Lecture 1: Lie Groups, Lie Algebras, and Homogeneous Spacesbenjaminlharris.weebly.com/uploads/1/0/9/...notes.pdf · 1. Lecture 1: Lie Groups, Lie Algebras, and Homogeneous Spaces

PADERBORN MINICOURSE

1. Lecture 1: Lie Groups, Lie Algebras, and HomogeneousSpaces

1.1. Subsets of Rn with symmetry. In mathematics, we often studysubsets of X ⊂ Rn or subsets Y ⊂ Cn.

Example 1.1. The n-sphere Sn ⊂ Rn is defined to be the set of realsolutions to the single polynomial equation

x21 + x22 + · · ·+ x2n = 1.

We may also view S2n−1 ⊂ Cn as the solution to the single polynomialequation

z1z1 + z2z2 + · · ·+ znzn = 1.

Example 1.2. The real (p, q)-hyperboloid Hp,q ⊂ Rn (n = p + q) isdefined to be the set of real solutions to the single polynomial equation

x21 + · · ·+ x2p − x2p+1 − · · · − x2n = 1.

The complex (p, q)-hyperboloid HCp,q ⊂ Cn (n = p+ q) is defined to be

the set of complex solutions to the single polynomial equation

z1z1 + · · ·+ zpzp − zp+1zp+1 − · · · − znzn = 1.

Next, if X ⊂ Rn or Y ⊂ Cn is a subset, then we may consider thegroup GX of real linear symmetries of X and the group GY of complexlinear symmetries of Y . More precisely, define

GX := g ∈ GL(n,R) | g · v ∈ X whenever v ∈ Xif X ⊂ Rn and

GY := g ∈ GL(n,C) | g · v ∈ Y whenever v ∈ Y if Y ⊂ Cn.

Exercise 1.3. (a) Define O(n) ⊂ GL(n,R) to be the collection of ma-trices A such that ATA = I. We call O(n) the orthogonal group.Show that O(n) is the group of linear symmetries of the n-sphereSn ⊂ Rn.

(b) Define U(n) ⊂ GL(n,C) to be the collection of matrices A suchthat A∗A = I. We call U(n) the unitary group. Show that U(n)is the group of (complex) linear symmetries of the 2n − 1 sphereS2n−1 ⊂ Cn.

1

2 PADERBORN MINICOURSE

(c) Define

Ip,q :=

1 0 · · · 0 00 1 · · · 0 0· · · · · · · · · · · · · · ·0 0 · · · −1 00 0 · · · 0 −1

where the n by n matrix Ip,q has p ones along the diagonal, qminus ones along the diagonal, zeroes in all non-diagonal entries,and n = p+q. Then define O(p, q) ⊂ GL(n,R) (n = p+q) to be thecollection of matrices A such that AT Ip,qA = Ip,q. We call O(p, q)the indefinite orthogonal group with signature (p, q). Show thatO(p, q) is the group of linear symmetries of the (real) hyperboloidHp,q ⊂ Rn.

(d) Define U(p, q) ⊂ GL(n,C) (n = p + q) to be the collection ofmatrices A such that A∗Ip,qA = Ip,q. We call U(p, q) the indefiniteunitary group with signature (p, q). Show that U(p, q) is the groupof (complex) linear symmetries of the (complex) hyperboloidHC

p,q ⊂Cn.

If one takes a generic subset X ⊂ Rn or Y ⊂ Cn, then there may beno nontrivial linear symmetries of X or Y . However, some of the mostinteresting subsets X ⊂ Rn and Y ⊂ Cn have lots of linear symmetries.In these notes, we will assume that X (respectively Y ) is a homogeneousspace for GX (resp. GY ). This means that for every x, y ∈ X, thereexists g ∈ GX such that g · x = y.

Exercise 1.4. (a) Show that Sn ⊂ Rn is a homogeneous space for thegroup O(n) and S2n−1 ⊂ Cn is a homogeneous space for the groupU(n).

(b) (Harder) Show that Hp,q ⊂ Rn (n = p+ q) is a homogeneous spacefor the group O(p, q) and HC

p,q ⊂ Cn (n = p+ q) is a homogeneousspace for the group U(p, q).

In this series of lectures, we are interested in studying analysis onspaces X ⊂ Rn (or Y ⊂ Cn) that are homogeneous spaces for thecorresponding linear group of symmetries GX (or GY ). Our first stepis to better understand linear subgroups G ⊂ GL(n,R) (and linearsubgroups G ⊂ GL(n,C)).

1.2. Linear Groups, Lie Algebras, and the Exponential Map.Let gl(n,R) (resp. gl(n,C)) denote the collection of all n by n matriceswith real (resp. complex) entries. The exponential map

exp: gl(n,R) −→ GL(n,R)

PADERBORN MINICOURSE 3

is defined to be

expX :=∞∑k=0

Xk

k!.

The same formula defines the exponential map

exp: gl(n,C) −→ GL(n,C).

Exercise 1.5. (a) Show expX converges for all X ∈ gl(n,R) and X ∈gl(n,C).

(b) If S ∈ GL(n,C), show exp(SXS−1) = S exp(X)S−1.(c) If λ1, . . . , λn are the (generalized) eigenvalues of X ∈ gl(n,C),

show that eλ1 , . . . , eλn are the (generalized) eigenvalues of expX ∈GL(n,C). (Hint: Utilize Jordan normal form and part (b)).

(d) Conclude etr(X) = det(exp(X)), and conclude expX ∈ GL(n,R)(resp. expX ∈ GL(n,C)) for all matrices X ∈ gl(n,C) (resp.X ∈ gl(n,C)).

Definition 1.6. A real linear group is a closed subgroupG ⊂ GL(n,R).A complex linear group is a closed subgroup G ⊂ GL(n,C).

Definition 1.7. The Lie algebra of a real linear group G ⊂ GL(n,R)(resp. complex linear group G ⊂ GL(n,C), denoted g ⊂ gl(n,R) (resp.g ⊂ gl(n,C)) is the collection of X ∈ gl(n,R) (resp. X ∈ gl(n,C))such that exp(tX) ∈ G for all t ∈ R. If X ∈ g, we call the collectionof exp(tX) with t ∈ R the one-parameter subgroup of G correspondingto X.

