Dirac cohomology for degenerate affine Hecke-Clifford algebras

In this paper, we study the Dirac cohomology theory on a class of algebraic structures. The main examples of this algebraic structure are the degenerate affine Hecke-Clifford algebra of type A_n-1 by Nazarov and of classical types by Khongsap-Wang. The algebraic structure contains a remarkable subalgebra, which usually refers to Sergeev algebra for type A_n-1. We define an analogue of the Dirac operator for those algebraic structures. A main result is to relate the central characters of modules of those algebras with the central characters of modules of the Sergeev algebra via the Dirac cohomology. The action of the Dirac operator on certain modules is also computed. Results in this paper could be viewed as a projective version of the Dirac cohomology of the degenerate affine Hecke algebra.