I would like to know about the motivation behind the Kazhdan–Lusztig involution on an Iwahori–Hecke algebra.

I'll borrow the conventions from Libedinsky's Gentle introduction to Soergel bimodules I: The basics. The *Iwahori–Hecke algebra* $\mathcal{H}$ of a Coxeter system $(W,S)$ is the $\mathbb{Z}[v,v^{-1}]$-algebra with generators $h_s$ for $s\in S$ and relations

- $h_s^2 = (v^{-1}-v)h_s +1$ for all $s\in S$
- $\underbrace{h_sh_rh_s\cdots}_{m_{rs}} = \underbrace{h_rh_sh_r\cdots}_{m_{rs}}$ for all $s,r\in S$.

The *Kazhdan–Lustig involution* is the $\mathbb{Z}$-algebra involution $d\colon \mathcal{H}\
\to\mathcal{H}$, defined by $d(h_s)=h_s^{-1}=h_s+v-v^{-1}$ and $d(v)=v^{-1}$.

Question 1: What is the motivation for considering the Kazhdan–Lusztig involution? Is there a motivation that is intrinsic to Coxeter systems and Iwahori–Hecke algebras?

Let me clarify my aim: I know that this involution leads to the definition of the Kazhdan–Lusztig basis, and that many representation-theoretical wonders ensue. But I am a topologist by nature and I can't claim to fully appreciate these applications. But I am familiar with Coxeter groups and Iwahori–Hecke algebras. So I am looking for some motivation, if it can be given, on the level that I do understand! Perhaps a sub-question might help:

Question 2: Let $\mathcal{A}\subseteq \mathcal{H}$ denote the $\mathbb{Z}$-submodule fixed by the Kashdan-Lusztig involution. This is a $\mathbb{Z}[v+v^{-1}]$-algebra with basis given by the Kazhdan–Lusztig basis. What is known about $\mathcal{A}$?

slightlylarger ring of coefficients, because $\mathcal{H} = \mathbb{Z}[v^{\pm 1}] \otimes_{\mathbb{Z}[v+v^{-1}]} \mathcal{A}$. $\endgroup$ – Johannes Hahn Oct 16 '20 at 13:37