Non-trivial $t$-intersecting families for the distance-regular graphs of bilinear forms

Abstract: 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 $\dim(A\cap B)\geq t$ 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 ${V\brack n,0}$.