Infinite DimensionalRepresentations of Real Reductive

Groups, Spring 20161. Locally Convex topological Vector Spaces

Definition (1). A vector space V is a locally convex topological vectorspace if it has a topology generated by translations of balanced absorbentconvex sets.

• C convex, ∀x, y ∈ C, 0 ≤ t ≤ 1, (1− t)x+ ty ∈ C.• C balanced, ∀x ∈ C, |λ| ≤ 1, λx ∈ C.• C absorbent ∪t>0tC = V.

Definition (2). A seminorm on V is a map p : V −→ R, such that

(1) p(v) ≥ 0,∀v ∈ V,(2) p(x+ y) ≤ p(x) + p(y),(3) p(λx) = |λ|p(v).

A vector space V is called locally convex if its topolgy is defined by afamily of seminorms pα.

For such a space, Hausdorff ⇐⇒ pα(v) = 0 ∀α, implies v = 0.

Theorem. The two definitions are equivalent.

See https://en.wikipedia.org/wiki/Locally convex topological vector spacefor a more ample explanation and references.

We will mostly deal with Banach and Hilbert spaces.

2. Differentiable and Analytic maps

Let f : U ⊂ Rn −→ V where V is a Banach space. Define

∂

∂tif(x) = lim

t→0

f(x+ ti)− f(x)

ti

Definition. f is called Ck with k ≥ 0 in the usual way, if all partialsup to order k exist and are continuous; similarly for C∞.f is called analytic, if there are vm ∈ V with m = (m1, . . . ,mn) a

multi-index such that for any x0 ∈ U there is a small ball such that

f(x) =∑

vm(x− x0)m such that∑m

|vm| · |x− x0|m <∞.

The notation is (x− x0)m := (x1 − x01)m1 . . . (xn − x0

n)mn.1

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3. Representations

3.1. The groups G are at least Haudorff topological groups, locallycompact, with countable base and countable at ∞. They will be realLie groups, so in fact analytic manifolds. Connected groups admitHaar measures, i.e. left invariant Borel measures dlg. A Haar measureis unique up to a constant factor. Similarly for right invariant Haarmeasures drg. The relation between them is given by the modulus func-tion δ(g). For a connected Lie group, the Haar measure is given by tbeleft or right invariant form.

REFERENCE: F. Warner, Lie groups.

3.2. Finite Dimensional Representations. Let Φ : G −→ H bea continuous group homomorphism between two real Lie groups. Inparticular, a continuous representation (π, V ) of a (connected for sim-plicity) real Lie group G where dimV < ∞ is of this form, π : G −→GL(V ). In the general Lie group homomorphism case, Φ is real analytic(by the Baker-Campbell-Hausdorff theorem). There is a commutativediagram

g h

G H

dΦ

expG expH

Φ

So we can study finite dimensional representations Φ by studying therepresentations of the Lie algebras g, or the universal enveloping algebraU(g)..

Theorem. Suppose φ : g −→ h is a Lie algebra homomorphism and Gis simply connected. Then there is a Lie homomorphism Φ : G −→ Hsuch that dΦ = φ.

This does not exactly work for infinite dimensional representations.For example the left and right regular representations of a group G onL2(G) or some of the Lp−spaces will not differentiate:

Lgf(x) := f(g−1x), Rgf(x) := f(xg).

The model on how to study such representations are Fourier Analysison L2(S1) and L2(R).

3.3. Infinite Dimensional Representations.

Definition. A group homomorphism π : G −→ V into a completelocally convex topological space will be called

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(1) continuous Π : G× V −→ V is continuous(2) strongly continuoous ∀v ∈ V, g → π(g)v is continuous(3) weakly continuous ∀v ∈ V, ` ∈ V ∗, g → `(π(g)v) is continu-

ous.(4) norm continuous for V a Banach space, π : G −→ Aut(V )

where Aut(V ) has the norm topology given by ||T || := sup|v|≤1 |Tv|.

V ∗ is the continuous dual of V.It is straightforward that

(4) =⇒ (1) =⇒ (2) =⇒ (3).

The fact that

(1)⇐⇒ (2)⇐⇒ (3)

is also true, but somewhat harder.We will be concentrating on the case of V a Banach space and π

norm continuous.REFERENCE G. Warner vol. I.

3.4.

Definition. For G a Lie group, we say v ∈ V ∞ if the map g → π(g)vis C∞. Similarly v ∈ V ω means g 7→ π(g)v is analytic.

If X ∈ g := Lie(G), then we define

π(X)v :=d

dt

∣∣∣∣t=0

π(exp tX)v.

Identify g with the space of left-invariant vector spaces. Then the uni-versal enveloping algebra U(g) is the space of left-invariant differentialoperators; V ∞ is a representation of U(g). V ∞ is also a representationof G:

π(X)π(g) = π(g)π(Adg−1X)v.

Let X1, . . . , Xn be a basis of g. In this discussion we have used thefact that the maps

X −→ g exp(X),

(t1, . . . tn) −→ g exp(t1X1) · · · · · exp(tnXn)

give charts for the Lie group. Similarly for multiplying with g on theright.

Theorem. V ω ⊂ V ∞ ⊂ V are dense G−invariant subspaces.

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3.5. V ∞ ⊂ V is dense. Let f ∈ Cc(G). Then we can define

π(f)v :=

∫G

f(g)π(g)v dlg.

This is well defined and bounded:∣∣∣∣∫G

f(g)π(g)v dlg

∣∣∣∣ ≤ supg∈suppf |π(g)v|∫G

|f(g)| dlg.

Lemma. If f ∈ C∞c (G), then π(f)v ∈ V ∞.

Proof. This follows from differentiating inside the integral and the factthat

π(g)π(f) = π(Lgf),

where by definition Lgf(x) := f(g−1x) (and Rgf(x) := f(xg)).

Corollary. V ∞ is G−stable.

Definition. The Garding space V g is the span of all the π(f)v withf ∈ C∞c (G) and v ∈ V .

It is clear that V g ⊂ V ∞ and V g is G−stable.

Proposition. G has an approximate identity i.e. a sequence offunctions fn ∈ C∞c (G) such that

(1) suppfn ⊂ Un where Un are a sequence of locally compact neigh-borhoods of the identity satisfying ∩Un = e,

(2) fn ≥ 0 and∫Gfn(g)dg = 1.

We can then check that π(fn)v → v :

|π(fn)v − v| =∣∣∣∣∫G

fn(x)(π(x)v − v) dx

∣∣∣∣ ≤ ∫G

|π(g)v − v|fn(g) dg ≤

≤ supg∈Un|π(g)v − v|

∫G

fn(g) dg,

and the continuity of π completes the proof.Since π(fn)v ∈ V g ⊂ V ∞, the density statement follows.

3.6. Convolutions. If f1, f2 ∈ Cc(G) the convolution f1 ∗f2 i definedas

f1 ∗ f2(g) =

∫G

f1(x)f2(x−1g) dx =

∫G

f1(gy−1)f2(y) dy.

Note that L : G −→ End[Cc(G)] given by Lg(f)(x) := f(g−1x) is arepresentation, and this formula is simply L(f1)f2. Also dx and dy areleft-invariant measures only.

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Proposition.

(1) π(f1) π(f2) = π(f1 ∗ f2).(2)

∣∣∫Gf1 ∗ f2(g) dg

∣∣ ≤ ∫G|f1∗f2(g)| dg ≤

∫G|f1(x)| dx·

∫G|f2(y)| dy.

So there are extensions to L1.

3.7. Note on integration. Let G be a Hausdorff, locally compacttopological group, countable at ∞ countable base etc, and V a locallyconvex complete Hausdorff topological space. Let Cc(G, V ) be contiu-ous compact supported functions from G to V.

Theorem. There exists a continuous map called integration,∫: Cc(G, V ) −→ V, f 7→

∫G

f(x) dx

satisfying

(1) p((∫

Gf(x) dx

)≤∫Gp(f(x)) dx for any seminorm p,

(2) if L : V1 −→ L2 is continuous linear, then

L

(∫G

f(x) d1x

)=

∫G

L(f(x) d2x.

