# The proof of the splitting principle in equivariant K-theory via flag manifolds

In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely:

Let $j: T\rightarrow U$ be the inclusion of the maximal torus in the unitary group $U=U(n)$. Fro any compact $U$-space $X$ let $$j^*:K_U(X)\rightarrow K_T(X)$$ be the restriction map. Then $j^*$ has a (functorial) left inverse $$j_*:K_T(X)\rightarrow K_U(X).$$

The construction of $j_*$ is done by repeatedly using the Dolbeault operator on projective spaces.

In a following remark Atiyah wrote: "However we can equally well define it at one go by using the Dolbeault complex of the flag manifold $U/T$. The important point is that the sheaf cohomology of $U/T$—like that of any rational variety—has the same properties as for projective space. Thus, more generally, we can replace $U$ in (4.9) by any compact connected Lie group: it being well known that $G/T$ has the structure of a homogeneous rational algebraic variety."

My question is: has the proof using the flag manifold been written down? Are there any references?

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