Representation theory of $\mathrm{GL}(n,\mathbb{K})$ where $\mathbb{K}$ is a (not necessarily algebraically closed) field of characteristic zero I would like to know about irreducible representations of $\mathrm{GL}(n,\mathbb{K})$ where $\mathbb{K}$ is a field of characteristic zero, and topics such as irreducible representations of  $\mathrm{GL}(n,\mathbb{K})$, Young tableaux, Schur functors, Schur-Weyl duality etc. 
Unfortunately, essentially every representation theory textbook I have looked at only discusses representations over $\mathbb{C}$, and does not give any indication whether any of the results generalize to other fields, and if they do, how to extract such information from the complex rep theory. 
Is there anywhere I could learn about representation theory of the general linear group over other fields of characteristic zero, which are not necessarily algebraically closed? Even just $\mathbb{K}=\mathbb{R}$ would suffice since that is the case I am most interested in.
 A: Appendix A to chapter one of Macdonald's book Symmetric functions and Hall polynomials treats the polynomial representations of $\mathrm{GL}(V)$ in this generality. By definition, a representation is polynomial if its matrix coefficients are polynomial functions (not rational functions, so for instance the inverse of the determinant is not a polynomial representation) of the entries of the matrix $g \in \mathrm{GL}_n(F) \cong \mathrm{GL}(V)$ (which is independent of the choice of basis implicit in this isomorphism).
Briefly, just as for the symmetric group one may take the base field to be any characteristic $0$ field, there is no significant difference between polynomial representations of $\mathrm{GL}_n(\mathbf{Q})$ and $\mathrm{GL}_n(F)$ for any field $F$ of characteristic $0$. All the irreducible objects for the latter arise via base change from irreducible objects for the former. 
Namely, for each partition $\lambda$ with at most $n$ parts the Schur functor $F_\lambda$ applied to $\mathbf{Q}^n$ gives the irreducible polynomial representation $F_\lambda(\mathbf{Q}^n)$, on which $g \in \mathrm{GL}_n(\mathbf{Q})$ acts by $F_\lambda(g)$ (here we use that $F_\lambda$ is a functor), and up to isomorphism these account for all the irreducible polynomial representations.
On the other hand, if you are interested in other sorts of representations (e.g., unitary representations of $\mathrm{GL}_n$ considered as a Lie group), there are significant differences between the theories e.g. for $\mathbf{R}$ and $\mathbf{C}$. If that is the case, I suggest starting with the books by Knapp and Vogan.
