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摘要：Schur's Q-functions are a family of symmetric functions first introduced by Schur to describe irreducible spin characters of symmetric groups. Here we mainly construct a lift of Schur's Q-functions to the peak algebra of the symmetric group, called the noncommutative Schur Q-functions, and extract from them a new natural basis with several nice properties such as the positive right-Pieri rule, combinatorial expansion, etc. Dually, we get a basis of the Stembridge algebra of peak functions refining Schur's P-functions in a simple way, and expect that such basis has a correspondence in the representation theory of 0-Hecke-Clifford algebras. We also define another kind of lift to the Malvenuto-Reutenauer algebra via the Sagan-Worley insertion.