For a finite group, there are finitely many irreducible representations of complex numbers. What if the field is changed to some other fields? Like real numbers, p-adic field, finite field?
In particular, for a (finite) Galois group over a p-adic field, and consider p-adic Galois representation. Are there finitely many irreducible representations? If there are, can we actually construct some kind of varieteis s.t. the geometric representations (etale cohomology) coming from these varieties are exactly the irreducible ones?
And what if we replace the finite groups to other groups? Say, profinite groups, or even Lie groups, algebraic groups with non-discrete topologies?