Let V be an \((n+\ell )\) -dimensional vector space over a finite field, and W a fixed \(\ell \) -dimensional subspace of V. Write \({V\brack n,0}\) to be the set of all n-dimensional subspaces U of V satisfying \(\dim (U\cap W)=0\) . A family \(\mathcal {F}\subseteq {V\brack n,0}\) is t-intersecting if \(\dim (A\cap B)\ge t\) for all \(A,B\in \mathcal {F}\) . A t-intersecting family \(\mathcal {F}\subseteq {V\brack n,0}\) is called non-trivial if \(\dim (\cap _{F\in \mathcal {F}}F)<t\) . In this paper, we describe the structure of non-trivial t-intersecting families of \({V\brack n,0}\) with large size. In particular, we show the structure of the non-trivial t-intersecting families with maximum size, which extends the Hilton–Milner theorem for \({V\brack n,0}\) .
