  For a derived stack obtained as a quotient of a derived affine scheme by a
reductive group, we show that shifted symplectic structures can be
characterized by the Cartan-de Rham complex. For non-reductive groups, we also
show the analogous statement for Getzler's extension of the Cartan-de Rham
complex. Dually, we construct a Cartan model for polyvector fields on global
quotients by reductive groups, and show that shifted Poisson structures can be
characterized by it.
