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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 $\mathcal{A}$?

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  • $\begingroup$ Q2 is a bit too vague for me. What specifically do you want to know about this algebra? In is in many respects basically the same the Hecke algebra, except with slightly larger 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
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I'm mostly a combinatorialist who doesn't completely understand this stuff, so I may have something slightly wrong, but...

When $W$ is a Weyl group, the Kazhdan--Lusztig involution is (the $K$-theoretic image of) Verdier duality on the bounded derived category of constructible $B$-equivariant sheaves on the flag variety $G/B$.

The references that will have this right are Springer's Quelques applications de la cohomologie d'intersection and Reitsch's An introduction to perverse sheaves.

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It is a genius definition, and I am still understanding. But let me state my thoughts.

In the representation theory of Lusztig type, the role of the parameter q (sometimes v) are different

  1. v is the nature representation of $\mathbb{C}^\times$

  2. v is the degree shift for a complex

  3. v is the degree shift for a graded module

One reason to consider the involution-invariant element is that such elements are ''central''. If you know some representation theory of associative algebras or category O, you will understand that a characteristic to be simple, is to be self-adjoint (isomorphic to the dual of itself). The same case for perverse sheves.

So, the process of ''involution-invariantifying'' a basis, is similar to the process of Gram--Schmidt, but for representations/complexes, say, finding the only unknown simple/central part one by one.

But all above is only a philosophy, the proof of Kazhdan--Lusztig conjuecture is far from trivial.

Add: usually, the natural basis are not ''central'', since most of them comes from induction from other stuff easy to be understood. This is in philosophy due to the fact that our field is more than one element. If you understand q^2 as the number of elements of the field, this explanation would be of more comforts.

By the way, there is a lot of stuff of geometric origin coincides with Hecke algebra. It seems that any algebra with a basis parameterized by Weyl group are more or less relative to Hecke algebra.

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I think it's quite natural.

  1. $W$ is built up by simple reflections $s_i$.
  2. KL involution is the $\mathbb{Z}$-linear map sending $\delta_{s}$ to $\delta^{-1}_{s^{-1}}$ and $v$ to $v^{-1}$.. you basically invert every thing.
  3. KL-basis, while exists, consists of elements that are fixed. So after expanding in terms of the KL basis, the fixed ones are those with coefficients $f(v)$ such that $f(v) = f(v^{-1})$.
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