  Let A(n) be the smooth dual of the p-adic group G=GL(n). We create on A(n)
the structure of a complex algebraic variety. There is a morphism of A(n) onto
the Bernstein variety Omega G which is injective on each component of A(n). The
tempered dual of G is a deformation retract of A(n). The periodic cyclic
homology of the Hecke algebra of G is isomorphic to the periodised de Rham
cohomology supported on finitely many components of A(n).
