Shifts in the decomposition of Bott-Samelson bimodules

Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that $W\subset\operatorname{GL}(V)$ is a finite reflection group, and let $S\subset W$ be a set of simple reflections so that $(W,S)$ is a finite Coxeter system. For any sequence of simple reflections, say $s_1,\ldots,s_N$, define the associated Bott-Samelson bimodule $$BS(s_1,\ldots,s_N)=R\otimes_{R^{s_1}}\cdots\otimes_{R^{s_N}} R(N)$$ where the $(N)$ at the end shifts the module down by $N$ (i.e. $M(N)^i=M^{i+N}$). By a theorem of Soergel, there exist indecomposable bimodules $\left\{\left.B_x\right|x\in W\right\}$, and every Bott-Samelson bimodule decomposes into a direct sum of shifted copies of these indecomposables, i.e. $$BS(s_1,\ldots,s_N)\cong\bigoplus_{x\in W}\bigoplus_{j\in\mathbb{Z}} B_x^{a_x^j}(j)$$

where the non-negative integer $a_x^j$ is the multiplicity of the shifted bimodule $B_x(j)$.

Is there a relationship between the shifts that occur in this decomposition and Kazhdan-Lusztig polynomials of $W$?

Are there any results about the shifts that can occur in this decomposition?

Thanks to some beautiful results of Soergel and Elias and Williamson, we can translate these questions into questions about multiplication in the Hecke algebra. Specifically, Soergel's categrorification theorem says that there is a $\mathbb{Z}[v,v^{-1}]$ algebra homomorphism $\mathcal{E}\colon\mathcal{H}\rightarrow\langle\mathcal{R}\rangle$ from $\mathcal{H}$, the Hecke algebra of $W$, to the split Grothendieck group of $\mathcal{R}$, the category of finitely generated $R$ bimodules. I should say here: a Laurent polynomial $\sum_{j\in\mathbb{Z}}a_j v^j\in\mathbb{Z}_{\geq 0}[v,v^{-1}]$ acts on a bimodule $M$ by $$\sum_{j\in\mathbb{Z}}a_j v^j\cdot M=\bigoplus_{j\in\mathbb{Z}}M^{a_j}(j).$$ In particular, in the decomposition of the Bott-Samelson bimodule above, we could write $$\left(\dagger\right) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ BS(s_1,\ldots,s_N)\cong \bigoplus_{x\in W}\left(\sum_{j\in\mathbb{Z}}a_x^jv^j\right)\cdot B_x.$$

Now Elias and Williamson have proved Soergel's conjecture, which states that for each $x\in W$, $$\mathcal{E}\left(C_x'\right)=B_x$$ where $\left\{\left.C_x'\right|x\in W\right\}$ is the Kazhdan-Lusztig basis for $\mathcal{H}$. Moreover, for any sequence $s_1,\ldots,s_N$ of simple reflections, we can compute the product in the Hecke algebra $$\left(\dagger\dagger\right) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C'_{s_1}\cdots C'_{s_N}=\sum_{x\in W}p_{x,(s_1,\ldots,s_N)}(v)\cdot C_x'$$ for some Laurent polynomials $p_{x,(s_1,\ldots,s_N)}(v)\in\mathbb{Z}[v,v^{-1}]$. Then by Soergel's categorification and Elias and Williamson's results, $\left(\dagger\right)$ is the image of $\left(\dagger\dagger\right)$ under Soergel's categorification homomorphism, hence $$p_{x,(s_1,\ldots,s_N)}(v)=\sum_{j\in\mathbb{Z}}a_x^jv^j.$$

Now my question becomes:

Is there a relationship between the polynomials $p_{x,(s_1,\ldots,s_N)}(v)$ and the Kazhdan-Lusztig polynomials?

What, if anything, is known about these (Laurent) polynomials $p_{x,(s_1,\ldots,s_N)}(v)$ in general??
• I assume you know that (1) they are symmetric under exchanging $v$ for $v^{-1}$ (or is that for the $C$ basis - in any case under the right conventions), and (2) from the degree bounds in the definition of the $C'$ basis you get a degree bound for your $p$'s, and this degree bound is achieved. I would suggest looking forward in Mathscinet from Deodhar's 1990 paper in Geom. Dedicata where he suggests the (so far successful only in a few cases) idea of finding a formula for $C'_w$ by trying to pick out the right terms in the product of $C'_{s_i}$. – Alexander Woo Jan 8 '16 at 6:55
• For example, Billey and Warrington characterize the cases in $S_n$ where $p_{x,(s_1,\ldots,s_N)}=0$ for all $x\neq s_1\cdots s_N$. – Alexander Woo Jan 8 '16 at 6:57