Lusztig' Character Formula

Let $(W,S)$ be a Coxeter system, It admits a presentation with generators $S$ and relations

  • $\forall s\in S$, $s^{2}=e$,
  • $\forall s,t\in S$, $(st)^{m_ {st}}=e$ braid relation.

For example, for $W=S_ {n}$, $m_ {s_ {i},s_ {j}}=3$.

Definition 1. Hecke algebra associated to $(W,S)$ is the $\mathbb{Z}[v,v^{-1}]$-algebra $\mathscr{H}$ with basis $\{H_ {w}\}_ {w\in W}$, and the multiplication is determined by

  • $(H_ {s}+vH_ {e})(H_ {s}-v^{-1}H_ {e}) = 0$ for $s\in S$,
  • $H_ {x}H_ {y}=H_ {xy}$ if $\ell(xy)=\ell(x)+\ell(y)$.

Remark 2. We have $H_ {s}^{-1}=H_ {s}+(v-v^{-1})$.

Therefore, we get $\mathscr{H}:=\mathscr{H}_ {W_ {\text{fin}}}$ and $\mathscr{H}_ {\text{aff}}$ for $W_ {\text{aff}}$.

Kazhdan-Lusztig basis

Definition 3. The Kazhdan-Lusztig involution is the unique ring homomorphism $\iota$ of $\mathscr{H}$ satisfying

  • $\iota(v)= v^{-1}$,
  • $\iota(H_ {x})=(H_ {x^{-1}})^{-1}.$
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Theorem 4. For any $w\in W$,there is a unique element $\underline{H_ {w}}\in \mathscr{H}$ (or denoted by $C_ {w}$) such that

  1. $\iota_ {\underline{H_ {w}}}=\underline{H_ {w}}$,
  2. $\leftarrow {H_ {w}}= H_ {w}+\sum_ {y\in W,y<w} v\mathbb{Z}[v]H_ {y}$.

Write $\underline{H_ {x}}=\sum_ {y\in W}h_ {y,x}H_ {y}$, where $h_ {y,x}$ is called Kazhdan-Lusztig polynomials.

Lusztig conjecture

Assume that $p\geq h$. Fix $\lambda\in C\cap \mathbb{X}$, where $C$ is the fundamental $p$-alcove. Then for any $w\in {}^{f}W_ {\text{aff}}$, such that $$\langle w\cdot_ {p}\lambda+\rho,\alpha^{\vee} \rangle\leq p(p-h+2)$$ for any $\alpha\in R^{+}$ (called Jantzen’s condition), one expects to have $$[L(w\cdot_ {p}\lambda)]=\sum_ {y\in {}^{f}W_ {\text{aff}}} (-1)^{\ell(y)+\ell(w)}h_ {w_ {0}y,w_ {0}w}(1)[N(y\cdot_ {p}\lambda)].$$

Definition 5. We set ${}^{f}W_ {\text{aff}}$ the subset of elements which are minimal coset representatives of $W_ {\text{fin}}\backslash W_ {\text{aff}}$.

Remark 6.

  1. By theory of translation functors, the choice of $\lambda$ does not matter, i.e. the conjecture holds for one choice of $\lambda$ if and only if it holds for all $\lambda$, so we can let $\lambda=0$.
  2. In the formula, the coefficients don’t depend on $p$. In particular, the formula says the characters of simple $G$-modules “don’t depend on $p$”.
  3. To explain Jantzen’s condition: write a dominant weight $\mu=\mu_ {0}+p\mu_ {1}$, then we have the Steinberg tensor product formula $$L(\mu_ {0})\otimes\mathrm{Fr}^{\ast}(\mu_ {1}).$$ If $\mu$ satisfies Jantzen’s condition, then $\langle \mu_ {1},\alpha^{\vee} \rangle <p-h+2$ and $\langle \mu,\alpha^{\vee} \rangle\leq p$, therefore $\mu_ {1}\in C$, which implies that $L(\mu_ {1})\cong N(\mu_ {1})$. Therefore, Jantzen’s condition means that when we apply Steinberg’s tensor product formula, the module $L(\mu_ {1})$ is an induced coWeyl module.

History

Lusztig’s idea (early 1990’s):

  1. to show characters of certain simple $G$-modules are equal to similar characters for quantum groups at root of unity.
  2. Build a bridge relating quantum groups at root unity and a category of representations of affine Lie algebras over $\mathbb{C}$.
  3. Build a localization theory for affine Lie algebras relating their representations to some category of $\mathscr{D}$-modules on affine flag variety.
  • Solution to 2 is carried out by Kazhdan-Lusztig and Lusztig.
  • Solution to 3 is carried out by Kashiwara-Tanisaki.
  • Step 1 is more subtle then expected.
    • Anderson-Jantzen-Soegel proved it but assuming $p$ bigger than some non-explicit bound depending on $R$, i.e. given a root data $\Delta$, there exists a bound $N(\Delta)$, such that if $\text{char}(\mathbb{k})>N(\Delta)$, then the conjecture holds.
    • Late 2000s, Fiebig reproved AJS result using “combinatorial category”, which gives some idea of what the bound on $p$ should be.
    • Williamson’s counter-example shows the original conjecture is false.