Hopf algebra and the duality operation for glₙ(Fq)

In this paper we study the space C(glₙ(Fq)) of complex invariant functions on glₙ(Fq), through a Hopf algebra viewpoint. First, we consider a variant notion of Zelevinsky's PSH algebra defined over the real numbers R. In particular, we show that two specific R-lattices inside the complex Hopf algebra ₙC(glₙ(Fq)) are real PSH algebras, and that they do not descend to Z. Then, among consequences, we prove that every element in C(glₙ(Fq)) is a linear combination of Harish-Chandra inductions of Kawanaka's pre-cuspidal functions, and give a conceptual characterisation of duality operation for glₙ(Fq), which in turn allows us to give a new proof of a classical result of Kawanaka.