Schur Orthogonality

Abstract

In this chapter and the next two, we will consider the representation theory of compact groups. Let us begin with a few observations about this theory and its relationship to some related theories.

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In this chapter and the next two, we will consider the representation theory of compact groups. Let us begin with a few observations about this theory and its relationship to some related theories.

If V is a finite-dimensional complex vector space, or more generally a Banach space, and $\pi: G\longrightarrow \mathrm{GL}(V )$ a continuous homomorphism, then (π,V) is called a representation. Assuming $\dim (V ) < \infty$, the function $\chi _{\pi }(g) = \text{tr}\;\pi (g)$ is called the character of π. Also assuming dim(V) < ∞, the representation (π,V) is called irreducible if V has no proper nonzero invariant subspaces, and a character is called irreducible if it is a character of an irreducible representation.

[If V is an infinite-dimensional topological vector space, then (π,V) is called irreducible if it has no proper nonzero invariant closed subspaces.]

A quasicharacter χ is a character in this sense since we can take $V = \mathbb{C}$ and π(g)v = χ(g)v to obtain a representation whose character is χ.

The archetypal compact Abelian group is the circle $\mathbb{T} =\big\{ z \in {\mathbb{C}}^{\times }\,\big\vert \,\vert z\vert = 1\big\}.$ We normalize the Haar measure on $\mathbb{T}$ so that it has volume 1. Its characters are the functions $\chi _{n}: \mathbb{T}\longrightarrow {\mathbb{C}}^{\times }$, $\chi _{n}(z) = {z}^{n}$. The important properties of the χ _n are that they form an orthonormal system and (deeper) an orthonormal basis of ${L}^{2}(\mathbb{T})$.

More generally, if G is a compact Abelian group, the characters of G form an orthonormal basis of L ²(G). If f ∈ L ²(G), we have a Fourier expansion,

$$\displaystyle{ f(g) =\sum _{\chi }a_{\chi }\,\chi (g),a_{\chi } =\int _{G}f(g)\overline{\chi (g)}\,\mathrm{d}g, }$$

(2.1)

and the Plancherel formula is the identity:

$$\displaystyle{ \int _{G}\vert f(g){\vert }^{2}\,\mathrm{d}g =\sum _{ \chi }\vert a_{\chi }{\vert }^{2}. }$$

(2.2)

These facts can be directly generalized in two ways. First, Fourier analysis on locally compact Abelian groups, including Pontriagin duality, Fourier inversion, the Plancherel formula, etc. is an important and complete theory due to Weil [169] and discussed, for example, in Rudin [140] or Loomis [121]. The most important difference from the compact case is that the characters can vary continuously. The characters themselves form a group, the dual group $\hat{G}$, whose topology is that of uniform convergence on compact sets. The Fourier expansion (2.1) is replaced by the Fourier inversion formula

$$\displaystyle{f(g) =\int _{\hat{G}}\hat{f}(\chi )\,\chi (g)\,\mathrm{d}\chi,\quad\hat{f}(\chi ) =\int _{G}f(g)\,\overline{\chi (g)}\,\mathrm{d}g.}$$

The symmetry between G and $\hat{G}$ is now evident. Similarly in the Plancherel formula (2.2) the sum on the right is replaced by an integral.

The second generalization, to arbitrary compact groups, is the subject of this chapter and the next two. In summary, group representation theory gives a orthonormal basis of L ²(G) in the matrix coefficients of irreducible representations of G and a (more important and very canonical) orthonormal basis of the subspace of L ²(G) consisting of class functions in terms of the characters of the irreducible representations. Most importantly, the irreducible representations are all finite-dimensional. The orthonormality of these sets is Schur orthogonality; the completeness is the Peter–Weyl theorem.

These two directions of generalization can be unified. Harmonic analysis on locally compact groups agrees with representation theory. The Fourier inversion formula and the Plancherel formula now involve the matrix coefficients of the irreducible unitary representations, which may occur in continuous families and are usually infinite-dimensional. This field of mathematics, largely created by Harish-Chandra, is fundamental but beyond the scope of this book. See Knapp [104] for an extended introduction, and Gelfand, Graev and Piatetski-Shapiro [55] and Varadarajan [165] for the Plancherel formula for $\mathrm{SL}(2, \mathbb{R})$.

Although infinite-dimensional representations are thus essential in harmonic analysis on a noncompact group such as $\mathrm{SL}(n, \mathbb{R})$, noncompact Lie groups also have irreducible finite-dimensional representations, which are important in their own right. They are seldom unitary and hence not relevant to the Plancherel formula. The scope of this book includes finite-dimensional representations of Lie groups but not infinite-dimensional ones.

