Categorification of the genus two DAHA

Abstract

We construct three derived endofunctors on the derived category of graded integrable representations of the current algebra of a flat degeneration of $D(2|1,\alpha )$, and prove that their classes in the Grothendieck group are the genus two Macdonald operators. Computing the graded $\mathrm{Ext}$ pairing explicitly, we deduce the self-adjointness of the genus two Macdonald operators and the orthogonality of genus $2$ Macdonald polynomials with respect to the certain skew-bilinear form.