  The determination of scalars involved in Lusztig's conjecture for finite
reductive groups $G(F_q)$ was achieved by Waldspurger in the case of symplectic
groups or orthogonal groups, under the condition that $p,q$ are large enough.
Here $p$ is the characteristic of the finite field $F_q$. In this paper, we
determine the scalars in the case of symplectic groups with $p = 2$, by
applying the theory of symmetric spaces over a finite field due to Kawanaka and
Lusztig. We also obtain a partial result in the case of special orthogonal
groups with $p = 2$.
