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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 representation theory 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 fµ(U)jU ⊇ X; U openg = sup fµ(K)jK ⊆ X; K compactg : A theorem that we will take for granted is that if G is a locally compact topological group, 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 = jx; y 2 ; 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 compact group a quasicharacter χ is a × continuous homomorphism χ : G −! C . It is a character (or is unitary) if jχ(g)j = 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 f1g, 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. 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). 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 vector space 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 dimension. Obviously W is irreducible. The orthogonal complement 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 2 V and L 2 V∗. The function 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: 0 1 0 1 0 1 π11(g) ··· π1n(g) a1 a1 π(g)v = B . C B . C ; v = B . C = Pn a e @ . A @ . A @ . A 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 linear combination 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 2 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 2 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 operator, also known as a G-equivariant map T : V1 −! V2 or (since V1 and V2 are sometimes called G-modules) a G-module homomorphism, is a linear transformation T : V1 −! V2 such that T ◦ π1(g) = π2(g) ◦ T for g 2 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 scalar λ 2 C such that T(v) = λv for all v 2 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 Hilbert space 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 2 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 2 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.
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