Consider the permutation action of $PGL_2(\mathbb F_q)$ on $\mathbb P^1(\mathbb F_q)$ by fractional linear transformations. We can consider the associated (complex) representation of dimension $q+1$.

What can we say about the irreducible representations occurring in this representation? This is probably doable because our representation is an induced representation. What I am really interested in is the following question:

What can we say about the invariant polynomials for this representation?