When G is unimodular,

∫G

f(x) dx =

∫G

f(x−1) dx, and when it is

compact, we can normalize so that∫Gv dx = v, ∀v ∈ V .

REFERENCE Lang ...

3.8. V ω ⊂ V is dense. This was first introduced and proved byHarish-Chandra in TAMS I and II for reductive groups only (see latersections for the notions of reductive, semisimple groups etc). It wasproved for arbitrary Lie groups by Nelson, Annals of Math 1959. Theproof is similar in spirit to the previous section; use an approximationof the identity with analytic functions. This requires the heat kernel.

ForG = Rn, the heat kernel is the functionK(t, x, y) := 1(4πt)n/2

e−|x−y|2/(4t).

It satisfies

(1) ∂K∂t

(t, x, y) = ∆xK(t, x, y), ∆x =∑∂2x.

(2) limt→0+

∫Rn K(t, x, y)f(y) dy = f(x) for any function f ∈ C∞c (Rn).

In other words it is a fundamental solution.

The paper of Nelson considers the heat kernel for an arbitrary (con-nected) Lie group.

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4. Reductive Groups

Let G be an arbitrary (connected) real Lie group with Lie algebra g.

Theorem (Ado). Every finite dimensional Lie algebra g has a faithfulfinite dimensional representation ρ : g −→ gl(V ).

Theorem (Levi). Any finite dimensional Lie algebra g is the semidirectproduct of a semisimple Lie algebra and a solvable ideal (called thesolvable radical).

Definition. A real Lie group is called reductive if

(1) g := Lie(G) is reductive. This means g = c + [g, g] where thesecond term is semisimple. Semisimple means a direct sum ofsimple ideals, and simple means dimension > 1 and no nontriv-ial ideals.

(2) G has a maximal compact subgroup K ⊂ G given by the fixedpoints of an involution θ. Furthermore

a: Adg : g −→ g is inner,b: G has a faithful finite dimensional representation (not al-

ways assumed)

Properties include (some of them are consequences, others are as-sumptions)

(1) g = k + s, the ±1 eigenspaces of θ. Write θ for the involutionon the group as well as the algebra. Then exp : s −→ G is anisomorphism on a submanifold S, such that K × s −→ G givenby (k,X) 7→ k expX is a diffeomorphism.

(2) Let h ⊂ g be a Cartan subalgebra (a maximal abelian subalge-bra formed of ad−semisimple elements. Then H := CentG(h)is abelian. Every such subalgebra/subgroup is conjugate underK to one which is θ−stable.

Main Example: G = GL(n,R) ⊃ K = O(n), θ(g) := Tg−1.

A large class is obtained as follows. Let Gc ⊂ GL(n,C) be an affinealgebraic group defined over R, ie. a group given by the zeroes ofpolynomial equations on gl(n,C). We say Gc is symmetric, if it isstabilized by g 7→ Tg.

A real reductive group G is a finite cover of an open subgroup ofthe real points of such a Gc.

The class of real points of a reductive connected linear algebraicgroup form a subclass with special properties.Examples: G = GL(n,R), SL(n,R), U(p, q), O(p, q), So(p, q),Sp(2n,R), Sp(2p, 2q) and also U∗(2n) and O∗(2n). So(p, q)e the iden-tity component of So(p, q) is in this class, but not the real points of a

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reductive ... group. Its representation theory is somewhat less reason-able. See Helgason for a list of all simple Lie algebras and their realforms.

REFERENCE Helgason 1962 or later edition.

4.1. SL(2,R). This is a typical case of the real points of a semisimplelinear algebraic group. The irreducible finite dimensional representa-

tions are parametrized by n ∈ N. If g =

[a bc d

], with det g = ad−bc =

1,

V = spanxkyn−2k, πn(g)xkyn−2k = (ax+ cy)k(bx+ dy)n−2k.

The maximal compact subgroup is

r(θ) =

[cos θ − sin θsin θ cos θ

], and it

has eigenvalues ei(n−2k)θ.

Since π1(SL(2,R)) = π1(SO(2)) = Z, there is a nontrivial cover foreach n ∈ N. The universal cover is realized as

˜SL(2,R) = (g, φ) : eφ(z) = cz + dwhere φ : P+ −→ C is a holomorphic function from the upper half-plane P+.

The n−fold cover is

˜SL(2,R)n = (g, φ) : φ(z)n = cz + d.Note that SL(2,R) acts on P+ by conformal transformations:

g · z :az + b

cz + d.

The stabilizer of i is K, and the stabilizer of ∞ is

B = AN =

[et 00 e−t

]·[

1 x0 1

],

the action is transitive, and you can use it to prove that G = B ·K.

Let φn : ˜Sl(2,R)n −→ SL(2,R) be the cover map. The ˜Sl(2,R) arenot linear groups for n > 1. Any finite dimensional representation πdifferentiates to a finite dimensional representation dπ of the Lie al-gebra sl(2,R). It complexifies to a representation of sl(2,C). Sincethe Lie group of sl(2,C) is SL(2,C) which is connected simply con-nected, it exponentiates to a representation π of SL(2,C) which wecan restrict to SL(2,R); it follows that dπ = dπ. Via the exponential

map there are open sets (isomorphic under φn) e ∈ U ⊂ ˜Sl(2,R)n ande ∈ U ⊂ Sl(2,R) so that π = π φn. But then this holds for the

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entire groups because they are generated by open sets containing theidentities. So π cannot be genuine.

More explicitly one can to examine the action of K in more detail us-

ing the realization of the cover. Let K, B be the inverse images. We can

replace B by a copy isomorphic to B by choosing φb(z) := e−t/n. Then

G = B · K, and kerφn ⊂ K. Both K and K are isomorphic to S1, butφn is raising to power n. The inverse image of r(θ) is (r(θ), φθ) where φθis determined by its value at i, satisfying φθ(i)

n = i sin θ + cos θ = eiθ.To see this, let φ0(i) := 1, and deform continuously in θ. The exponen-tial map is expG : θ 7→ (r(nθ), φθ). The points (r(2iπ/n), φ2iπ/n) arethe kernel of φn. The values on any irreducible representation can becomputed from the Lie algebra via the exponential map; they are theidentity, so π is trivial on the kernel and factors to SL(2,R). it followsthat this holds for any finite dimensional representation,

4.2. The Metaplectic Group. Let Mp(2,R) := ˜SL(2,R)2. TheMetaplectic/Oscillator/Segal-Shale-Weil representation of the Lie alge-bra is as follows. Let

H := spanpn(x)e−x2/2

be the vector space spanned by the Hermite functions. Then

E =

[0 10 0

]←→ i

2mx2 ,

H =

[1 00 −1

]←→ x∂x +

1

2,

F =

[0 01 0

]←→ i

2∂2x,

defines a representation of the Lie algebra. It does not exponentiate to

Sl(2,R). However there is an inner product 〈f1, f2〉 :=

∫Rf1(x)f2(x) dx

for which sl(2,R) acts by skew symmetric matrices, ie. they satisfy

〈X · f1, f2〉+ 〈f1, X · f2〉 = 0.

It follows that this representation exponentiates to a representation ofa group, not SL(2,R), but Mp(2,R) acting on the completion of H,which is L2(R). The group action can be written down for the groupB of upper triangular matrices, but is rather complicated otherwise.[

et 00 e−t

]· f(x) := et/2f(etx)

[1 t0 1

]· f(x) := eit

2/2f(x).

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The action of the maximal compact group is given by integral operatorswith some constants that are a bit difficult to deal with. One of themis the usual Fourier transform:

eπ/2(E−F ) ←→ FT

This representation appears in Mathematical Physics, and in Numbertheory via the Theta-function. It is easier to study the representationon the Lie algebra.

