Soegel's Modular Category $\mathcal{O}$

Soegel’s Modular Category $\mathcal{O}$

Assume that

  • $p>h$.
  • $G$ semi-simple, simply-connected.
  • Let $A$ be the Serre subcategory of $\mathrm{Rep}(G)$, generated by the simples $L(\lambda)$ such that $\lambda\in \mathbb{X}_ {+}$ and $\lambda\uparrow p\rho$ (i.e. $\exists w, w\cdot_ {p}\lambda = p\rho$ and can decompose $w$ such that each step it increases the Bruhat order).
  • Let $B$ be the Serre subcategory of $\mathrm{Rep}(G)$ generated by simples $L(\lambda)$ with $\lambda\in\mathbb{X}_ {+}$ with $\lambda\uparrow p\rho$ and $\lambda\ni\{(p-1)\rho+W_ {\text{fin}}\rho\}$.

Definition 1. Soegel’s modular category $\mathcal{O}$ is defined as $\mathcal{O}_ {\mathbb{k}}:=A/B$.

Let $\mu$ be the unique element in $C\cap W_ {\text{aff}}\cdot_ {p} (p\rho)$. Then $w\mapsto w\cdot_ {p}\mu$ induces a bijection $${}^{f}W_ {\text{aff}}\xrightarrow{\cong} \mathbb{X}_ {+}\cap W_ {\text{aff}}\cdot_ {p} (p\rho),$$ which identifies the Bruhat order on left hand side with $\uparrow$ order on right hand side.

If we let $w\in W_ {\text{aff}}$ be the unique element such that $w\cdot_ {p}\mu=p\rho$, then the objects in $A$ are in canonical bijection with $\{y\in{}^{f}W_ {\text{aff}}:y\leq w\}$.

Let $\omega:=w^{-1}t_ {\rho}\in W_ {\text{ext}}$.

Lemma 2. The element $\omega$ is maximal in the coset $wW^{\omega}$, where $W^{\omega}:=\omega W \omega^{-1}$. We identify hte poset $W^{\omega}$ with $W$ via $x \mapsto \omega x \omega^{-1}$. We have for $x\in W$, $w\omega x\omega^{-1}\cdot_ {p}\mu = t_ {\rho}x\cdot_ {p} 0 = (p-1)\rho+x\rho$.

So for any $x\in W$, we denote by $N_ {x},M_ {x}, L_ {x}$ the images of the modules $N((p-1)\rho+x(\rho))$, $M((p-1)\rho+x(\rho))$ and $L((p-1)\rho+x(\rho))$.

We can then study the composition factors $[N_ {y}: L_ {x}]=[M_ {y}:L_ {x}]$ for $x,y\in W:=W_ {\text{fin}}$.

Proposition 3 (Consequence of Lusztig’s conjecture of $\mathcal{O}$). Assuming Lusztig’s conjecture, one has $[N_ {y}: L_ {x}] = h_ {y,x}(1)$.