In this chapter and the next two, we will be mainly concerned with compact groups. In this chapter, all representations will be complex and finite-dimensional except when explicitly noted otherwise.

By an inner product on a complex vector space, we mean a positive definite Hermitian form, denoted $\left \langle \;,\;\right \rangle$. Thus, $\left \langle v,w\right \rangle$ is linear in v, conjugate linear in w, satisfies $\left \langle w,v\right \rangle = \overline{\left \langle v,w\right \rangle }$, and $\left \langle v,v\right \rangle > 0$ if v≠0. We will also use the term inner product for real vector spaces—an inner product on a real vector space is a positive definite symmetric bilinear form. Given a group G and a real or complex representation $\pi: G\longrightarrow \mathrm{GL}(V )$, we say the inner product $\left \langle \;,\;\right \rangle$ on V is invariant or G-equivariant if it satisfies the identity

$$\displaystyle{\left \langle \pi (g)v,\pi (g)w\right \rangle = \left \langle v,w\right \rangle.}$$

FormalPara Proposition 2.1.

If G is compact and (π,V ) is any finite-dimensional complex representation, then V admits a G-equivariant inner product.

FormalPara Proof.

Start with an arbitrary inner product $\left \langle \left \langle \,,\,\right \rangle \right \rangle$. Averaging it gives another inner product,

$$\displaystyle{\left \langle v,w\right \rangle =\int _{G}\left \langle \left \langle \pi (g)v,\pi (g)w\right \rangle \right \rangle \mathrm{d}g,}$$

for it is easy to see that this inner product is Hermitian and positive definite. It is G-invariant by construction. □

FormalPara Proposition 2.2.

If G is compact, then each finite-dimensional representation is the direct sum of irreducible representations.

FormalPara Proof.

Let (π,V) be given. Let V ₁ be a nonzero invariant subspace of minimal dimension. It is clearly irreducible. Let $V _{1}^{\perp }$ be the orthogonal complement of V ₁ with respect to a G-invariant inner product. It is easily checked to be invariant and is of lower dimension than V. By induction $V _{1}^{\perp } = V _{2} \oplus \cdots \oplus V _{n}$ is a direct sum of invariant subspaces and so $V = V _{1} \oplus \cdots \oplus V _{n}$ is also. □

A function of the form $\phi (g) = L\big(\pi (g)\,v\big)$, where (π,V) is a finite-dimensional representation of G, v ∈ V and $L: V \longrightarrow \mathbb{C}$ is a linear functional, is called a matrix coefficient on G. This terminology is natural, because if we choose a basis e ₁,…,e _n , of V, we can identify V with ${\mathbb{C}}^{n}$ and represent g by matrices:

$$\displaystyle{\pi (g)v = \left (\begin{array}{ccc} \pi _{11}(g) &\cdots & \pi _{1n}(g)\\ \vdots & & \vdots \\ \pi _{n1}(g)&\cdots &\pi _{nn}(g) \end{array} \right )\left (\begin{array}{c} v_{1}\\ \vdots \\ v_{n} \end{array} \right ),v = \left (\begin{array}{c} v_{1}\\ \vdots \\ v_{n} \end{array} \right ) =\sum _{ j=1}^{n}v_{ j}e_{j}.}$$

Then each of the n ² functions π _ij is a matrix coefficient. Indeed

$$\displaystyle{\pi _{ij}(g) = L_{i}\big(\pi (g)e_{j}\big),}$$

where $L_{i}(\sum _{j}v_{j}e_{j}) = v_{i}$.

FormalPara Proposition 2.3.

The matrix coefficients of G are continuous functions. The pointwise sum or product of two matrix coefficients is a matrix coefficient, so they form a ring.

FormalPara Proof.

If v ∈ V, then $g\longrightarrow \pi (g)v$ is continuous since by definition a representation $\pi: G\longrightarrow \mathrm{GL}(V )$ is continuous and so a matrix coefficient $L\big(\pi (g)\,v\big)$ is continuous.

If $(\pi _{1},V _{1})$ and $(\pi _{2},V _{2})$ are representations, $v_{i} \in V _{i}$ are vectors and $L_{i}: V _{i}\longrightarrow \mathbb{C}$ are linear functionals, then we have representations π₁ ⊕ π₂ and $\pi _{1} \otimes \pi _{2}$ on $V _{1} \oplus V _{2}$ and V ₁ ⊗ V ₂, respectively. Given vectors v _i ∈ V _i and functionals $L_{i} \in V _{i}^{{\ast}}$, then $L_{1}\big(\pi (g)v_{1}\big) \pm L_{2}\big(\pi (g)v_{2}\big)$ can be expressed as $L\big((\pi _{1} \oplus \pi _{2})(g)(v_{1},v_{2})\big)$ where $L: V _{1} \oplus V _{2}\longrightarrow \mathbb{C}$ is $L(x_{1},x_{2}) = L_{1}(x_{1}) \pm L_{2}(x_{2})$, so the matrix coefficients are closed under addition and subtraction.

