Gauss sums on GL2(Z/plZ)GL2(Z/plZ)

We determine explicitly the Gauss sumτl(χ,e)=∑X∈GL2(Z/plZ)χ(X)e(TrX) on the general linear group GL2(Z/plZ)GL2(Z/plZ) for every irreducible character χ   of GL2(Z/plZ)GL2(Z/plZ) and a nontrivial additive character e   of Z/plZZ/plZ, where p is an odd prime and l is an integer ⩾2. 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.