Type D Fock Spaces and Quantum Symmetric Pairs

A celebrated construction of the natural representation of the quantized affine algebra U_v(\widehat \mathfraks\mathfrakl_\ell ) is the Fock space by Misra and Miwa. This relates the classes of Weyl modules for U_q(\mathfraks\mathfrakl_N) where q is a root of unity to the action of U_v(\widehat \mathfraks\mathfrakl_\ell ) on the Fock space. For quantum group U_q(\mathfraks\mathfrako_2N) of type D, an embedding of the Grothendieck group of finite dimensional U_q(\mathfraks\mathfrako_2N) -modules into generalized Fock spaces was constructed and an action of an affine quantum symmetric pair on these spaces was defined. This relates the action of the affine quantum symmetric pair to the linkage principal for quantum groups at a root of unity and tensor product multiplicities in analogy to that for finite dimensional module categories of the quantum group U_q(\mathfraks\mathfrakl_N) .