Last revised 9:28 p.m. June 5, 2019

Continuous representations

Bill CasselmanUniversity of British Columbiacass@math.ubc.ca

This essay contains somewhat drymaterial, mostly useful inmotivating eventually a certain crucial but at firstsight somewhat technical transition from representations of groups to representations of Lie algebras. Partsof it will also be used in the theory of automorphic forms. I have made some effort to reduce everything towell known facts inmeasure theory and topology. The standard reference for thematerial here is [Borel:1972].I have also used [Weil:1965].

Contents

1. Continuous representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Representation of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. Interlude: Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54. Representations of a compact group I. Finite­dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65. Representations of a compact group II. Infinite­dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . 126. Smooth representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157. Representations of G and of (g, K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189. Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209. Appendix. Tensors and homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

This topic necessarily involves rather general topological vector spaces (as opposed to, say, only Hilbertspaces). In this paper, a TVS will always be assumed to be locally convex and Hausdorff. But as a rule, alltopological vector spaces occurring in representation theory are also quasi­complete. Quasi­completenessis at first sight a rather technical condition, but in fact very practical. A quasi­complete TVS V is one forwhich integrals of V ­valued functions are well defined, and in which derivatives can be characterized in aparticularly useful way. Nearly all TVS encountered in the real world are quasi­complete, and it is rare thatone has to thinkmuchabout it. For example, Frechet spaces andLF spaces are quasi­complete, and so are theirduals. The standard references on quasi­complete 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 Gungor forpointing 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 of1, and which is a countable union of compact subsets. Fix on G a right invariant Haar measure.

A representation of G on a finite­dimensional space V is simply a continuous homomorphism from G toGL(V ). But infinite­dimensional representations requiremore care. Suppose V to be aHausdorff topologicalvector space. A continuous representation of G on V is a homomorphism π from G to the group of lineartransformations 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 naiveone if V has finite dimension.

Continuous representations 2

The defining condition means that whenever we are given g in G, v in V , and a semi­norm ρ of V then wecan find a neighbourhood X of 1 in G and a semi­norm σ such that

∥∥π(xg)(v + u) − π(g)v∥∥

ρ< ε

whenever x lies in X and ‖u‖σ < 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 semi­norm of V , there exists a semi­norm σ such that

∥∥π(x)v∥∥

ρ≤ ‖v‖σ (x ∈ X) .

The first condition, in our circumstances, means

(a’) Suppose v in V . For every continuous semi­norm ρ and ε > 0 there exists a neighbourhood X of 1in G such that ‖π(g)v − v‖ρ < 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 insome sense invariant under G, and this is often transparently true. In fact, under a mild restriction on V (thatit 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, supposeg, v, and ρ given. Then

∥∥π(xg)(v + u) − π(g)v∥∥

ρ=

∥∥π(xg)v − π(g)v + π(x)π(g)u∥∥

ρ

≤∥∥π(x)v1 − v1

∥∥ρ

+∥∥π(x)π(g)u

∥∥ρ

(v1 = π(g)v) .

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

‖π(x)v1 − v1‖ρ < ε/2 ,

and then by (b) we can find σ such that

‖π(x)π(g)u‖ρ ≤ ‖u‖σ < ε/2

for x in X , if ‖u‖σ < ε/2.

The space C(G, V ) is that of all continuous functions on G with values in V . Its topology is defined bysemi­norms

‖F‖Ω,ρ = supg∈Ω

∥∥F (g)∥∥

ρ

where Ω is a compact subset of G and ρ a continuous semi­norm on V .

Given a continuous representation (π, V ) there exists a canonical embedding of V into C(G, V ), the space ofall 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 embeddingof 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.

Continuous representations 3

Proof. Condition (a) means that the image of V lies in C(G, V ). Condition (b) means that the map from Vto C(G, V ) is continuous.

If (π, V ) is a continuous representation of G, then a prioriwe have two topologies on V , the original one andthat 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 offunctions on itself and on quotient spaces H\G, as well as on certain spaces of induced representations. Thegroup G acts on these by means of the right regular , and in some cases left regular , representations:

RgF (x) := F (xg), LgF (x) := F (g−1x) .

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 bespaces 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 unionof a countable number of compact subsets. The TVS C(X) is that of all continuous functions on X . For eachcompact subset Ω in X define the semi­norm

‖f‖Ω = supx∈Ω

|f(x)| .

These make X into a Frechet space.

The TVS Cc(X) is that of all continuous functions on X whose support is a relatively compact open subsetof X . If Y is a relatively compact open subset, let Cc(Y ) be the subspace of all f in C(Y ) that vanish onthe boundary of Y . Such a function may be extended uniquely to a function in C(X) vanishing outside Y .Define the semi­norm

‖f‖Y = supx∈Y

|f(x)| .

This makes Cc(X) into an LF space.

Both spaces are quasi­complete, 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 Cc(H\G) are continuous.

Proofs should be evident.

1.5. Example. Suppose G to be unimodular, H a closed subgroup. Let δH be the modulus character of H ,and let Ωc (for real c) be the space of all continuous functions F : G → C such that F (hg) = δc

H(h)f(g) forall h in H , g in G. and which are of compact support modulo H . There exists on Ωc a positive, continuous,G­invariant linear functional, which I’ll express as integration. Thismay be embedded in a space of integrable

elements Ω1. Let Ω1/2 be the space of functions on G such that f(hg) = δ1/2H (h)f(g). This implies that f2

lies in Ω1.

