Graded sum formula for $\tilde{A}_1$-Soergel calculus and the nil-blob algebra

We study the representation theory of the Soergel calculus algebra $A_w := \mbox{End}_{{\mathcal D}_{(W,S)}} (\underline{w})$ over $\mathbb C$ in type $\tilde{A}_1$. We generalize the recent isomorphism between the nil-blob algebra ${\mathbb{NB}}_n$ and $A_w$ to deal with the two-parameter blob algebra. Under this generalization, the two parameters correspond to the two simple roots for $\tilde{A}_1$. Using this, together with calculations involving the Jones-Wenzl idempotents for the Temperley-Lieb subalgebra of $\mathbb{NB}_n$, we obtain a concrete diagonalization of the matrix of the bilinear form on the cell module $\Delta_w(v)$ for $A_w$. The entries of the diagonalized matrices turn out to be products of roots for $\tilde{A}_1$. We use this to study Jantzen type filtrations of $\Delta_w(v)$ for $A_w$. We show that at enriched Grothendieck group level the corresponding sum formula has terms $\Delta_w(s_{\alpha }v)[ l(s_{\alpha }v)- l(v)]$, where $[ \cdot ]$ denotes grading shift.