Continuous representations Bill Casselman

Continuous representations Bill Casselman

University of British Columbia [email protected]

This essay contains somewhat dry material, mostly 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].

Contents

1. Continuous representations . . . 1

2. Representation of measures . . . 4

4. Interlude: Fourier series . . . 5

4. Representations of a compact group I. Finite-dimensional . . . 6

5. Representations of a compact group II. Infinite-dimensional . . . 12

6. Smooth representations . . . 15

7. Representations of G and of (g, K) . . . 18

9. Realization . . . 20

9. Appendix. Tensors and homomorphisms . . . 21

10. References . . . 22 This topic necessarily involves rather general ^t opological ^v ector ^s paces (as opposed to, say, only Hilbert spaces). In this paper, a TVS will always be assumed to be locally convex and Hausdorff. But as a rule, all topological vector spaces occurring in representation theory are also quasi-complete. Quasi-completeness is at first sight a rather technical condition, but in fact very practical. 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. For example, Fr´echet spaces and LF spaces are quasi-complete, and so are their duals. The standard references on quasi-complete spaces are [Treves:1967], §VI.5 of [Bourbaki:Integration], and §III.8 of [Bourbaki:TVS].

Note: I decline ‘TVS’ as I do ‘sheep’ or ‘fish’: ‘one TVS’, ‘two TVS’, etc. I wish to thank Murat G ¨ung¨or for pointing out to me some gaps in an earlier exposition on smooth representations.

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 Hausdorff topological vector space. A continuous representation of G on V is a homomorphism π from G to the group of linear transformations of V such that

G × V → V : (g, v) 7−→ π(g)v is continuous.

By considering an explicit matrix representation, one can see easily that this definition agrees with the naive

one if V has finite dimension.

The defining condition means that whenever we are given g

in G, v

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(xg

)(v

+ u) − π(g

)v

ρ

< ε whenever x lies in X and kuk

< δ.

It is often annoying to check the definition 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, v in V , and ρ a semi-norm of V , there exists a semi-norm σ such that π(x)v

≤ kvk

(x ∈ X) .

The first condition, in our circumstances, means

(a’) Suppose v

in V . For every continuous semi-norm ρ and ε > 0 there exists a neighbourhood X of 1 in G such that kπ(g)v

− v

k

< 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 g

, v

, and ρ given. Then

π(xg

)(v

+ u) − π(g

)v

ρ

=

π(xg

)v

− π(g

)v

+ π(x)π(g

)u

≤

π(x)v

− v

ρ

+

π(x)π(g

)u

(v

= π(g

)v

) . According to (a’) we can find a neighbourhood X of 1 in G such that

kπ(x)v

− v

k

< ε/2 , and then by (b) we can find σ such that

kπ(x)π(g

)uk

≤ kuk

< ε/2 for x in X, if kuk

< ε/2.

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 k

= sup

g∈Ω

F (g)

where Ω is a compact subset of G and ρ a continuous semi-norm on V .

Given a continuous representation (π, V ) there exists a canonical embedding of V into C(G, V ), the space of all continuous functions on G with values in V :

v 7−→ [g 7−→ π(g)v] .

1.2. Corollary. The representation (π, V ) is a continuous representation of G if and only if this embedding of V into C(G, V ) is continuous.

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 ). 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:

R

F (x) := F (xg), L

F (x) := F (g

x) .

On G itself these commute, hence define an action of G × G on various spaces of functions on G.

1.3. Example. I’ll not give an exhaustive list of examples, but offer two types as models. They will all be spaces of functions on quotients H\G in which H is a closed subgroup of G.

I recall first a bit of topology. First, suppose X to be any locally compact Hausdorff space which the union of a countable number of compact subsets. The TVS C(X) is that of all continuous functions on X. For each compact subset Ω in X define the semi-norm

kf k

= sup

x∈Ω

|f (x)| .

These make X into a Fr´echet space.

The TVS C

(X) is that of all continuous functions on X whose support is a relatively compact open subset of X. If Y is a relatively compact open subset, let C

(Y ) be the subspace of all f in C(Y ) that vanish on the boundary of Y . Such a function may be extended uniquely to a function in C(X) vanishing outside Y . Define the semi-norm

kf k

= sup

x∈Y

|f (x)| . This makes C

(X) into an LF space.

Both spaces are quasi-complete, as are their duals.

1.4. Proposition. Suppose H a closed subgroup of G. The right regular representations of G on C(H\G) and C

(H\G) are continuous.

Proofs should be evident.