Exercise 1.8. (a) Show that the Lie algebra of O(n) is o(n), the col-lection of X ∈ gl(n,R) satisfying XT+X = 0. This is the collectionof skew symmetric matrices.(Hint: Differentiate the condition exp(tX)T exp(tX) = I and eval-uate at t = 0).

(b) Show that the Lie algebra of U(n) is u(n), the collection of X ∈gl(n,C) satisfying X∗ = −X. This is the collection of skew Her-mitian matrices.

In order to study a linear group G, it is very useful to study its Liealgebra g. There are many examples of this phenomenon; let us firstmention a fundamental one.

Proposition 1.9. (1) The Lie algebra g ⊂ gl(n,R) (resp. g ⊂gl(n,C)) is a vector subspace.

(2) Every closed linear group G ⊂ GL(n,R) or G ⊂ GL(n,C) is asmooth manifold.


(3) Let e ∈ G denote the identity element. Then there exist anopen subset 0 ∈ Ω ⊂ g and an open subset e ∈ Ξ ⊂ G such thatexp: Ω→ Ξ is a diffeomorphism.

We omit the proofs. See for instance the introduction of [Kna05].Part (c) of this proposition tells us that the Lie algebra together withthe exponential map give a coordinate chart for G near the identity el-ement e ∈ G. In particular, we may identify g ' TeG. We remark thatthis is actually how you prove part (b), that G is a smooth manifold.

Exercise 1.10. (a) Show that ιC : GL(m,C) → GL(2m,R) is a lineargroup. Moreover, show that it is the subgroup of matrices thatcommute with m by m block matrix

J :=

(0 −II 0

).

(b) Show that the Lie algebra of GL(m,C) is the image of ιC : gl(m,C) ⊂gl(2m,R), the collection of all 2m by 2m real matrices commutingwith J .

(c) Check that exp(ιC(X)) = ιC(expX). In words, exponentiating acomplex matrix and then making it into a real matrix of twice thesize is the same as making it a complex matrix of twice the sizeand then exponentiating the corresponding real matrix.

(d) If G ⊂ GL(m,C) is a complex linear group, show that we may useιC to view G as a real linear group. Check that the Lie algebra ofG viewed as a real linear group is ιC applied to g ⊂ gl(n,C).

The previous exercise allows us to view all complex linear groups asreal linear groups. For this reason, we will, from now on, only studyreal linear groups, and we will refer to them simply as linear groups inan effort to simplify our terminology.

Now, let us return to the setting of Section 1.1.

Proposition 1.11. Let X ⊂ Rn be a smooth submanifold. Let G be alinear group of symmetries acting transitively on X, and let g denotethe Lie algebra of G. Fix x ∈ X, let Gx ⊂ G denote the stabilizer ofx in G, and let gx ⊂ g denote its Lie algebra. Then there is a naturalisomorphism

(1) g/gx∼−→ TxX.

Exercise 1.12. (a) First define a smooth map px : G→ X by g 7→ g·x.Show that p is a smooth submersion at x = e.(Hint: Suppose not. Then p must map an open neighborhoode ∈ U ⊂ G into a closed, codimension one submanifold of X. Now,


show that G may be written as a countable union of open sets ofthe form gU for some g ∈ G. Then conclude that X must be acountable union of codimension one submanifolds of X. This is acontradiction.)

(b) Check that the derivative of px is the surjection g→ TxX by

X 7→ d

dt

∣∣∣t=0

exp(tX) · x.

Check that the kernel of this map is gx, and deduce (1).

Next, suppose G ⊂ GL(n,R) is a linear group, and suppose g ⊂gl(n,R) is its Lie algebra. Notice that G acts on itself by conjugation

c(g) : x 7→ gxg−1 for all g, x ∈ G.Differentiating this map at e ∈ G, we obtain a linear map

Ad(g) : g ' TeG→ g ' TeG.

Exercise 1.13. (a) Check that Ad(g)X = gXg−1 is simply conjuga-tion by matrices.

(b) Check that G→ GL(g) by g 7→ Ad(g) is a group homomorphism.

We call Ad(g) the adjoint action of g ∈ G on g.

1.3. Invariant Densities on Compact Homogeneous Spaces. LetG be a linear group, and let X ⊂ Rn be an m-dimensional smooth man-ifold on which G acts smoothly and transitively. We want to integratefunctions f on X in such a way that translation by g ∈ G does noteffect the result, ie ∫

X

f(x) =

∫X

f(g · x).

Of course, in order to make this precise, we must integrate againstsomething. In a first course on smooth manifolds, you are usuallytaught to integrate top-dimensional differential forms. Recall that atop-dimensional differential form ω ∈ Ωm(X) is a function

ωx : TxX⊕m → C

for each x ∈ X satisfying

ωx(AX1, . . . , AXm) = det(A)ωx(X1, . . . , Xm)

for all X1, . . . , Xm ∈ TxX and A ∈ GL(TxX). Moreover, ωx must varysmoothly as a function of x ∈ X. The trouble with integrating top-dimensional forms is that they require a choice of orientation on X.For our applications, it is easier to remove this ambiguity and insteadintegrate densities on X. A smooth density ν ∈ D(X) is a function

νx : TxX⊕m → C


for each x ∈ X satisfying

νx(AX1, . . . , AXm) = | det(A)|ωx(X1, . . . , Xm)

for all X1, . . . , Xm ∈ TxX and A ∈ GL(TxX). Moreover, νx mustvary smoothly as a function of x ∈ X. For a more systematic treat-ment of densities and how to integrate them see Chapter 16 of [Lee12].Alternately, you can use what you know about differential forms andcommon sense.

Let us get back to our integration question. We wish to find a smoothdensity ν ∈ D(X) such that

(2)

∫X

f(x)dν =

∫X

f(g · x)dν.

