Legendre's symbol in Schrödinger model for the Weil representation I have a question concerning the Schrödinger model for the Weil representation over a finite field $\mathbb{F}_q$. 
The way to present the action of the Weil representation $\omega$ of  $Sp(2n,\mathbb{F}_q)$ in this model, is to give the action of matrices $\left(\begin{smallmatrix}a & 0\\ 0&^{t}a^{-1}\end{smallmatrix}\right)$ in a standard Levi, $\left(\begin{smallmatrix}1 & u\\ 0& 1\end{smallmatrix}\right)$ of a radical unipotent, and finally of  matrices of the form $\left(\begin{smallmatrix}0 & -1\\ 1 & 0 \end{smallmatrix}\right)$. The question I have is about the Legendre symbol appearing in the formula for the action of elements of the Levi. The formula is 
$$
\omega\left(\begin{matrix}a & 0\\ 0&^{t}a^{-1}\end{matrix}\right)f(x)=\left(\frac{det(a)}{q}\right)f(^tax)
$$
I do not understand where the Legendre symbol $\left(\frac{\cdot}{q}\right)$ comes from. In the p-adic setting this is replaced by $|det(a)|^{1/2}$, and the reason is so that the action of this Levi is unitary. I hope someone can clarify this doubt.
Thanks in advance.
Greetings.
 A: As a way to "discover" the quadratic/norm characters appearing in the finite-field Segal-Shale-Weil/oscillator repn, as well as in that over local fields (with even-dimensional vector space so that the repn is a genuine linear repn rather than projective), one can imagine that a more naive guess for the group action by parabolic $P$ and Weyl element $w$ is correct, up to some correction factors. Then the correction factors can be found by re-expressing a product $g_1g_2$ with $g_1={p_1}wn_1$ and $g_2={p_2}wn_2$ of big-cell group elements as a big-cell element $g_1g_2=p_3wn_3$, and writing out the requirement that the repn be a group homomorphism.
The discrepancy is an exponential sum, which is relatively easily shown to be the character. The computations are a bit tedious, but straightforward.
The Stone-vonNeumann theorem shows that the actual range of possibilities is just that of a choice of constant, so any single computation determining the "adjustment" is sufficient.
The gory details are written out for the finite-field case in http://www.math.umn.edu/~garrett/m/repns/05_finite_heisenberg_ssw.pdf