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5. Representations of Compact Lie Groups

5.1. Compact Lie Groups. The Lie algebra decomposes k = c+ kssthe central and semisimple part. The Lie algebra admits an invariantnegative definite form B(X, Y ) which coincides with the Cartan-Killingform on kss. If K is a connected semisimple simply connected Lie group,then the universal cover is also compact. In particular the simplyconnected cover of a compact connected Lie group fits in a sequence

1 −→ κ −→ (R+)n × K −→ K −→ 1

with κ a discrete subgroup.We will restrict attention to connected semisimple groups at some

point; the modifications for a general group follow from the structureabove in a straightforward manner.

5.2. Structure of Compact Lie Groups. Assume the group K isconnected semisimple; the general case is not much harder. Let k bethe Lie algebra. The Cartan-Killing form B(X, Y ) := tr(adX adY ) isnegative definite. Let t ⊂ k be aCartan subalgebra . This means t is amaximal abelian subalgebra; it is automatic that adX is semisimple forany X ∈ t (in fact for any X ∈ k). Let T ⊂ K be the Cartan subgroupcorresponding to t. In this case T = CK(t) as well. Let g := kC. Thisis a semisimple Lie algebra, and h := (t)C is a Cartan subalgebra (amaximal abelian subalgebra formed of ad−semisimple elements). TheCartan subgroup is denoted H. We write h = t + a := t + it.

Proposition. There exists a connected group G with Lie algebra g suchthat K ⊂ G.

This follows from the fact that K has a faithful finite dimensionalrepresentation; a consequence of the Peter-Weyl theorem in the nextsection.

REFERENCE Helgason, Brocker-tomDieck.Let gα := X ∈ g : [h,X] = α(h)X ∀h ∈ h where α : h −→ C is a

linear functional. When gα 6= 0, α is called a root; the set of all rootsis denoted ∆(g, h).Structure of Semisimple Lie Algebras

(1) dim gα ≤ 1, [gα, gβ] ⊂ gα+β.(2) B |a abbreviated ( , ), is positive definite. Let hα be such that

B(h, hα) = α(h). Denote by α∨ := 2α/(α, α). The reflection as-sociated to α is sα(h) := h−α(h)hα∨ . Let W := NG(h)/CG(h) =NK(a)/CK(a). This is a finite group, generated by the sα.

(3) The set h′ := h ∈ a : α(h) 6= 0 is a finite union of opensets called chambers. They are in 1 − 1 correspondence with

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W . A choice C of chambers gives rise to a positive root systemP . Then ∆ = P ∪−P. Each psotive root system contains a setof simple roots Π ⊂ P, the roots in P which cannot be writtenas integer sums of roots in P.

The weights are the span of λ ∈ h∗ which take integer valueson the hα∨ . The fundamental weights $α are the dual function-als to the hα∨ with α ∈ Π.

(4) Choose 0 6= Xα ∈ gα. There is a choice such that eachXα, hα∨ , X−α is isomorphic to an sl(2,C), denoted sl(2,C)α.By a theorem of Weyl, there is a choice of Xα such that (Xα −X−α), i(Xα +X−α) ∈ k. This amounts to saying that if Nα,β aredefined by [Xα, Xβ] = Nα,βXα+β, then there is a choice suchthat Nα,β are integers satisfying Nα,β = −N−β,−α.

REFERENCE Helgason

Theorem. There is a 1− 1 correspondence between irreducible finitedimensional representations of connected simply connected K and

Λ ∈ h∗ : Λ(hα∨) ∈ NAny such Λ is a linear combination of $α with nonnegative integercoefficients.

Let VΛ be the finite dimensional irreducible representation associatedto Λ. Let ρ := 1

2

∑α∈P α. Then

dimVΛ =

∏α∈P (Λ + ρ, α∨)∏

α∈P (ρ, α∨).

The operator Ω ∈ U(k) essentially acts by the scalar ||Λ + ρ||2 − ||ρ||2.

REFERENCES Dixmier, JacobsonEXAMPLE: SU(2) ⊂ Sl(2,C), SU(m) ⊂ Sl(m,C).

5.3. Peter-Weyl Theorem. Let (ρ, V ) be a finite dimensional repre-sentation. Then (ρ∗, V ∗) is also a representation, where V ∗ is the lineardual space, and ρ∗(g)(`)(v) := `(ρ(g−1)v). The hermitian dual (ρh, V h)is the representation with the same formula, but V h is the complexspace of conjugate linear maps. A nondegenerate invariant hermitianproduct can be thought of as a choice of K−equivariant automorphism

ι : V −→ V h.

Every linear map f : V −→ W induces a map fh : W h −→ V h viafh(λ)(v) := λ(f(v)). In particular, ιh = ι is the skew-hermitian aspectof the hermitian form. Assume (V, 〈 , 〉) and (W, 〈 , 〉) are hermitianspaces. Then HomC[V,W ]h ∼= Hom[W h, V h] ∼= Hom[W,V ] inherits anaction of K×K, and an invariant hermitian form 〈A,B〉 := tr(ABh).

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If the hermitian products are positive definite, so is the ensuing oneon HomC[V, V ]. In particular, if (π, V ) is unitary, (π∗, V ∗) does toobecause,

V ∗ = Hom[V,C]

We obtain a K ×K−equivariant unitary map

Hom[V, V ] −→ C(K), A 7→ fA(g) := tr(A π(g)).

A matrix entry is a function of the form fA(g) := tr(A π(g)). Themore standard way is to fix an invariant inner product on (π, V ), andto write fx1,y1(g) := 〈π(g)x1, y1〉. This identification has to do with thefollowing fact. Let V conj be the vector space V with action of C givenby multiplication by the complex conjugate,

λ ·conj v := λ · v.V conj identifies with V ∗ with the usual complex multiplication via theinner product. Complex multiplication in V h and V ∗ is the usual mul-tiplication coming from V.

Theorem (Schur orthogonality relations). Let (ρ1, V1) and (ρ2, V2) beirreducible finite dimensional modules with invariant inner products〈 , 〉1,2, and let x1, y1 ∈ V1 and x2, y2 ∈ V2. Then∫K

〈π1(k)x1, y1〉·〈π1(k)x2, y2〉 dk =

0 if V1 6' V2,

1dimV

, 〈x1, x2〉 · 〈y1, y2〉 if V1 ' V2

.

In particular, if V1, V2 are inequivalent,

∫tr(πi(k)) · tr(πj(k)) dk = δi,j.

Furthermore, let φi(k) := 1dimVi

tr(πi(k)). Then φi ∗ φj = δi,jφi. Moregeneral, if fv,w is a matrix entry, from Vj, then φi ∗ fv,w = δi,jfv,w.

Theorem (Peter-Weyl). The image of ⊕V ∈KHomC[V, V ] is dense inC(K) in the L∞ and L2−topology. The individual spaces are orthogo-nal. The image coincides with the space of K−finite vectors.

Definition. Let (π, V ) be a continuous representation on a locally con-vex topological space. The space of K−finite vectors is

V K := v ∈ V | dim π(K)v <∞.The Lie algebra k acts on V K .

Corollary. Let (π, V ) be a representation on a complete locally convexspace. Then V K ⊂ V is dense.

If V is irreducible then dimV <∞.Proof. Recall fn the approximate identity. Then π(fn)v → v. Everyfn is approximated by K−finite functions.

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Theorem. Assume K is a connected compact Lie group and (π, V )is a Banach space representation. Then the Fourier series of every vconverges absolutely to v.

Example: G = S1. If f ∈ C∞(S1), then the Fourier coefficientsatisfies

an(f) =

∫ 2π

0

e−inxf(x) dx =1

n

∫ 2π

0

e−inxf ′(x) dx = . . .

So there is a constant Ck > 0 depending on fk(x) such that

|an| ≤Cknk.

For a general group, the Fourier series of v ∈ V is defined as follows.Let (ξ, E) be a finite dimensional irreducible representation of K withcharacter χE. The Schur orthogonality relations show that

φE ∗ φE = φE.

