Theta intertwining sheaves

The theta correspondence of Roger Howe gives a way to connect representations of different classical groups. We aim to geometrize the theta correspondence for groups over finite fields in the spirit of Lusztig's character sheaves. Given a reductive dual pair (G1,G2) acting on a symplectic space V, we will introduce a class of simple perverse sheaves on V, equivariant under the action of G1×G2, that we call theta intertwining sheaves. Roughly, these sheaves geometrize projectors onto simple constituents of the Weil representation under G1×G2. Moreover, they induce a correspondence between character sheaves on G1 and G2.

Based on joint work in progress with S.Gurevich