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Haar measure Schur orthogonality characters

Lecture 2: Schur Orthogonality

Daniel Bump

April 16, 2020 Haar measure Schur orthogonality characters

Compact Groups

In this lecture we will talk about results that are valid for compact groups, whether Lie groups or other. Compact groups include: Compact Lie groups such as U(n) or O(n); Finite groups;

Profinite groups such as GL(n, Zp) where Zp is the ring of p-adic integers, or (possibly infinite) Galois groups. For compact groups, the is very similar to finite groups. According to the Peter-Weyl theorem, every irreducible representation is finite-dimensional and unitary, and every unitary representation is a direct sum of irreducible representations. These properties fail for noncompact groups such as GL(n, R). Haar measure Schur orthogonality characters

Haar measure

Let G be a locally compact topological space with a Borel measure µ. The measure µ is called regular if for every measurable subset X we have

µ(X) = inf {µ(U)|U ⊇ X, U open} = sup {µ(K)|K ⊆ X, K compact} .

A theorem that we will take for granted is that if G is a locally compact topological , then G admits a regular Borel measure dµL that is invariant under left translation. Compact sets have finite volume in this left Haar measure.

Similarly there is a right Haar measure dµR. Haar measure Schur orthogonality characters

Unimodularity

Left and right Haar measures are the same in many important cases, but for an example where they are different, let

 y x   G = |x, y ∈ , y > 0 . 1 R

Then −2 −1 dµL = y dx dy, dµL = y dx dy. On the other hand, left and right Haar measures are equal for compact groups, and for many other important groups such as GL(n, R) or SL(n, R), O(n), Sp(2n, R). The group G is unimodular if µL = µR. Haar measure Schur orthogonality characters

The modular quasicharacter

If G is a locally a quasicharacter χ is a × continuous homomorphism χ : G −→ C . It is a (or is unitary) if |χ(g)| = 1. Proposition Let G be a locally compact group. There is a quasicharacter × δ : G −→ R+ such that

dµL = δ(g) dµR.

This δ is called the modular quasicharacter Haar measure Schur orthogonality characters

Proof

Indeed, conjugation is an automorphism of G, so it takes left Haar measure to another left Haar measure. Thus there exists a constant δ(g) > 0 such that Z Z −1 f (g hg) dµL(h) = δ(g) f (h) dµL(h). G G

From this formula it is easy to check that δ(g1g2) = δ(g1)δ(g2). Since dµL is left invariant this means Z Z f (hg) dµL(h) = δ(g) f (h) dµL(h) G G

so δ(g)µL is a right Haar measure. Haar measure Schur orthogonality characters

Compact groups are unimodular

If G is compact, it is natural to normalize the measure so that µL(G) = 1. Proposition A compact group is unimodular.

To prove that left and right Haar measures are equal on a compact group G consider note that δ(G) is a compact × subgroup of R+. The only such compact subgroup is {1}, so µL is trivial and therefore G is unimodular. Haar measure Schur orthogonality characters

Viewpoint

The representation theory of a compact group is so similar to the representation theory of a finite group that we may say that the main difference is that the general compact group will have infinitely many irreducible representations.

Schur orthogonality has two aspects. coefficients of irreducible representations give an orthonormal of L2(G); Characters of irreducible representations give an of class functions in L2(G). Haar measure Schur orthogonality characters

Maschke’s Theorem

Let (π, V) be a finite-dimensional representation of the compact group G. By an inner product on a we mean a positive definite Hermitian form hx, yi. Theorem Then V has an inner product h, i that is invariant under the group action, i.e.

hπ(g)x, π(g)yi = hx, yi.

The representation (π, V) is completely reducible, meaning that V is a direct sum of irreducible submodules. Haar measure Schur orthogonality characters

Proof

To prove this, we start with an arbitrary inner product hh , ii on V. We make it invariant by averaging: Z hx, yi = hhπ(g)x, π(g)yii dg. G It is easy to check that this is an inner product, and by averaging we have made it invariant, i.e. hπ(g)x, π(g)yi = hx, yi.

