Let $G=GL(2,p)$ be the group of linear transformations. It acts on the set $X=F\_p^2$.
Then $C^X$ is a linear representation of $G$. What is the decomposition of this representation into irreducible representations?


*Note*: Let $d\_i$ denote the number of isomorphic irreducible representations. Then $\sum d\_i^2$ is equal to the number of orbits of $G$ when $G$ acts on $X\times X$. The number of orbits of $G$ in $X\times X$ is equal to $p+1$. 



Decomposition of GL(2,p) into irreducible representations
**Note:**  *This was written up concurrently with David's answer, but wasn't proofread and didn't get past the captcha stage due to technical problems.* 


It is more common to consider the representation of $G=GL(2,\mathbb{F}\_p)$ on $\mathbb{C}^Y$, where $Y=X\setminus 0$. This is a direct sum over all multiplicative characters $\chi$ of $\mathbb{F}\_p$ of the induced representations of $G$ in the homogeneous functions of homogeneity degree $\chi:$


$$\mathbb{C}^Y=\bigoplus\_{\chi\in \mathbb{F}\_p^\ast}(\mathbb{C}^Y)\_{\chi}, \quad (\mathbb{C}^X)\_{\chi}=\{f:Y\to \mathbb{C}: f(\lambda a)=\chi(\lambda)f(a)\ \text{for all}\ \lambda\in \mathbb{F}\_p \},$$


and $\mathbb{C}^X=\mathbb{C}\oplus\mathbb{C}^Y,$ where the first summand corresponds to the functions supported at $0\in\mathbb{F}\_p^2.$ 


Each induced representation has dimension $p+1.$ For the trivial character, the induced representation contains a trivial subrepresentation of $GL(2,\mathbb{F}\_p)$ (constants), with irreducible $p$-dimensional quotient (Steinberg representation). All other induced representations are irreducible and pairwise non-isomorphic (they all have different central characters). The situation is a bit more complicated for restrictions to $H=SL(2,\mathbb{F}\_p)$ (denoted the same by abuse of notation): for any $\chi,\ (\mathbb{C}^Y)\_{\chi}\simeq (\mathbb{C}^Y)\_{\chi^{-1}}$ and if $p$ is odd and $\psi$ is a character of order $2$ then the representation $(\mathbb{C}^Y)\_{\psi}$ is a direct sum of two non-isomorphic $(p+1)/2$-dimensional representations, all other representations remain irreducible and pairwise non-isomorphic. These facts can be verified *ad hoc* by the "sum of squares" calculation or, more systematically, by using the Mackey theory for finite groups. This accounts for close to the half of irreducible representations of $H$; to get the rest, one needs to induce from the non-split torus $\simeq \mathbb{F}\_{p^2}^\ast/\mathbb{F}\_{p}^\ast. $