Similarly, we have a linear functional L ₁ ⊗ L ₂ on V ₁ ⊗ V ₂ satisfying

$$\displaystyle{(L_{1} \otimes L_{2})(x_{1} \otimes x_{2}) = L_{1}(x_{1})L_{2}(x_{2})}$$

and

$$\displaystyle{(L_{1} \otimes L_{2})\big((\pi _{1} \otimes \pi _{2})(g)(v_{1} \otimes v_{2})\big) = L_{1}\big(\pi _{1}(g)v_{1}\big)\,L_{2}\big(\pi _{2}(g)v_{2}\big),}$$

proving that the product of two matrix coefficients is a matrix coefficient. □

If (π,V) is a representation, let V ^∗ be the dual space of V. To emphasize the symmetry between V and V ^∗, let us write the dual pairing $V \times {V }^{{\ast}}\longrightarrow \mathbb{C}$ in the symmetrical form $L(v) = \left [[v,L\right ]]$. We have a representation $(\hat{\pi },{V }^{{\ast}})$, called the contragredient of π, defined by

$$[\![v,\hat{\pi }(g)L]\!] = [\![\pi ({g}^{-1})v,L]\!].$$

(2.3)

Note that the inverse is needed here so that $\hat{\pi }(g_{1}g_{2}) =\hat{ \pi }(g_{1})\hat{\pi }(g_{2}).$

If (π,V) is a representation, then by Proposition 2.3 any linear combination of functions of the form $L\big(\pi (g)\,v\big)$ with v ∈ V, L ∈ V ^∗ is a matrix coefficient, though it may be a function ${L^{\prime}}\big(\pi^{\prime}(g)\,v^{\prime}\big)$ where (π′,V ′) is not (π,V ), but a larger representation. Nevertheless, we call any linear combination of functions of the form $L\big(\pi (g)\,v\big)$ a matrix coefficient of the representation (π,V ). Thus, the matrix coefficients of π form a vector space, which we will denote by $\mathcal{M}_{\pi }$. Clearly, $\dim (\mathcal{M}_{\pi }) \leqslant \dim {(V )}^{2}.$

FormalPara Proposition 2.4.

If f is a matrix coefficient of (π,V ), then $\check{f}(g) = f({g}^{-1})$ is a matrix coefficient of $(\hat{\pi },{V }^{{\ast}})$ .

FormalPara Proof.

This is clear from (2.3), regarding v as a linear functional on V ^∗. □

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

$$\displaystyle{\big(\lambda (g)f\big)(x) = f({g}^{-1}x),\quad\big(\rho (g)f\big)(x) = f(xg).}$$

FormalPara Theorem 2.1.

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 matrix coefficient of a finite-dimensional representation.

FormalPara Proof.

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; in fact, a space of dimension at most $\dim {(V )}^{2}$. Thus, (iii) implies (i) and (ii).

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. Indeed, define a functional $L: V \longrightarrow \mathbb{C}$ by L(ϕ) = ϕ(1). Clearly, $L\big(\rho (g)f\big) = f(g)$, so f is a matrix coefficient, as required. Thus (ii) implies (iii).

Finally, if the functions λ(g)f span a finite-dimensional space, composing these functions with $g\longrightarrow {g}^{-1}$ gives another finite-dimensional space which is closed under right translation, and $\check{f}$ defined as in Proposition 2.4 is an element of this space; hence $\check{f}$ is a matrix coefficient by the case just considered. By Proposition 2.4, f is also a matrix coefficient, so (i) implies (iii). □

If $(\pi _{1},V _{1})$ and ($\pi _{2},V _{2})$ are representations, an intertwining operator, also known as a G-equivariant map $T: V _{1}\longrightarrow V _{2}$ or (since V ₁ and V ₂ are sometimes called G-modules) a G-module homomorphism, is a linear transformation $T: V _{1}\longrightarrow V _{2}$ such that

$$\displaystyle{T \circ \pi _{1}(g) = \pi _{2}(g) \circ T}$$

for g ∈ G. We will denote by $\mathrm{Hom}_{\mathbb{C}}(V _{1},V _{2})$ the space of all linear transformations $V _{1}\longrightarrow V _{2}$ and by $\mathrm{Hom}_{G}(V _{1},V _{2})$ the subspace of those that are intertwining maps.