1.6. Proposition. The right­regular representations of G on Ω1(H\G) and Ω1/2(H\G) are continuous.

If (π, V ) is a continuous representation of G then its dual (π, V ) on the continuous linear dual of V is thatdefined by the condition ⟨

π(g)v, π(g)v⟩

= 〈v, v〉 .

In other words, ⟨π(g)v, v

⟩=

⟨v, π(g)−1v

⟩.

The continuous linear dual of a TVS may be assigned the weak topology, with norms ‖v‖v = |〈v, v〉| for v inV . It is straightforward to prove:

Continuous representations 4

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

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

Φv,v: g 7−→ 〈v, π(g)v〉

on G. If π is finite­dimensional and (ei) is a basis of V with dual basis ei, then 〈ej , ei〉 is in fact a matrix entry.

2. Representation of measures

In this section, the TVS V will be assumed to be quasi­complete.

Let Mc(G) be the space of bounded measures on G of compact support. It can be identified with the spaceof continuous linear functionals on the space C(G, C) of all continuous functions on G. If µ1 and µ2 are twomeasures in Mc(G) their convolution is the measure defined by the formula

〈µ1 ∗ µ2, f〉 =

∫

G×G

f(xy) dµ1dµ2 .

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 beembedded in Mc(G): f 7→ f dx. The definition of convolution of measures then agrees with the formula forthe convolution of two functions in Cc(G):

[f1 ∗ f2](y) =

∫

G

f1(x)f2(x−1y) dx

Because of the assumption about quasi­completeness, the space of the continuous representation (π, V )becomes a module over Mc(G) in accordance with the formula

π(µ)v =

∫

G

π(g)v dµ .

This integral is characterized uniquely by the condition that

〈v, π(µ)v〉 =⟨v,

∫

G

π(g)v dµ⟩

=

∫

G

⟨v, π(g)v

⟩dµ

for every continuous linear function v on V .

Suppose Ω to be the support of µ. Condition (b) of Proposition 1.1 implies that for every semi­norm ρ of Vthere exists a semi­norm σ such that

‖π(g)v‖ρ ≤ ‖v‖σ

for all g in Ω. Then

(2.1)

∥∥π(µ)v∥∥

ρ≤

∫

Ω

∥∥π(g)v∥∥

ρ|dµ|

≤

(∫

Ω

|dµ|

)‖v‖σ .

Hence:

2.2. Proposition. For every µ in Mc(G) the operator π(µ) is continuous.

The map µ 7−→ π(µ) is a ring homomorphism, since the composition of two operators π(µ1)π(µ2) can becalculated easily to agree with π(µ1 ∗ µ2). For the left regular representation, Lµf = µ ∗ f .

If f(x) is a function in Cc(G) then f(x) dx is a measure of compact support. I define a Dirac function on Gto be a function f in C∞c (G, R) satisfying the conditions

Continuous representations 5

(a) f(g) = f(g−1);(b) f(g) ≥ 0 for all 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. From (2.1) :

2.3. Proposition. Suppose f to be a Dirac function with compact support Ω, v in V . Let ρ be a semi­norm onV , and σ a semi­norm such that ‖π(x)v‖ρ ≤ ‖v‖σ for x in Ω. Then

‖π(f)‖ρ ≤ ‖v‖σ .

Proof Apply (2.1) .

2.4. Corollary. If vi is a finite set in V , ρ a semi­norm on V , ε > 0, and fn a Dirac sequence on G, and ρa semi­norm on V then for some N ∥∥π(fn)vi − vi

∥∥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 lookat a familiar case. It is a simple example, but definitely prototypical. Let S be the multiplicative group ofcomplex numbers z with |z| = 1. Fix as measure on S

dθ

2π=

dz

2πiz.

For each m in Z, the map taking z to zm is a differentiable one­dimensional representation of S. The classicaltheory of Fourier series asserts that every f in L2(S) can be expressed as a sum (called here a Fourier series)

f =∑

n

cnzn

in the sense that the associated finite sums

fN =∑

|n|≤N

cnzn

converge to f in L2(S). The sum is orthogonal, and consequently the coefficients are given by the formula

cn =1

2πi

∫

S

z−nf(z)dz

z.

In other words, the space L2(S) is the Hilbert direct sum of the spaces spanned by the characters zn. Thisinduces 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 cn rapidly decreasing as a function of n—for each k the sequence|n|kcn 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 quasi­complete TVS, let Πm be the projection operator

Πnv =1

2πi

∫

|z|=1

z−nπ(z)vdz

z.

The image is the subspace Vn of v in V such that π(z)v = znv for all z. Of course it may happen that Vn = 0.Let Vfin be the algebraic direct sum of the spaces Vn. This may also be characterized as the S­finite vectors inV , those contained in some finite­dimensional S­stable subspace.

3.1. Lemma. The subspace Vfin is dense in V .

Proof. According to the Hahn­Banach theorem, it suffices to show that any continuous linear functional Fon V that vanishes on all Vm vanishes everywhere. Using a Dirac sequence to approximate δ1, it must beshown that for any ϕ in C∞(S) and v in V we have 〈F, π(ϕ)v〉 = 0. But ϕ will be the limit of finite sums offunctions

∑cnzn, so (2.1) and the assumption on f , imply that 〈F, π(ϕ)v〉 = 0.

Continuous representations 6

4. Representations of a compact group I. Finite-dimensiona l

Representations of compact groups are models for all of representation theory and are also required as anecessary preliminary to much of it. In this section, let K be an arbitrary compact group with a countablebasis of neighbourhoods of 1. I do not assume it, at least at first, to be a Lie group. For example, the additivegroup of p­adic integers Zp = lim

←−Z/(pn) is allowable.