1.5. Example. Suppose G to be unimodular, H a closed subgroup. Let δ

be the modulus character of H, and let Ω

(for real c) be the space of all continuous functions F : G → C such that F (hg) = δ

(h)f (g) for all h in H, g in G. and which are of compact support modulo H. There exists on Ω

a positive, continuous, G-invariant linear functional, which I’ll express as integration. This may be embedded in a space of integrable elements Ω

. Let Ω

be the space of functions on G such that f (hg) = δ

(h)f (g). This implies that f

lies in Ω

.

1.6. Proposition. The right-regular representations of G on Ω

(H\G) and Ω

(H\G) are continuous.

If (π, V ) is a continuous representation of G then its dual (bπ, b V ) on the continuous linear dual of V is that

defined by the condition

b π(g)bv, π(g)v 

= hbv, vi .

In other words,

b π(g)bv, v 

=

v, π(g) b

v  .

The continuous linear dual of a TVS may be assigned the weak topology, with norms kbvk

= |hbv, vi| for v in

V . It is straightforward to prove:

1.7. Proposition. If (π, V ) is a continuous representation of G, the dual representation bπ is continuous in the weak topology on b V .

The matrix coefficient corresponding to the pair bv and v is the continuous function Φ

: g 7−→ hbv, π(g)vi

on G. If π is finite-dimensional and (e

) is a basis of V with dual basis be

, then hbe

, e

i is in fact a matrix entry.

2. Representation of measures

In this section, the TVS V will be assumed to be quasi-complete.

Let M

(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(G, C) of all continuous functions on G. If µ

and µ

are two measures in M

(G) their convolution is the measure defined by the formula

hµ

∗ µ

, f i = Z

G×G

f (xy) dµ

dµ

. The identity in M

(G) is the Dirac δ

taking f to f (1).

If G is assigned a Haar measure dx, the space C

(G) of continuous functions with compact support may be embedded in M

(G): f 7→ f dx. The definition of convolution of measures then agrees with the formula for the convolution of two functions in C

(G):

[f

∗ f

](y) = Z

G

f

(x)f

(x

y) dx

Because of the assumption about quasi-completeness, the space of the continuous representation (π, V ) becomes a module over M

(G) in accordance with the formula

π(µ)v = Z

G

π(g)v dµ . This integral is characterized uniquely by the condition that

hbv, π(µ)vi = D b v,

Z

G

π(g)v dµ E

= Z

G

b v, π(g)v  dµ for every continuous linear function bv 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)vk

≤ kvk

for all g in Ω. Then

(2.1)

π(µ)v

≤

Z

Ω

π(g)v

|dµ|

≤

Z

Ω

|dµ|

 kvk

. Hence:

2.2. Proposition. For every µ in M

(G) the operator π(µ) is continuous.

The map µ 7−→ π(µ) is a ring homomorphism, since the composition of two operators π(µ

)π(µ

) can be calculated easily to agree with π(µ

∗ µ

). For the left regular representation, L

f = µ ∗ f .

If f (x) is a function in C

(G) then f (x) dx is a measure of compact support. I define a Dirac function on G

to be a function f in C

(G, R) satisfying the conditions

(a) f (g) = f (g

);

(b) f (g) ≥ 0 for all g;

(c) the integral R

G

f (g) dg is equal to 1.

A Dirac sequence is a sequence of Dirac functions f

with support tending to 1. From (2.1) :

2.3. Proposition. Suppose f to be a Dirac function with compact support Ω, v in V . Let ρ be a semi-norm on V , and σ a semi-norm such that kπ(x)vk

≤ kvk

for x in Ω. Then

kπ(f )k

≤ kvk

.

Proof Apply (2.1) .

2.4. Corollary. If {v

} is a finite set in V , ρ a semi-norm on V , ε > 0, and {f

} a Dirac sequence on G, and ρ a semi-norm on V then for some N

π(f

)v

− v

ρ

< ε 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 definitely prototypical. Let S be the multiplicative group of complex numbers z with |z| = 1. Fix as measure on S

dθ 2π = dz

2πiz .

For each m in Z, the map taking z to z

is a differentiable one-dimensional representation of S. The classical theory of Fourier series asserts that every f in L

(S) can be expressed as a sum (called here a Fourier series)

f = X

n

c

z

in the sense that the associated finite sums

f

= X

|n|≤N

c

z

converge to f in L

(S). The sum is orthogonal, and consequently the coefficients are given by the formula c

= 1

2πi Z

S

z

f (z) dz z .

In other words, the space L

(S) is the Hilbert direct sum of the spaces spanned by the characters z

. This induces an isomorphism of L

(Z) with L

(S). The smooth functions in the representation of S on L

(S) correspond to the sequences (c

) with c

rapidly decreasing as a function of n—for each k the sequence

|n|

c

is bounded. As we’ll see later, these are the smooth vectors of the representation.

