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](https://doi.org/10.1017/S1446788700026379)

[2]  I. M. Isaacs, Characters of solvable and symplectic groups, Amer. J. Math., 95 (1973), 594-635. [DOI](https://doi.org/10.2307/2373731)

[3]  R. E. Howe, On the characters of Weil’s representations, Trans. Amer. Math. Soc., 177 (1973), 287-298. [DOI](https://doi.org/10.1090/S0002-9947-1973-0316633-5)

[4]  P. Gérardin, Weil representations associated to finite fields, J. Algebra, 46 (1977),
54-101. [DOI](https://doi.org/10.1016/0021-8693(77)90394-5)

[5]  H. N. Ward, Representations of symplectic groups, J. Algebra, 20 (1972), 182-195. [DOI](https://doi.org/10.1016/0021-8693(72)90098-1)