A Marstrand projection theorem for lines

Fix integers $1<k<n$. For $V\in G(k,n)$, let $P_V:{\mathbb{R}}^n\to V$ be the orthogonal projection. For $V\in G(k,n)$, define the map $${\pi }_V:A(1,n)\to A(1,V)⨆V\mathrm{.}$$ $$\mathrm{\ell }\mapsto P_V(\mathrm{\ell })\mathrm{.}$$ For any $0<a<\text{dim}(A(1,n))$, we find the optimal number $s(a)$ such that the following is true. For any Borel set $\mathit{A}\subset A(1,n)$ with $\text{dim}(\mathit{A})=a$, we have $$\text{dim}({\pi }_V(\mathit{A}))=s(a),\text{for a.e. }V\in G(k,n)\mathrm{.}$$ When $A(1,n)$ is replaced by $A(0,n)={\mathbb{R}}^n$, it is the classical Marstrand projection theorem, for which $s(a)=\min \{k,a\}$. A new ingredient of the paper is the Fourier transform on affine Grassmannian.