@Qiaochu: The category of vector bundles is very useful sometimes. However a major drawback (for me) is that in the category of vector bundles there is no "negative vector bundles". In other words, we can only take direct sum of vector bundles but cannot take "direct difference" of vector bundles. In K-theory, taking difference is very important, especially in its connection with index theory. So that's why I ask for a better categorification.

@ Dustin: Thank you very much for your explanation! It is very illustrative and I will read that paper you mentioned.

@Dustin: Thank you for your answer! I think it is what I am looking for. Could you give more details? For example, is the "complex K-theory spectrum" just the set of Fredholm operators in a seperable Hilbert space?

@ Mariano You are right. It can be proved that for two Borel subalgebra $\mathfrak{b}$ and $\mathfrak{b}' $ , the resulting quotient $\mathfrak{b}/[\mathfrak{b}, \mathfrak{b}]$ and $\mathfrak{b}'/[\mathfrak{b}', \mathfrak{b}']$ are canknically isomorphic. It's in Representation Theory and Complex Geometry Chapter 3 by Chriss/Ginzburg. Oop, still there are choices.

This is a great point! Maybe the remaining problem to me is that can we relate this deck transformation with the set of $G$-orbits on $X \times X$, as Jim Humphreys pointed out in his answer.

@Jim Maybe we can look at the set of $G$ -orbits of $X \times X$ and say that "this is the Weyl group". But can we define a multiplication just on this set of $G$-orbits? If we can, then this is what I am seeking for: an intrinsic definition of Weyl group.

I get your point. Yes it is impossible to require that there is no fixed point.