Are the Atiyah-Bott-Shapiro Orientation and the Anderson-Brown-Peterson Splitting compatible in any sense?
The first guess is that the ABS-Orientation is related to the projections on the $BO\langle 4(n(J))\rangle$, respectively $BO\langle 4(n(J))-2\rangle$ factors.
According to Michael Hopkins, Mark Hovey, Spin cobordism determines real K-theory, Mathematische Zeitschrift 210.1 (1992): 181-196, 4th page of the pdf file, the Atiyah-Bott-Shapiro Orientation is just one of the Anderson-Brown-Peterson Splitting map $\pi ^0$. Now, according to the paper by Anderson-Brown-Peterson, the discussion between Theorem 1.3 and Corollary 1.4, this map is just the projection to the "bottom summand" $BO\langle 0 \rangle $.
