I heard that Deligne-Lusztig theory gives us geometric way to construct representations of finite algebraic groups over finite fields, such as $\mathrm{GL}_{2}(\mathbb{F}_{q})$, which arises from etale cohomology of some variety over a finite field. I learned about representation theory of $\mathrm{GL}_{2}(\mathbb{F}_{q})$ from Bump's automorphic representation book, and I know the classification of all the irreducible representation of the group. Especially, some of the representations are induced representation from Borel subgroup (or its sub/quotient representation), and the other representations are cuspidal representations.

Bump constructed cuspidal representations in a fairly nontrivial way via Weil representation. (Actually, he recovered all the representations including principal series representations from Weil representation.) For $E/\mathbb{F}_{q}$ a quadratic extension, let $\chi$ be a character of $E^{\times}$ which does not factor through the norm map $N:E^{\times} \to F^{\times}$. Then we define $$ W(\chi) = \{ \Phi : E\to \mathbb{C}\,:\, \Phi(yx) = \chi(y)^{-1}\Phi(x), \quad y\in \ker N\} $$ and define an action of $\mathrm{GL}_{2}(\mathbb{F}_{q})$ on $W(\chi)$ by $$ \left( \pi \begin{pmatrix} a & \\ & a^{-1} \end{pmatrix} \Phi\right)(x) = \Phi(ax) \\ \left( \pi \begin{pmatrix} 1 &z\\0&1 \end{pmatrix} \Phi\right)(x) = \psi(zN(x))\Phi(x) \\ \left( \pi\begin{pmatrix} & 1 \\ -1 & \end{pmatrix}\Phi\right)(x) = \hat{\Phi}(x) \\ \left( \pi\begin{pmatrix} a & \\ & 1 \end{pmatrix}\Phi\right)(x) = \chi(b) \Phi(bx) $$ where $\hat{\Phi}(x)$ is a Fourier transform of $\Phi$ (which is defined in a suitable way).

Since I can't find any intuition of Weil representation, I want to know where this Weil representation comes from. Especially, I hope this construction follows from Deligne-Lusztig theory in a very natural way. If that is true, it will be great if there is any reference that computes the representations very explicitly (at least for $\mathrm{GL}_{2}(\mathbb{F}_{p})$ so we can see that Weil representation is actually comes from Deligne-Lusztig theory (if it is correct). Thanks in advance.


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