finite upper half-plane model for the $\text{GL}_2(\Bbb{F}_q)$ Weil representation Let $\Bbb{F}_q$ be a finite field with $q$ elements, let $\Bbb{F}_{q^2}$
be its quadratic extension, and consider the finite "upper" half space
${\frak{H}}_q := \Bbb{F}_{q^2} - \Bbb{F}_q$. Apeing a construction found in one of S. Lang's immortal works one can show for $g \in  \text{GL}_2(\Bbb{F}_q)$ and $f: {\frak{H}}_q \longrightarrow \Bbb{C}$ that $\sigma_\chi$ given by
\begin{equation} \Big( \sigma_\chi(g) f \Big)(z) \ =\ f \Bigg( {az + b \over {cz +d} }\Bigg) \, \chi\big(cz + d \big)\end{equation}
defines a $q(q-1)$ dimensional representation of $\text{GL}_2(\Bbb{F}_q)$ on the vector space
$\Bbb{C}\big[ {\frak{H}}_q \big]$ consisting of all complex-valued function on ${\frak{H}}_q$ where 
\begin{equation} g^{-1} \ = \ \begin{pmatrix} a & b \\ c & d\end{pmatrix} \end{equation}
and where $\chi: \Bbb{F}_{q^2}^* \longrightarrow \Bbb{C}^*$ is a fixed (indecomposable) multiplicative character. 
Is it possilbe to identify the Weil representation of $\text{GL}_2(\Bbb{F}_q)$
corresponding to $\chi$ as a direct summand of $\Bbb{C}\big[ {\frak{H}}_q \big]$
and if so, with which multiplicity does it occur ?  
yours,
Ines
 A: It feels like it should be. In a sense it's really almost there! But alas, it's not.
Set $G=GL_2(\mathbb{F}_q)$. Fix $\epsilon\in\mathfrak{H}_q$, such that $\epsilon^2\in\mathbb{F}_q^\times$. We define the usual action of $G$ on $\mathfrak{H}_q$ by:
$$\left( \begin{array}{ccc}
a & b \\
c & d \end{array} \right)(z)=\frac{az+b}{cz+d}$$
and also the "automorphic factor":
$$j_\chi(\left( \begin{array}{ccc}
a & b \\
c & d \end{array} \right), z)=\chi(cz+d)$$
Exercise 0.0 (you're probably familiar with this): Show that the above is an action, i.e. $gg'(z)=g(g'(z))$
Exercise 0.1 Show that $j_\chi$ is a cocycle in the sense that: $j_\chi(gg',z)=j_\chi(g,g'(z))j_\chi(g',z)$
Let $T$ be the stabiliser of $\epsilon\in\mathfrak{H}_q$.
Exercise 1: Show that $T$ is the subgroup consisting of the matrices
$$\left( \begin{array}{ccc}
a & b\epsilon^2 \\
b & a \end{array} \right),\ \ (a,b)\not=0$$
Exercise 2: Show that 
$$d:\left( \begin{array}{ccc}
a & b\epsilon^2 \\
b & a \end{array} \right)\mapsto a+\epsilon b$$
is an isomorphism $T\cong\mathbb{F}_{q^2}^\times$.
Define $\varphi:\mathbb{C}[\mathfrak{H}_q]\rightarrow\mathbb{C}[G]$ as:
$$\varphi(f)(g)=f(g(\epsilon))j_\chi(g,\epsilon).$$
Endow $\mathbb{C}[G]$ with an action of $G$ on the left, by:
$$(gf)(h):=f(g^{-1}h).$$
Exercise 3: Show that $\varphi$ is an intertwining operator, i.e: $g\varphi(f)=\varphi(\sigma_\chi(g)f)$.
Exercise 3$\frac{1}{2}$: Show that $\varphi$ is injective.
Define $\text{Ind}_T^G\ \chi\circ d$ to be the subspace of $\mathbb{C}[G]$ of functions:
$$\{ F\in\mathbb{C}[G]\ |\ F(gt)=F(g)\chi(d(t)), \text{for all } g\in G, t\in T \}$$
Exercise 4: Show that $\varphi(\mathbb{C}[\mathfrak{H}_q])=\text{Ind}_T^G\ \chi\circ d$.
So in order to answer the question, we must compute $(W(\chi), \text{Ind}_T^G\ \chi\circ d)$. By Frobenius reciprocity, this is equal to $(Res_T\ W(\chi), \chi\circ d)_T$
Given the character of $Res_T\ W(\chi)$, this is a very easy computation. It is well known that:
$$Tr_{W(\chi)}(\left( \begin{array}{ccc}
a & 0 \\
0 & a \end{array} \right))=(q-1)\chi(a),\ \ a\not=0$$
and
$$Tr_{W(\chi)}(\left( \begin{array}{ccc}
a & b\epsilon^2 \\
b & a \end{array} \right))=-\chi(a+\epsilon b)-\chi(a-\epsilon b),\ \ b\not=0$$
So, finally:
Exercise 5: Show that $(W(\chi), \sigma_\chi)=0$.
If you've come this far, it is now also possible to prove:
Exercise 6: Let $\theta\in\widehat{\mathbb{F}_{q^2}^\times}$ be a generator. Show that $(W(\chi\theta^{i(q-1)}), \sigma_\chi)=1$ for $1\le i\le q$.
And we get:
$$\sigma_\chi=\bigoplus_{1\le i\le q} W(\chi\theta^{i(q-1)})$$
