It is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group that takes all reflections to –1.
Over a finite field of characteristic p, the Steinberg representation has degree equal to the largest power of p dividing the order of the group.
For the general linear group GL(2), the dimension of the Jacquet module of a special representation is always one.
Matsumoto (1969), Shalika (1970), and Harish-Chandra (1973) introduced Steinberg representations for algebraic groups over local fields.
Casselman (1973) showed that the different ways of defining Steinberg representations are equivalent.