Some aspects of representations of compact groups are simpler than most of the material in this essay, largelybecause finite­dimensional vector spaces are well understood. I make no claim to originality. The expositionpretty much follows a straightforward line, but because it is rather long I have divided it into two majorpieces, mostly separating finite dimensions from infinite dimensions, and each of these in turn into smallerpieces. The theory of continuousfinite dimensional representations of compact groups differs from the theoryof representations of finite groups primarily only in so far as integration replaces finite sums, so this first partis especially straightforward.

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

4.1. Proposition. The group K is unimodular.

This means that every right­invariant Haar measure is also left­invariant.

Proof. A left­invariant Haar measure dℓx is unique up to constants. For k in K , dℓxk is also left­invariant,hence dℓxk = δ(k) dℓx 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.

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

SEMI-SIMPLICITY. A continuous finite­dimensional (cfd ) representation of K on the space V is simply acontinuous 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 =

∫

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 ifπ(1)v = v. The kernel of π(1) is a closed K­stable summand of V and

V = V K ⊕ Kerπ(1) .

In other words, π(1) is the unique K­equivariant projection onto the subspace V K of K­fixed vectors, whichis a closed subspace of V .

Proof. We have

v =

∫

K

(I − π(k)

)v dk +

∫

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 Kerπ(1) is the closure of the spans of the π(k)v − v.

4.3. Corollary. If0 −→ U −→ V −→ W −→ 0

is an exact sequence of continuous representations of K then

0 −→ UK −→ V K −→ WK −→ 0

Continuous representations 7

is also exact.

! If W has finite dimension the map from V K to WK splits continuously, as does anycontinuous map from V onto a finite­dimensional vector space. But this is not generally trueotherwise.

This decomposition is a special case of a more general result. One major point of this section is to find ageneralization 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 K­representations, 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−→

∫

K

π(k) f(π(k−1)w

)dk

is a K­equivariant 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 finite­dimensional, but it has auseful 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.

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

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.

HOM. There are several ways to construct new representations from old ones. The twoprincipal ones involvetensor products and spaces of homomorphisms.

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(k1) f π1(k−12 )

(b) f 7−→ π2(k2) f π1(k−11 )

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

For either choice, the space HomK(V1, V2) of K­equivariant linear maps from V1 to V2 is the subspace ofinvariants 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 V , for which

〈π(k)v, v〉 = 〈v, π(k−1)v〉 .

Continuous representations 8

In other words, π(k) = tπ(k−1). Duality is an involution, since if V is finite­dimensional the canonical mapfrom V to the dual of its dual is an isomorphism. Already the case of K = Z/(3) will show that the dual ofa 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 −→ HomC(W, U) −→ HomC(W, V ) −→ HomC(W, W ) −→ 0

of K × K representations. According to Corollary 4.3, 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.

TENSOR PRODUCTS. If (π1, V1) and (π2, V2) are two cfd representations, the tensor product representationof 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 irreducible cfd representations of Ki, thenπ1 ⊗ 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 K­stable subspace of V1 ⊗ V2, and U 6= 0 an irreducibleK1­stable subspace of V . As a representation of K1, V is a direct sum of copies of V1, so that by Corollary4.7 the representation on U is isomorphic to that on V1, and we may as well assume that U = V1. and thecanonical map from U ⊗ Hom(U, V ) to V is an isomorphism. For the same reasons, the canonical map fromU ⊗HomK(U, W ) toW is an isomorphism. For the same reasons again V is isomorphic toU ⊗HomK(U, V ).But the embedding of W into V induces an embedding of HomK(U, W ) into HomK(U, V ). The second isisomorphic to V2 as a representation of K2, and since V2 is irreducible this embedding is an isomorphism.Hence W = V .

Similarly, if V is any irreducible representation of K1 ×K2 and U 6= 0 is any irreducible K1­stable subspace,then V is isomorphic to U ⊗ Hom(U, V ), and Hom(U, V ) is an irreducible representation of K2.

If we choose bases (ei) and (fj) for U and V , as well as the dual bases (ei) and (fi), the matrix Ej,i of Tei,fj

has zero entries everywhere except a 1 in column i and row j. What this means is that the map taking thematrix (Fj,i) in Hom(U, V ) to ∑

Fj,i · ei ⊗ ej

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

∑i ei ⊗ ei (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 seriousdifficulty that occurs when V has infinite dimension. One has to be especially careful for topological vectorspaces. Only for nuclear vector spaces is there a reasonable identification of a (topological) tensor productwith 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, and symmetric in the sense that

v1 • v2 = v2 • v1 .

Continuous representations 9

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 convexopen 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 scalarmultiplication:

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 ofK­invariantHermitian formson V has dimension one over R. The subset of positive definite ones is a real ray.

The last assertionmeans that there exists at least one positive definiteHermitian formonV that isK­invariant,and all others are positive multiples of it. Consequently, every cfd representation is unitary .

Proof. The only non­trivial point to verify is that there always exists a positive definite invariant Hermitianform on V . But if one starts with an arbitrary positive definite form on V its K­average will still be positivedefinite, because of convexity.

4.10. Proposition. If (π1, V1) and (π2, V2) are two irreducible cfd representations, the space of K­invariantHermitian pairings of V1 with V2 is trivial if π1 is not isomorphic to π2.

Proof. Any Hermitian pairing is a conjugate­linear map from V1 to V2. Because of irreducibility, it must bean isomorphism. But the previous result tells us that there exist conjugate­linear K­isomorphism of V2 withV2, hence a linear K­isomorphism of V1 with V2.

