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Convolution of matrix coefficients on a compact group

Example

Assume the Axiom of Choice. Let $G$ be a compact Hausdorff group with its normalized Haar probability measure $μ$ (Normalized Haar probability on a compact group). Then the constant function $1$ satisfies $1 * 1 = 1$ . Moreover, for continuous finite-dimensional irreducible unitary representations $π$ on $V$ and $σ$ on $W$ of $G$ (Subrepresentations, direct sums of representations, and irreducibility) with first-variable-linear coefficients $f_{u, v}^{π} (x) : = ⟨ π (x) u, v ⟩ (u, v \in V), f_{w, z}^{σ} (x) : = ⟨ σ (x) w, z ⟩ (w, z \in W),$ the convolution of coefficients is zero when $π ≇ σ$ . If they are equivalent, choose any unitary intertwiner $T : W \to V$ satisfying $T σ (x) = π (x) T$ . Then $f_{u, v}^{π} * f_{w, z}^{σ} = \frac{⟨ u, T z ⟩}{\dim π} f_{T w, v}^{π} .$ This formula is independent of the choice of $T$ . When $σ = π$ on the same space, one may take $T = I_{V}$ .

Facts & Assumptions

Given: A compact Hausdorff group $G$ with its normalized Haar probability measure $μ$ , continuous finite-dimensional irreducible unitary representations $π$ on $V$ and $σ$ on $W$ , and AC.

[F1]

A compact Hausdorff group has a left Haar probability measure, and that measure is right invariant; a compact group is unimodular, so $Δ_{G} \equiv 1$ and inversion preserves $μ$ (Normalized Haar probability on a compact group, Compact, discrete and abelian groups are unimodular, Unimodular locally compact group, Haar change of variables under inversion).

[F2]

For $f, g \in C_{c} (G)$ convolution is $(f * g) (x) = \int_{G} f (y) g (y^{- 1} x) d μ (y)$ , and convolution on $L^{1} (G)$ is the unique $C$ -bilinear extension with $∥ f * g ∥_{1} \leq ∥ f ∥_{1} ∥ g ∥_{1}$ , jointly continuous and agreeing with the $C_{c}$ formula (Compactly supported convolution on a group, Convolution on L1 of a locally compact group, Submultiplicativity of convolution in the L1 norm).

[F3]

$C_{c} (G)$ is dense in $L^{1} (G)$ and in $L^{2} (G)$ ; on the probability space $G$ one has $∥ h ∥_{1} \leq ∥ h ∥_{2}$ for $h \in L^{2} (G)$ , and $L^{p} (G) = L^{p} (G, μ; C)$ with $∥ h ∥_{p} = (\int_{G} ∣ h ∣^{p} d μ)^{1 / p}$ (Completeness of the complex Haar L1 and L2 spaces and density of Cc, Complex Haar L^p spaces and compactly supported functions).

[F4]

$L^{2} (G)$ carries the first-variable-linear inner product $⟨ h, k ⟩ = \int_{G} h \overline{k} d μ$ , whose induced norm is $∥ \cdot ∥_{2}$ (The complex $L^{2}$ pairing on equivalence classes).

[F5]

Left and modular right translations are strongly continuous on $L^{2} (G)$ ; since $Δ_{G} \equiv 1$ here, the plain right translation $h \mapsto h (\cdot g)$ is strongly continuous (Strong continuity of left and modular right translations on L1 and L2, [F1]).

[F6]

A subrepresentation of a finite-dimensional representation is a linear subspace carried into itself by every $ρ (g)$ , and irreducibility means the nonzero space has no proper nonzero subrepresentation; an intertwiner is a linear map with $f ρ (g) = σ (g) f$ for all $g$ , and equivalence means the existence of an invertible intertwiner (Subrepresentations, direct sums of representations, and irreducibility, Intertwiners, the spaces ${Hom}_{G} (V, W)$ and ${End}_{G} (V)$ , equivalent representations, and faithful representations, A finite-dimensional representation $ρ : G \to GL (V)$ over a field, and its degree).

[F7]

Every endomorphism of a nonzero finite-dimensional complex vector space has an eigenvalue in $C$ (Every endomorphism of a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue).

[F8]

For a continuous finite-dimensional irreducible unitary representation the coefficient $f_{u, v}^{π} (x) = ⟨ π (x) u, v ⟩$ is continuous because $x \mapsto π (x) u$ and the inner product are continuous, and $∣ f_{u, v}^{π} (x) ∣ \leq ∥ u ∥ ∥ v ∥$ by Cauchy–Schwarz and unitarity, so it lies in $L^{1} (G)$ and $L^{2} (G)$ because $μ$ is a probability measure.