For the remainder of this chapter, unless otherwise stated, G will denote a compact group.

FormalPara Theorem 2.2 (Schur’s lemma).

• (i) Let (π ₁ ,V ₁ ) and $(\pi _{2},V _{2})$ be irreducible representations, and let $T: V _{1}\longrightarrow V _{2}$ be an intertwining operator. Then either T is zero or it is an isomorphism.

• (ii) Suppose that (π,V ) is an irreducible representation of G and $T: V \longrightarrow V$ is an intertwining operator. Then there exists a scalar $\lambda \in \mathbb{C}$ such that T(v) = λv for all v ∈ V.

FormalPara Proof.

For (i), the kernel of T is an invariant subspace of V ₁, which is assumed irreducible, so if T is not zero, $\ker (T) = 0$. Thus, T is injective. Also, the image of T is an invariant subspace of V ₂. Since V ₂ is irreducible, if T is not zero, then im(T) = V ₂. Therefore T is bijective, so it is an isomorphism.

For (ii), let λ be any eigenvalue of T. Let $I: V \longrightarrow V$ denote the identity map. The linear transformation $T - \lambda I$ is an intertwining operator that is not an isomorphism, so it is the zero map by (i). □

We are assuming that G is compact. The Haar volume of G is therefore finite, and we normalize the Haar measure so that the volume of G is 1.

We will consider the space L ²(G) of functions on G that are square-integrable with respect to the Haar measure. This is a Hilbert space with the inner product

$$\displaystyle{\left \langle f_{1},f_{2}\right \rangle _{{L}^{2}} =\int _{G}f_{1}(g)\,\overline{f_{2}(g)}\,\mathrm{d}g.}$$

Schur orthogonality will give us an orthonormal basis for this space.

If (π,V) is a representation and $\left \langle \;,\;\right \rangle$ is an invariant inner product on V, then every linear functional is of the form $x\longrightarrow \left \langle x,v\right \rangle$ for some v ∈ V. Thus a matrix coefficient may be written in the form $g\longrightarrow \left \langle \pi (g)w,v\right \rangle$, and such a representation will be useful to us in our discussion of Schur orthogonality.

FormalPara Lemma 2.1.

Suppose that $(\pi _{1},V _{1})$ and $(\pi _{2},V _{2})$ are complex representations of the compact group G. Let $\left \langle \;,\right \rangle$ be any inner product on V ₁ . If $v_{i},w_{i} \in V _{i}$ , then the map $T: V _{1}\longrightarrow V _{2}$ given by

$$\displaystyle{ T(w) =\int _{G}\left \langle \pi _{1}(g)w,v_{1}\right \rangle \pi _{2}({g}^{-1})v_{ 2}\,\mathrm{d}g }$$

(2.4)

is G-equivariant.

FormalPara Proof.

We have

$$\displaystyle{T\big(\pi _{1}(h)w\big) =\int _{G}\left \langle \pi _{1}(gh)w,v_{1}\right \rangle \pi _{2}({g}^{-1})v_{ 2}\,\mathrm{d}g.}$$

The variable change $g\longrightarrow g{h}^{-1}$ shows that this equals π₂(h)T(w), as required. □

FormalPara Theorem 2.3 (Schur orthogonality).

Suppose that $(\pi _{1},V _{1})$ and $(\pi _{2},V _{2})$ are irreducible representations of the compact group G. Either every matrix coefficient of π ₁ is orthogonal in L ² (G) to every matrix coefficient of π ₂ , or the representations are isomorphic.

FormalPara Proof.

We must show that if there exist matrix coefficients $f_{i}: G\longrightarrow \mathbb{C}$ of π _i that are not orthogonal, then there is an isomorphism $T: V _{1}\longrightarrow V _{2}$. We may assume that the f _i have the form $f_{i}(g) = \left \langle \pi _{i}(g)w_{i},v_{i}\right \rangle$ since functions of that form span the spaces of matrix coefficients of the representations π _i . Here we use the notation $\left \langle \;,\;\right \rangle$ to denote invariant bilinear forms on both V ₁ and V ₂, and $v_{i},w_{i} \in V _{i}$. Then our assumption is that

$$\displaystyle{\int _{G}\left \langle \pi _{1}(g)w_{1},v_{1}\right \rangle \left \langle \pi _{2}({g}^{-1})v_{ 2},w_{2}\right \rangle \,\mathrm{d}g =\int _{G}\left \langle \pi _{1}(g)w_{1},v_{1}\right \rangle \overline{\left \langle \pi _{2}(g)w_{2},v_{2}\right \rangle }\,\mathrm{d}g\neq 0.}$$

Define $T: V _{1}\longrightarrow V _{2}$ by (2.4). The map is nonzero since the last inequality can be written $\left \langle T(w_{1}),w_{2}\right \rangle \neq 0$. It is an isomorphism by Schur’s lemma. □

This gives orthogonality for matrix coefficients coming from nonisomorphic irreducible representations. But what about matrix coefficients from the same representation? (If the representations are isomorphic, we may as well assume they are equal.) The following result gives us an answer to this question.

