Weil representations of finite symplectic groups and finite odd-dimensional orthogonal groups

In [8], Srinivasan decomposes the uniform projection of the character of Weil representation ωG,G′ of a finite reductive dual pair (G,G′) in terms of Deligne-Lusztig virtual characters when the dual pair is one of the following: (1) two general linear groups; (2) two unitary groups; (3) a symplectic group and an even-dimensional orthogonal group, assuming the order of the finite field is large enough. In this article we obtain the decomposition for the remaining case, i.e., the dual pair of a symplectic group and an odd-dimensional orthogonal group. In particular, we have the decomposition of the uniform projection of the character of the Weil representation of a finite symplectic group.