This question comes from my notes, heavily edited, thus slightly unusual structure.




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For Lie groups one can reformulate character theory as saying that 



> 
> **C** ⊗ K(`G`\ `pt`) = **C**[`T`/`W`] = **C**[ `X`\* ]W
> 
> 
> 


where `G` is the complex Lie group, `W` its Weyl group, `T` its torus. (subquestion: is this correct?) 


For example, one can write the character theory of a torus and `SL``2` as 



> 
> K(`G``m`\ `pt`) = **Z**[`q`, `q`-1]     `and`     K(`SL`2 \ `pt`) = **Z**[`q` + `q`-1].
> 
> 
> 


**Question:** it's interesting that we're allowed to write **Z** in the examples above; I wonder if this works for any `G` in the formula above, e.g. if the isomorphism is valid over **Z** or **Q** rather then **C**?



Character theory over integers
The isomorphism works over Z. The proof is that the basis change between W-symmetrized monomials and characters is upper-triangular with 1's on the diagonal.


