  For any finite group G it is an interesting question to ask which ordinary
irreducible representations of G remain irreducible in a given characteristic
p. We answer this question for p=2 when G is the proper double cover of the
alternating group. As a key ingredient in the proof, we prove a formula for the
decomposition numbers in Rouquier blocks of double covers of symmetric groups,
in terms of Schur P-functions.
