Just a comment, here are some original references for the construction of the Weil representation.
[1] B. Bolt, T. G. Room and G. E. Wall, On the Clifford collineation, transform and similarity groups. I, J. Austral. Math. Soc., 2 (1961-62), 60-79. DOI
[2] I. M. Isaacs, Characters of solvable and symplectic groups, Amer. J. Math., 95 (1973), 594-635. DOI
[3] R. E. Howe, On the characters of Weil’s representations, Trans. Amer. Math. Soc., 177 (1973), 287-298. DOI
[4] P. Gérardin, Weil representations associated to finite fields, J. Algebra, 46 (1977), 54-101. DOI
[5] H. N. Ward, Representations of symplectic groups, J. Algebra, 20 (1972), 182-195. DOI