Convolution of Almost Characters

1. Let G be a finite group. Let VecG be the category whose objects are C-vector bundles on G which are G-equivariant for the conjugation action of G on G. For U ∈ VecG let Ug denote the fibre of U at g ∈ G; the G-equivariant structure on U is given by isomorphisms ψ x : Uy −→ Uxyx−1 for any x, y ∈ G. For U,U ′ ∈ VecG we define U ∗ U ′ ∈ VecG as follows: for y ∈ G, we have (U ∗ U )y = ⊕(g,g′)∈G×G;gg′=yUg ⊗ U ′ g′ ; the G-equivariant structure on U ∗ U ′ is defined by ψ ′ x : ⊕(g,g′)∈G×G;gg′=yUg ⊗ U ′ g′ −→ ⊕(h,h′)∈G×G;hh′=xyx−1Uh ⊗ U ′ h′