# Integral $u$-deformed involution modules

## TL;DR

This paper extends Lusztig's $u$-deformed involution modules from the indeterminate $u$ to any nonzero complex number outside \\{0,1,-1\\}, for all Coxeter systems, broadening the applicability of the module construction.

## Contribution

It proves Lusztig's speculation that the $u$-deformed involution module construction can be generalized to any nonzero scalar outside \\{0,1,-1\\} over arbitrary fields.

## Key findings

- Confirmed the extension of the involution module to any scalar in the specified set.
- Established the isomorphism of the module with a submodule of formal sums over the Coxeter group.
- Generalized Lusztig's construction beyond the original indeterminate parameter.

## Abstract

Let $(W,S)$ be a Coxeter system and $\ast$ an automorphism of $W$ with order $\leq 2$ and $S^{\ast}=S$. Lusztig and Vogan ([11], [14]) have introduced a $u$-deformed version $M_u$ of Kottwitz's involution module over the Iwahori-Hecke algebra $\mathscr{H}_{u}(W)$ with Hecke parameter $u^2$, where $u$ is an indeterminate. Lusztig has proved that $M_u$ is isomorphic to the left $\mathscr{H}_{u}(W)$-submodule of ${\hat{\mathscr{H}}}_u$ generated by $X_{\emptyset}:=\sum_{w^*=w\in W}{u^{-\ell(w)}T_w}$, where ${\hat{\mathscr{H}}}_u$ is the vector space consisting of all formal (possibly infinite) sums $\sum_{x\in W}{c_xT_x}$ ($c_x\in\mathbb{Q}(u)$ for each $x$). He speculated that one can extend this by replacing $u$ with any $\lambda\in \mathbb{C}\setminus\{0,1,-1\}$. In this paper, we give a positive answer to his speculation for any $\lambda\in K\setminus\{0,1,-1\}$ and any $W$, where $K$ is an arbitrary ground field.