If (π, V ) is any continuous representation of S on a quasi-complete TVS, let Π

be the projection operator Π

v = 1

2πi Z

|z|=1

z

π(z)v dz z .

The image is the subspace V

of v in V such that π(z)v = z

v for all z. Of course it may happen that V

= 0.

Let V

be the algebraic direct sum of the spaces V

. 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 V

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 V

vanishes everywhere. Using a Dirac sequence to approximate δ

, it must be shown that for any 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 functions P

c

z

, so (2.1) and the assumption on f , imply that hF, π(ϕ)vi = 0.

4. Representations of a compact group I. Finite-dimensional

Representations of compact groups are models for all of representation theory and are also required 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 least at first, to be a Lie group. For example, the additive group of p-adic integers Z

= lim

←−

Z /(p

) is allowable.

Some aspects of representations of compact groups are simpler than most of the material in this essay, largely because finite-dimensional vector spaces are well understood. I make no claim to originality. The exposition pretty much follows a straightforward line, but because it is rather long I have divided it 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 only in so far as integration replaces finite sums, so this first part is especially straightforward.

I should say right at the beginning that every irreducible continuous representation of a compact group is finite-dimensional. This will be proved much later (as Corollary 5.8).

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 d

x is unique up to constants. For k in K, d

xk is also left-invariant, hence d

xk = δ(k) d

x for some positive constant δ(k). The map taking k to δ(k) is a continuous homomor- phism from K to the multiplicative group of positive real numbers, hence trivial.

• Assign K, once and for all, an invariant measure dk of total measure 1.

SEMI-SIMPLICITY

. A continuous finite-dimensional ( cfd ) representation of K on the space V is simply a continuous map from K to GL(V ).

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

K

π(k)v dk . 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

⊕ Ker π( 1 ) .

In other words, π( ¹ ) is the unique K-equivariant projection onto the subspace V

of K-fixed vectors, which is a closed subspace of V .

Proof. We have

v = Z

K

I − π(k)
v dk +

Z

K

π(k)v dk . 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

−→ V

−→ W

−→ 0

is also exact.

!

If W has finite dimension the map from V

to W

splits continuously, as does any continuous map from V onto a finite-dimensional vector space. But this is not generally true otherwise.

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.

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. Choose any linear splitting f : W → V , which is necessarily continuous. Then f : w 7−→

Z

K

π(k) f π(k

)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 End

(V ) = C.

That is to say that it is irreducible if and only if it is indecomposable.

4.7. Corollary. Suppose π

, π

to be two irreducible cfd representations of K. Then Hom

(V

, V

) =

 0 unless π

is isomorphic to π

C if π

= π

.

Proof. Should be clear, since any kernel or cokernel must be trivial.

HOM

. There are several ways to construct new representations from old ones. The two principal ones involve tensor products and spaces of homomorphisms.

Suppose (π

, V

) and (π

, V

) to be two cfd representations. There are two natural ways to define a repre- sentation of K × K on the vector space Hom

(V

, V

), the difference being a matter of order:

(a) f 7−→ π

(k

) f π

(k

) (b) f 7−→ π

(k

) f π

(k

)

I choose (a), for reasons explained in the appendix. Of course this can hardly be a deep matter, but it has a few awkward consequences regarding other notation. The awkwardness seems to be arise chiefly because of the well-fixed convention according to which we apply a sequence of maps in order from right to left, rather then the other way around.

For either choice, the space Hom

(V

, V

) of K-equivariant linear maps from V

to V

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 dual representation of K on b V , for which

hb π(k)bv, vi = hbv, π(k

)vi .

In other words, bπ(k) =

π(k

). Duality is an involution, since if V is finite-dimensional the canonical map from V to the dual of its dual is an isomorphism. Already the case of K = Z/(3) will show that the dual of a representation π is not generally isomorphic to π.

It is important to realize that the two representations defined by (a) and (b) are not the same. In some sense, one is the dual of the other.

The representation on homomorphisms gives us a different way to understand the proof of Proposition 4.4.

From the exact sequence in Corollary 4.3 we obtain an exact sequence

0 −→ Hom

(W, U ) −→ Hom

(W, V ) −→ Hom

(W, W ) −→ 0 of K × K representations. According to Corollary 4.3, this gives rise to an exact sequence

0 −→ Hom

(W, U ) −→ Hom

(W, V ) −→ Hom

(W, W ) −→ 0 Any lifting of the identity map in the last term is a splitting.

TENSOR PRODUCTS

. If (π

, V

) and (π

, V

) are two cfd representations, the tensor product representation of K × K is defined by the formula

[π

⊗ π

](k

, k

): v

⊗ v

7−→ π

(k

)v

⊗ π

(k

)v

.

