Irreducible representations of simple complex groups

Question

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.

Answer 1

If $\rho:G(\mathbf{C}) \rightarrow {\rm{GL}}(V)(\mathbf{C})$ is such an abstract linear representation, by irreducibility (or mere semisimplicity) the Zariski closure $H \subset {\rm{GL}}(V)$ of the image has connected reductive identity component (due to Lie-Kolchin). But $G(\mathbf{C})$ is perfect with simple quotient modulo its finite center, so $H^0$ is semisimple, and then even simple in the sense of algebraic groups. The group $G(\mathbf{C})$ is generated by its "unipotent" subgroups that have no nontrivial finite quotient, so $H$ is even connected.

Thus, your question amounts to asking if $G(\mathbf{C}) \rightarrow H(\mathbf{C})$ arises from a $\mathbf{C}$-homomorphism $G \rightarrow H$ up to a field map $\mathbf{C} \rightarrow \mathbf{C}$. This is a (very) special case of the setting that was completely analyzed in an optimal way by Borel and Tits in their paper Homomorphismes "abstraits" de groupes algébriques simples in Annals of Math 97 (1973), pp. 499-571. There they work over general infinite fields, with connnected semisimple groups that are absolutely simple. The Introduction presents their main results (A) and (B) which address your question (in view of the initial part of this answer). See 8.18 for a certain standard counterexample over $\mathbf{R}$ and 8.19 for a conjecture in wider generality which requires some caveats in characteristic 2 (the necessity of which is explained in a Remark in the Introduction of Igor Rapinchuk's 2013 paper "On abstract representations of the groups of rational points of algebraic groups and their deformations" in Algebra and Number Theory, vol. 7).

Answer 2

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).