I'm having a surprisingly hard time finding references for some facts about $GL_k$-equivariant cohomology of the space of $k\times n$ matrices. Specifically, I believe the following things to be true:

- The ring $H^*_{GL_k}(\mathrm{Mat}_{k\times n})$ is isomorphic to the ring $\Lambda_k$ of symmetric polynomials in $k$ variables.
- Under this isomorphism, the $GL_k$-equivariant cohomology class of a Schubert variety $\Omega_\lambda$ in $\mathrm{Mat}_{k\times n}$ maps to the Schur polynomial $s_\lambda$.
- If we restrict to the subvariety $\mathrm{Mat}_{k\times n}^\circ$ of full-rank $k\times n$ matrices, the corresponding pullback map $$H^*_{GL_k}(\mathrm{Mat}_{k\times n})\to H^*_{GL_k}(\mathrm{Mat}_{k\times n}^\circ)=H^*(Gr(k,n))$$ is the one that kills the Schur polynomials corresponding to partitions $\lambda$ with $\lambda_k>n-k$.

Does anyone know either a reference or a simple argument for these? I think (1) should follow simply from the fact that $\mathrm{Mat}_{k\times n}$ is contractible, and that the isomorphism is even canonical. I also think that (1) and (2) together should imply (3) without too much effort, but I'm not quite comfortable enough with these ideas to fill in all the details.