4.8. Proposition. If K

, K

are compact groups and (π

, V

) are irreducible cfd representations of K

, then π

⊗ π

is irreducible, and every irreducible representation of K

× K

is of this form.

Proof. Let K = K

× K

. Suppose V 6= 0 to be any K-stable subspace of V

⊗ V

, and U 6= 0 an irreducible K

-stable subspace of V . As a representation of K

, V is a direct sum of copies of V

, so that by Corollary 4.7 the representation on U is isomorphic to that on V

, and we may as well assume that U = V

. and the canonical map from U ⊗ Hom(U, V ) to V is an isomorphism. For the same reasons, the canonical map from U ⊗ Hom

(U, W ) to W is an isomorphism. For the same reasons again V is isomorphic to U ⊗ Hom

(U, V ).

But the embedding of W into V induces an embedding of Hom

(U, W ) into Hom

(U, V ). The second is isomorphic to V

as a representation of K

, and since V

is irreducible this embedding is an isomorphism.

Hence W = V .

Similarly, if V is any irreducible representation of K

× K

and U 6= 0 is any irreducible K

-stable subspace, then V is isomorphic to U ⊗ Hom(U, V ), and Hom(U, V ) is an irreducible representation of K

.

If we choose bases (e

) and (f

) for U and V , as well as the dual bases (be

) and ( b f

), the matrix E

of T

has zero entries everywhere except a 1 in column i and row j. What this means is that the map taking the matrix (F

) in Hom(U, V ) to

X F

· e

⊗ be

is inverse to the one I have defined. In the case where the two spaces are the same, the identity map from V to itself is the image of the sum P

i

e

⊗ be

(which is therefore independent of the choice of basis).

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 v

⊗ v

7−→ v

v

from V

× V

to C, linear in the first factor, conjugate linear in the second, and symmetric in the sense that

v

v

= v

v

.

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.

If V is a complex vector space, the conjugate vector space is the same space, but with the conjugate scalar multiplication:

c

v = c · v .

In these terms, a Hermitian form is a linear map from V to the conjugate of its linear dual.

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 positive definite 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 positive definite 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, because of convexity.

4.10. Proposition. If (π

, V

) and (π

, V

) are two irreducible cfd representations, the space of K-invariant Hermitian pairings of V

with V

is trivial if π

is not isomorphic to π

.

Proof. Any Hermitian pairing is a conjugate-linear map from V

to b V

. Because of irreducibility, it must be an isomorphism. But the previous result tells us that there exist conjugate-linear K-isomorphism of V

with V b

, hence a linear K-isomorphism of V

with V

.

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 bv in b V there exists a vector ϕ(bv) in V such that

u

ϕ(bv) = hbv, ui

for all u in V . The map bv 7→ ϕ(bv) is conjugate-linear. We can assign a positive definite Hermitian form on b V by the formula

u b b

b v = ϕ(bv)

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

and

= c ·

, then

(4.11) b

= c

· b

.

Given a Hermitian form on V , one can then define one on b V ⊗ V according to the formula (bv

⊗ v

)

(bv

⊗ v

) = (bv

b

b v

) · (v

v

) .

Because of (4.11) :

4.12. Lemma. This definition of Hermitian inner product on b V ⊗ V is canonical.

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 [ρ

f ](k) = [L

R

f ](k) = f (k

kk

) .

If (π, V ) is a cfd representation, the matrix coefficient associated to a pair (bv, v) is the continuous function

µ

(k) = hbv, π(k)vi .

Matrix coefficients define an equivariant map

V ⊗ V −→ C(G), bv ⊗ v 7−→ hbv, π(k)vi . b Swapping the roles of V and b V , we get also an equivariant map

V ⊗ b V −→ C(G), v ⊗ bv 7−→ hb π(k)bv, vi = hbv, π(k

)vi . We can then define an invariant pairing of these spaces by integration:

Z

K

hbv

, π

(k)v

ihbv

, π(k

)v

i .

Assume π to be irreducible, both π

= π. How does this pairing of b V ⊗ V with V ⊗ b V compare with the natural one? According to Corollary 4.7, one must be a scalar multiple of the other. More precisely:

4.13. Proposition. (Schur orthogonality) Suppose π

, π

to be two irreducible representations of K. For v

, b v

in V

, b V

Z

K

hbv

, π

(k)v

ihbv

, π(k

)v

i dk =

 



0 unless π

is isomorphic to π

1

dim(π) hbv

, v

ih b v

, v

i if π = π

= π

Proof. It remains only to determine the constant c

such that Z

K

hbv

, π(k)v

ihbv

, π(k

)v

i dk = c

hbv

, v

ih b v

, v

i . Choose a base (e

) and its dual (be

). The equation above implies that

Z

K

X

j

hbe

, π(k)e

ihbe

, π(k

)e

i dk = c

· X

j

hbe

, e

ihbe

, e

i = c

dim(π) · hbe

, e

i .

