Projective unitary representation in the Shale paper

A projective unitary representation assigns each $g\in G$ a unitary ray $\overline{U}(g)=\{\alpha U(g):|\alpha|=1\}$ , with multiplication holding up to a phase (a cocycle). Equivalently, it is a homomorphism

\overline U:G\longrightarrow PU(H)=U(H)/U(1)

for a complex Hilbert space $H$ . After choosing a unitary representative $U_g$ of each projective class, there is a function $\omega:G\times G\to U(1)$ such that

U_gU_h=\omega(g,h)U_{gh}.

Associativity forces the cocycle identity

\omega(g,h)\omega(gh,k)=\omega(h,k)\omega(g,hk).

Changing the representatives $U_g$ changes $\omega$ by a coboundary but leaves $\overline U$ unchanged.

Remarks

Key properties (paper use):

Shale's implementers $Y(T)$ are unique only up to phase, so $\overline{Y}$ is projective.
In finite dimensions, $\overline{Y}$ lifts to a genuine double-valued unitary representation (§5).
More generally, a projective representation lifts to an ordinary unitary representation precisely when its multiplier class is trivial; it may instead lift after passing to a central extension of $G$ .

Examples

Spin representations are projective representations of rotation groups and ordinary representations of their double covers.