Kazhdan-Lusztig basis elements appearing in product with distinguished involution

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

Let (W,S) be the affine Weyl group of a reductive group G, and let {Cw} be the Kazhdan-Lusztig C-basis (an answer in terms of the Cw-basis is of course equally useful, though). Let H be the corresponding Hecke algebra over Z[q,q1], and write hx,y,z for the structure constants of H in the Cw-basis.

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

Consider the structure constants hw,d,z for zΓ. What is known about them? For example, is it known in general for which z they are nonzero?

For example, in the case when d=s is a simple reflection, we have
CwCs=(q1/2+q1/2)Cw,
so that hw,d,w=(q1/2+q1/2) and hw,d,z=0 otherwise.

In the other extreme, let W0W be the finite Weyl group, with longest element w0. Then the lowest two-sided cell contains W0-many distinguished involutions, all conjugate to w0. In this case for w in the same left cell as w0, CwCw0 is again proportional to Cw, and hw,w0,w is itself proportional to the number of Fq-points of the finite-dimensional flag variety of G.

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

Answer

Attribution
Source : Link , Question Author : Stefan Dawydiak , Answer Author : Community

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