Notice (2) holds if, and only if ν is a G-invariant density on X.Therefore, we must ask whether or not there exists a G-invariant den-sity on X. Let us begin with the special case X = G. There is a leftaction of G on D(G) by

(L∗gω)x(X1, . . . , Xm) := ωg−1x((Lg−1)∗X1, . . . , (Lg−1)∗Xm)

and there is a right action of G on D(G) by

(R∗gω)x(X1, . . . , Xm) := ωxg((Rg)∗X1, . . . , (Rg)∗Xm).

To define a left G-invariant density ν on G, first let us define ν at theidentity e ∈ G. In particular, we fix a non-zero νe : TeG

⊕m → C satis-fying νe(AX1, . . . , AXm) = | detA|νe(X1, . . . , Xm) for all X1, . . . , Xm ∈TeG and A ∈ End(TeG). Then we define

νx := L∗x−1νe.

Question: Is this density also right G-invariant?If so, then we would have

(R∗xL∗xνe)(X1, . . . , Xm)

?= νe(X1, . . . , Xm).

The left hand side is equal to

νe((Lx−1)∗(Rx)∗X1, . . . , (Lx−1)∗(Rx)∗Xm)

=νe(Ad(x−1)X1, . . . ,Ad(x−1)Xm)

=| det(Ad(x−1))|νe(X1, . . . , Xm).

In particular, there exists a simultaneously left and right G-invariantdensity ν ∈ D(G) only if | det(Ad(x))| = 1 for all x ∈ G.

Exercise 1.14. Show that there exists a left and right G-invariantdensity ν ∈ D(G) if, and only if | det(Ad(x))| = 1 for all x ∈ G.


Exercise 1.15. (a) Define δ(x) := | det(Ad(x))|. We call δ : G → R+

the modular function of G.(b) Show that δ is a continuous group homomorphism.(c) If G is compact, show that the image of δ is compact.(d) Deduce that δ = 1 if G is a compact linear group. In particular,

if G is a compact linear group, then every left G-invariant densityis also right G-invariant. Further, a non-zero left and right G-invariant density on G is unique up to multiplication by a scalar.

Now, let us go back to the general case. Assume X ⊂ Rn is a smoothmanifold with a smooth, transitive G-action we call a. Then G acts onD(X) by

(a(g)∗ · ν)x(X1, . . . , Xm) := νa(g−1)x(a(g−1)∗X1, . . . , a(g−1)∗Xm).

If x ∈ X, letGx := g ∈ G | g · x = x

denote the stabilizer of x in G. If x ∈ X and g ∈ Gx, then

(a(g)∗νx)(X1, . . . , Xm) = νx(a(g−1)∗X1, . . . , a(g−1)∗Xm)

= | det(a(g−1)∗)|νx(X1, . . . , Xm).

By (1), we may identify TxX ' g/gx.

Exercise 1.16. (a) If g ∈ Gx ⊂ G, check that the adjoint action,Ad(g), on g preserves gx and descends to an action of g on g/gx.

(b) Check that the induced action of Ad(g) for g ∈ Gx on g/gx is thesame as the action a(g)∗ on g/gx.

(c) Using part (b) deduce

| det(a(g−1)∗)| = δG(g−1)/δGx(g−1).

where δG denotes the modular function for G and δGx denote themodular function for Gx. Deduce that if X admits a non-zero G-invariant density, then

δG|Gx = δGx

for all x ∈ X.(d) Now, translate by an arbitrary g ∈ G. Check that X admits a

non-zero G-invariant density if, and only if for some (equivalentlyall) x ∈ X, we have

δG|Gx = δGx .

If this density exists, check that it is unique up to multiplicationby a scalar.

Exercise 1.15 and Exercise 1.16 imply the following key fact.


Proposition 1.17. Suppose X ⊂ Rn is a smooth manifold and G isa compact linear group acting smoothly and transitively on X. Thenthere exists a non-zero density ν ∈ D(X) that is invariant under theaction of G. Moreover, this density is unique up to multiplication by anon-zero complex scalar.

We say that a density ν ∈ D(X) is positive if

νx(X1, . . . , Xm) ∈ R>0

whenever X1, . . . , Xm ∈ TxX is a linearly independent set.

Exercise 1.18. (a) If ν ∈ D(X) is non-zero and G-invariant, showthat c · ν is positive for some scalar c ∈ C.

(b) If ν1, ν2 ∈ D(X) are positive and G-invariant, show ν1 = c · ν2 forsome c ∈ R>0.

Therefore, we may Proposition 1.17 also holds with “non-zero den-sity” replaced by “positive density” and “non-zero complex scalar” re-placed by “positive real scalar”.

2. Lecture 2: Characters of Compact Linear Groups

2.1. Representations of Compact Linear Groups.

Definition 2.1. A representation of a (real) linear group G is a pair(π, V ) where

(a) V is a topological complex vector space(b) π : G× V → V is a continuous group action of G on V .

We usually write π(g)v instead of π(g, v) for this group action.

Definition 2.2. A representation (π, V ) of a (real) linear group G isunitary if

(a) V is Hilbert space with a fixed inner product (·, ·).(b) The action of G on V preserves the inner product, ie

(π(g)v, π(g)w) = (v, w)

for all v, w ∈ V and g ∈ G.

Example 2.3. Let X ⊂ Rn be a smooth manifold with a smooth,transitive action of a linear group G, and let ν be a smooth, positiveG-invariant density on X. Then

L2(X) :=

f : X → C measurable |

∫X

|f(x)|2dν <∞


is a Hilbert space with inner product

(f1, f2) :=

∫X

f1(x)f2(x)dν.

Since ν is unique up to a positive scalar, L2(X) is independent of thechoice of ν. Further, there is a continuous action of G on L2(X) by

(lgf)(x) := f(g−1 · x).

Since ν is G-invariant, one checks that this action preserves the innerproduct on L2(X). In particular, (l, L2(X)) is a unitary representationof G.

Definition 2.4. We say that two unitary representations (π1, V1) and(π2, V2) of a linear group G are isomorphic if there exists a Hilbertspace isomorphism

T : V1∼−→ V2

satisfying T (π1(g)v1) = π2(g)T (v1).