FormalPara Theorem 2.4 (Schur orthogonality).

Let (π,V ) be an irreducible representation of the compact group G, with invariant inner product $\left \langle \;,\;\right \rangle.$ Then there exists a constant d > 0 such that

$$\displaystyle{ \int _{G}\left \langle \pi (g)w_{1},v_{1}\right \rangle \overline{\left \langle \pi (g)w_{2},v_{2}\right \rangle }\,\mathrm{d}g = {d}^{-1}\left \langle w_{ 1},w_{2}\right \rangle \left \langle v_{2},v_{1}\right \rangle. }$$

(2.5)

Later, in Proposition 2.9, we will show that $d =\dim (V )$.

FormalPara Proof.

We will show that if v ₁ and v ₂ are fixed, there exists a constant $c(v_{1},v_{2})$ such that

$$\displaystyle{ \int _{G}\left \langle \pi (g)w_{1},v_{1}\right \rangle \overline{\left \langle \pi (g)w_{2},v_{2}\right \rangle }\,\mathrm{d}g = c(v_{1},v_{2})\left \langle w_{1,}w_{2}\right \rangle. }$$

(2.6)

Indeed, T given by (2.4) is G-equivariant, so by Schur’s lemma it is a scalar. Thus, there is a constant c = c(v ₁,v ₂) depending only on v ₁ and v ₂ such that T(w) = cw. In particular, $T(w_{1}) = cw_{1}$, and so the right-hand side of (2.6) equals

$$\displaystyle{\left \langle T(w_{1}),w_{2}\right \rangle =\int _{G}\left \langle \pi (g)w_{1},v_{1}\right \rangle \left \langle \pi ({g}^{-1})v_{ 2},w_{2}\right \rangle \,\mathrm{d}g,}$$

Now the variable change $g\longrightarrow {g}^{-1}$ and the properties of the inner product show that this equals the left-hand side of (2.6), proving the identity. The same argument shows that there exists another constant $c^{\prime}(w_{1},w_{2})$ such that for all v ₁ and v ₂ we have

$$\displaystyle{\int _{G}\left \langle \pi (g)w_{1},v_{1}\right \rangle \overline{\left \langle \pi (g)w_{2},v_{2}\right \rangle }\,\mathrm{d}g = c^{\prime}(w_{1},w_{2})\left \langle v_{2,}v_{1}\right \rangle.}$$

Combining this with (2.6), we get (2.5). We will compute d later in Proposition 2.9, but for now we simply note that it is positive since, taking $w_{1} = w_{2}$ and v ₁ = v ₂, both the left-hand side of (2.5) and the two inner products on the right-hand side are positive. □

Before we turn to the evaluation of the constant d, we will prove a different orthogonality for the characters of irreducible representations (Theorem 2.5). This will require some preparations.

FormalPara Proposition 2.5.

The character χ of a representation (π,V ) is a matrix coefficient of V.

FormalPara Proof.

If v ₁,…,v _n is a matrix of V, and L ₁,…,L _n is the dual basis of V ^∗, then $\chi (g) =\sum _{ i=1}^{n}L_{i}\big(\pi (g)v_{i}\big)$. □

FormalPara Proposition 2.6.

Suppose that (π,V ) is a representation of G. Let χ be the character of π.

• (i) If g ∈ V then $\chi ({g}^{-1}) = \overline{\chi (g)}$ .

• (ii) Let $(\hat{\pi },{V }^{{\ast}})$ be the contragredient representation of π. Then the character of $\hat{\pi }$ is the complex conjugate $\overline{\chi }$ of the character χ of G.

FormalPara Proof.

Since π(g) is unitary with respect to an invariant inner product $\left \langle \;,\;\right \rangle$, its eigenvalues t ₁,…,t _n all have absolute value 1, and so

$$\displaystyle{\text{tr}\;\pi {(g)}^{-1} =\sum _{ i}t_{i}^{-1} =\sum _{ i}\overline{t_{i}} = \overline{\chi (g)}.}$$

This proves (i). As for (ii), referring to (2.3), $\hat{\pi }(g)$ is the adjoint of π(g)⁻¹ with respect to the dual pairing $\left [[\;,\;\right ]]$, so its trace equals the trace of π(g)⁻¹. □

The trivial representation of any group G is the representation on a one-dimensional vector space V with π(g)v = v being the trivial action.

