STRANGE EXPECTATIONS IN AFFINE WEYL GROUPS

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Problem: Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, affine Weyl group $\widetilde W=W\ltimes \check Q$, Coxeter number $h$, highest root $\theta$, fundamental alcove

$$\mathcal A=\{x:\langle x,\alpha_i\rangle\ge0,\ \langle x,\theta\rangle\le1\},$$

and $\check\rho=\sum_i\check\omega_i$. For $b$ coprime to $h$, let $\widetilde w_b\in\widetilde W$ be the unique affine element satisfying

$$\widetilde w_b(\check\rho/h)=b\check\rho/h.$$

Using the paper’s convention that “dominant” means $\widetilde w(\mathcal A)$ lies in the dominant chamber, Conjecture 6.5 says:

$$\widetilde w_b \text{ is a maximum in weak order among all dominant } \widetilde w\in\widetilde W \text{ such that } \widetilde w^{-1}(0)\in \mathcal S(b).$$

Here $\mathcal S(b)$ is the $b$-Sommers region from Definition 6.1 of the paper.

Result: The conjecture is true; in fact $\widetilde w\le \widetilde w_b$ for every such dominant $\widetilde w$.

Let $\lambda=\check\rho/h$. A standard Weyl–Worpitzky covering says

$$W(\lambda-\overline{\mathcal A})\cap \overline C \subseteq \{y\in\overline C:0\le \langle y,\alpha_i\rangle\le 1/h\ \forall i\}.$$

Equivalently, after scaling by $b$,

$$W\bigl(b(\lambda-\overline{\mathcal A})\bigr)\cap\overline C \subseteq \{y\in\overline C:0\le \langle y,\alpha_i\rangle\le b/h\ \forall i\}. \tag{1}$$

Now take a dominant $\widetilde w$ with $q=\widetilde w^{-1}(0)\in\mathcal S(b)$. The known Sommers property of $\widetilde w_b$ gives

$$\widetilde w_b(\mathcal S(b))=b\mathcal A.$$

Write $z=\widetilde w_b(q)\in b\mathcal A$. Since $\widetilde w_b(\lambda)=b\lambda$,

$$b\lambda-z$$

is $W$-conjugate to $\lambda-q$. Also $b\lambda-z\in b(\lambda-\mathcal A)$. Therefore, by (1), the dominant representative of the $W$-orbit of $\lambda-q$, namely $\widetilde w(\lambda)$, satisfies

$$0\le \langle \widetilde w(\lambda),\alpha_i\rangle\le b/h \quad\forall i.$$

Hence for every positive root $\alpha=\sum_i a_i\alpha_i$,

$$0\le \langle \widetilde w(\lambda),\alpha\rangle \le \frac b h \operatorname{ht}(\alpha).$$

For a dominant affine element $u$, its inversion set is

$$\operatorname{Inv}(u) = \{-\alpha+k\delta:\alpha\in\Phi^+,\ 1\le k\le \lfloor\langle u(\lambda),\alpha\rangle\rfloor\}.$$

Since $b$ is coprime to $h$, $b\,\operatorname{ht}(\alpha)/h\notin\mathbb Z$, so

$$\lfloor\langle \widetilde w(\lambda),\alpha\rangle\rfloor \le \left\lfloor\frac{b\,\operatorname{ht}(\alpha)}h\right\rfloor = \lfloor\langle \widetilde w_b(\lambda),\alpha\rangle\rfloor .$$

Thus

$$\operatorname{Inv}(\widetilde w)\subseteq \operatorname{Inv}(\widetilde w_b),$$

which is exactly $\widetilde w\le \widetilde w_b$ in weak order.

Also $\widetilde w_b^{-1}(0)\in\mathcal S(b)$, because $\widetilde w_b(\widetilde w_b^{-1}(0))=0\in b\mathcal A$. Hence $\widetilde w_b$ itself belongs to the set and is its maximum.

Audit: the hypotheses used are precisely irreducibility, crystallographic type, and $\gcd(b,h)=1$, the latter being part of the definition of $\mathcal S(b)$ in the paper. The dominance convention is the standard affine-Grassmannian one. No extra finiteness or simply-laced assumption is introduced.

Citation: Definitions and Sommers-region property: Stucky–Thiel–Williams, “Strange expectations in affine Weyl groups,” arXiv:2309.14481.
Weyl–Worpitzky covering: Athanasiadis, “On a refinement of the generalized Catalan numbers for Weyl groups,” Trans. Amer. Math. Soc. 357 (2005).