its closure in V is a bijection
between (g, K)-stable subspaces of V(K) and closed G-stable subspaces of V ;
(c) the map taking (π, V ) to V(K) is a bijection between irreducible unitary representations of G and
irreducible Harish-Chandra modules with a positive definite metric invariant with respect to (g, K);
(d) there exist exact functorial assignments of smooth representations of G to Harish-Chandra modules,
inverse to the restriction map.
I’ll look at (a) in the next section. As for the rest, I’ll just look at them briefly without going into detail.
I’ll sketch the proof of (b) in a moment.
[Borel:1972] proves (c).
There are several ways to assign continuous representations of G to Harish-Chandra modules in a functorial
manner. One is in a sense minimal, the other perhaps the most natural. The original ideas were in joint work
by Nolan Wallach and myself. This is explained in [Casselman:1990]. A different approach is to be found in
[Bernstein-Krötz:2010].
I now sketch the proof of (b).
7.4. Proposition. If Φ is a distribution on G which is left- or right-K -finite as well as Z(g)-finite, then it is a
real-analytic function on G.
Proof. Let π be the left or right regular representation of G, depending on the assumption regarding K .
Since Φ is the sum of its K -isotypic components, one may as well assume that Φ is itself an isotypic component,
hence that π(CK )Φ = λΦ for some λ, where CK is the Casimir element of U (k). Since CK commutes with
all of Z(g), π(XCK )Φ = π(CK )π(X)Φ = λπ(X)Φ for all X in Z(g). Since Φ is Z(g)-finite
n
Y
1
π(C) − µi Φ = 0
Continuous representations
20
for some set of scalars µi , where now C is the Casimir element of U (g). Let ω = C + 2CK , and let
Φk =
k
Y
1
and in particular Φ0 = Φ. Thus
leading to
π(C) − µi Φ (0 ≤ k ≤ n)
π(C) − µk Φk−1 = Φk
π(C) − µn Φn−1 = 0
π(ω) − (2λ + µk ) Φk−1 = Φk
π(ω) − (2λ + µn ))Φn−1 = 0 .
Since ω is an elliptic operator on G with analytic coefficients, each Φk is analytic.
7.5. Corollary. If (π, V ) is an admissible representation of G, v a K -finite vector and v
b in Vb any continuous
linear functional on V , then the matrix coefficient hπ(g)v, vbi is analytic.
7.6. Theorem. If (π, V ) is admissible, the map taking W to W is a bijection of (g, K)-stable subspaces of V
and closed G stable subspaces of V .
Proof. The difficult point, and the one that convinces most strongly that representations of (g, K) are a
reasonable thing to serve as formal substitutes for representations of G, is the claim that if W is a (g, K)stable subspace then its closure W is G-stable. By the Hahn-Banach theorem, in order to show
that W is
G-stable, it suffices to show that if F is a linear function on V that vanishes on W , then F π(g)w = 0 for all w
in W . Because K meets all components of G and W is K -stable, it suffices to show this for g in the connected
component of G. But according to Corollary 7.5 this is an analytic function of g , and therefore it suffices to
show that the coefficients of its Taylor series at 1 vanish. However, these coefficients are determined by the
constants F (π(X)w), which vanish by assumption.
8. Realization
I call a Harish-Chandra module (π, v) realizable if it is the representation of (g, K) on the K -finite vectors
in admissible continuous representation of G. As I have said in the previous section, all Harish-Chandra
modules are realizable. I also mentioned there that there is in fact a canonical way to realize every HarishChandra module, but that seems not to be so important as the simple existence of some realization.
The original result about realizability was one by Harish-Chandra, which asserted that ever irreducible
admissible representation of(g, K) could be realized as subquotient of some principal series. A later and
more interesting proof of this was found by [Beilinson-Bernstein:1982]. I myself might have been the first the
prove that every finitely generated (g, K)-module was realizable. This proof showed that formal solutions of
the differential equations satisfied by matrix coefficients were in fact converging solutions, but this proof was
never published. Other proofs have been found since. The existence of matrix coefficients is presumably the
most important fact about realizable representations,a because the asymptotic behaviour of matrix coefficients
is an extremely important part of the theory. Jacquet and Langlands essentially postulated this in the their
book on GL2 , where it is disguised in terms of an action of the Hecke algebra.
Continuous representations
21
9. References
1. A. Beilinson and J. Bernstein, ‘A generalization of Casselman’s submodule theorem’, pp. 35–52 in
Representation Theory of Reductive Groups, edited by P.C. Trombi, Birkhauser, Boston, 1982.
2.
J. Bernstein and B. Krötz, ‘Smooth Fréchet globalizations of Harish-Chandra modules’, preprint
arXiv.org:0812.1684, 2010.
3. Armand Borel, Représentations de groupes localement compacts, Lecture Notes in Mathematics 276,
Springer, 1972.
4. Nicholas Bourbaki, Integration I, Springer. Translated from the original French edition, which is dated
1959–1965.
5. ——, Topological vector spaces, Springer, translated from the French original dated 1981.
6. Jacques Dixmier and Paul Malliavin, ‘Factorisations de fonctions et de vecteurs indéfiniment différentiables’,
Bulletin des Sciences Mathématiques 102 (1978), 305–330. 1978
7. Daniel Mostow, ‘Self-adjoint groups’, Annals of Mathematics 62 (1955), 44–55.
8. Michael Reed and Barry Simon, Methods of mathematical physics I. Functional analysis, Academic
Press, 1972.
9. L. A. Rubel, W. A. Squires, and B. A. Taylor, ‘Irreducibility of certain entire functions with applications
to harmonic analysis’, Annals of Mathematics 108 (1978), 553–567.
10. André Weil, L’intégration dans les groupes topologiques, Hermann, 1965.