Non-trivial cross-$t$-intersecting families for vector spaces with the maximum sum of sizes

Let $V$ be an $n$-dimensional vector space over a finite field. Suppose that $\mathcal{F}$ and $\mathcal{G}$ are non-empty families of $k$-subspaces and $\mathrm{\ell }$-subspaces of $V$, respectively. They are said to be cross-$t$-intersecting if $\dim (F\cap G)\ge t$ for any $F\in \mathcal{F}$ and $G\in \mathcal{G}$, and are further called non-trivial if $\dim ({\cap }_{F\in \mathcal{F}}F)<t$ and $\dim ({\cap }_{G\in \mathcal{G}}G)<t$. In this paper, we characterize the non-trivial cross-$t$-intersecting families with the maximum sum of sizes. When $t=1$, our result serves as the $q$-analog of the theorems in [9,11].