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.