# Etymology of cuspidal representations

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.

• 1. Harish-Chandra. 2. Take a look at Gelfand-Graev-PS "Representation theory and automorphic functions" to understand the connection with the cusps of lattices in $PSL_2(\mathbb{R}).$ – Victor Protsak Jan 14 '14 at 7:35
• The cuspidal representations are essentially the representations that are actually native to the group - they aren't induced from subgroups like the principal or special series. People seem to use "cuspidal" and "supercuspidal" pretty interchangably when talking about groups such as $GL_2(\Bbb{Q}_p)$, but I believe that the "correct" terminology is that the representations with all Jacquet modules trivial are cuspidal while the representations which cannot be obtained through the constructions of the principal and special series are supercuspidal. – user19918273 Jan 14 '14 at 7:39
• It's also worth noting that for a supercuspidal representation of $GL_2(\Bbb{Q}_p)$, its matrix coefficients are all supercusp forms - I belive this is why Harish-Chandra gave them that name in the first place. – user19918273 Jan 14 '14 at 7:41
• Can't edit the first comment for some reason but I suppose I should clarify that when I say "aren't induced from subgroups" I actually meant "aren't induced from lower dimensional subgroups", so in the $GL_2$ case that's just "aren't induced from characters". And of course, cuspidal = supercuspidal for $GL_2$, but there are cases where you should differentiate between the two. – user19918273 Jan 14 '14 at 7:47
• @Peter- Thank you, this is very helpful. – Dylan Yott Jan 14 '14 at 7:52

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.

To reinforce what's already been said and add some references, I'd emphasize first that Harish-Chandra's "philosophy of cusp forms" was indeed a driving force in the study of representations over both finite and local fields. Borel organized the 1965 AMS summer institute at Boulder around these ideas and the emerging Langlands program. Certainly the term "cusp" arose in the early work on arithmetic groups and fundamental domains, leading indirectly to the idea of "discrete series" of representations. This starts with $\mathrm{SL}_2(\mathbb{R})$ and was developed very generally by Harish-Chandra for real semisimple Lie groups. He and many others also opened the way to study of such groups over local fields as well as over finite fields. His conference talk Eisenstein series over finite fields (in the volume Functional Analysis and Related Fields, Springer, 1970) reflects his interest in carrying classical ideas over to such fields.

Concerning the character theory of finite groups of Lie type, there is of course a long history going back to work of Frobenius and Schur, in which it became clear that parabolic induction alone would not produce all characters.

In his influential 1955 paper on characters of finite general linear groups, J.A. Green was able to use combinatorial methods to fill in the missing characters. But his student Bhama Srinivasan had more difficulty with the case of $\mathrm{Sp}_4(\mathbb{F}_q)$, which helped to shift attitudes toward the systematic study of "discrete series" or "cuspidal characters" in terms of Harish-Chandra's philosophy. Here one associates series of characters to the various types of finite tori, ranging from split to anisotropic ("compact"). The latter lead to cuspidal characters, and the 1976 paper by Deligne-Lusztig made possible the use of sophisticated geometric methods for their study. (Lusztig himself had been influenced by Green in his earlier construction of some discrete series characters satisfying "cusp conditions" for general linear groups, in his paper Ann. of Math. Studies 81 (1974).)

Along the way there was also a special year at IAS in 1968-69, organized by Borel and influenced by Harish-Chandra, Springer, and others. In particular, Springer had two relevant series of lectures written up in Lecture Notes in Mathematics 131 (Springer, 1970), including Cusp forms for finite groups. It's difficult to disentangle all the overlapping contributions here, but the terminology (codified in Roger Carter's 1985 book) is appropriate even if indirect.

As noted already, terminology for the groups over local fields developed along similar philosophical lines, but with the addition of terms like "supercuspidal" which doesn't come up over finite fields.

• @MarcPalm , Jim Humprheys I was wondering for a long time - what is the relation between two points of view on cuspidal representations presented in your answers, it would be quite helpful if it can be explained... – Alexander Chervov Jan 18 '14 at 13:19
• @AC: The relationship between working over finite fields or local fields (including the reals) is part of the puzzle, I guess, which experts know more about than I do. But all of this work related to Harish-Chandra's philosophy contributes to the hoped-for "unity" of mathematics. Of course, there are more direct connections between $p$-adic rings and finite residue fields, but involving $p$-modular representations rather than ordinary characters. (For finite groups these theories too are intricately related.) – Jim Humphreys Jan 18 '14 at 18:27