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

We study the representation theory of the Soergel calculus algebra $A_w:=En{d}_{{\mathcal{D}}_{(W,S)}}(\underset{‾}{w})$ over $\mathbb{C}$ in type ${\tilde{A}}_1$. We generalize the recent isomorphism between the nil-blob algebra ${\mathbb{N}\mathbb{B}}_n$ 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 ${\mathbb{N}\mathbb{B}}_n$, we obtain a concrete diagonalization of the matrix of the bilinear form on the cell module ${\mathrm{\Delta }}_w(v)$ 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 ${\mathrm{\Delta }}_w(v)$ for $A_w$. We show that at enriched Grothendieck group level the corresponding sum formula has terms ${\mathrm{\Delta }}_w(s_{\alpha }v)[l(s_{\alpha }v)-l(v)]$, where $[\cdot ]$ denotes grading shift.