TITLE: Irreducible representations of simple complex groups
QUESTION [8 upvotes]: Let $G$ be a simple complex algebraic group. What are its complex irreducible finite-dimensional representations?
Before you start voting to close the question, I never said "rational". I am asking about abstract representations of $G$ as a group.
I can a make an obvious conjecture but I have no idea how to prove it, even for $SL_2$. Yet, it could be something well-known. Pick a rational irreducible representation $(V,\rho)$ and a field endomorphism $\tau: {\mathbb C} \rightarrow {\mathbb C}$. The field endomorphism extends to an endmorphism of $G$: just apply $\tau$ to each matrix entry of of $g\in G\subseteq GL_n(\mathbb C)$. Thus, we get a twisted rational representation $(V,\tau\circ \rho)$ that clearly remains irreducible.
STUPID WABBIT CONJECTURE: all irreducible representations are twisted rational representations.

REPLY [5 votes]: Just to complete the answer, every complex irreducible representation of $G(\mathbb{C})$ ($G$ is simply connected) is an $r$-fold  tensor product of the form $\rho _1\circ \sigma _1\otimes \cdots \otimes \rho _r \circ \sigma _r$, where each $\rho _i$ is an algebraic irreducible representation of $G$ and $\sigma _i$ are $distinct$ embeddings of the field $\mathbb{C}$ into itself. This readily follows from the results of Borel-Tits (see Theorem (10.3)  of the article  of Borel-Tits mentioned by nfdc23).