A differential form ω∈Ω dR p(G)\omega \in \Omega_{dR}^p(G) on a Lie group GG is called left invariant if for every g∈Gg \in G it is invariant under the pullback of differential forms

(1)(L g) *ω=ω (L_g)^* \omega = \omega

along the left multiplication action

L g: G ⟶ G x ↦ g⋅x \array{ L_g \colon & G &\longrightarrow& G \\ & x &\mapsto& g \cdot x }

Analogously a form is right invariant if it is invariant under the pullback by right translations R gR_g.

More generally, given a differentiable (e.g. smooth) group action of GG on a differentiable (e.g. smooth) manifold MM

G×M ⟶ρ M (g,x) ↦ g⋅x \array{ G \times M & \overset{\rho}{\longrightarrow} & M \\ (g,x) &\mapsto& g \cdot x }

then a differential form ω∈Ω dR p(M)\omega \in \Omega^p_{dR}(M) is called invariant if for all g∈Gg \in G

ρ(g) *(ω)=ω. \rho(g)^\ast(\omega) \;=\; \omega \,.

This reduces to the left invariance (1) for M=GM = G and ρ\rho being the left multiplication action of GG on itself.

Invariant vector field

For a vector field XX one instead typically defines the invariance via the pushforward (TL g)X=(L g) *X(T L_g) X = (L_g)_* X. Regarding that L gL_g and T gT_g are diffeomorphisms, both pullbacks and pushfowards (hence invariance as well) are defined for every tensor field; and the two requirements are equivalent.

References

See most textbooks on on Lie theory, e.g.:

Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [ams:gsm-34]
François Bruhat (notes by S. Ramanan): Lectures on Lie groups and representations of locally compact groups, Tata Institute Bombay (1958, 1968) [pdf, pdf]