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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 $r\in[0,1]$. In particular, these space approximate the classifying space of $Gl(k,\Bbbk)$ as $n\to\infty$.