Virtual Workshop on Recent Developments in Geometric Representation Theory

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 $(G_1, G_2)$ acting on a symplectic space $V$, we will introduce a class of simple perverse sheaves on $V$, equivariant under the action of $G_1 \times G_2$, that we call theta intertwining sheaves. Roughly, these sheaves geometrize projectors onto simple constituents of the Weil representation under $G_1 \times G_2$. Moreover, they induce a correspondence between character sheaves on $G_1$ and $G_2$. Based on joint work in progress with S.Gurevich