Definition 2.5. If (π, V ) is a unitary representation of a linear groupG, then an invariant subspace of V is a closed subspace W ⊂ V suchthat π(g)w ∈ W for all g ∈ G and w ∈ W . We say that a unitaryrepresentation (π, V ) is irreducible if the only invariant subspaces of Vare 0 and V .

Suppose Vα, (·, ·)αα∈A is a family of Hilbert spaces with fixed innerproducts. Then we may form the inner product space⊕

α∈A

Vα.

This is the collection of sums∑

α∈A vα where vα ∈ Vα and vα = 0 forall but finitely many α. The inner product on the direct sum is givenby (∑

α∈A

vα,∑α∈A

wα

)=∑α∈A

(vα, wα)α .

The completion of this inner product is a Hilbert space which we denoteby

V :=⊕

α∈AVα.

Further, if (πα, Vα) is a unitary representation of a linear group G,then π :=

⊕πα acts on

⊕α∈A Vα preserving the inner product, and

then π extends to an action of G on V . This makes (π, V ) into aunitary representation of G.

Proposition 2.6. Let G be a compact linear group.


(a) If (π, V ) is an irreducible, unitary representation of G, then V isfinite dimensional.

(b) Let G denote the collection (isomorphism classes) of irreducible,unitary representations of G. If (π, V ) is a unitary representa-tion of G, then there exists a multiplicity m(τ, V ) ∈ N for every

(τ,Wτ ) ∈ G and a G-equivariant isomorphism

V∼−→⊕

(τ,Wτ )∈GW⊕m(τ,V )τ .

See for instance Section 3.2 of [Sep07] for proofs. We omit them here.

Exercise 2.7. (a) Suppose (π, V ) is a finite dimensional representa-tion of a compact linear group G. Show that (π, V ) is a unitaryrepresentation of G.(Hint: Fix an inner product (·, ·) on V . Then define a new innerproduct

(v, w)G :=

∫g∈G

(π(g)v, π(g)w)dν

where ν is a left and right G-invariant density on G. Show (·, ·)Gis a G-invariant inner product on V .)

(b) If (π, V ) is an irreducible, unitary representation of a compact lin-ear group G, show that the G-invariant inner product on V isunique up to multiplication by a positive scalar.(Hint: Define the contragradient representation of V to be V ∗, thecollection of complex linear functionals on V , with the group action

(π∗(g) · l)(v) := l(π(g−1)v).

Show that a G-invariant inner product (·, ·) gives rise to a G-equivariant, conjugate linear isomorphism V → V ∗. Given twoG-invariant inner products, one can compose one of these isomor-phisms with the inverse of the other to obtain an isomorphism ofrepresentations V → V . Show that this isomorphism has to bea scalar times the identity since (π, V ) is irreducible. Then workbackwards to show that the two inner products must be scalarmultiples of one another.)

In the next lecture, we will see how harmonic analysis on X is reallyabout decomposing L2(X) into irreducible unitary representations ofG. In order to write down this decomposition explicitly, we will needthe notion of a character.


2.2. Characters of Compact Linear Groups.

Definition 2.8. Let (π, V ) be an irreducible, unitary representationof the compact linear group G. The character of (π, V ) is defined tobe

Θπ(x) := Tr(π(x))

for x ∈ G.

Exercise 2.9. (a) Show

Θπ(gxg−1) = Θπ(x)

for all g, x ∈ G.(b) Show that Θπ is a smooth function on G.

Define SU(2) = A ∈ GL(2,C) | A∗A = I & detA = 1.Exercise 2.10. (a) Show

SU(2) =

(α −ββ α

)| α, β ∈ C, |α|2 + |β|2 = 1

.

(b) Show that the Lie algebra of SU(2) is

su(2) = A ∈ SU(2) | A∗ + A = 0

=

(ia β−β −ia

)| a ∈ R, β ∈ C

.

Let Pol(C2) denote the collection of polynomials on C2. More pre-cisely, we can name two variables, x and y, and Pol(C2) ' C[x, y]. LetPold(C2) denote the vector space of degree d complex-valued polyno-mials in the variables x and y. Now, SU(2) acts on Pold(C2) by

(πd(A) · p)(x, y) := p(A−1x,A−1y).

Exercise 2.11. Show that (πd,Pold(C2)) is a representation of SU(2).

Define

T :=

tθ :=

(eiθ 00 e−iθ

)| θ ∈ R

⊂ SU(2).

Observe T ' S1 is isomorphic to the unit circle.

Exercise 2.12. (a) Show that a basis for Pold(C2) is

xd, xd−1y, . . . , xyd−1, yd.(b) Show πd(tθ) · (xkyd−k) = ei(d−2k)θ. Conclude

Θπd(tθ) = eidθ + ei(d−2)θ + · · ·+ e−idθ

=ei(d+1)θ − e−i(d+1)θ

eiθ − e−iθ.


Exercise 2.13. (a) Show that every element x ∈ SU(2) may be writ-ten as x = gtθg

−1 for some tθ ∈ T .

Let νSU(2) denote the unique positive SU(2) left and right invariantdensity on SU(2) satisfying∫

SU(2)

dνSU(2) = 1.

Let νT denote the unique positive T left and right invariant density onT satisfying ∫

T

dνT = 1.

In coordinate, νT = |dθ|2π

.

Exercise 2.14. (a) Identify su(2) ' R3 via the coordinates(ix y + iz

−y + iz −ix

).

Let SU(2) act on su(2) ' R3 by the adjoint action (matrix conju-gation). Show that SU(2) acts transitively on the spheres x2 +y2 +z2 = r for r > 0.

(b) Show that the stabilizer of the point (1, 0, 0) is T . Therefore, wemay identify SU(2)/T ' S2 ⊂ R3. Deduce that there exists anSU(2)-invariant positive density νSU(2)/T on SU(2)/T ' S2 satisfy-ing ∫

SU(2)/T

dνSU(2)/T = 1.

