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

Irreducibility of Infinite Dimensional Steinberg Modules of Reductive Groups with Frobenius Maps