FormalPara Proposition 2.7.

If (π,V ) is an irreducible representation and χ its character, then

$$\displaystyle{\int _{G}\chi (g)\,\mathrm{d}g = \left \{\begin{array}{cl} 1&\text{if}\,\pi\,\text{is\,the\,trivialrepresentation;}\\ 0 &\text{otherwise.} \end{array} \right.}$$

FormalPara Proof.

The character of the trivial representation is just the constant function 1, and since we normalized the Haar measure so that G has volume 1, this integral is 1 if π is trivial. In general, we may regard $\int _{G}\chi (g)\,\mathrm{d}g$ as the inner product of χ with the character 1 of the trivial representation, and if π is nontrivial, these are matrix coefficients of different irreducible representations and hence orthogonal by Theorem 2.3. □

If (π,V) is a representation, let V ^G be the subspace of G-invariants, that is,

$$\displaystyle{{V }^{G} = \left \{v \in V \,\vert \,\pi (g)v = v\text{for all}\,g \in G\right \}.}$$

FormalPara Proposition 2.8.

If (π,V ) is a representation of G and χ its character, then

$$\displaystyle{\int _{G}\chi (g)\,\mathrm{d}g =\dim ({V }^{G}).}$$

FormalPara Proof.

Decompose V = ⊕ _i V _i into a direct sum of irreducible invariant subspaces, and let χ _i be the character of the restriction π _i of π to V _i . By Proposition 2.7, $\int _{G}\chi _{i}(g)\,\mathrm{d}g = 1$ if and only if π _i is trivial. Hence $\int _{G}\chi (g)\,\mathrm{d}g$ is the number of trivial π _i . The direct sum of the V _i with π _i trivial is V ^G, and the statement follows. □

If (π₁,V ₁) and (π₂,V ₂) are irreducible representations, and χ₁n and χ₂ are their characters, we have already noted in proving Proposition 2.3 that we may form representations π₁ ⊕ π₂n and π₁ ⊗ π₂ on V ₁ ⊕ V ₂ and V ₁ ⊗ V ₂n. It is easy to see that $\chi _{\pi _{1}\oplus \pi _{2}} = \chi _{\pi _{1}} + \chi _{\pi _{2}}$ and $\chi _{\pi _{1}\otimes \pi _{2}} = \chi _{\pi _{1}}\chi _{\pi _{2}}$. It is not quite true that the characters form a ring. Certainly the negative of a matrix coefficient is a matrix coefficient, yet the negative of a character is not a character. The set of characters is closed under addition and multiplication but not subtraction. We define a generalized (or virtual) character to be a function of the form χ₁ − χ₂, where χ₁ and χ₂ are characters. It is now clear that the generalized characters form a ring.

FormalPara Lemma 2.2.

Define a representation $\varPsi: \mathrm{GL}(n, \mathbb{C}) \times \mathrm{GL}(m, \mathbb{C})\longrightarrow \mathrm{GL}(\varOmega )$ where $\varOmega = \mathrm{Mat}_{n\times m}(\mathbb{C})$ by $\varPsi (g_{1},g_{2}): X\longrightarrow g_{2}Xg_{1}^{-1}$ n. Then the trace of $\varPsi (g_{1},g_{2})$ is $\mathrm{tr}(g_{1}^{-1})\,\mathrm{tr}(g_{2})$ n.

FormalPara Proof.

Both $\mathrm{tr}\;\varPsi (g_{1},g_{2})$ and $\mathrm{tr}(g_{1}^{-1})\,\mathrm{tr}(g_{2})$ are continuous, and since diagonalizable matrices are dense in $\mathrm{GL}(n, \mathbb{C})$ we may assume that both g ₁ and g ₂ are diagonalizable. Also if γ is invertible we have $\varPsi (\gamma g_{1}{\gamma }^{-1},g_{2}) =\varPsi (\gamma,1)\varPsi (g_{1},g_{2})\varPsi {(\gamma,1)}^{-1}$ so the trace of both $\mathrm{tr}\;\varPsi (g_{1},g_{2})$ and $\mathrm{tr}(g_{1}^{-1})\mathrm{tr}(g_{2})$ are unchanged if g ₁ is replaced by γg ₁γ⁻¹. So we may assume that g ₁ is diagonal, and similarly g ₂. Now if $\alpha _{1},\ldots,\alpha _{n}$ and $\beta _{1},\ldots,\beta _{m}$ are the diagonal entries of g ₁ and $g_{2}^{-1}$, the effect of Ψ(g ₁,g ₂) on X ∈Ω is to multiply the columns by the $\alpha _{i}^{-1}$ and the rows by the β _j . So the trace is $\mathrm{tr}(g_{1}^{-1})\mathrm{tr}(g_{2})$. □

FormalPara Theorem 2.5 (Schur orthogonality).

