Minimal Integral Models for Principal Series Weil Characters

We prove a conjecture of Udo Riese about the minimal ring of definition for principal series Weil characters of ${\text{SL}}_2({\mathbb{F}}_p)$, for $p$ an odd prime. More precisely, we show that the $(p+1)\mathrm{/}2$-dimensional Weil characters can be realized over the ring of integers of $$\mathbb{Q}$(\sqrt{\varepsilon p})$,where$\varepsilon =(-1)^{(p-1)/2}$, and we provide explicit integral models over these quadratic rings. We do so by studying the Galois action on the integral models of Weil characters recently discovered by Yilong Wang.