  Let G be a reductive connected group over an algebraic closure of a finite
field. I define a tensor structure on the category of perverse sheaves on G
which are direct sums of unipotent character sheaves in a fixed two-sided cell,
in accordance with a conjecture I have made in 2004. I also show that that the
resulting monoidal category is equivalent to the centre of a monoidal category
which I defined in 1997 (a categorical version of the J-ring attached to the
same two-sided cell), thus verifying a conjecture of Bezrukavnikov, Finkelberg,
Ostrik. A possible interpretation of unipotent characters associated to a
finite noncrystallographic Coxeter group is given.