But the left hand side is the (i, ℓ) entry of π(g)π(g

) = I , so c

· dim(π) = 1.

This Proposition can be reformulated as a basic fact in harmonic analysis on K. Matrix coefficients define an equivariant map

(4.14) End( b V ) = b V ⊗ V −→ C(K) ,

and we have also the map

(4.15) C(G) −→ End( b V ), f 7−→ b π(f ) .

4.16. Corollary. The composition of (4.15) with (4.14) amounts to scalar multiplication by 1/ dim(π).

Proof. The endomorphism bπ is characterized by the equation

hb π(f )b u, ui = Z

K

f (k)hb π(k)b u, ui dk .

But if f is the matrix coefficient corresponding to bv ⊗ v this becomes the left hand side of Schur’s formula.

The representation of K × K on C(K) extends to a continuous representation of K × K on the Hilbert space L

(K) with the invariant Hilbert norm

f

f

= Z

K

f

(k)f

(k) dk .

The canonical positive definite Hermitian form on b V ⊗ V and the Hilbert norm restricted to the matrix coefficient image must be scalar multiples of each other. Explicitly:

4.17. Corollary. (Unitary Schur orthogonality) Suppose π

, π

to be two irreducible representations of K. For v

, bv

in V

, b V

Z

K

hbv

, π(k)v

ihbv

, π(k)v

i dk =

 



0 unless π

is isomorphic to π

1

dim(π) (bv

⊗ v

)

(bv

⊗ v

) if π = π

= π

This corollary might give some intuition to Lemma 4.12.

If we set the f

to be of the form u 7→ hbv, uiv

in Proposition 4.13, we deduce:

4.18. Proposition. (Trace form of Schur orthogonality) If (π, V ) is an irreducible cfd representation of K then for f

, f

in End

(V )

Z

K

trace π(k)f

trace σ(k

)f

dk =

( 1

dim(π) · trace π(f

)π(f

)

if π ∼ = σ

0 otherwise.

CHARACTERS

. Let

be the character of π, the function

TR_π

(k) = trace π(k)
.

If we set f

= f

= I in Proposition 4.18, we see that Z

K

TR_σ

(k)

(k

) dk = n 0 if σ is not isomorphic to π 1 if π ∼ = σ.

Since π(k) is unitary, the trace of π(k

is the conjugate of that of π(k). Hence:

4.19. Corollary. The characters of irreducible representations from an orthonormal set in L

(K).

Recall that for ny function f on G, f

(g) = f (g

). For each irreducible cfd representation (π, V ) define the function

ξ

= dim(π) ·

.

It will replace 1 when dealing with π. It is useful to keep in mind that

=

.

Suppose (π, U ) to be an irreducible cfd representation and V an arbitrary cfd. The ^isotypic π-component is the canonical image of the embedding of U ⊗ Hom(U, V ) into V . Every equivariant map from π factors through it.

4.20. Corollary. Suppose (π, V ) to be an irreducible cfd representation of K. Then

(a) the function ξ

lies in the centre of C(K);

(b) it is idempotent with respect to convolution;

(c) if π is irreducible then π(ξ

) = I;

(d) if π, ρ are two non-isomorphic irreducible representations of K then π(ξ

) = 0;

(e) if π is irreducible, π(ξ

) amounts to projection onto the isotypic π-component.

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(ξ

)u = dim(π) Z

K

TR_π

(k

)π(k)u dk and then continue

hb u, π(ξ

)ui = dim(π) Z

K

TR_π

(k

) hb u, π(k)ui dk

= dim(π) Z

K

trace π(k

)I

TR

(π(k)Φ

) dk

= trace Φ

= hb u, ui .

5. Representations of a compact group II. Infinite-dimensional

The principal results of this section are:

(a) every irreducible continuous representation of a compact group K is finite-dimensional;

(b) every continuous representation of K is a topological direct sum of isotypic components associated to irreducible representations;

(c) integration against ξ

is a projection onto the π-component.

The second of these implies 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, V

), let L

(K) be the subspace of its matrix coefficients, the canonical image of b V

⊗ V

in C(K) ⊆ L

(K). The first important result to be proved here is that L

(K) is the Hilbert direct sum of the subspaces L

(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 right-K-finite functions in L

(K) are dense.

