Double Dyck Path Algebra Representations From DAHA

The double Dyck path algebra $\mathbb{A}_{q,t}$ was introduced by Carlsson-Mellit in their proof of the Shuffle Theorem. A variant of this algebra, $\mathbb{B}_{q,t}$, was introduced by Carlsson-Gorsky-Mellit in their study of the parabolic flag Hilbert schemes of points in $\mathbb{C}^2$ showing that $\mathbb{B}_{q,t}$ acts naturally on the equivariant $K$-theory of these spaces. The algebraic relations defining $\mathbb{B}_{q,t}$ appear superficially similar to those of the positive double affine Hecke algebras (DAHA) in type $GL$, $\mathscr{D}_n^{+}$, introduced by Cherednik. In this paper we provide a general method for constructing $\mathbb{B}_{q,t}$ representations from DAHA representations. In particular, every $\mathscr{D}_n^{+}$ module yields a representation of a subalgebra $\mathbb{B}_{q,t}^{(n)}$ of $\mathbb{B}_{q,t}$ and special families of compatible DAHA representations give representations of $\mathbb{B}_{q,t}$. These constructions are functorial. Lastly, we will construct a large family of $\mathbb{B}_{q,t}$ representations indexed by partitions using this method related to the Murnaghan-type representations of the positive elliptic Hall algebra introduced previously by the author.