Proposition 2.15 (Weyl). For all continuous functions f ∈ C(SU(2)),we have(3)∫SU(2)

f(x)dνSU(2)(x) =1

2

∫T

|(eiθ−e−iθ)|2∫SU(2)/T

f(gtθg−1)dνSU(2)/T (g)dνT (θ).

This formula is called Weyl’s integral formula for G = SU(2). DefineSU(2)′ := SU(2) \ e and T ′ := T \ e. Then the map

c : SU(2)/T × T ′ → SU(2)′

by (g, t) 7→ gtg−1 is a 2-1 local diffeomorphism. This explains the12

in the formula. Then one can pullback the density νSU(2) under c

and compute directly that the pullback is equal to |eiθ − e−iθ|2 timesνSU(2)/T × νT . We omit the details of the calculation. In Lecture 5, wewill discuss the generalization of this formula to an arbitrary compact,


connected Lie group. The function eiθ − e−iθ is especially interestingand important. More on this later.

Exercise 2.16. (a) Check directly that (3) holds for f(x) = 1.(b) Use the Weyl’s integral formula (3) to prove∫

SU(2)

|Θπd |2 = 1.

(c) Use Weyl’s integral formula (3) to prove∫SU(2)

ΘπmΘπn = 1

if m 6= n.

The calculations in the above exercise are a special case of a generalphenomenon. Let G be a compact linear group, and fix a G-invariant,positive density ν on G satisfying

∫Gdν = 1.

Proposition 2.17 (Schur Orthogonality for Characters). Suppose (π1, V1)and (π2, V2) are irreducible, unitary representations of a compact lineargroup G. Then

(4)

∫G

Θπ1Θπ2dν =

1 if π1 ' π20 if π1 6= π2.

We omit the proof. See for instance Section 4.2 of [Kna05] or Section3.1 of [Sep07] for proofs.

Exercise 2.18. (a) Assume π1, . . . , πk are distinct irreducible, unitaryrepresentations of G. One can define the character Θπ of

π 'k⊕i=1

π⊕mii

in the same way as in the irreducible case. Show Θπ =∑k

i=1miΘπi .(b) Utilize (4) to show ∫

G

|Θπ|2dν =k∑i=1

m2i .

(c) Use part (b) of this Exercise together with part (b) of Exercise2.16 to deduce that the representations (πd,Pold(C2)) of SU(2) areirreducible.


3. Lecture 3: Harmonic Analysis on Compact, ConnectedLinear Groups

3.1. Fourier Analysis on S1. Before discussing harmonic analysis oncompact, connected linear groups, let us discuss Fourier analysis on S1.We view

S1 :=eiθ | 0 ≤ θ < 2π

⊂ C×.

For every n ∈ Z, we have the one-dimensional representation (πn,Cn)of S1 by

πn(eiθ) · z := einθz.

Exercise 3.1. Show that every irreducible, unitary representation ofS1 is one-dimensional. (Hint: By Proposition 2.6, we know that everyirreducible, unitary representation of S1 is finite-dimensional. Now,if (π, V ) is a finite-dimensional irreducible, unitary representation ofS1 and dimV > 1, show that there must exist eiθ ∈ S1 such thatπ(eiθ) is not a scalar times the identity matrix. Then find a non-trivialeigenspace 0 6= W ( V . Show that W is an invariant subspace,contradicting the irreducibility of (π, V ).)

Notice that the character of πn is

Θn(eiθ) = einθ.

Next, assume G is a compact linear group. Let Diff(G) ⊂ End(C∞(G))denote the algebra of smooth differential operators on G. Notice thatG acts on C∞(G) on the left and right by

(L∗gf)(x) := f(g−1x), (R∗gf)(x) := f(xg).

Next, G acts on Diff(G) ⊂ End(C∞(G)) on the left and right by

((Lg)∗D)f := D(L∗g−1f), ((Rg)∗D)f := D(R∗g−1f).

We say that a differential operator D ∈ Diff(S1) is invariant if

(Lg)∗D = (Rg)∗D = D

for all g ∈ G. We denote the algebra of left and right G-invariantdifferential operators on G by Diff(G)G.

Exercise 3.2. (a) Show

Diff(S1)S1 ' C

[d

dθ

]'

N∑k=0

akdk

dθk| ak ∈ C, N ∈ N

.


(Hint: Observe Diff(S1) '∑N

k=0 fkdk

dθk| fk ∈ C∞(S1), N ∈ N

.

Translate on the left or right by an arbitrary eiϕ and then induc-tively apply the differential operator to functions that are “locally”equal to 1, θ, θ2, etc.)

(b) Show by direct calculation that the characters Θn = einθ are all

eigenfunctions for every differential operator D ∈ Diff(S1)S1.

(c) Suppose (π, V ) is an irreducible unitary representation of S1 withcharacter Θ, and assume that the action of S1 on V is smooth.Show that Θ is an eigenfunction for d

dθ.

(d) Show that all eigenfunctions Θ of ddθ

on S1 are of the form Ceinθ

for some C ∈ C and n ∈ Z.(e) Deduce that every (smooth) irreducible character of S1 is equal to

Θn for some n ∈ Z.

Theorem 3.3 (Abstract Harmonic Analysis on S1). There is an iso-morphism of unitary representation of S1

(5) L2(S1)∼−→⊕n∈Z

Cn.

If G is a compact linear group and f, g ∈ C(G) are two continuousfunctions, then we define the convolution of f and g to be

(f ∗ g)(x) :=

∫G

f(xy−1)g(y)dν(y) =

∫G

f(y)g(y−1x)dν(y)

where ν is a positive density on G satisfying∫Gdν = 1.

Theorem 3.4 (Fourier Analysis on S1). If f ∈ C∞(S1), then

(6) f =∑n∈Z

f ∗Θn.

The sum converges absolutely.

Exercise 3.5. Show f ∗ Θn = aneinθ for some constant an ∈ C. We

call an the nth Fourier coefficient of f .(Hint: Differentiate f ∗Θn with respect to d

dθ.)

One observes that the decompositions (5) and (6) diagonalize the

action of the algebra Diff(S1)S1 ' C

[ddθ

]on L2(X).