So π(φE) is a projector onto the space

spanT (f)T∈HomK [E,V ],f∈E.

The Fourier series of v is∑π(φE)v.

The following facts are used in the proof:

(1) The irreducible representations of K are parametrized by a lat-tice in a vector space of dimension dimK. Actually it is a cone.

(2) Let X1, . . . , Xn be an orthonormal basis of the Lie algebra. Theoperator Ω = 1 −

∑X2i ∈ U(k) is central and acts by a scalar

c(ξ) > 0 on E which grows like a polynomial. For any integers >> 0 the series∑

ξ∈K

c(ξ)−s dim(Eξ)2 <∞.

This implies that the Fourier series of v converges to a v0 ∈ V, and thatall the Fourier components of v − v0 are zero. The key inequality is

|π(φE)v| =∣∣∣∣∫K

φE(k)π(k) dk

∣∣∣∣ ≤ c(ξ)−s∫K

|φE(k)| · |π(Ωsk)v| dk.

Then |φE(k)| ≤ (dimE)2, and the rest of the integral is bounded bya constant that only depends on v. (You may need another φE usingφE ∗ φE = φE in the formula to replace v by the differentiable π(φE)von the right side of the formula).

Remains to show that if v ∈ V satisfies π(φE)v = 0 for all E ∈ K,then v = 0. This follows from the fact that π(f)v = 0 for any K−finitefunction, and then by the Peter-Weyl theorem, π(f)v = 0 for any

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f ∈ C(K). The existence of the approximate identity completes theproof.

REFERENCES Brocker-tomDieck

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6. Representations on Banach and Hilbert spaces

6.1. Let (π, V ) be a continuous (in the operator norm mostly) rep-resentation of a reductive group G. We define V K to be the space ofanalytic K−finit vectors. V K defined earlier consists of analytic vec-tors for K, but it is not clear they are analytic for V. The previoussections show that g acts on V K , and every vector v ∈ V has a Fourierseries which converges absolutely to v. Furthermore using the projec-tions π(φE),

V K = ⊕Vξ;the Vξ are called the ξ−isotypic components of V. Every vector v ∈ Vξis contained in a finite dimensional representation of K which in turnis a direct sum of irreducible representations isomorphic to Vξ. In thecase of a Hilbert space, the Vξ are orthogonal.

Proposition. Assume G is connected. LetW ⊂ V K be a U(g)−invariantsubspace. Then W := Cl(W) is a G−invariant subspace of V, andWK =W .

Proof. Suppose there is x0 ∈ G and v′ ∈ W such that v := π(x0)v′ /∈ W.By the Hahn-Banach theorem there is a continuous functional ` suchthat `(W ) = 0 but `(v) = 1. Let

f(g) := `(π(g)v′).

Then f is analytic, and π(u)f(e) = 0 for all u ∈ U(g). This impliesthat f ≡ 0, in contradiction to f(x0) = 1.

This establishes a 1-1 correspondence/functor between Banach spacerepresentations, and representations of g with an action of K. Preciselythe ensuing modules are called (g, K)−modules. The terminology isthat g = (g0)C, the complexification of a real reductive Lie algebra g0

of a real redfctive group G with maximal compact subgroup K.

6.2.

Definition. A representation of (g, K) on a Banach/Hilbert space Vis called a (g, K)−module if it has an action πK of K and an actionπg of g which are compatible:

(1) π(k)π(X) = π(AdkX)π(k),(2) V =

⊕Vξ,

(3) the differential of πK coincides with the action πg |k of k ⊂ g.

The representation is called admissible if dimVξ <∞ for all ξ ∈ K.

16

Remarks. Given a (g, K)−module, it is not clear that there is a Ba-nach space representation that it comes from; results of Casselmann,Kashiwara, Schmid, Wallach, independently and together, show thatthere are several possible completions of V . For the case when Vadmits an invariant positive definite inner product, Harish-Chandrashows that there exists a (unique) irreducible unitary representationwhose K−finite vectors is the original module.

6.3. The Lie algebra g is the complexification of g0 := Lie(G).

Definition. A hermitian form on a representation (π, V ) of g is calledinvariant if it satisfies

〈π(X)v1, v2〉+ 〈v1, π(X)v2〉, ∀X ∈ g0.

For U(g), define an anti-automorphism by X† := −X where conjuga-tion is with respect to the real Lie algebra g0 ⊂ g. Then invariancemeans

〈π(u)v1, v2〉 = 〈v1, π(u†)v2〉, ∀u ∈ U(g).

We will study admissible modules. To make any headway we willhave to study irreducible rerpesentations (for G and for (g, K)) andinvolve the action of center of the universal envelopoing algebraZ := U(g)G.

EXAMPLE: Sl(2,R).

6.4. Unitary Irreducible Modules. We need some results aboutthe spectral decomposition of bounded self-adjoint operators on Hilbertspaces (separable).

REFERENCE Simon-ReedA bounded operator T : H −→ H is called self-adjoint if

〈Tv1, v2〉 = 〈v1, T v2〉.In general, for a bounded operator one defines T ∗ by the formula

〈Tv, w〉 = 〈v, T ∗w〉.If f(x) is a real valued polynomial function, the definition of f(T ) isclear, and it is also bounded self-adjoint. Using Stone-Weierstrass, thedefinition extends to continuous functions. We get a functional fromCC(R) (or rather continuouos functions on the spectrum of T whcih isa compact set), and the Riesz representation theorem insures that it isgiven by a Borel measure. Thus the definition of f(T ) can be extendedto Borel measurable functions. The spectral projection of T is thefamily PΩ = χΩ(T ) where χΩ is the indicator function of the Borelmeasurable set Ω. These operators satisfy the following:

17

(1) PΩ is a projection.(2) P∅ = 0, P(−a,a) = Id for some a > 0.

(3) If Ω = ∪Ωn disjoint union, then lim∑N

1 PΩi = PΩ.

Such a family is called a projection valued measure. For each v ∈ H,one can define a measure by the formula λ→ 〈v, Pλv〉. One can attachto such a family a self adjoint operator by the formula

〈v, Tv〉 :=

∫Rλ d(〈v, Pλv).

In summary,

Theorem. There is a 1-1 correspondence between projection valuedmeasures and bounded self-adjoint operators. In this correspondence,

PΩ ←→< v, Tw >=

∫Rλ d(v, Pλw).

6.5. Schur’s Lemma.

Theorem (Schur’s lemma for a Hilbert space representation).Let (π,H) be an irreducible representation on a Hilbert space. Denote

by HomG[H,H] the space of bounded operators commuting with theaction of H. Then

HomG[H,H] = CId.

Proof. Any T = T+T ∗

2− iT−T ∗

2i, so a linear combination of self-adjoint

bounded operators If T is G−equivariant, so is T ∗. It is thereforeenough to prove the statement for bounded self-adjoint operators. Inthis case, since π(g)T = Tπ(g), the same holds for the projectors PΩ.Since kerPΩ and imPΩ are closed invariant subspaces, PΩ = Id or 0.We know P(−a,a) = Id. Divide the interval in two equal parts. One ofthe P ′s = Id by (3). We obtain a point λ such that Pλ = Id. ThenT = λId.

This implies that the center of the group acts by scalars on a unitaryirreducible representation. This is not a rich enough set of operatorsto derive any substantial conclusion. The enveloping algebra U(g) hasa larger center Z := U(g)G ⊂ U(g)g. For a connected group, U(g)G =U(g)g. We will often assume this, the modifications for the more generalcase are not very hard.

These operators do not act on the whole space. What we can say isthat for z ∈ Z, z∗ ∈ Z as well. Here z∗ is defined algebraically via theanti involution in Section 6.3. They satisfy

〈π(z)v1, v2〉 = 〈v1, π(z∗)v2〉 ∀v1, v2 ∈ HK .

18

6.6. The next theorem is closely related to the Jacobson density the-orem and the analogous result of von Neumann for C∗−algebras.

