EKR Results for Some Cuspidal Representations

Abstract

Let ${\mathcal{S}}_{\chi }$ be a cuspidal representation of ${\mathrm{GL}}_2(q)$. A subset $S\subseteq {\mathrm{GL}}_2(q)$ is $t$-intersecting if ${\dim }_{\mathbb{C}}\{v\in {\mathcal{S}}_{\chi }:g_{1}v=g_{2}v\}⩾t$ for any $g_1,g_2\in S$. For $q=p^f$, where $p$ is an odd prime, $f⩾2$, and $1⩽t<p$ is an integer, we determine the maximum size of an $t$-intersecting subset when $\chi (-1)=-1$, and when $\chi (-1)=1$ with $t⩽2$. The proof combines explicit fixed-space dimensions, attaining subgroup constructions, and weighted spectral bounds for normal Cayley graphs.