# Irreps of $S_3=GL(2,2)$. Who is cuspidal?

$S_3 = GL(2, F_2)$. It has $3$ irreps - trivial, sign, standard 2d.

Can one help me to understand general words about Irreps of $G(F_q)$ ("principal series", "parabolically induced", "cuspidal", "complimentrary series", "Steinberg irrep") in this particular example ? I mean who of these irreps is who ?

Notes on the subject I am looking:

Paul Garrett: http://www.math.umn.edu/~garrett/m/v/toy_GL2.pdf

Etingof&Students: http://arxiv.org/abs/0901.0827

Amritanshu Prasad http://www.imsc.res.in/~amri/html_notes/notes.html#notesch2.html

## 1 Answer

Here, the "parabolically-induced" (="principal series") repn's standard model is just left $P$-invariant ($P$=upper-triangular) functions on the group $G$, since there is only the trivial character on diagonal things $M=\{1\}$. This repn is $|G|/|P|$-dimensional, that is, $(3\cdot 2)/2=3$, contains the trivial repn, obviously, and the 2-dimensional complement is the "Steinberg", by conventional naming. Evidently, the single "supercuspidal" (meaning sums to $0$ over the unipotent radical $N$ of $P$) is the "sign" repn. A peculiar outcome!