Intertwiners and $\text{SL}(2, \mathbb{F}_q)$, vector space decomposition of $\mathbb{C}\{X\}$?

Question

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$$\mathbb{C}\{X\}^\chi := \{f \in \mathbb{C}\{X\} : f(z \cdot x) = \chi(z) \cdot f(x), \text{ for all }z \in \mathbb{F}^\times\}.$$How do I see that there is a vector space direct sum decomposition$$\mathbb{C}\{X\} = \oplus_{\chi \in \widehat{H}} \mathbb{C}\{X\}^\chi,$$where we put $H := \mathbb{F}^\times$?