Monodromy of elliptic curve convolution, seven-point sheaves of G_2-type and motives of Beauville type

We study the Tannakian properties of the category of perverse sheaves on elliptic curves endowed with the convolution product. We establish that for certain sheaves with unipotent local monodromy over seven points the corresponding Tannaka group is isomorphic to G_2. This monodromy approach generalizes a result of Katz on the existence of G_2-motives in the middle cohomology of deformations of Beauville surfaces.