# ag.algebraic geometry – What are the irreps in this canonical action of \$PGL_2(F_q)\$?

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?