Let $AG_k(\mathbb{R}^N)$ be the "affine Grassmannian" consisting of $k$-dimensional hyperplanes (i.e. affine subspaces) in $\mathbb{R}^N$. Is there any relation between $AG_k(\mathbb{R}^N)$ and the usual Grassmannian? Is $AG_k(\mathbb{R}^N)$ a classifying space of any Lie groups?

Let $AG_k(\mathbb{C}^N)$ be the "affine Grassmannian" consisting of $k$-dimensional complex hyperplanes in $\mathbb{C}^N$. Is there any relation between $AG_k(\mathbb{C}^N)$ and the usual complex Grassmannian? Is $AG_k(\mathbb{C}^N)$ a classifying space of any Lie groups?