One can find proofs of Theorem 3.4 and Theorem 3.3 in many places.For instance, see Chapter 8 of [Fol07].


3.2. Harmonic Analysis on Compact Linear Groups. Let G bea compact linear group. Fix a positive G-invariant density ν on Gsatisfying

∫Gdν = 1. Recall

L2(G) :=

f : G→ C measurable |

∫G

|f(x)|2dν(x) <∞

is a Hilbert space with inner product

(f1, f2) :=

∫G

f1(x)f2(x)dν(x).

Further, G acts on L2(G) on the left by

(lg · f)(x) := f(g−1x).

And G acts on L2(G) on the right by

(rg · f)(x) := f(xg).

Exercise 3.6. Check that both the left and right action of G on L2(G)preserve the inner product. More precisely, show

(lgf1, lgf2) = (f1, f1) for all g ∈ G, f1, f2 ∈ L2(G)

and

(rgf1, rgf2) = (f1, f1) for all g ∈ G, f1, f2 ∈ L2(G).

The actions l and r commute. Therefore, (l × r, L2(G)) is a unitaryrepresentation of G×G.

Exercise 3.7. Suppose (σ, U) and (τ,W ) are two irreducible, unitaryrepresentations of G.

(a) Show (σ ⊗ τ, U ⊗W ) is a unitary representation of G×G.(b) Show (σ ⊗ τ, U ⊗W ) is an irreducible representation of G×G.

(Hint: Suppose V ⊂ U ⊗W is a non-trivial, proper invariant sub-space. Restrict V to a representation G× e ⊂ G×G and showV ' U ⊗W1 for some 0 6= W1 ( W . Observe that W1 is notinvariant under e ×G and deduce a contradiction.)

Exercise 3.8. Show that every irreducible, unitary representation ofG × G is of the form (σ ⊗ τ, U ⊗W ) for two irreducible, unitary rep-resentations (σ, U) and (τ,W ) of G.(Hint: Assume (π, V ) is an irreducible, unitary representation of G×G.First, restrict to G × e and show V ' U ⊗W where U is an irre-ducible, unitary representation of G × e and G × e acts triviallyon W . Then restrict to e ×G.)


If (π, V ) is a unitary representation of G, then the contragradientrepresentation is (π∗, V ∗) where

V ∗ := l : V −→ C complex linearis the dual space of V and

(π∗(g)l)(v) := l(π(g−1)v).

By Proposition 2.6, we know that there is a decomposition of L2(G)into irreducible, unitary representations of G×G. The precise descrip-tion is due to Peter and Weyl.

Theorem 3.9 (Abstract Harmonic Analysis on Compact Linear Groups).There is an isomorphism of unitary representations of G×G

L2(G)∼−→⊕

(τ,Wτ )∈GWτ ⊗W ∗

τ .

The Peter-Weyl Theorem describes abstract harmonic analysis on

L2(G). For a proof, see for instance ?? and ??. If (τ,Wτ ) ∈ G, letΘτ (g) := Tr(τ(g)) denote the character of (τ,Wτ ).

Theorem 3.10 (Fourier Analysis on Compact Linear Groups). Let Gbe a compact linear group. If f ∈ C∞(G), then

f =∑

(τ,Wτ )∈G

(dimWτ )f ∗Θτ .


For a proof, see for instance ?? of [?]. As before, let Diff(G)G

denote the left and right G-invariant differential operators on G. First,this algebra of differential operators is commutative. Second, for everyirreducible character Θτ of G, there exists an algebra homomorphism

χτ : Diff(G)G → Csuch that

(7) DΘτ = χτ (D)Θτ .

Finally, if G is a compact, connected linear group, then the map τ 7→ χτis injective. In particular, these facts imply that the decomposition(3.10) diagonalizes the action of Diff(G)G on C∞(G). Unfortunately,we do not have time to prove these facts here.

Let X ⊂ Rn be an m dimensional smooth submanifold, and assumethat G acts smoothly and transitively on X. Let ν denote a G-invariantdensity on X with

∫Xdν = 1, and define

L2(X) :=

f : X → C measurable |

∫X

|f(x)|2dν(x) <∞


to be the Hilbert space with inner product

(f1, f2) :=

∫X

f1(x)f2(x)dν(x).

Notice that G acts on L2(X) by

(g · f)(x) := f(g−1 · x).

Exercise 3.11. Check that this action of G on L2(X) defines a unitaryrepresentation of G on L2(X).

Fix x ∈ X, and define

H := Gx = g ∈ G | g · x = x

be the stabilizer of x in G. Then the pullback of the map

G→ G/H ' X

yields a map

ιX : L2(X)→ L2(G).

Exercise 3.12. (a) Let L2(G)H denote the collection of L2-functionson G that are invariant under the right action of H. Show

ιX(L2(X)) ⊂ L2(G)H .

(b) Notice that L2(G)H is a unitary representation of G under the leftaction of G. Show that ιX is an isomorphism of L2(X) onto L2(G)H

as G-representations.

We may utilize Exercise 3.12 to give an analogue of Theorem 3.9 forthe homogeneous space X.

Corollary 3.13 (Abstract Harmonic Analysis on Compact Homoge-neous Spaces). Suppose X ⊂ Rn is a smooth submanifold, suppose G isa compact linear group acting smoothly and transitively on X, and writeX ' G/H. Then there is an isomorphism of unitary G-representations

L2(X)∼−→

⊕(τ,Wτ )∈G

Wτ ⊗ (W ∗τ )H .

Exercise 3.14. Deduce Corollary 3.13 from Theorem 3.9 and Exercise3.12.

Corollary 3.15 (Fourier Analysis on Compact Homogeneous Spaces).Suppose X ⊂ Rn is a smooth submanifold, suppose G is a compact


linear group acting smoothly and transitively on X, and write X 'G/H. If f ∈ C∞(X) ' C∞(G)H , then

f =∑

(τ,Wτ )∈G

(dimWτ )f ∗Θτ .


Exercise 3.16. If f ∈ C∞(X) ' C∞(G)H , show

f ∗Θτ = Θτ ∗ f ∈ C∞(G)H ' C∞(X).

