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Let G be a reductive connected group over an algebraic closure of a finitefield. I define a tensor structure on the category of perverse sheaves on Gwhich 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 theresulting monoidal category is equivalent to the centre of a monoidal categorywhich I defined in 1997 (a categorical version of the J-ring attached to thesame two-sided cell), thus verifying a conjecture of Bezrukavnikov, Finkelberg,Ostrik. A possible interpretation of unipotent characters associated to afinite noncrystallographic Coxeter group is given.

Truncated convolution of character sheaves