Suppose (π, V ) to be an irreducible cfd representation, u • v an invariant positive definite Hermitian form onV . Since it is non­degenerate, for each v in V there exists a vector ϕ(v) in V such that

u •ϕ(v) = 〈v, u〉

for all u in V . The map v 7→ ϕ(v) is conjugate­linear. We can assign a positive definite Hermitian form on Vby the formula

u • v = ϕ(v) •ϕ(u)

which is conjugate­linear in the second factor. How does this depend on the choice of Hermitian innerproduct? If we are given two Hermitian inner products •1 and •2 = c · •1, then

(4.11) •2 = c−1 · •1 .

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

(v1 ⊗ v1) • (v2 ⊗ v2) = (v1 • v2) · (v1 • v2) .

Because of (4.11) :

4.12. Lemma. This definition of Hermitian inner product on 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 wayto interpret this in a moment.

MATRIX COEFFICIENTS. The product K × K acts on C(K), the space of continuous functions on K , by

[ρk1,k2f ](k) = [Lk1

Rk2f ](k) = f(k−1

1 kk2) .

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

µv⊗v(k) = 〈v, π(k)v〉 .

Continuous representations 10

Matrix coefficients define an equivariant map

V ⊗ V −→ C(G), v ⊗ v 7−→ 〈v, π(k)v〉 .

Swapping the roles of V and V , we get also an equivariant map

V ⊗ V −→ C(G), v ⊗ v 7−→ 〈π(k)v, v〉 = 〈v, π(k−1)v〉 .

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

∫

K

〈v1, π1(k)v1〉〈v2, π(k−1)v2〉 .

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

4.13. Proposition. (Schur orthogonality) Suppose π1, π2 to be two irreducible representations of K . For vi,vi in Vi, Vi

∫

K

〈v1, π1(k)v1〉〈v2, π(k−1)v2〉 dk =

0 unless π1 is isomorphic to π2

1dim(π)

〈v1, v2〉〈v2, v1〉 if π = π1 = π2

Proof. It remains only to determine the constant cπ such that

∫

K

〈v1, π(k)v1〉〈v2, π(k−1)v2〉 dk = cπ 〈v1, v2〉〈v2, v1〉 .

Choose a base (ei) and its dual (ei). The equation above implies that

∫

K

∑

j

〈ei, π(k)ej〉〈ej , π(k−1)eℓ〉 dk = cπ ·∑

j

〈ei, eℓ〉〈ej , ej〉 = cπ dim(π) · 〈ei, eℓ〉 .

But the left hand side is the (i, ℓ) entry of π(g)π(g−1) = I , so cπ ·dim(π) = 1.

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

(4.14) End(V ) = V ⊗ V −→ C(K) ,

and we have also the map

(4.15) C(G) −→ End(V ), f 7−→ π(f) .

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

Proof. The endomorphism π is characterized by the equation

〈π(f)u, u〉 =

∫

K

f(k)〈π(k)u, u〉 dk .

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

Continuous representations 11

The representation ofK ×K on C(K) extends to a continuous representation of K ×K on the Hilbert spaceL2(K) with the invariant Hilbert norm

f1 • f2 =

∫

K

f1(k)f2(k) dk .

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

4.17. Corollary. (Unitary Schur orthogonality) Suppose π1, π2 to be two irreducible representations ofK . Forvi, vi in Vi, Vi

∫

K

〈v1, π(k)v1〉〈v2, π(k)v2〉 dk =

0 unless π1 is isomorphic to π2

1dim(π)

(v1 ⊗ v1) • (v2 ⊗ v2) if π = π1 = π2

This corollary might give some intuition to Lemma 4.12.

If we set the fi to be of the form u 7→ 〈v, u〉vi in Proposition 4.13, we deduce:

4.18. Proposition. (Trace form of Schur orthogonality) If (π, V ) is an irreducible cfd representation of K thenfor f1, f2 in EndC(V )

∫

K

trace(π(k)f1

)trace

(σ(k−1)f2

)dk =

1

dim(π)· trace

(π(f1)π(f2)

)if π ∼= σ

0 otherwise.

CHARACTERS. Let TRπ be the character of π, the function

TRπ(k) = trace(π(k)

).

If we set f1 = f2 = I in Proposition 4.18, we see that

∫

K

TRσ(k)TRπ(k−1) dk =

0 if σ is not isomorphic to π1 if π ∼= σ.

Since π(k) is unitary, the trace of π(k−1 is the conjugate of that of π(k). Hence:

4.19. Corollary. The characters of irreducible representations from an orthonormal set in L2(K).

Recall that for ny function f on G, f∨(g) = f(g−1). For each irreducible cfd representation (π, V ) define thefunction

ξπ = dim(π) ·TRπ .

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

π = TRπ .

Suppose (π, U) to be an irreducible cfd representation and V an arbitrary cfd. The isotypic π­componentis the canonical image of the embedding of U ⊗ Hom(U, V ) into V . Every equivariant map from π factorsthrough it.

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

Continuous representations 12

(a) the function ξπ lies in the centre of C(K);(b) it is idempotent with respect to convolution;(c) if π is irreducible then π(ξπ) = I ;(d) if π, ρ are two non­isomorphic irreducible representations of K then π(ξρ) = 0;(e) if π is irreducible, π(ξπ) amounts to projection onto the isotypic π­component.

Proof. The first is true of any conjugation­invariant function in C(K). The second follows from Schurorthogonality. For the last we start with the formula

π(ξπ)u = dim(π)

∫

K

TRπ(k−1)π(k)u dk

and then continue

〈u, π(ξπ)u〉 = dim(π)

∫

K

TRπ(k−1) 〈u, π(k)u〉 dk

= dim(π)

∫

K

trace(π(k−1)I

)TR(π(k)Φu⊗u) dk

= traceΦu⊗u

= 〈u, u〉 .

