Invariant Inner Products

The existence of invariant inner products is crucial for proving complete reducibility and understanding unitary representations.

DefinitionG-Invariant Inner Product

Let $V$ be a representation of $G$ over $\mathbb{R}$ or $\mathbb{C}$ . An inner product $\langle \cdot, \cdot \rangle$ on $V$ is $G$ -invariant if: $\langle \rho(g)(v), \rho(g)(w) \rangle = \langle v, w \rangle$ for all $g \in G$ and $v, w \in V$ .

Equivalently, each $\rho(g)$ is a unitary (or orthogonal) transformation with respect to the inner product.

TheoremExistence of Invariant Inner Products

Let $V$ be a finite-dimensional representation of a finite group $G$ over $\mathbb{R}$ or $\mathbb{C}$ . Then there exists a $G$ -invariant inner product on $V$ .

Proof: Start with any inner product $\langle \cdot, \cdot \rangle_0$ on $V$ . Define: $\langle v, w \rangle = \frac{1}{|G|} \sum_{g \in G} \langle \rho(g)(v), \rho(g)(w) \rangle_0$

This is positive definite (since $\langle \cdot, \cdot \rangle_0$ is) and $G$ -invariant by construction.

ExampleApplications

Complete Reducibility: With a $G$ -invariant inner product, if $W \subseteq V$ is a subrepresentation, then the orthogonal complement $W^\perp$ is also a subrepresentation. Thus $V = W \oplus W^\perp$ decomposes.

Unitary Representations: A representation with a $G$ -invariant inner product is called unitary (over $\mathbb{C}$ ) or orthogonal (over $\mathbb{R}$ ). All finite-dimensional representations of finite groups and compact groups are unitarizable.

Matrix Form: If we choose an orthonormal basis with respect to a $G$ -invariant inner product, then each $\rho(g)$ is represented by a unitary matrix: $\rho(g)^* \rho(g) = I$ .

DefinitionUnitary Representation

A representation $(\rho, V)$ over $\mathbb{C}$ is unitary if there exists an inner product on $V$ such that: $\rho(g)^* = \rho(g)^{-1} = \rho(g^{-1})$ for all $g \in G$ , where $\rho(g)^*$ denotes the adjoint with respect to the inner product.

Remark

For compact groups (including finite groups), every finite-dimensional representation is equivalent to a unitary representation. This is proved using Haar measure to average an arbitrary inner product. The unitary perspective provides powerful tools: orthogonality of matrix coefficients, Peter-Weyl theorem, and connections to harmonic analysis.

For non-compact groups, not all representations are unitarizable. Understanding which representations admit invariant inner products is a deep question in harmonic analysis and the theory of automorphic forms.

The existence of invariant inner products dramatically simplifies the structure theory. With a fixed invariant inner product, we can use orthogonal projections, compute direct sum decompositions explicitly, and apply tools from functional analysis and spectral theory.