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Equivariant K-theory for products of groups?

It is exactly the content of a paper of Minami (At least for compact Lie groups) A Künneth formula for equivariant $K$-theory. Haruo Minami. Osaka J. Math. 6(1): 143-146 (1969). Following ideas of ...
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Frobenius pushforward of an equivariant tautological bundle on the flag variety

EDIT. Corrected the statement ($\sigma$ should be $p-1$ times what I wrote) and answered the question in the comment. In general, the push-forward of a line bundle on the flag variety $G/B$ will not ...

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