[A1]

AC is assumed in the choice-function form of the cited definition; it is inherited from the Haar-measure, completeness and strong-continuity suppliers [F1], [F3] and [F5], and is first used in step 1.1 through [F3] (The Axiom of Choice).

Verification

technique · direct

1.1

Coefficients are square integrable. By [F8] each coefficient $f_{u, v}^{π}$ and $f_{w, z}^{σ}$ is continuous with $∣ f_{u, v}^{π} ∣ \leq ∥ u ∥ ∥ v ∥$ and $∣ f_{w, z}^{σ} ∣ \leq ∥ w ∥ ∥ z ∥$ , so both lie in $L^{\infty} (G) \subseteq L^{2} (G)$ with $∥ f_{u, v}^{π} ∥_{2}, ∥ f_{w, z}^{σ} ∥_{2} < \infty$ because $μ$ is a probability measure; the description of $L^{2} (G)$ used here is the one of [F3], available under the AC of [A1].

A1F3F8

1.2

The integral formula computes convolution for $L^{2}$ -functions. Let $f, g \in L^{2} (G)$ and put $h (x) : = \int_{G} f (y) g (y^{- 1} x) d μ (y)$ . For each $x$ the integral converges absolutely with $∣ h (x) ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ by Cauchy–Schwarz [F4]. The function $h$ is continuous: for $x, x_{0} \in G$ , Cauchy–Schwarz and the measure-preserving substitution $y \mapsto w = y^{- 1} x_{0}$ (measure preserving by the unimodularity and inversion invariance of [F1]) give $∣ h (x) - h (x_{0}) ∣ \leq ∥ f ∥_{2} ∥ g (\cdot x_{0}^{- 1} x) - g ∥_{2} \to 0$ as $x \to x_{0}$ by strong continuity of right translation [F5]. Also $∥ h ∥_{2} \leq ∥ f ∥_{2} ∥ g ∥_{2}$ : the pointwise Cauchy–Schwarz bound and $μ (G) = 1$ give $\int_{G} ∣ h (x) ∣^{2} d μ (x) \leq \int_{G} (\int_{G} ∣ f (y) ∣^{2} d μ (y)) (\int_{G} ∣ g (y^{- 1} x) ∣^{2} d μ (y)) d μ (x) = ∥ f ∥_{2}^{2} ∥ g ∥_{2}^{2}$ by left invariance of $μ$ [F1]. Finally, on $C_{c} (G) \times C_{c} (G)$ the map $(f, g) \mapsto h$ is the $C_{c}$ convolution by [F2], and both $(f, g) \mapsto h$ and the $L^{1}$ convolution are continuous bilinear maps from $L^{2} (G) \times L^{2} (G)$ into $L^{1} (G)$ : the first because $∥ h ∥_{1} \leq ∥ h ∥_{2} \leq ∥ f ∥_{2} ∥ g ∥_{2}$ , the second because $∥ f * g ∥_{1} \leq ∥ f ∥_{1} ∥ g ∥_{1} \leq ∥ f ∥_{2} ∥ g ∥_{2}$ , using $∥ t ∥_{1} \leq ∥ t ∥_{2}$ on the probability space $G$ and [F2], [F3]; so density of $C_{c} (G)$ in $L^{2} (G)$ [F3] forces $h = f * g$ for all $f, g \in L^{2} (G)$ .

F1F2F3F4F5

1.3

The averaged operator. Define $A : W \to V$ by $A (ξ) : = \int_{G} ⟨ σ (y)^{- 1} ξ, z ⟩ π (y) u d μ (y)$ , the integral being computed componentwise in a basis of $V$ ; the integrand is continuous on the compact group, so the integral exists and $A$ is linear in $ξ$ . Then $A$ intertwines $σ$ with $π$ : for $g \in G$ and $ξ \in W$ , substituting $y = g y^{'}$ and using invariance of $μ$ gives $A (σ (g) ξ) = \int_{G} ⟨ σ (y)^{- 1} σ (g) ξ, z ⟩ π (y) u d μ (y) = \int_{G} ⟨ σ (g^{- 1} y)^{- 1} ξ, z ⟩ π (y) u d μ (y) = \int_{G} ⟨ σ (y^{'})^{- 1} ξ, z ⟩ π (g y^{'}) u d μ (y^{'}) = π (g) A (ξ)$ , using $σ (y)^{- 1} σ (g) = σ (y^{- 1} g)$ and the substitution $y = g y^{'}$ .

