Cross $t$-intersecting families for symplectic polar spaces

Let $\mathcal{P}$ be a symplectic polar space over a finite field ${\mathbb{F}}_q$, and ${\mathcal{P}}_m$ denote the collection of all $k$-dimensional totally isotropic subspace in $\mathcal{P}$. Let ${\mathcal{F}}_1\subset {\mathcal{P}}_{m_1}$ and ${\mathcal{F}}_2\subset {\mathcal{P}}_{m_2}$ satisfy $\dim (F_1\cap F_2)\ge t$ for any $F_1\in {\mathcal{F}}_1$ and $F_2\in {\mathcal{F}}_2$. We say they are cross $t$-intersecting families. Moreover, we say they are trivial if each member of them contains a fixed $t$-dimensional totally isotropic subspace. In this paper, we show that cross $t$-intersecting families with maximum product of sizes are trivial. We also describe the structure of non-trivial $t$-intersecting families with maximum product of sizes.