Algebra, Vol.1 by Redei L.

By Redei L.

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The next theorem [28] is an easy consequence of Atiyah-Segal-Singer theorem. Theorem 29 If E is a G-equivariant vector bundle over a spin manifold M , then Tr Q(M, E) = i− dim M/2 bch(E, A). 2). Atiyah-Segal-Singer formula is the fixed point formula above for X = 0: (16) (2π)− dim M (s)/2 chs (E, A)(0) Tr Q(M, E)(s) = i− dim M/2 M (s) 1/2 Ds (M/M (s))(0)J 1/2 (M (s))(0) . 3. In the particular case of the bouquet of Chern characters bch(E, A), we know that the result of integration b bch(E, A) is a global C ∞ function on G as it is the trace of the finite dimensional representation Q(M, E) of G.

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