Extremal cross $t$-intersecting families under $t$-covering number constraints for vector spaces

Let $V$ be an $n$-dimensional vector space over the finite field ${\mathbb{F}}_q$, and $\left[\genfrac{}{}{0px}{}{V}{k}\right]$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F}\subseteq \left[\genfrac{}{}{0px}{}{V}{k}\right]$ and $\mathcal{G}\subseteq \left[\genfrac{}{}{0px}{}{V}{\mathrm{\ell }}\right]$ are said to be cross $t$-intersecting if $\dim (F\cap G)\ge t$ for all $F\in \mathcal{F}$ and $G\in \mathcal{G}$. In this paper, we determine the extremal structures when $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\mathrm{\mid }\mathcal{G}\mathrm{\mid }$ attains the maximum value under the conditions $\dim \left({\cap }_{F\in \mathcal{F}}F\right)<t$ and $\dim \left({\cap }_{G\in \mathcal{G}}G\right)<t$.