Proof. Let V = L

(K), v in V . Let f be a self-adjoint Dirac function such that

L

v − vk < ε. Since K is compact, the operator L

is compact, and its eigenspaces V

for eigenvalues ω

6= 0 are finite-dimensional.

We have

L

v − vk

= kv

k

+ X

(ω

− 1)

kv

k

< ε

.

Each of the spaces V

is finite-dimensional and stable under right multiplication by K. If the set of ω

is finite, then

v − P v

= kv

k < ε. Otherwise, say ω

< 1/2 for i > n, and

v − X

i≤n

v

< 2ε .

5.2. Lemma. Any right-K-finite function in L

(K) is continuous.

Proof. Suppose F is contained in the finite-dimensional right-K-stable subspace V ⊆ L

(K). The image of

C(K) in End

(V )is closed, but by Corollary 2.4 the identity operator I is in its closure. That image therefore

contains I, so for some f in C(K) we have R

F = F . But it is simple to verify that if f lies in C(K) then R

F is continuous.

5.3. Lemma. If π is a cfd representation of K, the image of any K-equivariant map from V

to L

(K) is contained in L

(K).

Proof. By the preceding Lemma, the image of V

is contained in C(K). A simple version of Frobenius reciprocity asserts that

Hom

V, C(K)

= Hom

(V, C) = b V . But this means precisely that it is in the image of b V ⊗ V .

5.4. Proposition. If π and σ are not isomorphic the spaces L

(K) and L

(K) are orthogonal.

Proof. This is an immediate consequence of unitary Schur orthogonality.

For F in L

(K), let F

= F ∗ ξ

.

5.5. Theorem. (Peter-Weyl) For any F in L

(K) and ε > 0 there exists a finite subset Π of irreducible cfd representations of K such that

F − X

π∈Π

F

< ε .

In other words, L

(K) is the Hilbert direct sum of the spaces L

(K).

This is immediate from the preceding Lemmas.

5.6. Proposition. Let (π, V ) be any continuous representation of K, (σ, U ) an irreducible cfd representation, V

the image of the operator π(ξ

). Then:

(a) the image of any K-map from U to V is contained in V

; (b) the space V

is canonically isomorphic to U ⊗ Hom

(U, V ).

Proof. The first follows from the previous result, since any K-equivariant map from U to V commutes with ξ

. The second follows from the identification

U ⊗ Hom

(U, V ) ∼ = Hom

( b U ⊗ U, V )

where K acts on the second factor in the right-hand term. This is because of Corollary 4.20 together with the observation that V ∼ = Hom

(M

(K), V ) with K acting on M

on the right.

In particular, if C

(K) is the image of b V ⊗ V in C(K), it is the image of R

, which is also the image of convolution with dim(π)

b

. Keep in mind that

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.

THE FOURIER TRANSFORM OF CERTAIN DISTRIBUTIONS

. One can expect some sort of Fourier decom- position to hold for some elements in the linear dual of C(K). For example, if K = S and f lies in C

(K)

then the series X

f b

z

converges uniformly to f (z). Exactly what is true for general K depends on the particular group K. But the

following result is suggestive. I take it from §1 of [Kottwitz:2006].

For any k in K the associated orbital integral is the ‘distribution’

hO

, f i = Z

K

f (x

gx) dx

5.9. Proposition. For every k in K

O

= X

π

χ

(g)χ

.

More explicitly Z

K

f (x

gx) dx = X

π

χ

(g)χb

(f ) for any K-finite f on K.

This formula has no obvious meaning for functions in L

(K), because a priori it requires f to be continuous.

For specific groups K and one might expect a formula like this to hold for more general f , one for which the sum converges but is no longer finite.

Proof. The group K × K acts on C(K) according to formula ϕ 7−→ L

R

ϕ . The tensor product C(K) ⊗ C(K) therefore also acts:

[ρ

f ](k) = Z

K

Z

K

f (k

)f (k

)ϕ(k

kk

) dk

dk

.

But if we change variables k

= k

k

h the right hand side becomes Z

K

Z

K

f (k

)f (k

k

h)ϕ(h) dk

dk

. This is an integral operator with kernel

K(k, ℓ) = Z

K

f

(x)f

(k

xℓ) dx .

It is a trace class operator, and its trace is the integral of the kernel over the diagonal:

(5.10) trace ρ

= Z

K

K(k, k) dk = Z

K

Z

K

f (x)f

(y

xy) dy dx .

In the formula we wish to prove, both sides are continuous functions of g. It is therefore sufficient to see that they agree when integrated against an arbitrary continuous function on K. But this follows from (5.10) . The only completely satisfactory proof of the validity of the formula for the trace as a diagonal integral that I know of is that in [Bryslawn:1992].

