Hopf algebra and the duality operation for $\mathfrak{gl}_n(\mathbb{F}_q)$

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