  Let G be a connected reductive group over an algebraic closure of a finite
field Fq. In this paper it is proved that the infinite dimensional Steinberg
module of kG defined by N. Xi in 2014 is irreducible when k is a field of
positive characteristic and char k is not char Fq. For certain special linear
groups, we show that the Steinberg modules of the groups are not quasi-finite
with respect to some natural quasi-finite sequences of the groups.
