# 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.

A Weil representation in general is a specific representation of a metaplectic group (double cover of $$\text{Sp}_{2n}$$). It is discovered by Shale, Segal and Weil, so that it is also called as Shale-Segal-Weil representation or oscillator representation, and the intuition comes from physics I guess. Unfortunately I do not know much physics to comment on the physics origin. The specific equations you wrote down form a specific realization (Schrodinger model) of the Weil representation, and there are several other realizations of the Weil representation like Fock model. The Weil representation is however actually something quite canonical if you think about it abstractly.
A quick way of defining Weil representation is as follows. Let $$W$$ be a symplectic vector space over a local field $$k$$, then one can define the Heisenberg group $$H(V)$$ which is a central extension of $$V$$ by $$k$$ with group multiplication given by $$(v_{1},t_{1})(v_{2},t_{2})=(v_{1}+v_{2},t_{1}+t_{2}+\langle v_{1},v_{2}\rangle/2)$$. Then, the Stone-von Neumann theorem says that, for any nontrivial unitary character $$\chi$$ of $$k$$, there is a unique irreducible unitary representation $$\pi_{\chi}$$ of $$H(V)$$ with central character $$\chi$$ (the center of $$H(V)$$ is identified $$k$$). On the other hand, $$\text{Sp}(V)$$ acts on $$H(V)$$ by $$g\cdot(v,t)=(gv,t)$$, so by the uniqueness of the representation, the $$\text{Sp}(V)$$-action gives intertwining operators, unique up to scalar, of $$\pi_{\chi}$$. This is a projective representation of $$\text{Sp}(V)$$, and this eventually gives rise to a genuine representation, the Weil representation, of a two-fold cover of $$\text{Sp}(V)$$, the metaplectic group.
The way of constructing representations of $$\text{GL}_{2}(\mathbb{F}_{q})$$ in Bump's book is rather along the lines of what's called as theta correspondence or Howe duality. The case of $$\text{GL}_{2}$$ (or rather $$\text{SL}_{2}$$) is quite special because $$\text{SL}_{2}=\text{Sp}_{2}$$, and the theta correspondence is rather a way of transferring between certain representations of symplectic group and certain representations of orthogonal group. A rough idea is as follows: if $$V$$ is a symplectic vector space and $$W$$ is an orthogonal vector space (i.e. a vector space with a symmetric bilinear pairing), then the Weil representation $$\omega$$ on $$\text{Sp}(V\otimes W)$$ restricted to $$\text{Sp}(V)\times\text{O}(W)$$ decomposes as a direct sum $$\bigoplus \sigma\otimes\tau$$, and this $$\sigma\leftrightarrow\tau$$ gives a bijection between representations of $$\text{Sp}(V)$$ appearing in $$\omega$$ and those of $$\text{O}(W)$$ appearing in $$\omega$$. Of course not all representations occur in a single theta correspondence, so this is why in the case dealt in Bump's book one has to use both split and nonsplit torus. In general if you go up to higher rank one has to consider more orthogonal groups of different ranks. I am not entirely certain, but as in principle one has to use many different theta correspondences to build all (super)cuspidal representations, I doubt there is a clean picture related to Deligne-Lusztig varieties.