# How I could define a inner product in the characters in $SL(2, \mathbb R)$

I have homework in a course on Lie groups, in which I must show that $(\pi_m,\mathbb C_m[x,y])$ are the only irreducible representations of finite dimension in SL(2,ℝ). Here $C_m[x,y])$ is the vector space of homogeneous polynomials, and $\pi_m(g)$ acts via $\pi_m(g)(f(x,y))=f((x,y)\cdot g)$. It was not too complicated to calculate the characters, taking the basis $\{p_k:0\leq k \leq n\}$ with $p_k = x^ky^{n−k}$, since $\pi_m(exp(H))p_k=exp(d\pi_m(H))p_k=exp(2k−n)p_k$, but the question is how one could define an inner product for these matrices. But, I don’t have the volume form of SL(2,ℝ) to define the integral $\langle f,g \rangle=\int_Gf(z)⋅\overline{g(z)}dz$