In particular, the Fourier coefficients of f ∈ C∞(X) are functions onX.

3.3. The Example G = SU(2) and X = S2. Recall G = SU(2) actson its Lie algebra su(2) ' R3 by the adjoint (conjugation) action. Asin Exercise 2.14, we may impose coordinates on R3 so that the orbitsof G = SU(2) are the spheres x2 + y2 + z2 = r. Take the unit sphereX = S2.

Exercise 3.17. Show that the stabilizer of (1, 0, 0) under the actionof SU(2) on S2 is T ' S1.

First, let us decompose L2(X) into irreducible representations ofSU(2) abstractly. Recall from Lecture 2 that we constructed the irre-ducible representations (πd,Pold(C2)) of SU(2). Let us shorten thenotation and write Vd := Pold(C2). It turns out that every irre-ducible, unitary representation of SU(2) is of the form (πd, Vd) for somed = 0, 1, 2, . . .. Then, by Corollary 3.13, we have an isomorphism ofSU(2)-representations

L2(S2) '⊕

d∈N∪0

Vd ⊗ (V ∗d )T .

Exercise 3.18. (a) Use Exercise 2.12 to show (Vd)T 6= 0 if, and only

if d ∈ 2N ∪ 0 is even. Further, when d is even, show (Vd)T is a

one-dimensional space.(b) Show that Vd irreducible implies V ∗d is irreducible. Since Vd is the

unique irreducible, unitary representation of SU(2) of dimension d,conclude Vd ' V ∗d . Conclude (V ∗d )T 6= 0 if, and only if d ∈ 2N∪0.Further, when d is even, conclude that (V ∗d )T is one-dimensional.

The above exercise implies that there is an abstract isomorphism ofunitary G-representations

L2(S2) '⊕

d∈2N∪0

Vd.


Next, we wish to write down the Fourier coefficients of f ∈ C∞(S2).We know from Corollary 3.15 that our decomposition looks like

(8) f =∑

d∈2N∪0

f ∗Θd.

Recall from Exercise 2.12 that one can write down Θd more explicitlyas

Θd(gtθg−1) =

ei(d+1)θ − e−i(d+1)θ

eiθ − e−iθfor all g ∈ G = SU(2).

Finally, let us consider Diff(SU(2))SU(2).

Exercise 3.19. Show that SU(2) has a left and right SU(2)-invariant,Riemannian structure.(Hint: Fix a positive definite form B1(·, ·) on g ' TeG. Now, average B1

over G = SU(2) against the conjugation action to form an Ad(SU(2))invariant positive definite form B(·, ·) on g ' TeG. Translate this formaround SU(2) to give a left and right invariant Riemannian structureon SU(2)).

Next, we wish to form the Laplacian ∆SU(2) on SU(2) with respect tothis Riemannian metric. The Riemannian structure B(·, ·) on g yieldsisomorphisms

g⊗ g ' g⊗ g∗ ' HomC(g, g).

These isomorphisms are invariant under all linear transformations thatpreserve B(·, ·). Therefore, the identity element I in Hom(g, g) corre-sponds to an element in T ∈ g ⊗ g. If X1, X2, X3 is an orthonormalbasis of g, then this element is

X1 ⊗X1 +X2 ⊗X2 +X3 ⊗X3.

This element is invariant under all transformations that preserve B(·, ·).Then there is a map g ⊗ g → Diff(g) by X ⊗ Y 7→ d

dXddY

. The imageof T ∈ g⊗ g in orthonormal coordinates is

d2

dX21

+d2

dX22

+d2

dX23

.

Once we translate this over all of SU(2), we obtain the Laplacian ∆SU(2),and one sees that this differential operator is invariant under left andright translation by SU(2).

Proposition 3.20. We have

Diff(SU(2))SU(2) ' C[∆SU(2)].


Let ∆S2 denote the Laplacian on the sphere S2. It is the case thatDiff(S2)SU(2) ' C[∆S2 ]. Moreover, ∆S2 acts on each component of thedecomposition (8) by a different scalar. Therefore, the decomposition(8) diagonalizes the action of ∆S2 .

4. Lecture 4: Characters and Coadjoint Orbits

4.1. Coadjoint Orbits. Let G be a compact, connected linear groupwith Lie algebra g. Recall G acts on g by the adjoint (conjugation)action. Then G acts on

√−1g∗ := HomR(g, iR) by the dual action

(Ad∗(g) · ξ)(X) := ξ(Ad(g−1)X).

An orbit for this coadjoint action is called a coadjoint orbit for G. LetPol(√−1g∗) denote the collection of polynomials on

√−1g∗. The group

G acts on polynomials by

(g · p)(ξ) := ξ(Ad∗(g−1) · ξ).Let Pol(

√−1g∗)G denote the collection of G-invariant polynomials on√

−1g∗. If ξ ∈√−1g∗, define

Ωξ :=η ∈√−1g∗ | p(ξ) = p(η) for all p ∈ Pol(

√−1g∗)G

.

One checks that Ωξ is a G-invariant set. In particular, it is a union ofcoadjoint orbits. In fact, we always have that

(9) Ωξ = Oξ = Ad∗(G) · ξis a single coadjoint orbit.

Next, define

ad∗X ξ :=d

dt

∣∣∣t=0

Ad∗(exp(tX)) · ξ

for ξ ∈√−1g∗. Then

TξOξ ' ad∗X ξ | X ∈ g .The Kirillov-Kostant two form ω on Oξ is defined to be

ωξ(ad∗X ξ, ad∗Y ξ) := ξ([X, Y ])

where [X, Y ] := XY − Y X is the bracket operation on the Lie algebrag. It turns out ω is G-invariant and it gives O the structure of asymplectic manifold for every ξ ∈

√−1g∗. In particular, for every

ξ ∈√−1g∗, dimOξ = 2m is even dimensional. Further,

(10) νξ :=

∣∣∣∣ ω∧mξ

2π√−1

∣∣∣∣is a G-invariant density on Oξ.


Example 4.1. Add example SU(2).