Theorem (Schur’s Lemma, infinitesimal version). Assume (π,H) is anirreducible unitary representation. There is an algebra homomorphismχH : Z −→ C, such that

π(z) = χH(z)Id ∀z ∈ Z.

We have two operators S, T, satisfying 〈Tv1, v2〉 = 〈v1, Sv2〉 forv1, v2 ∈ H′ a dense subspace (H∞ or Hω) on which G still acts. Wewant to show that if π(g)T = Tπ(g) ∀g ∈ G, then T = λId.

Lemma. Assume (π,H) unitary irreducible Hilbert space representa-tion. Let A be the algebra generated by the π(g). It is invariant under∗. Let Z be a bounded operator on H, x, y ∈ H linearly independent,δ > 0. There is U ∈ A such that

||Ux− Zx||, ||Uy − Zy|| < δ.

We prove the lemma in the next section. The fact that A is invariantunder ∗ holds because of the assumption of unitarity. Suppose thereexist x, Tx ∈ H′, that are linearly independent. Let x1 = x, y1 = Txand x2 = y2 = x. The lemma implies that there is a sequence Uj ∈ Asuch that

Ujx→ x, UjTx→ x.

Then

〈x,w〉 = lim〈UjTx,w〉 = lim〈TUjx,w〉 = lim〈Ujx, Sw〉 = 〈Tx,w〉.Since H′ is dense, we conclude Tx = x, a contradiction. So for anyx ∈ H′, there is λx such that Tx = λxx. The linearity of T impliesλx+y = λx = λy ∀x, y ∈ H.

6.7. Proof of the Lemma. Set V = H⊕H. Let

B = (U,U) ; U ∈ A.This is a subalgebra of all the bounded operators on V ; it containsthe identity operator, and if u ∈ B then u∗ ∈ B as well. Define

B′ := T : V −→ V : Tb = bT ∀b ∈ B.We can compute what this is:

B′ = T : V −→ V : T (x, y) = (Zx,Zy)for some Z : H −→ H. This follows from Schur’s lemma for a Hilbertspace representation. The proof of the lemma is completed by an ob-servation of von Neumann:

19

If Id ∈ B and T ∈ B =⇒ T ∗ ∈ B, then

(B′)′v ⊂ Cl[Bv] ∀v ∈ V.

Indeed, let Ω := Cl[Bv]. Then V = Ω + Ω⊥, and both Ω and Ω⊥

are B−invariant. The orthogonal projection PΩ satisfies PΩ ∈ B′. IfT ∈ (B′)′, then TPΩ = PΩT, so T preserves Ω, since Id ∈ B impliesv ∈ Bv.

The proof is completed by applying the observation to

v = (x, y = Tx), Zx = x, ZTx = x.

Then

(Z,Z)(x, Tx) = (x, Tx) ∈ Cl[(Ux, UTx)].

REFERENCES Wallach

6.8. The Enveloping Algebra Case. The analogous theorem for(g, K)−modules is much easier. We are not assuming that the repre-sentation comes from an irreducible unitary representation on a Hilbertspace.

Theorem (Dixmier). Let (π, V ) be an irreducible module for U(g) ona space with countable base. Then HomU(g)[V, V ] = CId.

Proof. Suppose not. The kernel and image of T − λI are 0 and Vrespectively; T − λI is invertible. The same holds for any P (T ) whereP is a polynomial, and therefore for any f(T ) where f is a rationalfunction. Then f(T ) is a field which does not have a countable baseas a vector space, mapping into V which does; a contradiction. SoT − λId has a nontrivial kernel for some λ, and the kernel must be allof V.

Precisely, let 0 6= v ∈ V. Define an inclusion C(x) −→ V by

P

Q7→ P (T )Q(T )−1v.

C(x) does not have a countable base.

For an irreducible representation (π, V ) on a Banach space, we knowthat the (g, K)−module V K is irreducible. Since U(g) has a countablebase, and V = U(g)v0 for any 0 6= v0 ∈ V K , the theorem applies.

6.9. Square Integrable Representations. Let (π,H) be an irre-ducible Hilbert space representation. View L2(G, drg) as a representa-tion of G under Rg. Abbreviate it as L2 or L2(G).

20

6.10.

Definition. A matrix coefficient is a function of the form

fv1,v2(g) = 〈π(g)v1, v2〉We say (π,H) is square integrable if there are v1, v2 ∈ H such thatfv1,v2 ∈ L2(G, drg).

Lemma. If (π,H) is square integrable, then all matrix entries are inL2.

Proof. Fix w′, v′ so that fw′,v′ ∈ L2. Let

D′ := v ∈ H : fv,v′ ∈ L2.Since Rxfv,v′ = fπ(x)v,v′ , spanπ(g)w′ ⊂ D′, so D′ is dense in H. Simi-larlyD′ is invariant underG. In summary we can define aG−equivariantlinear map T : D′ −→ L2(G, drg) by

Tv := fv,v′ .

D′ admits an inner product,

(v1, v2) := 〈v1, v2〉+ 〈Tv1, T v2〉,which is continuous. Then D′ is complete with respect to ( , ). Indeedif vj → v is a Cauchy sequence in this topology, vj and Tvj are Cauchysequences. So vj → v and Tvj → f . But

|fvj ,v′(g)− fv,v′(g)| ≤ ||π(g)(vj − v)|| · ||v′|| ≤ ||vj − v|| · ||v′||,so fvj ,v′ → fv,v′ uniformly. Since also Tvj → f a.e,, we conclude f ∈ D′.

The inclusion i : D′ −→ H is a bounded linear map. Let i∗ be theadjoint. By the infinitesimal version of Schur’s lemma, we find thati∗ = aId. Then D′ = H. Furthermore, there is b > 0 such that

〈Tv, Tw〉 = b〈v, w〉

Then we can renormalize T by b1/2 to make it unitary.

Corollary. If (π,H) is square integrable, there is a unitary operatorT ∈ HomG[H, L2(G, drg)] with closed range and such that T (H) is asubspace of C(G) ∩ L2(G). Furthermore there is a d(π) > 0 such that∫

G

fv1,v2(g)fw1,w2(g) dg = d(π)−1〈v1, w1〉 · 〈v2, w2〉.

Matrix entries from inequivalent square integrable representations areorthogonal.

REFERENCES Wallach, Varadarajan

21

7. An Example

Let (X , dx) be a compact space with a Borel measure, and a contin-uous action of a group G (locally compact, Hausdorff etc) such that dxis invariant. Then G acts on L2(X ) by unitary transformations.

Example.

(1) X = K, where K is a compact group acting on X by left orright translations.

(2) G = Sl(2,R) and X = G/Γ where Γ is a discrete subgroup.Discrete subgroups such that G/Γ is compact exist.

7.1.

Theorem (Gelfand, Graev, Piatetskii-Shapiro).

L2(X ) =⊕

mπVπ

with 0 ≤ mπ <∞.

Proof. It is enough to prove the following:

Assume (π,H) is a unitary representation of G on a separable Hilbertspace H such that π(φ) is compact for any φ ∈ Cc(G). Then (π,H)splits into a direct sum of irreducible unitary representations with finitemultiplicities.

C(G) admits a † given by φ†(g) := φ(g−1). As before,

π(φ)v :=

∫G

φ(g)π(g)v dg.

Assume φ† = φ. Then π(φ) is bounded selfadjoint (and compact byassumption).

Recall that an operator is compact if it maps bounded sets intoprecompact sets. The basic property of a compact selfadjoint operatorT is that

H = H0 +⊕Hk

whereHk = v ∈ H : Tv = λkv

are finite dimensional (λk 6= 0). Furthermore, if H′ ⊂ H is a closedT−invariant subspace, then the analogous decomposition holds.

Let H1 be the minimal subspace which contains all the Hk,φ.CLAIM: H1 = H.