Let (π ₁ ,V ₁ ) and $(\pi _{2},V _{2})$ be representations of G with characters χ ₁ and χ ₂ . Then

$$\displaystyle{ \int _{G}\chi _{1}(g)\,\overline{\chi _{2}(g)}\,\mathrm{d}g =\dim \mathrm{Hom}_{G}(V _{1},V _{2}). }$$

(2.7)

If π ₁ and π ₂ are irreducible, then

$$\displaystyle{\int _{G}\chi _{1}(g)\overline{\chi _{2}(g)}\,\mathrm{d}g = \left \{\begin{array}{ccl} 1&&\text{if}\,\pi _{1}\mathop{\cong}\pi _{2};\\ 0 & &\text{otherwise.} \end{array} \right.}$$

FormalPara Proof.

Define a representation Π of G on the space $\varOmega = \mathrm{Hom}_{\mathbb{C}}(V _{1},V _{2})$ of all linear transformations $T: V _{1}\longrightarrow V _{2}$ by

$$\displaystyle{\varPi (g)T = \pi _{2}(g) \circ T \circ \pi _{1}{(g)}^{-1}.}$$

By lemma 2.2 and Proposition 2.6, the character of Π(g) is $\chi _{2}(g)\overline{\chi _{1}(g)}$. The space of invariants Ω ^G exactly of the T which are G-module homomorphisms, so by Proposition 2.8 we get

$$\displaystyle{\int _{G}\overline{\chi _{1}(g)}\,\chi _{2}(g)\,\mathrm{d}g =\dim \, \mathrm{Hom}_{G}(V _{1},V _{2}).}$$

Since this is real, we may conjugate to obtain (2.7). □

FormalPara Proposition 2.9.

The constant d in Theorem 2.4 equals dim (V ).

FormalPara Proof.

Let v ₁,…,v _n be an orthonormal basis of V, $n =\dim (V )$. We have

$$\displaystyle{\chi (g) =\sum _{i}\left \langle \pi _{i}(g)v_{i},v_{i}\right \rangle }$$

since $\left \langle \pi (g)v_{j},v_{i}\right \rangle$ is the $i,j$ component of the matrix of π(g) with respect to this basis. Now

$$\displaystyle{1 =\int _{G}\vert \chi (g){\vert }^{2}\,\mathrm{d}g =\sum _{ i,j}\int _{G}\left \langle \pi (g)v_{i},v_{i}\right \rangle \overline{\left \langle \pi (g)v_{j},v_{j}\right \rangle }\,\mathrm{d}g.}$$

There are n ² terms on the right, but by (2.5) only the terms with i = j are nonzero, and those equal d ⁻¹. Thus, d = n. □

We now return to the matrix coefficients $\mathcal{M}_{\pi }$ of an irreducible representation (π,V ). We define a representation Θ of G × G on $\mathcal{M}_{\pi }$ by

$$\displaystyle{\varTheta (g_{1},g_{2})f(x) = f(g_{2}^{-1}xg_{ 1}).}$$

We also have a representation Π of G × G on $\mathrm{End}_{\mathbb{C}}(V )$ by

$$\displaystyle{\varPi (g_{1},g_{2})T = \pi {(g_{2})}^{-1}T\,\pi (g_{ 1}).}$$

FormalPara Proposition 2.10.

If $f \in \mathcal{M}_{\pi }$ then so is $\varTheta (g_{1},g_{2})\,f$ . The representations Θ and Π are equivalent.

FormalPara Proof.

Let L ∈ V ^∗ and v ∈ V. Define $f_{L,v}(g) = L(\pi (g)v)$. The map $L,v\longmapsto f_{L,v}$ is bilinear, hence induces a linear map $\sigma: {V }^{{\ast}}\otimes V \longrightarrow \mathcal{M}_{\pi }$. It is surjective by the definition of $\mathcal{M}_{\pi }$, and it follows from Proposition 2.4 that if L _i and v _j run through orthonormal bases, then $f_{L_{i},v_{j}}$ are orthonormal, hence linearly independent. Therefore, σ is a vector space isomorphism. We have

$$\displaystyle{\varTheta (g_{1},g_{2})f_{L,v}(g) = L(g_{2}^{-1}gg_{ 1}v) = f_{\hat{\pi }(g_{2})L,\pi (g_{1})v}(x),}$$

