Books on reductive groups using scheme theory Prof. Conrad mentioned in a recent answer that most of the (introductory?) books on reductive groups do not make use of scheme theory.  Do any books using scheme theory actually exist?  Further, are there any books that use the functor of points approach?  Demazure-Gabriel's second book would have covered general group schemes in this way, but it was never written, and it's not clear whether or not it would have covered reductive groups anyway.  There is a lot of material in SGA 3 using more modern machinery to study group schemes, but I'm not aware of any significant treatment of reductive groups in that book (although I haven't read very much of it).  
Correction: Prof. Conrad has noted that SGA 3 does contain a significant treatment of reductive groups using modern machinery.
 A: Have you looked at Jantzen's Representations of Algebraic Groups? It's written in the functor / sheaf language and is about reductive groups.
Here's a link to the google books page: http://books.google.com/books?id=Liqas0afjR0C&lpg=PP1&dq=jantzen%20algebraic%20groups&pg=PP1#v=onepage&q=&f=false
A: Personally, I find the "classical" books (Borel, Humphreys, Springer) unpleasant to read because they work in the wrong category, namely, that of reduced algebraic group schemes rather than all algebraic group schemes. In that category, the isomorphism theorems in group theory fail, so you never know what is true. For example, the map $H/H\cap N\rightarrow HN/N$ needn't be an isomorphism (take $G=GL_{p}$, $H=SL_{p}$, $N=\mathbb{G}_{m}$ embedded diagonally). Moreover, since the terminology they use goes back to Weil's Foundations, there are strange statements like "the kernel of a homomorphism of algebraic groups defined over $k$ need not be defined over $k$". Also I don't agree with Brian that if you don't know descent theory, EGA, etc. then you don't "know scheme theory well enough to be asking for a scheme-theoretic treatment'. 
Which explains why I've been working on a book whose goal is to allow people to learn the theory of algebraic group schemes (including the structure of reductive algebraic group schemes) without first reading the classical books and with only the minimum of prerequisites (for what's currently available, see my website under course notes).  In a sense, my aim is to complete what Waterhouse started with his book.
So my answer to the question is, no, there is no such book, but I'm working on it....
A: Oh my goodness, SGA3 is an absolutely fundamental reference on the theory of reductive groups.  The significance of its treatment is tremendous.  But it freely assumes familiarity with the theory over an algebraically closed field.  I see that as no big deal.  It is like learning some basics about varieties before schemes:  totally reasonable, almost absurd to do otherwise (in terms of understanding where the ideas come from, having experience with real examples on which to test one's knowledge of subtleties, etc.) 
I think it is a perfectly good thing to first learn the theory over fields by reading one of these more "classical" books, since all of the serious work takes place there and the relativization brings in other tools to bootstrap from the field case (i.e., one cannot do anything serious without first setting up a good theory over fields, just like real theorems about abelian schemes rely on first doing the theory of abelian varieties).  That being said, when learning the theory over fields from these "classical" books, those who wish for a scheme-theoretic treatment can just re-interpret most of the proofs in terms of schemes if one wishes to do so (using knowledge of topics from descent theory, EGA, etc.) If that is too hard, it means one doesn't know scheme theory well enough to be asking for a scheme-theoretic treatment.  I should note that even Jantzen assumes familiarity with a fair amount of the theory over fields.  There is nothing wrong with that, in my opinion. 
