Chern character for the Schwartz algebra of p-adic GL(n)

We construct a Chern character map from the K-theory of the reduced C^* algebra of the p-adic GL(n) with values in the periodic cyclic homology of the Schwartz algebra of this group. We prove that this map is an isomorphism after tensoring with C by comparing an explicit formula, stated in the algebraic case by Cuntz and Quillen, with the classical Chern character. This Chern character is a crucial ingredient in the proof of the Baum-Connes conjecture for the p-adic GL(n) due to Baum, Higson and Plymen.