Weil representations over finite fields and Shintani lift

Let SpV(F)SpV(F) be the group of isometries of a symplectic vector space V over a finite field F   of odd cardinality. The group SpV(F)SpV(F) possesses distinguished representations—the Weil representations. We show that they are compatible with base change in the sense of Shintani for a finite extension F′/FF′/F. The result is also true for the group of similitudes of V.