Clearly H1 is invariant under all the π(φ), and so is H⊥1 . SupposeH1 6= H. Let 0 6= f ∈ H⊥1 . Then π(φ)f = 0 ∀φ ∈ Cc(G). Use anapproximate identity to show that f = 0.

22

CONCLUSION: Every (closed) invariant V ⊂ H must satisfy

V ∩Hk,φ 6= (0)

for some φ and some λk 6= 0.Fix a Hφ,k and let 0 6= H1 ⊂ Hφ,k be such that

(1) there is V ⊂ H invariant satsifying H1 = V ∩Hφ,k,(2) dim[H1] is minimal subject to (1).

Consider the set

H′ ⊂ H invariant : H′ ∩Hφ,k = H1.There is a minimal Hm in this set, the intersection of all the subspaces.CLAIM: Hm is irreducible.

If not, Hm = V +V ⊥ both invariant proper nonzero subspaces. Oneof them must intersect Hφ,k in H1, a contradiction to the minimalityof Hm.CONCLUSION: H =

⊕Hπ countable sum of irreducible unitary

representations.This follows by doing a transfinite induction.

Remains to show that any irreducible (π,Hπ) occurs only a finitenumber of times. Fix a Hπ irreducible. There is φ = φ† such thatHπ∩Hk,φ 6= (0). For any otherH`,H`∩Hk,φ is orthogonal toHπ∩Hk,φ.So there are only finitely many H` that intersect Hφ,k nontrivially. Butif Hπ ' H`, then π(φ) and `(φ) must be equivalent, so they havenontrivial intersection with Hk,φ.

REFERENCE Gelfand, Graev, Piatetskii-Shapiro.This is a good introduction to the beginings of automorphic forms.

Concentrates on GL(2) in detail.

8. Summary

Assume (π, V ) is a continuous Banach space representation of a realreductive group G with maximal compact subgroup K.

8.1. Banach Space.

Definition. A (g, K)−module is an algebraic module of g and thereforeU(g), such that

(1) the actions of g and K are compatible ie,

π(k)π(X) = πAdkX)π(k), dπK = πg.

(2) V =⊕

Vξ.

The module is called admissible if dimVξ <∞.

23

Theorem (Harish-Chandra).

V K := K − finite analytic vectors ⊂ V

is dense in V .V K =

⊕ξ∈K

Vξ, and the Fourier series of any C∞−vector converges to

the vector absolutely.

For admissible modules, the correspondence W ⊂ V ←→ WK fromclosed invariant subspaces of V to U(g)−submodules of V K is a bijec-tion. The inverse is W 7→ Cl(W).

For an admissible module, we can drop the analytic part in the def-inition of V K .

For examples/counterexamples see REFERENCE Soergel.

8.2. Hilbert Space.

Theorem (Harish-Chandra). Any irreducible unitary representation isadmissible. In general,

dimHξ ≤ C(dim ξ)2

for some C > 0. Any bounded operator T which commutes with π(g)∀g ∈ G is a multiple of a scalar.

There is a 1− 1 correspondence

Unitary irreducible (g, K)−modules ←→ Unitary irreducible G−modules

For any irreducible (g, K)−module (π, V ), there is a unique unitaryG−module π,H) such that HK ∼= V.

Any unitary admissible module is a direct sum of irreducible modules.

Two modules (πi,Hi) are called infinitesimally equivalent if HK1∼=

HK2 . infinitesimally equivalent unitary modules are equivalent.NOT TRUE for Banach space representations.

8.3. Example Sl(2,R). Recall G = KAN with

K = [cos θ − sin θsin θ cos θ

] A =

[et 00 e−t

] N =

[1 x0 1

]

Write B = [a x0 a−1

] with a ∈ R×. Gven any ν ∈ C, we can define

two characters of B, by the formulas

χ+,ν(b) = e(ν) log |a|, χ−,ν(a) = sgn(a)e(ν) log |a|

24

The Principal Series X(B,χ) are defined as the representations by lefttranslation on the spaces

f : G −→ C : f(gb) = χ(a−1)δ(a)−1f(g).A function in such a space is determined by its values on K. We assumethat the space is formed of functions which are L2 when restricted toK. So this is a Hilbert space, though the representation is not neces-sarily unitary. The K−finite vectors are vn(r(θ)) = einθ, so these areadmissible representations. These vectors are orhtogonal with respectto the inner product (with appropriate Haar measure on K).

8.4. We compute the infinitesimal action of U(g) on X(B,χ). Twobases of the complexified Lie algebra are

E =

[0 10 0

], H =

[1 00 −1

], F =

[0 10 0

],

e =1

2

[1 i−i 1

], h = −i

[0 1−1 0

], f =

1

2

[1 −i−i 0

].

It is an exercise to verify that

h · vn = nvn, e · vn =ν + 1 + n

2vn+2, f · vn =

ν + 1− n2

vn−2

Then X(B,χ+) =⊕

V2n and X(B,χ−) =⊕

V2n+1. The first one isirreducible except for ν = 2n + 1, the second one for ν = 2n. Therepresentations are unitary for ν ∈ iR with the given inner product.At the reducibility points there are either 2 or three irreducible factors.

Theorem. The factors of the principal series are all the admissibleirreducible (g, K)−modules.

The main point of this example is to highlight that for a general realreductive group, irreducible admissble representations are not muchmore complicated.

8.5. (g, K)−modules. Let U(g)K be the ring of operators invariantunder K; if K is connected, we can replace it by k. if (π, V ) is a(g, K)−module, then U(g)K acts on Vξ. In particular Z := U(g)G ⊂U(g)K acts on Vξ.

Theorem (1). If V is finitely generated, then Vξ is a finitely generatedZ−module.

Corollary (1). Any finitely generated (g, K)−module is admissible.

Theorem (2). There is C > 0 depending on g and the number ofgenerators such that dimVξ ≤ C(dim(ξ)2.

25

Corollary (2). If (π, V ) is irreducible, it is admissible.

Theorem. Any admissible irreducible (π, V ) is a subsquotient of aprincipal series IndGP [σ ⊗ ν].

In this theorem, P = MAN is a minimal parabolic subgroup, andσ ⊗ ν a representation of MA.

8.6. Characters. Recall ΩK = 1 −∑X2i with Xi an orthonormal

basis of k with respect to a nondegenerate invariant form of g nega-tive on k. This operator acts by a scalar c(ξ) ≥ 1 on any irreducibleK−module; so also on any Vξ. It satisfies

(1) There exist r, C > 0 such that dim ξ ≤ Cc(ξ)r.(2) For m >> 0,

∑c(ξ)−m <∞.

Definition. An operator A : H −→ H is called trace class if for anyo.n. basis ei, ∑

〈Aei, ei〉 <∞.A is summable with respect to ei if∑

i,j

|〈Aei, ej〉| <∞

In this case, tr(A) :=∑〈Aei, ei〉 exists, and is independent of the

choice of o.n. basis. It satisfies

tr(SAT ) = tr(ATS) = tr(TSA)

for any bounded operators S, T.Summable implies trace class.

Theorem (1, Harish-Chandra). Assume dimHξ < c dim(ξ)2. Thenπ(f) is trace class for any f ∈ C∞c (G). In particular

Θπ(f) := tr(π(f)

defines a G−invariant distribution. If Z acts by a character, then Θπ

is an eigendistribution for Z.

Sketch. A matrix entry of π is a function of the form fv,w(g) := 〈π(g)v, w〉.Choose an o.n. basis ei,ξ so that the ei,ξ form a basis of Vξ. Any ma-trix entry can be differentiated by ΩK on the right and on the left:

∂rL,ΩK∂rR,ΩK

fei,ξ,ej,η = c(ξ)rc(η)rfei,ξ,ej,η .

Then

|〈π(f)ei,ξ, ej,η〉| ≤ c(ξ)−rc(η)−r∫G

||π(x)|| ·∣∣∂rL,ΩK∂rR,ΩKf ∣∣ dx.

