In the literature on representation theory of $GL_2(\Bbb F_p)$ and $GL_2(\Bbb Q_p)$, the irreducible representations with trivial Jacquet module are often called "cuspidal" or "supercuspidal". Why are these representations called cuspidal and who started calling them that? Also, is there any difference between cuspidal and supercuspidal? I imagine this is related to modular forms but haven't been able to find any explanation in the literature. These representations seem mysterious to me and maybe understanding their etymology will give me a better sense of how they fit into the bigger picture.
Cuspidal Representations – Etymology and Meaning
etymologymodular-formsnt.number-theoryrt.representation-theory
Best Answer
Cuspidal modular forms vanish at the cusps of $SL_2(\mathbb{Z}) \backslash \mathbb{H}$.
Via Strong approximation you can lift the to functions on $SL_2(\mathbb{Q}) \backslash SL_2(\mathbb{A})$ because $$ SL_2(\mathbb{Z}) \backslash \mathbb{H} \cong SL_2(\mathbb{Q}) \backslash SL_2(\mathbb{A}) / \prod\limits_p SL_2(\mathbb{Z}_p) \times SO(2).$$
Let $N$ be the group of upper diagonal matrices. Being a cusp form here means $$ \int\limits_{N(\mathbb{Q}) \backslash N(\mathbb{A})} f(ng) d n=0 $$ for all $g$.
Being a cuspidal (or nowadays rather square-integrable modulo center) representation $(\pi,V)$ on $GL_2(F)$ for $F$ local means for all $g \in GL_2(F)$ $$ \int\limits_{N(F)} \pi(n) v dn =0.$$ So it is the analogue of a global concept. So cuspidal means the integral along unipotent subgroups vanishes.
Supercuspidal representation is a different concept. It means that the matrix coefficient hae compact support. They are among the cuspidal representations.