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I am looking for some simple and concrete -- but still non-trivial and illustrative -- applications of Atiyah-Bott localization in the context of equivariant cohomology.

Do you know any good ones?

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You want to read A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications, Atiyah and Bott, Ann. of Math., Vol. 88, No. 3 (Nov., 1968), pp. 451-491

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