5. Representations of a compact group II. Infinite-dimensio nal

The principal results of this section are:

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

irreducible representations;(c) integration against ξπ is a projection onto the π­component.

The second of these implies that the subspace of K­finite vectors is dense in any continuous representationof 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 Vπ ⊗ Vπ in C(K) ⊆ L2(K). The first important result to be proved here is that L2(K) isthe Hilbert direct sum of the subspaces L2

π(K). This will take place in several steps.

A K­finite vector in a representation of K is one that is contained in a finite­dimensional K­stable subspace.

5.1. Lemma. The right­K­finite functions in L2(K) are dense.

Proof. Let V = L2(K), v in V . Let f be a self­adjoint Dirac function such that∥∥Lfv − v‖ < ε. Since K is

compact, the operator Lf is compact, and its eigenspaces Vf,ωifor eigenvalues ωi 6= 0 are finite­dimensional.

We have ∥∥Lfv − v‖2 = ‖v0‖2 +

∑(ωi − 1)2‖vi‖

2 < ε2 .

Each of the spaces Vf,ωiis finite­dimensional and stable under right multiplication by K . If the set of ωi is

finite, then∥∥v −

∑vi

∥∥ = ‖v0‖ < ε. Otherwise, say ωi < 1/2 for i > n, and

∥∥∥v −∑

i≤nvi

∥∥∥ < 2ε .

5.2. Lemma. Any right­K­finite function in L2(K) is continuous.

Proof. Suppose F is contained in the finite­dimensional right­K­stable subspace V ⊆ L2(K). The image ofC(K) in EndC(V )is closed, but by Corollary 2.4 the identity operator I is in its closure. That image therefore

Continuous representations 13

contains I , so for some f in C(K) we have RfF = F . But it is simple to verify that if f lies in C(K) thenRfF is continuous.

5.3. Lemma. If π is a cfd representation of K , the image of any K­equivariant map from Vπ to L2(K) iscontained in L2

π(K).

Proof. By the preceding Lemma, the image of Vπ is contained in C(K). A simple version of Frobeniusreciprocity asserts that

HomK

(V, C(K)

)= HomC(V, C) = V .

But this means precisely that it is in the image of V ⊗ V .

5.4. Proposition. If π and σ are not isomorphic the spaces L2π(K) and L2

σ(K) are orthogonal.

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

For F in L2(K), let Fπ = F ∗ ξπ .

5.5. Theorem. (Peter­Weyl) For any F in L2(K) and ε > 0 there exists a finite subset Π of irreducible cfdrepresentations of K such that ∥∥∥F −

∑π∈Π

Fπ

∥∥∥ < ε .

In other words, L2(K) is the Hilbert direct sum of the spaces L2π(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,Vσ the image of the operator π(ξσ). Then:

(a) the image of any K­map 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 K­equivariant map from U to V commutes withξσ . The second follows from the identification

U ⊗ HomK(U, V ) ∼= HomK(U ⊗ U, V )

where K acts on the second factor in the right­hand term. This is because of Corollary 4.20 together with theobservation that V ∼= HomK(Mc(K), V ) with K acting on Mc on the right.

In particular, if Cσ(K) is the image of V ⊗ V in C(K), it is the image of Rξπ, which is also the image of

convolution with dim(π) TRπ. Keep in mind that TRπ itself is in this space.

5.7. Proposition. In any continuous representation of K the K­finite vectors are dense.

Proof. By the Hahn­Banach theorem, it suffices to show that any continuous linear function f on V thatvanishes on each K­stable finite­dimensional subspace vanishes on every v in V . Apply Peter­Weyl to thecontinuous function taking k to 〈f, π(k)v〉.

5.8. Corollary. Any irreducible continuous representation of K is finite­dimensional.

THE FOURIER TRANSFORM OF CERTAIN DISTRIBUTIONS. One can expect some sort of Fourier decom­position to hold for some elements in the linear dual of C(K). For example, if K = S and f lies in C1(K)then the series ∑

fnzn

converges uniformly to f(z). Exactly what is true for general K depends on the particular group K . But thefollowing result is suggestive. I take it from §1 of [Kottwitz:2006].

Continuous representations 14

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

〈Ok, f〉 =

∫

K

f(x−1gx) dx

5.9. Proposition. For every k in K

Ok =∑

πχπ(g)χπ .

More explicitly ∫

K

f(x−1gx) dx =∑

πχπ(g)χ

π(f)

for any K­finite f on K .

This formula has no obvious meaning for functions in L2(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 thesum converges but is no longer finite.

Proof. The group K × K acts on C(K) according to formula

ϕ 7−→ Lk1Rk2

ϕ .

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

[ρf1⊗f2f ](k) =

∫

K

∫

K

f(k1)f(k2)ϕ(k−11 kk2) dk1 dk2 .

But if we change variables k2 = k−1k1h the right hand side becomes

∫

K

∫

K

f(k1)f(k−1k1h)ϕ(h) dk1 dk2 .

This is an integral operator with kernel

K(k, ℓ) =

∫

K

f1(x)f2(k−1xℓ) dx .

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

(5.10) trace ρf1⊗f2=

∫

K

K(k, k) dk =

∫

K

∫

K

f(x)f2(y−1xy) dy dx .

