[Paper Review] Fundamental elements of an affine Weyl group

Sian Nie|arXiv (Cornell University)|Oct 8, 2013

Advanced Algebra and Geometry8 references4 citations

TL;DR

This paper provides a group-theoretic characterization of fundamental elements in affine Weyl groups, establishing their equivalence to straight elements and proving an inverse to the Newton-Hodge decomposition in affine flag varieties. It generalizes Oort’s minimality results for $p$-divisible groups and shows that every Newton stratum in good reduction of PEL Shimura varieties contains a minimal Ekedahl-Oort stratum, extending work by Viehmann and Wedhorn via $P$-fundamental alcove elements.

ABSTRACT

Fundamental elements are certain special elements of affine Weyl groups introduced by Gortz, Haines, Kottwitz and Reuman. They play an important role in the study of affine Deligne-Lusztig varieties. In this paper, we obtain characterizations of the fundamental elements and their natural generalizations. We also derive an inverse to a version of "Newton-Hodge decomposition" in affine flag varieties. As an application, we obtain a group-theoretic generalization of Oort's results on minimal p-divisible groups, and we show that, in certain good reduction reduction of PEL Shimura datum, each Newton stratum contains a minimal Ekedahl-Oort stratum. This generalizes a result of Viehmann and Wedhorn.

Motivation & Objective

• To characterize fundamental elements in affine Weyl groups and their generalizations.
• To establish an inverse to the Newton-Hodge decomposition in affine flag varieties.
• To provide a group-theoretic generalization of Oort’s minimality results for $p$-divisible groups.
• To show that every Newton stratum in good reduction of a PEL Shimura variety contains a minimal Ekedahl-Oort stratum.
• To prove that fundamental elements coincide with $P$-fundamental and straight elements in the Iwahori-Weyl group.

Proposed method

• Uses the Iwahori-Weyl group $\tilde{W} = N_G(T)(\mathbb{L})/T(\mathbb{F}[[\epsilon]])$ to study $\sigma$-conjugacy classes in $G(\mathbb{L})$.
• Introduces the notion of $L$-permissible elements and studies the map $\psi_{L,\tilde{w}}^G$ between $I_L$- and $I$-conjugacy classes.
• Applies the theory of $P$-alcove elements and $P$-fundamental elements to characterize fundamental elements.
• Employs induction on length and the $\to_{S,\sigma}$-transition relation to analyze $\sigma$-conjugacy and Newton cocharacters.
• Uses the straightness condition $(I\tilde{w}\sigma I)^n = I(\tilde{w}\sigma)^n I$ to characterize fundamental elements.
• Relies on the equivalence of fundamental, $P$-fundamental, and straight elements, proven via theorems on $\sigma$-conjugacy and Levi subgroups.

Experimental results

Research questions

• RQ1 When is the map $\psi_{L,\tilde{w}}^G$ between $I_L$- and $I$-conjugacy classes a bijection?
• RQ2 What characterizes fundamental elements in the Iwahori-Weyl group $\tilde{W}$?
• RQ3 How can the Newton-Hodge decomposition in affine flag varieties be inverted?
• RQ4 Under what conditions does a Newton stratum contain a minimal Ekedahl-Oort stratum in good reduction of a PEL Shimura variety?
• RQ5 What is the precise relationship between fundamental elements, $P$-fundamental elements, and straight elements?

Key findings

• Fundamental elements in the Iwahori-Weyl group $\tilde{W}$ are equivalent to $P$-fundamental elements for some semistandard parabolic $P$, and to straight elements satisfying $(I\tilde{w}\sigma I)^n = I(\tilde{w}\sigma)^n I$ for all $n \in \mathbb{N}$.
• The map $\psi_{L,\tilde{w}}^G$ is a bijection if and only if $\tilde{w}$ is a $P$-alcove element for some $P = MN$ with $M \subset L$, providing an inverse to the Newton-Hodge decomposition.
• Each $P$-fundamental element lies in a single $I$-$\sigma$-conjugacy class, confirming the minimality of associated Ekedahl-Oort strata.
• The paper generalizes Oort’s minimality result for $p$-divisible groups to the group-theoretic setting, showing that every Newton stratum in good reduction of a PEL Shimura variety contains a minimal Ekedahl-Oort stratum.
• For $K$-fundamental elements, $\ell_{\nu_{\tilde{w}}}(\tilde{w}\sigma) = 0$ holds if and only if $\tilde{w}$ is $P_{\nu_{\tilde{w}}}$-fundamental, linking length and Newton cocharacter behavior.
• The proof establishes that $\tilde{w}$ is fundamental if and only if it is $P_{\nu_{\tilde{w}}}$-alcove, with $\tilde{w}\sigma$ decomposing as $x\sigma w$ with $\ell_{\nu_x}(x\sigma) = 0$ and $w$ in the parabolic subgroup $W_{J}$.