32.1 Unitary Representations of Locally Compact Groups

Definition 32.1.1 (Unitary Representation).label Let $G$ be a locally compact group, then a unitary representation of $G$ is a pair $(H, \pi)$, where $H$ is a non-zero complex Hilbert space and $\pi: G \to U(B(H))$ is a strong-operator continuous homomorphism.

Definition 32.1.2 (Left Regular Representation).label Let $G$ be a locally compact group, then the mapping

is the left regular representation of $G$.

Definition 32.1.3 (Intertwining Operator).label Let $G$ be a locally compact group, $(H_{1}, \pi_{1})$ and $(H_{2}, \pi_{2})$ be unitary representations of $G$, and $T \in L(H_{1}; H_{2})$, then $T$ is an intertwining operator for $\pi_{1}$ and $\pi_{2}$ if the following diagram commutes

for all $x \in G$. The set $\mathcal{C}(\pi_{1}, \pi_{2})$ is the space of intertwining operators for $\pi_{1}$ and $\pi_{2}$.

Lemma 32.1.4.label Let $G$ be a locally compact group and $(H, \pi)$ be a unitary representation of $G$, then $\mathcal{C}(\pi, \pi) = \pi(G)'$, and hence a von Neumann algebra.

Definition 32.1.5 (Invariant Subspace).label Let $G$ be a locally compact group, $(H, \pi)$ be a unitary representation of $G$, $M \subset H$ be a closed subspace, and $P \in B(H)$ be the orthogonal projection onto $M$, then the following are equivalent:

1. (1)

   For each $x \in G$, $\pi(x)(M) \subset M$.

2. (2)

   $P \in \pi(G)'$.

If the above holds, then $M$ is a invariant subspace for $\pi$.

Definition 32.1.6 (Subrepresentation).label Let $G$ be a locally compact group, $(H, \pi)$ be a unitary representation of $G$, and $M \subset H$ be an invariant subspace, then the mapping

is a unitary subrepresentation of $G$.

If $\pi$ admits a non-trivial invariant subspace, then $\pi$ is reducible. Otherwise, it is irreducible.

Definition 32.1.7 (Direct Sum).label Let $G$ be a locally compact group, $\bracsn{(H_i, \pi_i)}_{i \in I}$ be unitary representations of $G$, then the unitary representation

is the direct sum of $\bracsn{(H_i, \pi_i)}_{i \in I}$, denoted $\bigoplus_{i \in I}\pi_{i}$.

Definition 32.1.8 (Cyclic).label Let $G$ be a locally compact group, $(H, \pi)$ be a unitary representation of $G$, and $\xi \in H$, then $\xi$ is a cyclic vector if $\text{span}\bracsn{\pi(x)\xi|x \in G}$ is dense in $H$. The representation $(H, \pi)$ is cyclic if it admits a cyclic vector.

Proposition 32.1.9.label Let $G$ be a locally compact group, $(H, \pi)$ be a unitary representation of $G$, and $M \subset H$ be an invariant subspace, then:

1. (1)

   $M^{\perp}$ is also an invariant subspace.

2. (2)

   $(H, \pi)$ is the direct sum of $(M, \pi_{M})$ and $(M^{\perp}, \pi_{M^\perp})$.

Moreover,

1. (4)

   $(H, \pi)$ is a direct sum of cyclic representations.

Lemma 32.1.10 (Schur).label Let $G$ be a locally compact group and $(H, \pi), (K, \tau)$ be unitary representations of $G$, then

1. (1)

   $(H, \pi)$ is irreducible if and only if the commutant $\pi(G)' = \complex I$.

2. (2)

   If $(H, \pi)$ and $(K, \tau)$ are irreducible, then $(H, \pi)$ and $(K, \tau)$ are unitarily equivalent if and only if $\mathcal{C}(\pi, \tau)$ is one-dimensional. Otherwise, $\mathcal{C}(\pi, \tau) = \bracsn{0}$.

Corollary 32.1.11.label Let $G$ be a locally compact group. If $G$ is abelian, then every irreducible representation of $G$ is one-dimensional.