26

Theorem (2, Harish-Chandra). Two irreducible representations (onHilbert spaces) are infinitesimally equivalent if and only if their char-acters coincide.

8.7. Example. We apply some of this theory to the case of L2(G/Γ)with Γ ⊂ G discrete cocompact. First some more facts about compactand trace class operators.

REFERENCE Reed-Simon, chapter VI.

Lemma (1). Let K(H) be the space of compact operators on the sep-arable Hilbert space H. Then K(H) is a closed ideal.

Lemma (2). Any trace class operator is compact.

Definition. We say that T is Hilbert-Schmidt (H-S)if∑||Tvj|| < ∞

for some o.n basis vj.

(1) If A is H-S, then so is A∗.(2) If A is H-S, then so are AB,BA for any bounded B.(3) T is of trace class if and only if it can be written as T = AB

with A,B H-S.

Theorem (Dixmier-Malliavin). Fort a Lie group G, any f ∈ C∞c (G)can be written as a sum φi ? ψi with φi, ψi ∈ C∞c (G).

We apply this to the case of X = G/Γ and (Π, L2(X )). Any π(φ) ∈C∞c (G) is compact because it is a kernel operator, Kφ(x, y) =

∑γ∈Γ φ(xγy−1.

Kernel operators are H-S. By the summary above, π(φ) is thereforetrace class.

The following is a formal computation which can be justified withappropriate estimates.

let ei be an o.n. basis of H := L2(X ). We can write K = Kφ as

K(x, y) =∑k,l

λk,lek(x)el(y),

with λk,l =

∫X×X

K(x, y)ek(x)el(y) dxdy. Then Kei(x) =∑λk,iek(x).

So ∑i

〈Kei, ei〉 =∑

λi,i.

On the other hand, ∑i

λi,i =

∫XK(x, x) dx.

27

In our case, we conclude

trΠ(φ) =

∫X

∑γ∈Γ

φ(xγx−1) dx.

On the left, we can write∑

mπtrπ(φ). On the right we can express

the sum as follows. Let Γ be a set of representatives of G−conjugacyclasses of elements in Γ. If γ1 and γ2 are conjugate, then the terms inthe sum are the same. Collecting such terms we get∫G/Γ

∑γ∈Γ

φ(xγx−1) dγ =

∫G/Γ

∑γ∈Γ

∑δ∈Γ/Γγ

φ(xδγδ−1x−1) dγ =

=∑γ∈Γ

∫G/Γγ

φ(xγx−1) dγ =

=∑γ∈Γ

∫G/Gγ

∫GΓ/Γγ

φ(xuγu−1x−1) dγ =∑γ∈Γ

vol(Gγ/Γγ)

∫G/Gγ

φ(xγx−1) dx.

The simplest example of this formula is the Poisson formula, G =(R,+) and Γ = Z. On the left hand side, we have the decomposition ofR/Z = S1. The space is the direct sum of the representations einθ.Taking traces we get

∑φ(n). On the other side we get

∑f(n).

28

9. Center of the Enveloping Algebra

9.1. Universal Enveloping Algebra. The universal enveloping al-gebra of a real Lie group can be identified with the left or right invariantdifferential operators. The algebraic definition comes from the follow-ing theorem.

Theorem. There is a unique (up to isomorphism) universal object(U(g), ι where U(g) is an associative algebra with unit, and iota : g −→U(g) is a Lie algebra homomorphism such that any Lie algebra mapφ : g −→ A into an associative algebra extends to a commutative dia-gram

gφ //

ι

A

U(g)

Φ

==

Properties:

(1) ι is an inclusion,(2) if X1, . . . Xn is an ordered basis of g, then Xa−1

1 . . . Xann is a basis

of U(g).(3) Equivalently, U(g) has a filtration by U(g)m which are gener-

ated by the monomials Y1·, · · · · Ym. The graded Gr(U(g) isavbelian and isomorphic to S(g).

(4) ad : g −→ Hom(g) given by adX(Y ) := [X, Y ] extends toa representation of g on S(g) and U(g). The canonical mapgr : U(g) −→ Gr(U(g)) ∼= S(g) is a Lie algebra isomorphism.

(5) The map

sym : S(g) −→ U(g), sym(Y1 · · · · ·Ym) :=1

m!

∑σ∈Sm

Yσ(1) · · · · ·Yσ(m)

is a g-isomorphism, inverse to gr.

Corollary. If G is connected,

S(g)G = S(g)G ∼= U(g)G = U(g)g.

9.2. Restricted Cartan Subalgebras. Let g be real reductive, cor-responding to the real reductive group G. We may assume connectedsemisimple for simplicity.

Let a ⊂ s be a maximal abelian subalgebra.

Proposition. Any two such subalgebras in s are conjugate under K.

29

Let m := Ck(a); this is a reductive subalgebra. If t is a Cartansubalgebra of m, then h := t + a is a Cartan subalgebra . The groupM := CK(a) has Lie algebra m. The restricted Weyl group is W :=NK(a)/CK(a).

9.3. Iwasawa Decomposition. We can define ∆(g, a) as in the caseof a complex algebra. These are called the restricted roots. We canchoose a positive system P. If α is a root, so is −α, and at most ±2αor ±1

2α are also roots. The restricted Weyl group acts transitively on

the positive systems chambers etc. Let n± be the sums of roots spacesfor ±P . Let A,N be the connected subgroups corresponding to a, n.Note that n− = θ(n).

Proposition. g = n + m + a + n−, and G ∼= KAN.

REFERENCE Helgason.

9.4. We first determine S(s)K . We will often identify S(V ) for a vectorsubspace of g with P (V ); this is because there is a K− invariant innerproduct on g.

Proposition.

S(s)K ∼= S(a)W .

Proof. We will do it in several steps.I. The retriction map φ : P (s) −→ P (a) is injective.

This follows from the fact that any X ∈ s can be embedded into arestricted Cartan subalgebra , and any two are conjugate by K.

So we need to show that the map is onto.II. P (s)K separates points.

if K ·h1∩K ·h2 = ∅, there existws a contunuous function f which is1 on K · h1 and 0 on K · h2. By the Stone-Weierstrass theorem, thereis a polynomial p which is within less than 1/2 of f on the compact

union of these sets. Then

∫K

p(k ·X) dk is the desired element.

III. Let S = P (a) ⊃ I = P (a)W ⊃ J = Imφ ∼= P (s)K , and letC ⊃ C(I) ⊃ C(J) be the quotient fields. Then S, P (s) are integrallyclosed. This follows from the fact that they satisfy unique factorization:if r := p/q is a reduced expression and satisfies an equation

rm + am−1rm−1 + . . . a0 = 0,

then any prime s|p must also divide q. But then r = p/q is not reducedunless q is a unit.IV. S is integral over I.

30∏w∈W (t− w · p) is a polynomial in t with coefficients in I, and has

p as a root.V. If p1, p2 ∈ P (s)K and p2 = p1 · q, then q ∈ P (s)K . Similar for I.

Integrate p− 2 = p1 · q over K or W respectively.VI. J ∼= P (s)K is integrally closed. If x ∈ C(J) is integral, thenit is integral in C(P (s)). By the previous, x = p2

p1with p1, q1 ∈ J ; so

x ∈ P (s), and then the previous point shows that it is in J.VII. P (a) is integral over J.

Consider the polynomial

det(λ− (adX)2

)= λr + pr−1λ

r−1 + . . . p`λ`.

It has coefficients in J. Restict this polynomial to a.IX. let h1, h2 ∈ a be such that p(h1) = p(h2) for any p ∈ J. Thenh2 ∈ W · h1.

From (III), there is k ∈ K such that k · h2 = ·h1. Consider Cg(h1).This is a θ−stable reductive subalgebra, and both a and k · a are re-stricted Cartan subalgebra ’s in Cg(h1). So there is an element k′ ∈K ∩ CG(h1) such that k′ka = a. Thus k′k ∈ NK(a) and k′k)h2 = h1 asclaimed.X. C/C(J) is a normal extension. Let

F (λ) := λ2r + p2r−2λr−1 + . . . p`λ

2`.

