Let $k$ be a field which I don't suppose to be algebraically closed. Then, the endomorphism ring of any irreducible representation of a finite group over $k$ is a division ring.


What is known about these division rings? When are they fields? When they are, are these fields Galois over $k$? At least, normal? When not, are the centers of the division rings Galois over $k$? What do we know about these centers?


If $k$ is a perfect field, do the irreducible representations themselves split into *distinct* irreducible representations over the algebraic closure of $k$, or can some be equal?


Sorry for this many questions that are probably well-known, but I can't find an introductory text on representation theory which takes non-algebraically closed fields seriously.



Galois theory of endomorphism rings of irreducible representations
Concerning 



> 
> are the centers of the division rings Galois over $k$
> 
> 
> 


A finite dim'l $k$-algebra $A$ is 
*split* provided $\operatorname{End}\_A(S) = k$
for every simple $A$-module $S$. 


[This terminology is consistent with that used
in other contexts -- $A$ is split
just in case the reductive quotient of the "unit group" $A^\times$ is a split reductive algebraic group over $k$.]


Suppose that $k$ is perfect and that $A\_\ell = A \otimes\_k \ell$ is a split $\ell$-algebra for a finite, separable extension $\ell \supset k$. Then we have the following:



> 
> $(\*)$ If $S$ is a simple $A$-module, the center $Z$ of the division $k$-algebra $\operatorname{End}\_A(S)$ is a subfield of $\ell$.
> 
> 
> 


To see this, I claim first that $\operatorname{End}\_A(S) \otimes\_k \ell$ is a split semisimple
$\ell$-algebra. Indeed, since $k \subset \ell$ is separable, the $A\_\ell$-module $S \otimes\_k \ell = S\_\ell$ is semisimple, say $S\_\ell = \bigoplus\_i S\_i$ as $A\_\ell$-module, where $S\_i$ is the $T\_i$-isotypic
component of $S\_\ell$ and where $T\_i$ are distinct simple $A\_\ell$-modules. Since
by assumption $\operatorname{End}\_{A\_\ell}(T\_i) = \ell$, we see
that $$\operatorname{End}\_A(S) \otimes\_k \ell \simeq \operatorname{End}\_{A\_\ell}(S\_\ell) = \prod\_i \operatorname{End}\_{A\_\ell}(S\_i)$$
is a product of full matrix algebras over $\ell$. Now observe that the center of a split semisimple $\ell$-algebra
is a a split commutative etale $\ell$-algbra $\ell \times \cdots \times \ell$. Assertion $(\*)$
now follows.


Apply this to $A = kG$ for a finite group $G$. By Torsten's comment (following Emerton's answer) we may suppose $k$ to be perfect. 
Then $A \otimes\_k \ell$ is split
for an *Abelian* extension $\ell$ of $k$ -- a suitable $\ell$ can be obtained by adjoining to $k$ enough roots of unity. It follows that $Z$ is always Galois over $k$ (for $A = kG$).


