If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G\_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}\_n$ over $\mathrm{Spec}(\mathbf{Z})$? Or is life not so easy? I feel like if I actually knew the construction of the irreps of $G\_{\mathbf{C}}$ (rather than treating the entire "highest weight" theory as a black box, which I've done up to this point in my life) I should be able to answer this. 


For $\mathrm{SL}(2)$ the answer is going to be yes, because the only irreps are symmetric powers, which can be realised over $\mathbf{Z}$. But for groups of rank greater than 1 I realise I am a bit unclear about the relationship between irreps (of the algebraic group) in char 0 and in char $p$; for example can the dimension of the irrep with some given highest weight jump as one moves from char 0 to char $p$? Even if it can, this doesn't mean my question has a negative answer.



Representations of complex semi-simple algebraic group "defined over $\mathbf{Z}$"?
This is an application of the geometric method of constructing highest-weight representations via line bundles on the "variety of Borel subgroups", together with the theory of split semisimple Chevalley groups (existence of Borel $\mathbf{Z}$-subgroups containing a split maximal $\mathbf{Z}$-torus, existence of $G/B$ as a projective $\mathbf{Z}$-scheme, etc.). 


More precisely, let $T$ be a split maximal $\mathbf{Z}$-torus and $B$ a Borel $\mathbf{Z}$-subgroup of $G$ containing $T$, so $B/U=T$ for the "relative unipotent radical" $U \subset B$ over $\mathbf{Z}$. Use $B$ to define a positive system of roots in $\Phi(G,T) = \Phi(G\_{\mathbf{C}}, T\_{\mathbf{C}})$, and let $\chi:T \rightarrow \mathbf{G}\_m$ be a dominant weight for this positive system of roots. View $\chi$ as a character on $B$ via the identification $B/U=T$, and let $V$ be a rank-1 $\mathbf{Z}$-lattice on which $B$ acts through $1/\chi$. Let $L(\chi) = G \times^B V$ be the associated line bundle on the projective $\mathbf{Z}$-scheme $G/B$. The $G\_{\mathbf{C}}$-representation space ${\rm{H}}^0(G/B, L(\chi))\_{\mathbf{C}} = {\rm{H}}^0(G\_{\mathbf{C}}/B\_{\mathbf{C}}, L(\chi)\_{\mathbf{C}})$ is the irreducible highest-weight representation for the weight $\chi$ (by the geometric theory in characteristic 0), and the finitely generated $\mathbf{Z}$-module ${\rm{H}}^0(G/B, L(\chi))$ admits a natural action by the $\mathbf{Z}$-flat $G$; the image of this $\mathbf{Z}$-module in the $\mathbf{C}$-version is the integral model that you seek.