LIE GROUPS

. So far, everything I have said is valid for all compact groups. Compact Lie groups are special.

5.11. Corollary. Any compact Lie group K has a faithful, continuous representation of finite dimension.

Proof. Let H be the intersection of all the kernels of finite-dimensional representations. It is closed in K. Let P

be the projection

[P

f ](k) = Z

H

f (hk) dh

onto the subspace of H-fixed functions in L

(K). The group H is normal and closed in K, and the subspace of its invariants closed in L

(K).

Suppose H 6= {1}. By Urysohn’s Lemma, there exists on K a non-negative continuous function f such that f (1) = 0, f (h) = 1 for some h in H. Consequently the open subset of f in L

(K) such that P

f − f 6= 0 is not empty. By the Peter-Weyl theorem, the space L

(K) is a direct Hilbert sum of K-stable finite-dimensional subspaces,a and these are hence dense in L

(K). Therefore there exist K-finite functions that are not constant on H, contradicting the definition of H.

We now know that H = {1}. Hence for every k 6= 1 in K there exists a representation π with π(k) 6= I.

Since K is a Lie group, there exists an open neighbourhood X of 1 which contains no subgroup other than {1}. Let Y be the complement of X in K. Since Y is compact, there exists a finite set Σ of representations π such that for every k in Y there exists some π in Σ with k / ∈ Ker(π). The kernel of Π = L

Σ

π then contains no element of Y , so it must be contained in U . It must therefore be {1}, so Π is a faithful representation of K.

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 GL

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).

Finally, I should say something about harmonic analysis. If f is in C

(K), I defined its Fourier transform to be the collection of operators bπ(f) as π varies over the set of irreducible representations of K. This does not make much sense, on the face of it. For one thing, the realization of an irreducible representation is by no means canonically defined. The only truly intrinsic object associated to a representation is its trace function on K. That realizations are not intrinsic is illustrated by rationality considerations—for example, the trace of the representation of SU

on C

takes real values, but this representation cannot be defined over R. But it does make some sense. For one thing, the image of bπ(f) in C(K) (via matrix coefficients) does not depend on a particular realization. And in many situations one is given, if not a canonical realization of irreducible representations, at least some very explicit ones, for which it becomes an interesting problem to determine matrix coefficients explicitly.

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 R

. The function F in C(Ω, V ) is said to be differentiable at x if

t→0

lim

F (x + tv) − F (x)

t = [∂

F ](x)

exists for all v in R

. It is said to be continuously differentiable in Ω if [∂

f ](x) is a continuous function on Ω × R

, which case it is a linear function of v and defines a continuous differential dF in C(G, Hom(R

, 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

t→0

lim

F (g exp(tX)) − F (g)

t = [R

F ](g)

exist for all g in G and X in g, and then that [R

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.

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(t

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) = t

∆(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 (β(t)) − f (0)

t = f (α(t) · exp(t

X(t))) − f (0) t

= f (α(t) · exp(t

X(t))) − f (α(t))

t + f (α(t)) − f (0)

t .

Therefore it suffices now to show that lim

f (α(t) · exp(t

X(t))) − f (α(t))

t = 0 .

Since

f (α(t) · exp(t

X(t))) − f (α(t))

t = t · f (α(t) · exp(t

X(t))) − f (α(t)) t

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−→ [R

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

([0, ε), V ) for some ε > 0, and

F

(s) = [R

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

F (s) − F (0) = Z

0

F

(u) du = s · Z

0

F

(θs) dθ .

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 R

[R

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 R

F is continuously differentiable of

order n. It is ^smooth if it is differentiable of all orders. Let C

(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 R

F : 6.3. Lemma. Suppose F to be a continuously differentiable function on G with values in V and bva continuous linear function on V . Let

ϕ(g) = hbv, F (g)i .

The function ϕ(g) is a continuously differentiable scalar function on G, and [R

ϕ](g) =

b v, [R

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

= {v ∈ V | g 7→ π(g)v is in C

(G, V )}

This space, assigned the norms π(X)v

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 C

(G) and v in V the vector π(f )v is differentiable of order m, and π(X)π(f ) = π(L

f ) .

for X in U

(g).

Proof. Only the last requires confirmation. We have π(f ) =

Z

G

f (x)π(x) dx π exp(tX)

π(f ) = Z

G

f (x)π exp(tX)x
dx

= Z

G

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

. 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 C

(G) in End(V ) is closed and by Corollary 2.4 therefore contains I. If π(f ) = I, then v = π(f )v for every v, which is therefore smooth.

Remark. If (π, V ) is a continuous representation, we have seen that the space spanned by vectors π(f )v

with f in C

(G) is contained in the space of smooth vectors. It is a remarkable result found in [Dixmier-

Malliavin:1978] that if V is Fr´echet these two spaces are the same.

