  For a finitary hereditary abelian category $\mathcal{A}$, we define a derived
Hall algebra of its root category by counting the triangles and using the
octahedral axiom, which is proved to be isomorphic to the Drinfeld double of
Hall algebra of $\mathcal{A}$. When applied to finite-dimensional nilpotent
representations of the Jordan quiver or coherent sheaves over elliptic curves,
these algebras provide categorical realizations of the ring of Laurent
symmetric functions and also double affine Hecke algebras.