Now let W be a nonzero invariant subspace of V of minimal . Obviously W is irreducible. The W⊥ of W is also invariant and by induction it is completely reducible. Hence V = W ⊕ W⊥ is completely reducible. Haar measure Schur orthogonality characters

Matrix coefficients

Fix a representation (typically irreducible) (π, V). Let v ∈ V and L ∈ V∗. The φ(g) = Lπ(g) v is called a matrix coefficient of π. This terminology is natural, because if we n choose a basis e1,..., en, of V, we can identify V with C and represent g by matrices:       π11(g) ··· π1n(g) a1 a1 π(g)v =  . .   .  , v =  .  = Pn a e  . .   .   .  j=1 j j . πn1(g) ··· πnn(g) an an

2 Then each of the n functions πij is a matrix coefficient. Indeed  πij(g) = Li π(g)ej , P  where Li j ajej = ai. Haar measure Schur orthogonality characters

Representative functions

A finite of matrix coefficients (of different representations) is called a representative function. Proposition Representative functions on G are continuous functions. The pointwise sum or product of two representative functions is a representative function, so they form a ring.

Representative functions are continuous since representations are continuous. If f1 and f2 are matrix coefficients of (π1, V1) and (π2, V2) then c1f1 + c2f2 is a matrix coefficient of V1 ⊕ V2 and f1f2 is a matrix coefficient of V1 ⊗ V2, from which it follows that the space of representative functions generated by matrix coefficients is closed under linear combinations and multiplication. Haar measure Schur orthogonality characters

Representative functions (continued)

We have actions of G on the space of functions on G by left and right translation. Thus if f is a function and g ∈ G, the left and right translates are

λ(g)f (x) = f (g−1x), ρ(g)f (x) = f (xg).

Theorem Let f be a function on G. The following are equivalent. (i) The functions λ(g)f span a finite-dimensional vector space. (ii) The functions ρ(g)f span a finite-dimensional vector space. (iii) The function f is a representative function. Haar measure Schur orthogonality characters

Proofs

It is easy to check that if f is a matrix coefficient of a particular representation V, then so are λ(g)f and ρ(g)f for any g ∈ G. Since V is finite-dimensional, its matrix coefficients span a finite-dimensional vector space, Thus, (iii) implies (i) and (ii).

We show (ii) =⇒ (iii); (i) is similar. Suppose that the functions ρ(g)f span a finite-dimensional vector space V. Then (ρ, V) is a finite-dimensional representation of G, and we claim that f is a matrix coefficient of this representation. Indeed, define a  functional L : V −→ C by L(φ) = φ(1). Clearly, L ρ(g)f = f (g), so f is a matrix coefficient, as required. Haar measure Schur orthogonality characters

Intertwining operators

If (π1, V1) and (π2, V2) are representations, an intertwining , also known as a G-equivariant map T : V1 −→ V2 or (since V1 and V2 are sometimes called G-modules) a G- homomorphism, is a linear transformation T : V1 −→ V2 such that T ◦ π1(g) = π2(g) ◦ T

for g ∈ G. We will denote by HomC(V1, V2) the space of all linear transformations V1 −→ V2 and by HomG(V1, V2) the subspace of those that are intertwining maps. Haar measure Schur orthogonality characters

Schur’s Lemma

Theorem (Schur’s lemma)

(i) Let (π1, V1) and (π2, V2) be irreducible representations, and let T : V1 −→ V2 be an intertwining operator. Then either T is zero or else it is an isomorphism. (ii) Suppose that (π, V) is an irreducible representation of G and T : V −→ V is an intertwining operator. Then there exists a λ ∈ C such that T(v) = λv for all v ∈ V. This is proved the same way as for representations of finite groups. Haar measure Schur orthogonality characters

L2(G)

We will consider the space L2(G) of functions on G that are square-integrable with respect to the Haar measure. This is a with the inner product Z hf1, f2iL2 = f1(g) f2(g) dg. G

Schur orthogonality will gives an orthonormal basis for L2(G) of matrix coefficients of irreducible representations.

