Q:

number of simple representations

For a linearly reductive Group $G$ over a field $k$ one has that the category of finite dimensional representations of $G$ is semisimple. What can one say about the number of simple representations? For what kind of groups the number is finite?

A:

Let $G$ be an affine linearly reductive group scheme over a field. Suppose that there are only finitely many simple representations (up to isomorphism) and let $X$ be the direct sum of them. Then every representation of $G$ is isomorphic to a subquotient (in fact, direct factor) of $X^n$ for some $n$. This implies that $G$ is finite (see, for example, Deligne and Milne, Tannakian Categories, 2.20).

