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$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K$ be a field of characteristic zero, and let $\overline{K}$ be an algebraic closure of $K$. Is it true that the irreducible, projective, rational representations $$ \SL_n(K) \rightarrow \GL_m(\overline{K}) $$ are just restriction to $\SL_n(K)$ of the Weyl modules plus field automorphisms? References and comments are most appreciative. Thanks!

Note: Rational representation == given by rational map of algebraic varieties.

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  • $\begingroup$ I guess it should follow from highest weight theory that all homomorphisms of algebraic groups $SL_n\to PGL_m$ are defined over $\mathbb Q$. (Since both groups are split.) Hence there is no Galois action to consider, only those Homs of algebraic groups. $\endgroup$
    – user473423
    Commented Nov 5, 2022 at 18:54

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