Finite-Weyl Lusztig data and the precise basic-set hypotheses

Work in the split generic finite-Weyl setting: $W$ is a finite Weyl group, $L:W\to {\mathbb{Z}}_{\ge 0}$ is a weight function, $A=\mathbb{Z}[v,v^{-1}]$, and $H=H_A(W,L)$ has the symmetric normalization $(T_s-v^{L(s)})(T_s+v^{-L(s)})=0$. Its KL basis is $C_w$. Write $K=\mathbb{Q}(v)$ and assume the generic algebra $H_K$ is split semisimple, as in the generic setup of Canonical basic sets for split Hecke specializations. This hypothesis supplies the generic labels; this item asserts no new theorem on generic rational splitting. Let $\tau (T_w)={\delta }_{w,1}$ be the canonical symmetrizing trace.

The $A$-linear algebra involution $†$ is $T_s\mapsto -T_s^{-1}$, and put $D_w=C_w^{†}$. Define structure constants by $C_x{C}_y={\sum }_z{h}_{x,y,z}{C}_z$; equivalently $D_x{D}_y={\sum }_z{h}_{x,y,z}{D}_z$. Define

$a(z)=\min \{n\ge 0:v^n{h}_{x,y,z}\in \mathbb{Z}[v]\text{ for all }x,y\in W\},{\gamma }_{x,y,z}=[v^0](v^{a(z)}{h}_{x,y,z^{-1}})\mathrm{.}$

The integer $a(z)$ exists since $W$ is finite and all $h$ are Laurent polynomials invariant under bar. Let $\mathrm{\Delta }(z)$ be the negative of the largest exponent in $\tau (C_z)$, and $n_z$ its nonzero coefficient. If $\tau (C_z)=0$, put $\mathrm{\Delta }(z)=\mathrm{\infty }$ and $n_z=0$. Let $\mathcal{D}=\{z:a(z)=\mathrm{\Delta }(z)\}$. In particular $\mathrm{\Delta }(1)=0$ and $n_1=1$; P3 below forces $a(1)=0$.

The following are explicit hypotheses, not assertions for every unequal weight. They are precisely the subset used in Geck section 5.2:

• P2: if $d\in \mathcal{D}$ and ${\gamma }_{x,y,d}\mathrm{\ne }0$, then $x=y^{-1}$.
• P3: for every $y$ there is a unique $d\in \mathcal{D}$ with ${\gamma }_{y^{-1},y,d}\mathrm{\ne }0$.
• P4: $h_{x,y,z}\mathrm{\ne }0$ implies $a(z)\ge a(x)$ and $a(z)\ge a(y)$.
• P5: in P3, ${\gamma }_{y^{-1},y,d}=n_d=\pm 1$.
• P6: every $d\in \mathcal{D}$ satisfies $d^2=1$.
• P7: ${\gamma }_{x,y,z}={\gamma }_{y,z,x}$.
• P8: ${\gamma }_{x,y,z}\mathrm{\ne }0$ implies $a(x)=a(y)=a(z)$.
• P15′: if $a(w)=a(y)$, then

${\sum }_u{h}_{x,u,y}{\gamma }_{w,x^{\mathrm{\prime }},u^{-1}}={\sum }_u{h}_{x,w,u}{\gamma }_{u,x^{\mathrm{\prime }},y^{-1}}\mathrm{.}$

All sums are finite. These hypotheses produce the asymptotic algebra and modular theorem in the following proved suppliers; they do not assume positivity or irreducibility of cell modules. The generic Schur element $c_E(v)$ is determined by ${\tau }_K={\sum }_E{c}_E(v{)}^{-1}{\mathrm{tr}}_E$. We use Geck's notation ${\alpha }_E\in \mathbb{N}$ and $f_E\in {\mathbb{Z}}_{>0}$ for the Schur exponent and leading integer in Theorem 4.4. The local leading-Schur lemma below proves, from the stated P hypotheses, that $c_E(v)$ has lowest term $f_E{v}^{-2{\alpha }_E}$ and that ${\alpha }_E$ equals the support degree of $V_E$ in $J$. A specialization $\theta :A\to k$ has $u=\theta (v)\in k^{\times }$, multiplicative parameter $\xi =u^2$, and is L-good when $\theta (f_E)\mathrm{\ne }0$ for every generic label $E$. Characteristic zero is always L-good. The modular theorem does not require $\theta (c_E(v))\mathrm{\ne }0$.

To compare normalizations, Geck's multiplicative generator is $S_s=v^{L(s)}{T}_s$ and obeys $(S_s-v^{2L(s)})(S_s+1)=0$; after specialization its parameter is ${\xi }^{L(s)}$. The asymptotic formulas below use Lusztig's revised convention $D_w\leftrightarrow t_w$ and coefficients $n_d$ in $\varphi$, with no diagonal sign change. Geck's signed starred convention is not mixed into these formulas.