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
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?


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

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