Riemann-Roch formula in KR-theory

The (Grothendieck-)Riemann-Roch formula due to Atiyah and Hirzebruch is a classical formula which describes the values of the push-forward in K-theory by means of the Chern character. The theme of my talk is a generalization of the Riemann-Roch formula in K-theory to KR-theory. KR-theory is a variant of an equivariant K-theory which covers complex K-theory and KO-theory as its special cases. The generalization in KR-theory makes use of a Chern character with values in the Borel equivariant cohomology with certain local coefficients.