Affine flag varieties of type D

The Hecke algebras and quantum group of affine type A admit geometric realizations in terms of complete flags and partial flags over a local field, respectively. Subsequently, it is demonstrated that the quantum group associated to partial flag varieties of affine type C is a coideal subalgebra of quantum group of affine type A. In this paper, we establish a lattice presentation of the complete (partial) flag varieties of affine type D. Additionally, we determine the structures of convolution algebra associated to complete flag varieties of affine type D, which is isomorphic to the (extended) affine Hecke algebra. We also show that there exists a monomial basis and a canonical basis of the convolution algebra, and establish the positivity properties of the canonical basis with respect to multiplication.