Central simple representations and superelliptic jacobians

Let p be an odd prime and f(x)  a polynomial of degree at least 5 with complex coefficients and without repeated roots. Suppose that all the coefficients of f(x) lie in a subfield K such that:

1)      K contains a primitive p-th root of unity;

2)      f(x) is irreducible over K;

3)      the Galois group Gal(f) acts doubly transitively on the set of roots of f(x);

4)      the index of every maximal subgroup of Gal(f) does not divide deg(f)-1.

Then the endomorphism ring of the Jacobian of the superelliptic curve y^p=f(x) is isomorphic to the p-th cyclotomic ring for all primes p not dividing the order of Gal(f). We outline the proof, which is based on ideas from representation theory that may be of certain independent interest (very simple representations and central simple representations of groups).