where we recall that $(\hat{\pi },{V }^{{\ast}})$ is the contragredient representation. This means that σ is a G × G-module homomorphism and so $\mathcal{M}_{\pi }\mathop{\cong}{V }^{{\ast}}\otimes V$ as G × G-modules. On the other hand we also have a bilinear map ${V }^{{\ast}}\times V \longrightarrow \mathrm{End}_{\mathbb{C}}(V )$ that associates with (L,v) the rank-one linear map $T_{L,v}(u) = L(u)v$. This induces an isomorphism ${V }^{{\ast}}\otimes V \longrightarrow \mathrm{End}_{\mathbb{C}}(V )$ which is G × G equivariant. We see that $\mathcal{M}_{\pi }\mathop{\cong}{V }^{{\ast}}\otimes V \mathop{\cong}\mathrm{End}_{\mathbb{C}}(V )$. □

A function f on G is called a class function if it is constant on conjugacy classes, that is, if it satisfies the equation $f(hg{h}^{-1}) = f(g)$. The character of a representation is a class function since the trace of a linear transformation is unchanged by conjugation.

FormalPara Proposition 2.11.

If f is the matrix coefficient of an irreducible representation (π,V ), and if f is a class function, then f is a constant multiple of χ _π .

FormalPara Proof.

By Schur’s lemma, there is a unique G-invariant vector in $\mathrm{Hom}_{\mathbb{C}}(V,V )$; hence. by Proposition 2.10, the same is true of $\mathcal{M}_{\pi }$ in the action of G by conjugation. This matrix coefficient is of course χ_π. □

FormalPara Theorem 2.6.

If f is a matrix coefficient and also a class function, then f is a finite linear combination of characters of irreducible representations.

FormalPara Proof.

Write $f =\sum _{ i=1}^{n}f_{i}$, where each f _i is a class function of a distinct irreducible representation (π _i ,V _i ). Since f is conjugation-invariant, and since the f _i live in spaces $\mathcal{M}_{\pi _{i}}$, which are conjugation-invariant and mutually orthogonal, each f _i is itself a class function and hence a constant multiple of $\chi _{\pi _{i}}$ by Proposition 2.11. □

Exercises

FormalPara Exercise 2.1.

Suppose that G is a compact Abelian group and $\pi: G\longrightarrow \mathrm{GL}(n, \mathbb{C})$ an irreducible representation. Prove that n = 1.

FormalPara Exercise 2.2.

Suppose that G is compact group and f: G ⟶ ℂ is the matrix coefficient of an irreducible representation π. Show that $g\longmapsto \overline{f({g}^{-1})}$ is a matrix coefficient of the same representation π.

FormalPara Exercise 2.3.

Suppose that G is compact group. Let C(G) be the space of continuous functions on G. If f ₁ and f ₂ ∈ C(G), define the convolution f ₁ ∗ f ₂ of f ₁ and f ₂ by

$$\displaystyle{(f_{1} {\ast} f_{2})(g) =\int _{G}f_{1}(g{h}^{-1})\,f_{ 2}(h)\,\mathrm{d}h =\int _{G}f_{1}(h)\,f_{2}({h}^{-1}g)\,\mathrm{d}h.}$$

• (i) Use the variable change h ⟶ h ⁻¹ g to prove the identity of the last two terms. Prove that this operation is associative, and so C(G) is a ring (without unit) with respect to covolution.

• (ii) Let π be an irreducible representation. Show that the space ℳ _π of matrix coefficients of π is a 2-sided ideal in C(G), and explain how this fact implies Theorem 2.3.

FormalPara Exercise 2.4.

Let G be a compact group, and let G × G act on the space ℳ _π by left and right translation: $(g,h)f(x) = f({g}^{-1}xh)$. Show that $\mathcal{M}_{\pi }\mathop{\cong}\hat{\pi } \otimes \pi$ as (G × G)-modules.

FormalPara Exercise 2.5.

Let G be a compact group and let g,h ∈ G. Show that g and h are conjugate if and only if χ(g) = χ(h) for every irreducible character χ. Show also that every character is real-valued if and only if every element is conjugate to its inverse.

FormalPara Exercise 2.6.

Let G be a compact group, and let V,W be irreducible G-modules. An invariant bilinear form B : V × W → ℂ is one that satisfies $B\big(g\cdot v,g\cdot w\big)=B(v,w)$ for g ∈ G, v ∈ V, w ∈ W. Show that the space of invariant bilinear forms is at most one-dimensional, and is one-dimensional if and only if V and W are contragredient.