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!
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).
