# Representations of Algebraic Groups

Suppose we have an algebraic group over an algebraic closed field of prime characteristic, for example we may think of a subgroup of $GL_n(\bar{\mathbb{F}}_q)$ (where $\bar{\mathbb{F}}_q$ is the algebraic closure of the field with $q$ elements) (that is the case I'm interested).

My question is what can be said, or exists any reference for the $\mathbb{C}$-representations of that group, this is the $\mathbb{C}$-vector spaces (finite or infinite dimensional) equiped with an linear action of the group.

• The first assertion is not correct: the group may not be defined over an algebraic extension of a finite field. – YCor Oct 20 '17 at 16:16
• Yes, sorry I meant to say for example. I mostly interested in the representation of subgroups of $GL_n(\bar{\mathbb{F}}_q)$ – João Dias Oct 20 '17 at 16:20

To say anything meaningful here, you should start with a connected algebraic group. Finite groups for example are technically affine algebraic groups, but their representations over $\mathbb{C}$ have a literature of their own.
The key reference, following years of partial exploration, involves connected semisimple algebraic groups: see the 1973 paper by Borel and Tits here. (They also provided a short overview of their initial results in English as early as the Tata Institute conference on algebraic geometry held in Bombay in 1968; the proceedings were published in 1969.) The basic answer in this case is that nothing interesting can be done if the group lives in prime characteristic but you want representations over $\mathbb{C}$. Typically, abstract homomorphisms such as representations are gotten by combining algebraic group morphisms with field embeddings.
[On the other hand, passing from representations of semisimple algebraic groups over $\mathbb{C}$ to analogous groups over an algebraically closed field of prime characteristic is quite fruitful. Here Chevalley pioneered a method based on an integral basis of the Lie algebra, which was soon refined by Steinberg and others. For a modern account, see the second edition of J.C. Jantzen's book Representations of Algebraic Groups, Amer. Math. Soc., 2003.]
ADDED: I should also comment on a couple of other issues. (1) While the main emphasis is most often on finite dimensional representations, as in the Borel-Tits work, it's hard to see an interesting example in which a (say connected) affine algebraic group over an algebraically closed field of prime characteristic acts on an infinite dimensional vector space over $\mathbb{C}$. As in the case of Lie groups, most infinite dimensional representations involve some added structure to the vector space. Are there examples to consider? (2) Algebraic groups which aren't semisimple pose other questions in the finite dimensional case. For example, a unipotent group in prime characteristic $p$ only has elements of $p$-power order, unlike unipotent groups in characteristic 0. So one has to consider Jordan-Chevalley decomposition in the algebraic group (as Borel-Tits and earlier work did), and whether this has any interaction with a hypothetical representation over $\mathbb{C}$. Again I'm doubtful that there are interesting examples to study. It would be helpful to provide more motivation for your line of questioning.
• For example: We know that if we have a abelian connected algebraic group $A$ over an extension of $\mathbb{F}_q$, we have norm maps: $N_m^n:A(q^m): A(q^n)\rightarrow A(q^m)$ whenever $m|n$. And it defines injective maps: $N_m^n^*:Hom(A(q^m),\mathbb{C})\rightarrow Hom(A(q^n),\mathbb{C})$ And in some cases (for example $A=\mathbb{F}$ the algebraic closure of $\mathbb{F}_q$) the image is $Hom(A(q^n(,\mathbb{C}^{F^m})$ So in this case my question it would be more how to relate $Hom(A(\mathbb{F}),\mathbb{C})$ with $Hom(A(q^m),\mathbb{C}$. – João Dias Oct 21 '17 at 17:26
• @Joao: Your question may involve more number theory than algebraic groups (in which case a tag 'nt.number-theory' might be helpful). One possible dividing line is the fact that the structure and representations of a given algebraic group or group-scheme over a field of prime characteristic tend to behave uniformly over an arbitrary algebraically closed field of that characteristic (not just over the algebraic closure of the prime field). Similarly, the field $\mathbb{C}$ should be replaceable by any algebraically closed field of characteristic 0. – Jim Humphreys Oct 21 '17 at 20:29