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 repre- sentations 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 C

(G) or L

(G). In short, they are basic objects in the theory.

If (π, V ) is any continuous representation of G, let V

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

there exists a function f in C

(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(C

(G))π(ξ) is a closed subalgebra of the finite-dimensional algebra End

(U ). But if we choose a Dirac sequence ϕ

in C

(G), then π(ξ)ϕ

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

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 finite- dimensional 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 represen- tations of SL

(R) associated to its action on P = P

(R). The spaces of analytic functions, smooth functions, and locally L

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 sl

, 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

is an admissible representation of (g, K).

In practice, in the theory of representations of the reductive group G, one works with admissible represen-

tations 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

that of the group? One problem that the representation of the Lie algebra alone can’t handle at all is the be- haviour 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 = SL

(R), and let P = P

(R).

The group G acts on the right this space through the action

v =

 x y



7−→ g

v .

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

(P) of real analytic functions, and L

(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 ε

of K where

ε:

 c −s

s c



7−→ c + is .

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

to its closure in V is a bijection between (g, K)-stable subspaces of V

and closed G-stable subspaces of V ;

(c) the map taking (π, V ) to V

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, and is discussed in [Kashiwara-Schmid:1994]. The other perhaps the most natural, in that it incorporates features most useful in applications. The original ideas were in joint work by Nolan Wallach and myself. This is explained in [Casselman:1990]. A different approach to the same construction is to be found in [Bernstein-Kr¨otz: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(C

)Φ = λΦ for some λ, where C

is the Casimir element of U (k). Since C

commutes with all of Z(g), π(XC

)Φ = π(C

)π(X)Φ = λπ(X)Φ for all X in Z(g). Since Φ is Z(g)-finite

Y

π(C) − µ

Φ = 0

for some set of scalars µ

, where now C is the Casimir element of U (g). Let ω = C + 2C

, and let

Φ

= Y

1

π(C) − µ

Φ (0 ≤ k ≤ n)

and in particular Φ

= Φ. Thus

π(C) − µ

Φ

= Φ

π(C) − µ

Φ

= 0 leading to

π(ω) − (2λ + µ

)

Φ

= Φ

π(ω) − (2λ + µ

))Φ

= 0 .

Since ω is an elliptic operator on G with analytic coefficients, each Φ

is analytic.

7.5. Corollary. If (π, V ) is an admissible representation of G, v a K-finite vector and bv in b V any continuous linear functional on V , then the matrix coefficient hπ(g)v, bvi 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 Harish- Chandra 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 GL

, where it is disguised in terms of an action of the Hecke algebra.

9. Appendix. Tensors and homomorphisms

My goal in this appendix is to formulate a number of results in linear algebra that are useful in representation theory, but that do not depend on properties of G or the dimension of V . In addition, I’ll prove a few that suggest what to do when these things do matter.

Throughout, G will be an arbitrary group and k an arbitrary field of characteristic 0. Representations of G will be on vector spaces over k. If V is a vector space over k, b V will be its k-linear dual. The space C(G) will be that of functions on G with values in k, C

(G) the subspace of functions with finite support.

• Suppose (π, V ) to be a representation of G. Then G maps into End(V ) through the map g 7→ π(g). If G × G acts on G according to the rule

g 7−→ g

gg

and on End(V ) according to the recipe

F 7−→ π(g

)F π(g

)

this map is equivariant. This motivates the definition of the representation of G × G on Hom

(V

, V

) when the (π

, V

) are representations of G:

(9.1) Hom

(g

, g

): F 7−→ π

(g

) F π

(g

) .

• Suppose (π, V ) to be a representation of G. There is a canonical map from V ⊗ b V to End(V ):

F

: w 7−→ hbv, wiv .

9.2. Lemma. The map taking w to F

is a G × G-equivariant isomorphism of V ⊗ b V with the ideal End(V ) of End(V ) consisting of maps of finite rank.

• Suppose F to be in End(V ). It may then be factored as F : V −→ U −→ V , and by duality gives rise to its transpose

t

F : b V −→ b U −→ b V , which is also of finite rank. For Φ in End( b V ) the expression

(9.3) hΦ, F i = trace (Φ

F )

is therefore well defined, and thus determines a map from End( b V ) to the dual of End(V ). It is G × G- equivariant. (Note: the pairing

trace (Φ

F ) of End(V ) and End(V ) is not invariant.)

There is similarly a canonical equivariant map

End( b V ) −→ dual of End(V ) .

• Given bv in b V , v in V , the associated matrix coefficient is the function

hbv, π(g)vi