Asymptotic Algebra

Abstract

The important asymptotic algebra is constructed in this chapter. It is the only chapter where Lusztig’s Conjecture (P15) is used.

Appendices

Notes

The construction of the asymptotic algebra is due to Lusztig [Lus12, §2] in the split case and again to Lusztig [Lus20] in the general case. See also [Lus13]. Example 19.3.3 is due to Lusztig [Lus21].

Exercises

Exercise 19.1

(Type ${\varvec{G_2}}$). As in Exercises 3.1, 4.1, 5.1, 6.1, 8.3, we assume here that $S=\{s, t\}$ with $m_{st}=6$, so that (W, S) is of type $G_2$. We set $\varphi (s)=a$ and $\varphi (t)=b$, and assume here that a, $b > 0$. Here is the corresponding Coxeter graph, together with the values of $\varphi$:

Determine, according to the values of a, b, the structure of the asymptotic algebra ${\mathscr {J}}$.

Exercise 19.2.

Let $\Gamma$ be a two-sided cell. Let

$$\tau _\Gamma : {\mathscr {J}}[\Gamma ] \longrightarrow {\mathbb {Z}}$$

be the ${\mathbb {Z}}$-linear map defined by

$$\tau _\Gamma ({\mathbf t}_w)=\delta _{w \in {\mathscr {D}}}$$

for all $w \in \Gamma$. Show that

$$\tau _\Gamma ({\mathbf t}_x {\mathbf t}_y)=\delta _{xy {\scriptscriptstyle {=}}1}$$

for all x, $y \in \Gamma$. In particular, if $\Gamma$ is finite, prove that ${\mathscr {J}}[\Gamma ]$ is a symmetric algebra.

Prove similar results for the algebra ${\mathscr {J}}[C \cap C^{-1}]$, where C is a left cell.