# Kazhdan-Lusztig basis elements appearing in product with distinguished involution

My apologies if the below is too malformed to make sense.

Let $$(W,S)(W,S)$$ be the affine Weyl group of a reductive group $$GG$$, and let $${Cw}\{C_w\}$$ be the Kazhdan-Lusztig $$CC$$-basis (an answer in terms of the $$C′wC'_w$$-basis is of course equally useful, though). Let $$HH$$ be the corresponding Hecke algebra over $$Z[q,q−1]\mathbb{Z}[q,q^{-1}]$$, and write $$hx,y,zh_{x,y,z}$$ for the structure constants of $$HH$$ in the $$CwC_w$$-basis.

It is know that $$WW$$ is partitioned into finitely-many two-sided cells, and finitely-many left cells, each of which contains a distinguished involution $$dd$$ (all these terms understood in the sense of Kazhdan-Lusztig and Lusztig). Let $$w∈Ww\in W$$ and let $$Γ\Gamma$$ be the left cell containing $$ww$$. Let $$dd$$ be the distinguished involution in $$Γ\Gamma$$.

Consider the structure constants $$hw,d,zh_{w,d,z}$$ for $$z∈Γz\in\Gamma$$. What is known about them? For example, is it known in general for which $$zz$$ they are nonzero?

For example, in the case when $$d=sd=s$$ is a simple reflection, we have
$$CwCs=−(q1/2+q−1/2)Cw, C_wC_s=-(q^{1/2}+q^{-1/2})C_w,$$
so that $$hw,d,w=−(q1/2+q−1/2)h_{w,d,w}=-(q^{1/2}+q^{-1/2})$$ and $$hw,d,z=0h_{w,d,z}=0$$ otherwise.

In the other extreme, let $$W0⊂WW_0\subset W$$ be the finite Weyl group, with longest element $$w0w_0$$. Then the lowest two-sided cell contains $$W0W_0$$-many distinguished involutions, all conjugate to $$w0w_0$$. In this case for $$ww$$ in the same left cell as $$w0w_0$$, $$CwCw0C_wC_{w_0}$$ is again proportional to $$CwC_w$$, and $$hw,w0,wh_{w,w_0,w}$$ is itself proportional to the number of $$Fq\mathbb{F}_q$$-points of the finite-dimensional flag variety of $$GG$$.

It seems too much to hope for that $$CwCdC_wC_d$$ will always be proportional to $$CwC_w$$, but perhaps something like this is almost true?