# Deligne-Lusztig theory and cuspidal representations of $\mathrm{GL}_{2}(\mathbb{F}_{q})$

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.