Gauss Sums on $GL_2(\mathbb{Z}/p^l\mathbb{Z})$

We determine explicitly the Gauss sums on the general linear group $GL_2(\mathbb{Z}/p^l\mathbb{Z})$ for all irreducible characters, where $p$ is an odd prime and $l$ is an integer > 1. While there are several studies of the Gauss sums on finite algebraic groups defined over a finite field, this paper seems to be the first one which determines the Gauss sums on a matrix group over a finite ring.