An Erdős-Ko-Rado result for some principal series representations

Let $V$ be an irreducible principal series representation of ${\mathrm{G}\mathrm{L}}_2(q)$ satisfying certain conditions. Two subsets $S_1,S_2\subseteq {\mathrm{G}\mathrm{L}}_2(q)$ are called cross-$t$-intersecting if $\dim \{v\in V:g_{1}v=g_{2}v\}⩾t$ for any $(g_1,g_2)\in S_1\times S_2$. In this paper, we determine $\max (\mathrm{\mid }{S}_1\mathrm{\mid }\cdot \mathrm{\mid }{S}_2\mathrm{\mid })$ where $S_1,S_2\subseteq {\mathrm{G}\mathrm{L}}_2(q)$ are cross-$1$-intersecting. Our proofs are based on eigenvalue techniques and the representation theory of ${\mathrm{G}\mathrm{L}}_2(q)$.