Invariant vector of a Lie algebra representation ($V^{\mathfrak{g}}={\bigcap }_X\ker \rho (X)$)

For a Lie algebra representation $\rho :\mathfrak{g}\to \mathfrak{gl}(V)$, an invariant vector is a vector $v$ annihilated by every represented Lie algebra element:The condition is zero infinitesimal action, not $\rho (X)v=v$. The invariant vectors form a vector subspace.

For a one-dimensional abelian Lie algebra acting on $K^2$ through $\rho (X)=\mathrm{diag}(0,1)$, the invariant vectors are exactly the first coordinate line. For the Adjoint representation, invariants are the center of a Lie algebra, because $\rho (X)v=[X,v]$.

If the representation differentiates a representation of a connected Lie group, Lie algebra invariants are also group invariants. Connectedness matters: a disconnected group can act nontrivially even when its Lie algebra acts trivially. The sign action of $\{1,-1\}$ has zero Lie algebra but no nonzero fixed vector.