Let V be a real, finite-dimensional vector space, and define√−1V ∗ := HomR(V,

√−1R).

Let M ⊂√−1V ∗ be a compact, smooth submanifold with a positive,

smooth density ν. Define the Fourier transform of M to be

F [M](X) :=

∫Me〈ξ,X〉dν(ξ).

Exercise 4.2. Show F [M] is a smooth function on V .(Hint: Differentiate under the integral sign).

Exercise 4.3. Suppose there is a compact, connected linear groupG acting smoothly and transitively on M. Show that F [M] is anAd∗(G)-invariant smooth function on V .

Exercise 4.4. (a) If v ∈ V , show

d

dvF [M](X) =

∫Me〈ξ,X〉〈ξ, v〉dν(ξ).

(b) In the first exercise, we have tacitly identified the vector v ∈ Vwith the differential operator d

dvon V and the degree one polyno-

mial 〈·, v〉 on√−1V ∗. Show that this identification extends to an

isomorphismPol(√−1V ∗)

∼−→ Diff0(V ).

where Diff0(V ) denote the collection of constant coefficient differ-ential operators on

√−1V ∗. We denote this map

p 7→ ∂(p).

(c) If p ∈ Pol(√−1V ∗), then

(∂(p)F [M])(X) =

∫Me〈ξ,X〉p(ξ)dν(ξ).

Let us return to the terminology of Section 4.1. Put M = Oξ, acoadjoint orbit for a compact linear group G, let ν be the density onOξ defined in (10), and put g = V . Let Diff0(g)G denote the alge-bra of constant coefficient Ad(G)-invariant differential operators on g.Exercise 4.2 and Exercise 4.3 imply that

F [Oξ]is a smooth, Ad(G)-invariant function on g. Further, Exercise (4.4)implies that there exists an algebra homomorphism

χξ : Diff0(g)G → C


such thatDF [Oξ] = χξ(D)F [Oξ].

Add Example: SU(2).

4.2. The Kirillov/Harish-Chandra Character Formula. Let Gbe a compact linear group, let g denote the Lie algebra of G, and letexp: g → G be the exponential map. Let νG denote a positive, leftand right G-invariant density on G, and let |dX| denote a positive,translation invariant density on g. One can pullback the density νG onG to a density exp∗ νG on g defined by

(exp∗ νG)(X1, . . . , Xn) := νG(exp∗X1, . . . , exp∗Xn)

if n = dim g and X1, . . . , Xn ∈ g. Since νG is a smooth, positive densityon G, exp∗ νG is a smooth, positive density on g. Therefore, there existsa smooth function jG on g such that

exp∗ νG = jG(X)|dX|.We multiply |dX| by a positive constant so that jG(0) = 1.

Exercise 4.5. Show jG(X) is an Ad(G)-invariant function.

Exercise 4.6. If G = SU(2), observe

t =

Xθ :=

(iθ 00 −iθ

)| θ ∈ R

is the Lie algebra of T . Show that for all X ∈ g = su(2), there existsg ∈ G and θ ∈ R such that

X = Ad(g)Xθ.

Example 4.7. If G = SU(2), then

jG(Ad(g)Xθ) =

(eiθ − e−iθ

iθ

)2

.

Lemma 4.8. There exists a unique analytic square root j1/2G of jG

satisfying j1/2G (0) = 1.

Example 4.9. If G = SU(2), then

j1/2G (Ad(g)Xθ) =

eiθ − e−iθ

iθ.

Definition 4.10. An invariant eigenfunction on G is a smooth func-tion Θ ∈ C∞(G) satisfying the following two properties.

(1) For all g, x ∈ G, we have

Θ(gxg−1) = Θ(g).


(2) There exists an algebra homomorphism

χG : Diff(G)G → Csatisfying

D ·Θ = χG(D)Θ

for all D ∈ Diff(G)G.

Definition 4.11. An invariant eigenfunction on g is a smooth functionθ ∈ C∞(g) satisfying the following two properties.

(1) For all g ∈ G and X ∈ g, we have

θ(Ad(g)X) = θ(X).

(2) There exists an algebra homomorphism

χg : Diff0(g)G → Csatisfying

D · θ = χg(D)θ

for all D ∈ Diff0(g)G.

Exercise 4.12. Construct an isomorphism

φ : Diff0(g)G → Diff(G)G.

(Hint: Move the Ad(G)-invariant differential operator on g ' TeGaround the Lie group G using left or right translation.)

Theorem 4.13 (Harish-Chandra). Let G be a compact linear groupwith Lie algebra g. If Θ is an invariant eigenfunction on G, then

θ := (Θ exp) · j1/2G

is an invariant eigenfunction on g.

Let (π, V ) be a finite dimensional representation of a compact lineargroup G, and let Θπ denote the character of π. Then Θπ is an invarianteigenfunction on G as we saw in Lecture 2. Now, we call

θπ := (Θπ exp) · j1/2G

the Lie algebra analogue of the character of π. By Theorem 4.13, θπis an invariant eigenfunction on g.

Exercise 4.14. Lie algebra analogue of characters of SU(2).

Theorem 4.15 (Harish-Chandra, Kirillov). Let G be a compact lineargroup. For every irreducible unitary representation (π, V ) of G, thereexists a coadjoint orbit Oπ ⊂

√−1g∗ such that

θπ = F [Oπ].

Exercise 4.16. SU(2) example


References

[Fol07] G.B. Folland, Real analysis, Wiley, 2007.[Kna05] A. Knapp, Lie groups beyond an introduction, Progress in Mathematics,

vol. 140, Birkhauser, Boston, 2005.[Lee12] J. Lee, Introduction to smooth manifolds, Graduate Texts in Mathematics,

vol. 218, Springer, 2012.[Sep07] M. Sepanski, Compact Lie groups, Graduate Texts in Mathematics, vol.

235, Springer, 2007.

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Lecture 1: Lie Groups, Lie Algebras, and Homogeneous Spacesbenjaminlharris.weebly.com/uploads/1/0/9/...notes.pdf · 1. Lecture 1: Lie Groups, Lie Algebras, and Homogeneous Spaces