In the paper "A new cohomology theory for orbifold" by Chen/Ruan, they define the orbifold cohomology group of an orbifold $X$ by
$H^d_{orb}(X)=\bigoplus_{(g) \in T} H^{d-2\iota_{(g)}}(X_{(g)})$
where $H^\ast(X_{(g)})$ is the singular cohomology of $X_{(g)}$ with real coefficients and $\iota(g)$ is a rational number.
Also, the orbifold Dolbeault cohomology group is defined by
$H^{p,q}_{orb}(X)=\bigoplus_{(g) \in T}H^{p-\iota_{(g)},q-\iota_{(g)}}(X_{(g)})$.
How are these cohomologies with rational coefficients defined?
EDIT: In Adem/Leida/Ruan's "Orbifolds and stringy topology" they similarly define the Chen-Ruan cohomology group for orbifold groupoid as
$H^d_{CR}=\bigoplus_{(g) \in T}H^d(X_{(g)})[-2\iota_{(g)}]=\bigoplus_{(g) \in T}H^{d-2\iota_{(g)}}(X_{(g)})$,
but I am also unfamiliar with the notation $H^d(X_{(g)})[-2\iota_{(g)}]$.