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 $W$?

Or even more broadly:

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?

Or, even more broadly:

What, if anything, is known about these (Laurent) polynomials $p_{x,(s_1,\ldots,s_N)}(v)$ in general??

Since Hecke algebras are such well studied objects, I'm guessing that somebody has studied these polynomials before. Unfortunately, I could not find any mention of them in the places that I looked--but I probably wasn't looking in the right places...so references are welcome (although my college doesn't have subscriptions to a lot of math journals, so I guess more detailed answers would be preferred :-)

Sorry for the long post--thanks for reading.