Pseudo-centralizers in affine Hecke algebras

We introduce and study a subalgebra $\mathcal{B}$ of the affine Hecke algebra, which arises from a centralizer construction in the double affine Hecke algebra, and which may be regarded as a $v$-deformation of the affine Fomin-Stanley subalgebra introduced by Lam as a combinatorial model for the affine Grassmannian homology ring. In types ${\mathsf{A}}_n$ and ${\mathsf{B}}_2$ and ${\mathsf{G}}_2$, we show that ${\mathcal{B}}_{\mathrm{a}\mathrm{f}\mathrm{f}}$ admits a canonical basis indexed by the cosets of the finite Weyl group in the affine Weyl group. We also discuss conjectural positivity properties of the canonical basis and explain how it can be used to study the center of the affine Hecke algebra.