I am interested in the manifold of affine subspaces of dimension $k$ of $\mathbb{R}^n$, which can be viewed as the homogeneous space $$ E(n)/(E(k)\times O(n-k)),$$ where $E$ refers to rigid motions and $O$ to orthogonal transformations. The Grassmannian of vector subspaces $O(n)/(O(k)\times O(n-k))$ is a symmetric space. Is there a chance that the affine Grassmanian also is?



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