Its roots are the restricted roots ±α ∈ ∆. C is obtained from adjoiningthese roots.XI. Any σ ∈ Aut(C)/C(J) must fix I.

The maph0 ∈ a 7→ (p 7→ pσ(h0))

is an algebra map of C, so of the form p 7→ p(h1) for some h1 ∈ a. Thisrestricts to a λa∗ of the form λ(h) = B(h, h1). (g assumed semisimple).Then

p(h1) = λ(p) = pσ(h0) = p(h0) ∀p ∈ J.By (IX), there is w ∈ W such that h1 = w · h0.

If q ∈ I, then

qσ(h0) = q(h1) = q(w · h0) = q(h0).

So σ fixes I, and therefore I = J.

9.5. Complex Algebras. In this case, g = u + iu where u is a com-pact form. So k = u and s = iu. Then S(s)k = S(g)g by complexifyingu.

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9.6. The Harish-Chandra Homomorphism.

9.7. Let g be complex semisimple (reductive, but for simplicity ... ).Recall g = n+ + h + n−. Then U(g) = U(h) + n−U(g) + U(g)n. Letq : U(g)h −→ U(h) be the projection according to the space

(n−U(g)

).

Lemma. q : U(g)h −→ U(h) is an algebra homomorphism.

Proof. This follows from the identity(n−U(g)

)h=(U(g)n

)h.

If a monomial element in U(g)h is of the form Xαu with Xα ∈ n, thenu itself must be of the form uHX−β with X−β ∈ n−.

Note that in sl(2), the element h2 +2(ef +fe) projects onto h2 +2h.This is not in S(h)W ; the Harish-Chandra homomorphism means toidentify U(g)g with S(h).

9.8. Recall ρ := 12

∑α∈P α. Let µ : S(h) −→ S(h) be the map defined

by µ(h) = h− ρ(h).

Theorem. The map γ = µ q : U(g)g −→ S(h)W is an algebra iso-morphism independent of the choice g = n + h + n−.

We have identified U(h) with S(h) because h is abelian.

Proof. We sketch the part that γ(Z) ⊂ S(h)W . Recall from the generalstructure theory that W = NG(h)/CG(h) is a finite group generated by

the reflections sα(x) = x − 2(x,α)(α,α)

α. Here α is a simple root associated

to ∆(n, h).Let λ ∈ h∗. We associate a 1−dimensional representation Cλ of

b := h + n to λ by the formula (H + X) · vλ = λ(H)vλ. The Vermamodule M(λ) is defined as

M(λ) := U(g)⊗U(b) Cλ−ρ.

As an n−−module it is U(n−)⊗C Cλ−ρ. The action of h is via commu-tation, and the action of n+ is also via brackets.FACT 1. M(λ) has a unique irreducible quotient called L(λ). Letα∨ := 2α

(α,α). Then sα(λ) = λ − (λα∨)α. Assume (λ, α∨) ∈ N for all

simple roots. We say λ is dominant integral; in this case the L(λ) giveall the irreducible finite dimensional representations of g.FACT 2. Let vλ−ρ := 1⊗ 11λ−ρ. If u ∈ U(g)h, then

u · vλ−ρ = q(u)(λ− ρ)vλ−ρ = γ(u)(λ)vλ−ρ.

In particular, this is the case for any z ∈ Z; in fact it acts by this scalaron all of M(λ).

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FACT 3. When λ is dominant integral, there is an inclusion M(wλ) ⊂M(λ) for any w ∈ W . It follows that

γ(z)(λ) = γ(z)(wλ) ∀w ∈ W.

Corollary.

Z(g)∼←→ h∗/W.

REFERENCES Dixmier, Wallach. The proofs in the two referencesare different.

The modules M(λ) will play an essential role later. They are partof a category similar to (g, K)−modules, called (g, B)−modules.

10. Admissible (g, K)−modules

10.1. Graded Modules. Let (π,X) be a (g, K)-module. We say it isfinitely generated if it is finitely generated as a U(g)−module. Assumeit is finitely generated by a K−invariant set X0. Recall that U(g) hasa filtration U(g)n defines by

U0 = C, U(g)n+1 = gU(g)n.

ThengrU(g) :=

⊕U(g)n/U(g)n−1

∼=⊕

Sn(g)

by the PBW-theorem. Set

Xn := U(g)nX0.

This is a filtration of X by K−invariant finite dimensional subspaces.We denote

gr(X) :=⊕

Xn/Xn−1.

This is a module for S(g) such that S(k) acts trivially; so it is aS(s)−module. The action of K is preserved however, and gr(X) isa finitely generated (S(g), K)−module.

10.2. Recall than X =⊕

X(ξ). Then Z ⊂ U(g)K , so it acts on eachV (ξ).

Theorem. If (π,X) is finitely generated, then X(ξ) is finitely gener-ated as a Z−module.

Proof. From the structure theorem of Z we know that gr(Z) mapsto S(g)G; identify S(g) with P (g). Then P (g)G will act on gr(X) byrestricting the polynomials to s. We know from the earlier proof thatS(s)K is finitely generated as a RessS(g)g−module. So it is enough toshow that X(ξ) is finitely generated as a S(s)k−module. The module

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X(ξ) is the image of HomC[Vξ, X]⊗ Vξ under the map T ⊗ v 7→ T (v).The module HomC[Vξ, X] is a finitely generated K⊗S(s)−module, andbecause S(s) is Noetherian, so is the module S(s) · HomK [Vξ, X]. Sothere are T1, . . . , Td ∈ HomK [Vξ, X] which generate this module. Sofor any T ∈ HomK [Vξ, X], there are p ∈ S(s) such that T =

∑piTi.

Integrating over K, we find that T =∑pKi Ti with pKi ∈ S(s)K as

claimed. It follows that X(ξ) is finitely generated.

Corollary. If (π,X) is finitely generated, and locally Z−finite, then itis admissible. This is true for any irreducible (g, K)−module.

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References

[B] T. Brocker, T. tomDieck, Representations of Compact Lie Groups[BW] A. Borel, N. Wallach, Continuous Cohomology Princeton University Press[Dx] J. Dixmier, Enveloping Algebras[FW] F. Warner, Foundations of Differentiable Manifolds and Lie Groups[NW] N. Wallach, Real Reductive Groups vol. I and II[GGP] I. Gelfand, Graev, I. Piatetskii-Shapiro, Generalized Functions, volume 6[GW] G. Warner, Harmonic Analysis on Semi-Simple Lie groups vol. I and II[He] S. Helgason, Differential geometry and Symmetric Spaces 1962[HC] Harish-Chandra, Representation of Semi-Simple Lie Group on a Banach space

I and II TAMS 1953 and 1955[HT] R. Howe, E.C. Tan Non-Abelian harmonic Analysis; Applications to Sl(2,R)[J] N. Jacobson, Lie Algebras[Kn1] A. Knapp, Representation Theory of Semi-Simple Lie groups; an Overview

based on Examples[Kn2] A. Knapp, Lie groups beyond an Introduction[KnV] A. Knapp, D. Vogan Cohomological Induction and Unitary Pepresentations

Princeton University Press[L] S. Lang, Analysis II[LV] G. Lion, M. Vergne The Weil Representation, Maslov Index, and Theta series[N] E. Nelson, Analytic Vectors, Ann. of Math. 1959[SR] B. Simon, M. Reed Methods of Modern Mathematica Physics volume I, Func-

tional Analysis[Soe] W. Soergel An irreducible not admissible Banach space representation of

Sl(2,R) Proc. AMS 104, 1988, 1322-1324[Va] V.S. Varadarajan, An Introduction to Harmonic Analysis on Semisimple Lie

Groups[V] D. Vogan, Representations of Real Reductive Lie Groups Birkhauser, 1981[W] N. Wallach, Real Reductive Groups, vol I and II, Academic Press 132 and 133