Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = \text{SL}_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes the origin $0$. Hence, $G$ acts on the set $X := \mathbb{F}^2 \setminus \{0\}$, the complement of the origin. For any group homomorphism $\chi: \mathbb{F}^\times \to S^1 \subset \mathbb{C}^\times$, in $\mathbb{C}\{X\}$, we define a subspace$$\mathbb{C}\{X\}^\chi := \{f \in \mathbb{C}\{X\} : f(z \cdot x) = \chi(z) \cdot f(x), \text{ for all }z \in \mathbb{F}^\times\}.$$My question is, do there exist irreducible representations of $G$ not occur in the decomposition of $\mathbb{C}\{X\}$? I would preferably like to see a way of attacking this which doesn't use too "high-tech" machinery.