A note on cusp forms and representations of ${\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)$

Cusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroup $\mathrm{\Gamma }(p)$, $p$ a prime, is acted by ${\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)$. Meanwhile, there is a finite field incarnation of the upper half-plane, the Deligne–Lusztig (or Drinfeld) curve, whose cohomology space is also acted by ${\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_p)$. In this note we compute the relation between these two spaces in the weight $2$ case.