If (π, V) is a representation and h , i is an invariant inner product on V, then every linear functional is of the form x −→ hx, vi for some v ∈ V. Thus a matrix coefficient of π may be written in the form g −→ hπ(g)w, vi. Haar measure Schur orthogonality characters

Averaging to obtain an intertwining operator

Lemma

Suppose that (π1, V1) and (π2, V2) are complex representations of the compact group G. Let h , i be any inner product on V1. If vi, wi ∈ Vi, then the map T : V1 −→ V2 defined by Z −1 T(w) = hπ1(g)w, v1i π2(g )v2 dg (1) G is G-equivariant.

Z  −1 T π1(h)w = hπ1(gh)w, v1i π2(g )v2 dg. G The variable change g −→ gh−1 shows that this equals π2(h)T(w), as required. Haar measure Schur orthogonality characters

Schur Orthogonality

Theorem (Schur orthogonality)

Suppose that (π1, V1) and (π2, V2) are irreducible representations of the compact group G. Either every matrix 2 coefficient of π1 is orthogonal in L (G) to every matrix coefficient of π2, or the representations are isomorphic.

Suppose matrix coefficients fi : G −→ C of πi are not orthogonal. Define T : V1 −→ V2 as above. Then hT(w1), w2i= 6 0 equals

R −1 R G hπ1(g)w1, v1i π2(g )v2, w2 dg = G hπ1(g)w1, v1i hπ2(g)w2, v2i dg By assumption this is nonzero so T 6= 0. It is an isomorphism by Schur’s lemma. Haar measure Schur orthogonality characters

Schur Orthogonality (continued)

So matrix coefficients for nonisomorphic irreducibles are orthogonal. For two matrix coefficients of the same representation: Theorem (Schur orthogonality)

Let (π, V) be an irreducible representation of the compact group G, with invariant inner product h , i . Then

Z 1 hπ(g)w1, v1i hπ(g)w2, v2i dg = hw1, w2i hv2, v1i . (2) G dim(V)

See Theorem 2.4 in our book for the proof. It uses the same method as the first statement of Schur orthogonality, but is slightly longer; particularly showing that the proportionality constant is 1/ dim(V) requires some work. Haar measure Schur orthogonality characters

An orthonormal system

Let (πλ, Vλ) be representatives of the isomorphism classes of irreducible representations. Let dλ = dim(Vλ). For each Vλ λ choose an inner product and an orthonormal basis vi (i = 1, ··· , dλ). Then it follows from the last two theorems that

λ p λ λ fi,j(g) = dλhπλ(g)vi , vj i

are an orthonormal system, consisting of representative functions. Theorem (Peter-Weyl) This orthonormal set is complete. That is, it spans L2(G) as a Hilbert space.

This is proved in Chapter 4 of the book. We won’t take the time to do it in the course. Haar measure Schur orthogonality characters

Class functions

To repeat: Schur orthogonality has two aspects. Matrix coefficients of irreducible representations give an orthonormal basis of L2(G); Characters of irreducible representations give an orthonormal basis of class functions in L2(G). A class function f (g) is one that is constant on conjugacy classes. The character of a representation is a class function.

The character χ of a representation (π, V) is a sum of matrix coefficients of V, so it is a representative function. Haar measure Schur orthogonality characters

Orthonormality of characters

Let (π1, V1) and (π2, V2) be the characters of two irreducible representations, and let χi be their characters. Theorem Z  ∼ 1 if π1 = π2; χ1(g)χ2(g) dg = G 0 otherwise

If π1 and π2 are not isomorphic, then χi are (sums of) matrix coefficients of nonisomorphic representations, and therefore the characters are orthogonal. If the representations are isomorphic, then some work is required to determine that Z 2 |χi(g)| dg = 1.

See Theorem 2.5 in the book.