  We define an inner product (suitably interpreted) on the K(n)-local spectrum
LG := L_{K(n)}BG_+, where G is a finite group or groupoid. This gives an inner
product on E^*BG_+ for suitable K(n)-local ring spectra E. We relate this to
the usual inner product on the representation ring when n=1, and to the
Hopkins-Kuhn-Ravenel generalised character theory. We show that LG is a
Frobenius algebra object in the K(n)-local stable category, and we recall the
connection between Frobenius algebras and topological quantum field theories to
help analyse this structure. In many places we find it convenient to use
groupoids rather than groups, and to assist with this we include a detailed
treatment of the homotopy theory of groupoids. We also explain some striking
formal similarities between our duality and Atiyah-Poincare duality for
manifolds.
