Let $G$ be a compact connected simple Lie group and let $V$ be a real $G$-representation. How does one go about computing the ring of $G$-invariant (polynomial?) functions $V \to \mathbb{R}$?
I've been told that this ring is always the coordinate ring of some algebraic variety, which sounds important.
I would really appreciate any references that address this, preferably a textbook that contains lots of down-to-earth examples. I should underscore that I'm a differential geometer whose interest is in concrete computations, not an algebraist seeking the most general powerful theorems possible.