Book for representation theory of finite groups over algebraically closed fields of characteristic $0$? Since I am writing on my thesis, I need more general literature about representation theory of finite groups over algebraically closed fields of characteristic $0$ and not only about the typical case of complex numbers. Does anyone of you have references I could use? Online scripts are also appreciated.
Moreover the most books uses the inner product space theory, so a generalization is not so easy without any references... New techniques are required.
Thank you!
 A: As JCAA says in the comments, Curtis and Reiner's Representation Theory of Finite Groups and Associative Algebras works quite generally (over not-necessarily-algebraically-closed fields and fields of positive characteristic even) and should be enough for your purposes, although I'm not familiar with the text so I can't say more than that.
What ends up happening that the representation theory of a finite group $G$ is the same over every algebraically closed field $K$ of characteristic $0$, and in particular is the same over $\overline{\mathbb{Q}}$ as any other such field. A clean way to see this is to use Maschke's theorem, which implies that the group algebra $\overline{\mathbb{Q}}[G]$ is semisimple (no inner products necessary). By the Artin-Wedderburn theorem it must therefore be a finite product of matrix rings $M_{n_i}(\overline{\mathbb{Q}})$ over $\overline{\mathbb{Q}}$, each of which corresponds to an irrep of dimension $n_i$ defined over $\overline{\mathbb{Q}}$, and then further extending scalars to any algebraically closed field $K$ of characteristic $0$ produces the same finite product of matrix rings $M_{n_i}(K)$ but over $K$. So the classification of irreducibles over the two fields coincides; more concretely, every representation defined over $K$ is actually defined over $\overline{\mathbb{Q}}$, and in fact (with slightly more work) over a finite extension of $\mathbb{Q}$ (that is, a number field).
Similarly the orthogonality relations hold with no modifications except that you use $\chi(g^{-1})$ instead of $\overline{\chi(g)}$ (these are the same over $\mathbb{C}$ since $\chi(g)$ is a sum of roots of unity).
