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?



Is this affine-subspace analogue of a Grassmannian a classifying space?
I assume you mean $k$-dimensional planes, not just hyperplanes. Then the space $AG\_k(\Bbbk^n)$ deformation retracts onto the usual Grassmannian by sending each $k$-plane $E$ to $rE$ at time $r\in[0,1]$. More precisely, if $E=a+V$ with $a\in\Bbbk^n$ and $V\subset\Bbbk^n$ a linear $k$-dimensional subspace, then $rE$ denotes the affin plane $ra+V$.


In particular, these space approximate the classifying space of $Gl(k,\Bbbk)$ as $n\to\infty$. Because the deformation retractions above are compatible with the natural inclusions $AG\_k(\Bbbk^n)\to AG\_k(\Bbbk^{n+1})$, their colimit $AG\_k(\Bbbk^\infty):=\lim\_\to AG\_k(\Bbbk^n)$ still has $\lim\_\to G\_k(\Bbbk^n)$ as a deformation retract. Therefore, $AG\_k(\Bbbk^\infty)$ is a model for $BGl(k,\Bbbk)$.


