exceptional cases in Kazhdan-Lusztig

Question

The Kazhdan-Lusztig story doesn't apply to the four exceptional cases $(E_6)_1$, $(E_7)_1$, $(E_8)_1$, $(E_8)_2$ (see this earlier question of mine).

What's special about those cases?

Answer 1

If I remember correctly, Kazhdan and Lusztig first pick an appropriate irreducible rigid object/module ${\bf V}^{\kappa}_{\lambda}$ in the tensor category $\tilde{\mathcal{O}}_{\kappa}$ of $\tilde{\frak{g}}$-modules (here $\kappa-h^{\vee}$ is the level). Then they use this object to prove rigidity of the whole tensor category.

But ${\bf V}^{\kappa}_{\lambda}$ is by definition an induced representation (Weyl module), where you induce from the finite-dimensional $\frak{g}$-module $L(\lambda)$. As you know, induced modules are reducible in general, so you have to choose $\ell$ negative enough in order to be irreducible (there is a simple condition for that).

For example, for $E_6$ you choose $L(\lambda)$ to be a smallest non-trivial irreducible representation (there are two of dimension $27$). Of course, you don't pick the trivial trivial representation; ${\bf V}_0^{\kappa}$ is the unit object in the category. The irreducibility condition implies $\kappa \leq -14$ [There is something wrong here; see Remark 1]. Finkelberg reverses $\kappa$ to $-\kappa$, so you get the level to be at least $14-12=2$.

Remark 1: What I said above is probably incorrect. In KL (Part I, Proposition 2.12) it was proved that ${\bf V}^\kappa_{\lambda}$ is irreducible if $\langle \lambda+ 2\rho, \lambda \rangle < -2\kappa$, where $\kappa \in \mathbb{Q}_{<0}$. For $E_6$ there are two $27$-dimensional modules $L(\omega_1)$ and $L(\omega_5)$ of dimension $27$. For $\omega_1$ (same for $\omega_5$) I'm getting $\langle \omega_1+2 \rho,\omega_1 \rangle=17.33..$ so $\kappa \leq -9$ already guarantees irreducibility. I guess one also has to consider the rigidity condition (Part IV).

Remark 2: There seems to be a mismatch between the values of $\kappa$ for which $\tilde{\mathcal{O}}_{-\kappa}$ is rigid in Finkelberg's paper and in KL. In 3.15 (Corollary 2) of KL, Part IV, they list $\kappa \geq 14,20,{\bf 26}$ for $E_{6,7,8}$ resp. while Finkelberg (in Section 2.6) claims $\kappa \geq 14,20,{\bf 33}$, referring to the same result of KL. It's either a typo or Finkelberg for some reason chooses more conservative lower bound $3+h^{\vee}$ which applies to all types. If $\kappa \geq 26$ for $E_8$ is correct, level 1 and 2 are non-issues.