In the formula we wish to prove, both sides are continuous functions of g. It is therefore sufficient to see thatthey 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 thatI 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 finite­dimensional representations. It is closed in K . LetPH be the projection

[PHf ](k) =

∫

H

f(hk) dh

Continuous representations 15

onto the subspace of H­fixed functions in L2(K). The group H is normal and closed in K , and the subspaceof its invariants closed in L2(K).

Suppose H 6= 1. By Urysohn’s Lemma, there exists on K a non­negative continuous function f such thatf(1) = 0, f(h) = 1 for some h in H . Consequently the open subset of f in L2(K) such that PHf − f 6= 0 isnot empty. By the Peter­Weyl theorem, the space L2(K) is a direct Hilbert sum ofK­stable finite­dimensionalsubspaces,a and these are hence dense inL2(K). Therefore there existK­finite functions that are not constanton 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 than1. Let Y be the complement of X in K . Since Y is compact, there exists a finite set Σ of representations πsuch that for every k in Y there exists some π in Σ with k /∈ Ker(π). The kernel of Π =

⊕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 ofK .

If K is a compact Lie group, then the Fourier decomposition of a smooth function f looks much like that of asmooth function on the unit circle, but I’ll not prove that here. There are two approaches, one concerned withthe 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 connectedcompact 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 areseveralways to see this, but one very satisfactory approach is to see that it is in fact an algebraic group, definedas 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 Cc(K), I defined its Fourier transform tobe the collection of operators π(f) as π varies over the set of irreducible representations of K . This does notmake much sense, on the face of it. For one thing, the realization of an irreducible representation is by nomeans canonically defined. The only truly intrinsic object associated to a representation is its trace functionon K . That realizations are not intrinsic is illustrated by rationality considerations—for example, the traceof the representation of SU2 on C2 takes real values, but this representation cannot be defined over R. But itdoes make some sense. For one thing, the image of π(f) in C(K) (via matrix coefficients) does not dependon a particular realization. And in many situations one is given, if not a canonical realization of irreduciblerepresentations, at least some very explicit ones, for which it becomes an interesting problem to determinematrix coefficients explicitly.

6. Smooth representations

In this section, let G be a Lie group.

Suppose V to be a quasi­complete TVS. There are two ways to define continuously differentiable functionson 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 tothe case of an open subset Ω in Rn. The function F in C(Ω, V ) is said to be differentiable at x if

limt→0

F (x + tv) − F (x)

t= [∂vF ](x)

exists for all v in Rn. It is said to be continuously differentiable in Ω if [∂vf ](x) is a continuous function onΩ×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 quasi­complete space, and often reducedirectly to well known results about the usual calculus by applying the Hahn­Banach theorem.

The second definition depends on the structure of G. It stipulates that

limt→0

F (g exp(tX)) − F (g)

t= [RXF ](g)

Continuous representations 16

exist for all g in G and X in g, and then that [RXF ](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 Vbe quasi­complete. The literature usually skips over the question while implicitly assuming it to be true, orattempts a proof carelessly.

Proof. Verifying one implication is much like verifying the other, so I’ll just do one. The proof reduces to twosteps. The first is a basic fact about the geometry of a neighbourhood of the identity ofG: If α(t) and β(t) aretwo 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(t2X(t)). This is because the exponential map is a smoothdiffeomorphism 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(β(t)) − f(0)

t=

f(α(t) ·exp(t2X(t))) − f(0)

t

=f(α(t) ·exp(t2X(t))) − f(α(t))

t+

f(α(t)) − f(0)

t.

Therefore it suffices now to show that

limt→0

f(α(t) ·exp(t2X(t))) − f(α(t))

t= 0 .

Sincef(α(t) ·exp(t2X(t))) − f(α(t))

t= t ·

f(α(t) ·exp(t2X(t))) − f(α(t))

t2

this will follow from:

6.2. Lemma. Locally on G we havef(g ·exp(sX) − f(g) = sϕ

where ϕ lies in the convex hull of the image of the map

[0, s] −→ V, u 7−→ [RXf ](g ·exp(uX)) .

Proof of the Lemma. Fix g, and consider the map

F (s) = f(g ·exp(sX)) .

By assumption, it is in C1([0, ε), V ) for some ε > 0, and

F ′(s) = [RXf ](g ·exp(sX))

since exp((s + h)X) = exp(sX) · exp(hX). According to the Fundamental Lemma for V ­valued functionsof one variable

F (s) − F (0) =

∫ s

0

F ′(u) du = s ·

∫ 1

0

F ′(θs) dθ .

The integral makes sense because V is assumed to be quasi­complete. A basic property of the integral is thatit 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 secondorder, in which case RX [RY F ] is a continuous function for all X and Y in g. Proceeding by induction, it issaid to be differentiable of order n + 1 if for each X in g the function RXF is continuously differentiable of

Continuous representations 17

order n. It is smooth if it is differentiable of all orders. Let Cm(G, V ) be the space of all functions on G withvalues in V that are continuously differentiable of orders ≤ m, C∞(G, V ) be the space of smooth functionsfrom G to V .

The following is immediate, and, because of the Hahn­Banach Theorem, characterizes the derivative RXF :

6.3. Lemma. Suppose F to be a continuously differentiable function onGwith values inV and v a continuouslinear function on V . Let

ϕ(g) = 〈v, F (g)〉 .

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

[RXϕ](g) =⟨v, [RXF ](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 continuousfunction on G with values in V . The vector v is said to be smooth if this map is smooth. The representationis 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 Cm(G, V )

This space, assigned the norms∥∥π(X)v

∥∥ρfor X of order at most m in U(g), is a quasi­complete 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 Cm

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

π(X)π(f) = π(LXf) .

for X in Um(g).