F1F6

2.1

The constant function. Since $μ (G) = 1$ and $1 \in L^{2} (G)$ by [F3], the integral formula of step 1.2 gives $(1 * 1) (x) = \int_{G} 1 d μ = 1$ for every $x$ , so $1 * 1 = 1$ .

F1F3step 1.2

2.2

The product identity. For $x \in G$ , the integral formula of step 1.2 applied to the coefficients of step 1.1 gives $(f_{u, v}^{π} * f_{w, z}^{σ}) (x) = \int_{G} ⟨ π (y) u, v ⟩ ⟨ σ (y^{- 1} x) w, z ⟩ d μ (y) = ⟨ \int_{G} ⟨ σ (y)^{- 1} σ (x) w, z ⟩ π (y) u d μ (y), v ⟩,$ where the second equality uses $σ (y^{- 1} x) = σ (y)^{- 1} σ (x)$ and the linearity of the inner product in its first variable; the right-hand side is $⟨ A (σ (x) w), v ⟩$ with $A$ as in step 1.3.

F4F6step 1.1step 1.2step 1.3

2.3

The trace of $A$ in the case $σ = π$ . Suppose $σ$ is the same representation $π$ on $V$ , of dimension $d$ , so that $A$ is an endomorphism of $V$ ; write $R_{y} (ξ) : = ⟨ π (y)^{- 1} ξ, z ⟩ π (y) u = ⟨ ξ, π (y) z ⟩ π (y) u$ , a rank-one operator with $tr R_{y} = ⟨ π (y) u, π (y) z ⟩ = ⟨ u, z ⟩$ by unitarity of $π (y)$ . Integrating the traces, $tr A = \int_{G} ⟨ u, z ⟩ d μ (y) = ⟨ u, z ⟩$ .

F1F4step 1.3

3.1

The Schur step. The kernel and the image of any intertwiner are subrepresentations, by [F6]; hence if $A \neq 0$ , irreducibility makes $A$ an isomorphism. Thus $π ≇ σ$ forces $A = 0$ . If $π ≅ σ$ , rescale an invertible intertwiner to a unitary one $T : W \to V$ : $T^{*} T$ commutes with $σ$ , hence is a positive scalar by the eigenvalue argument of [F7] and irreducibility. Then $A T^{- 1}$ commutes with $π$ and is scalar by the same argument. Replacing $z$ by $T z$ in the trace calculation of step 2.3 gives $tr (A T^{- 1}) = ⟨ u, T z ⟩$ , so $A = (⟨ u, T z ⟩ / \dim π) T$ .

F6F7step 1.3step 2.3

4.1

The coefficient products. The coefficients lie in $L^{2} (G) \subseteq L^{1} (G)$ by [F3] and [F8], so step 2.2 computes their convolution. If $π ≇ σ$ , $A = 0$ by step 3.1. Otherwise step 3.1 gives $A = (⟨ u, T z ⟩ / \dim π) T$ , hence step 2.2 gives $(f_{u, v}^{π} * f_{w, z}^{σ}) (x) = (⟨ u, T z ⟩ / \dim π) ⟨ π (x) T w, v ⟩$ . Any two unitary intertwiners differ by a scalar $c$ with $∣ c ∣ = 1$ , since their quotient commutes with $π$ and the scalar-endomorphism argument of step 3.1 applies. Replacing $T$ by $c T$ conjugate-scales the first factor and scales the coefficient by $c$ , so the product is independent of $T$ .

F2F3F8step 2.2step 3.1

5.1

Together with $1 * 1 = 1$ from step 2.1, step 4.1 proves the coefficient formula for inequivalent and equivalent irreducible representations. For $σ = π$ , taking $T = I_{V}$ gives $(⟨ u, z ⟩ / \dim π) f_{w, v}^{π}$ . ∎

step 2.1step 4.1

Verification notes

Hypotheses used. Continuity, irreducibility, unitarity in a finite dimension and compactness of $G$ enter; the coefficient $⟨ u, z ⟩$ pairs the "input" vector of the first coefficient with the "output" vector of the second, and $f_{w, v}^{π}$ uses the two remaining vectors, matching the classical matrix-unit rule.
Choice cost. [A1] is inherited from the suppliers [F1], [F3] and [F5] and is first used in step 1.1 through [F3], as declared in the fact itself; the eigenvalue theorem [F7], the averaging, the trace computation and the norm estimates add no further selection.

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Sources

Emmanuel Kowalski, Representation Theory of Groups, §§5.2–5.3 (standard reference, not scraped)
Lynn Loomis, An Introduction to Abstract Harmonic Analysis, §§30–31 (standard reference, not scraped)