Proof. Only the last requires confirmation. We have

π(f) =

∫

G

f(x)π(x) dx

π(exp(tX)

)π(f) =

∫

G

f(x)π(exp(tX)x

)dx

=

∫

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 subspaceof 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 haveπ([X, Y ]

)=

[π(X), π(Y )

],

on the subspace V (2). The map from g to continuous endomorphisms of V∞ associated to a continuousrepresentation of G is a representation of the Lie algebra g.

6.6. Proposition. Any continuous finite­dimensional representation of a Lie group is smooth.

Proof. If (π, V ) is continuous, the image of C∞c (G) in End(V ) is closed and by Corollary 2.4 thereforecontains 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)vwith f in C∞c (G) is contained in the space of smooth vectors. It is a remarkable result found in [Dixmier­Malliavin:1978] that if V is Frechet these two spaces are the same.

Continuous representations 18

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

In this section, let G be the group of R­rational points on a Zariski­connected reductive group defined overR, K a maximal compact subgroup of G. According to Cartan’s fixed point theorem, all choices of K areconjugate to each other by an element of the connected component of G. Suppose A to be the group ofR­rational points on a maximal split torus of G. According to §14 of [Borel­Tits:1972], A meets all connectedcomponents ofG. This implies the groupK meets all of the connected components ofG. (This is a special caseof an older result of [Mostow:1955] about arbitrary Lie groupswith a finite numberof 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 K­isotypic component has finitedimension. 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 variousdecompositions of natural representations of G, such as that of G × G on C∞c (G) or L2(G). In short, theyare basic objects in the theory.

If (π, V ) is any continuous representation of G, let V(K) be the subspace of K­finite vectors—that is to say,contained in a K­stable finite­dimensional subspace of V .

7.1. Proposition. If (π, V is an admissible continuous representation of G, then for every v in V(K) thereexists a function f in C∞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 K­isotypiccomponent U , then the image of π(ξ)π(C∞c (G))π(ξ) is a closed subalgebra of the finite­dimensional algebraEndC(U). But if we choose a Dirac sequence ϕi in C∞c (G), then π(ξ)ϕiπ(ξ) will converge to the identity ofU . 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 isstable 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 K­spaces g ⊗ U → π(g)U .

7.3. Corollary. Every K­finite vector in an admissible representation (π, V ) of G is also Z(g)­finite. If π isirreducible, 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 infinite­dimensional representations ofG that don’texist for finite­dimensional ones. The most serious arise because many different continuous representationsof G are all in some sense equivalent, even though analytically of a very different nature. Some are quitecomplicated, 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 samebeast. What they all have in common is the same underlying space of K­finite functions. This, happily, is arepresentation 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’lldenote 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 finite­dimensional representationsof 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 anadmissible 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 bejustified. 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 problemis addressed by including the action of K , which meets all connected components.

Before taking up the justification of the definition ofHarish­Chandra modules, letme point outwhy replacingthe 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 =

[xy

]7−→ g−1v .

It therefore acts on the left on various spaces of functions onP. Among these are the spaceC(P) of continuousfunctions, the space C∞(P) of smooth functions, the space Cω(P) of real analytic functions, and L2,loc(P) oflocally square­integrable functions. All of them contain exactly the same space of K­finite functions, whichas a representation of K is the direct sum of characters ε2n of K where

ε:

[c −ss c

]7−→ c + is .

As you can imagine, a great deal of simplification takes place if one looks just at this last space—analysis willbe replaced by algebra.

There are a number of facts that justify the switch from representations of G to those of (g, K). I state themhere briefly:

(a) every Harish­Chandra module is the restriction to K­finite vectors of some continuous representationof G;

(b) if (π, V ) is an admissible representation of G, the map taking U ⊆ V(K) to its closure in V is a bijectionbetween (g, K)­stable subspaces of V(K) and closed G­stable subspaces of V ;

(c) the map taking (π, V ) to V(K) is a bijection between irreducible unitary representations of G andirreducible Harish­Chandra modules with a positive definite metric invariant with respect to (g, K);

(d) there exist exact functorial assignments of smooth representations of G to Harish­Chandra 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 Harish­Chandra modules in a functorialmanner. One is in a sense minimal, and is discussed in [Kashiwara­Schmid:1994]. The other perhaps themost natural, in that it incorporates features most useful in applications. The original ideas were in jointwork by NolanWallach and myself. This is explained in [Casselman:1990]. A different approach to the sameconstruction is to be found in [Bernstein­Krotz:2010].

I now sketch the proof of (b).

7.4. Proposition. If Φ is a distribution on G which is left­ or right­K­finite as well as Z(g)­finite, then it is areal­analytic function on G.

Proof. Let π be the left or right regular representation of G, depending on the assumption regarding K .

SinceΦ is the sumof itsK­isotypic components, onemay aswell assume thatΦ 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 withall 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∏

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∏

1

(π(C) − µi

)Φ (0 ≤ k ≤ n)

and in particular Φ0 = Φ. Thus (π(C) − µk

)Φk−1 = Φk(

π(C) − µn

)Φn−1 = 0

leading to (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 K­finite vector and v in V any continuouslinear functional on V , then the matrix coefficient 〈π(g)v, v〉 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 Vand closed G stable subspaces of V .

Proof. The difficult point, and the one that convinces most strongly that representations of (g, K) are areasonable 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 G­stable. By the Hahn­Banach theorem, in order to show that W isG­stable, it suffices to show that ifF is a linear function on V that vanishes onW , thenF

(π(g)w

)= 0 for allw

in W . Because K meets all components ofG and W is K­stable, it suffices to show this for g in the connectedcomponent of G. But according to Corollary 7.5 this is an analytic function of g, and therefore it suffices toshow that the coefficients of its Taylor series at 1 vanish. However, these coefficients are determined by theconstants F (π(X)w), which vanish by assumption.

8. Realization

I call a Harish­Chandra module (π, v) realizable if it is the representation of (g, K) on the K­finite vectorsin admissible continuous representation of G. As I have said in the previous section, all Harish­Chandramodules 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 Harish­Chandra, which asserted that ever irreducibleadmissible representation of(g, K) could be realized as subquotient of some principal series. A later andmore interesting proof of this was found by [Beilinson­Bernstein:1982]. I myself might have been the first theprove that every finitely generated (g, K)­modulewas realizable. This proof showed that formal solutions ofthe differential equations satisfied by matrix coefficients were in fact converging solutions, but this proof wasnever published. Other proofs have been found since. The existence of matrix coefficients is presumably themost important fact about realizable representations,a because the asymptotic behaviour ofmatrix coefficientsis an extremely important part of the theory. Jacquet and Langlands essentially postulated this in the theirbook on GL2, where it is disguised in terms of an action of the Hecke algebra.

Continuous representations 21

9. Appendix. Tensors and homomorphisms

My goal in this appendix is to formulate a number of results in linear algebra that are useful in representationtheory, but that do not depend on properties of G or the dimension of V . In addition, I’ll prove a few thatsuggest 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, V will be its k­linear dual. The space C(G) willbe that of functions on G with values in k, Cc(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). IfG × G acts on G according to the rule

g 7−→ g1gg−12

and on End(V ) according to the recipe

F 7−→ π(g1)F π(g2)−1

this map is equivariant. This motivates the definition of the representation of G×G on HomF (V1, V2) whenthe (πi, Vi) are representations of G:

(9.1) Homπ1,π2(g1, g2): F 7−→ π2(g1)F π1(g

−12 ) .

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

Fv×v: w 7−→ 〈v, w〉v .

9.2. Lemma. The map taking w to Fw is a G × G­equivariant isomorphism of V ⊗ 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

tF : V −→ U −→ V ,

which is also of finite rank. For Φ in End(V ) the expression

(9.3) 〈Φ, F 〉 = trace (ΦtF )

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

trace (ΦF )

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

There is similarly a canonical equivariant map

End(V ) −→ dual of End(V ) .

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

〈v, π(g)v〉

Continuous representations 22

in C(G). The map V ⊗ V → C(G) is G × G­equivariant.

The following diagram is commutative:

V ⊗ V −→ C(G)↓ ↑

End(V ) −→ dual of End(V )

9.4. Proposition. The map taking v ⊗ v to µv⊗v is a K × K­equivariant map from V ⊗ V to C(G).

If π is irreducible, then according to Proposition 4.8 this is an embedding.

If we swap V and V , we get a variant:

µv⊗v(k) = 〈v, π(k)v, v〉 = 〈v, π(k−1)v〉 .

There is another way to define certain matrix coefficients. Define a map τ from EndC(V ) to C(K):

τ : f 7−→ [k 7−→ trace(π(k−1)f

)] .

9.5. Proposition. The diagram

V ⊗ V

C(G)

EndC(V )

F

µ

τ

is a commutative diagram of K × K­representations.

Proof. Because the trace of u 7→ 〈v, u〉v is 〈v, v〉.

10. References

1. James Arthur, David Ellwood, and Robert Kottwitz (editors), Harmonic analysis, the trace formula, andShimura varieties ,Clay Mathematical Proceedings 4, American Mathematical Society, 2006.

2. A. Beilinson and J. Bernstein, ‘A generalization of Casselman’s submodule theorem’, pp. 35–52 in Repre-sentation Theory of Reductive Groups , edited by P.C. Trombi, Birkhauser, Boston, 1982.

3. J. Bernstein andB.Krotz, ‘SmoothFrechet globalizationsofHarish­Chandramodules’, preprintarXiv.org:0812.1684,2010.

4. Armand Borel, Representations de groupes localement compacts , Lecture Notes in Mathematics 276,Springer, 1972.

5. Nicholas Bourbaki, Integration I , Springer. Translated from the original French edition, which is dated1959–1965.

6. ——, Topological vector spaces , Springer, translated from the French original dated 1981.

7. Christopher Brislawn, ‘Traceable integral kernels on countably generatedmeasure spaces’, Pacific Journalof Mathematics 150 (1991), 229–240.

8. Jacques Dixmier and Paul Malliavin, ‘Factorisations de fonctions et de vecteurs indefiniment differen­tiables’, Bulletin des Sciences Mathematiques 102 (1978), 305–330. 1978

9. Robert E. Kottwitz, ‘Harmonic analysis on reductive p­adic groups and Lie algebras’, pp. 393–522 in[Arthur et al.:2006].

Continuous representations 23

10. Daniel Mostow, ‘Self­adjoint groups’, Annals of Mathematics 62 (1955), 44–55.

11. Michael Reed and Barry Simon, Methods of mathematical physics I. Functional analysis , AcademicPress, 1972.

12. Masaaki Kashiwara and Wilfried Schmid, ‘Quasi­equivariant D­modules, equivariant derived category,and representations of reductive Lie groups’, pp. 457–488, in Lie Theory and Geometry, in Honor of BertramKostant , Progress in Mathematics 123, BirkhÃ⁄user, 1994.

13. Andre Weil, L’int egration dans les groupes topologiques , Hermann, 1965.