Geometric characterization of the group law in the Weyl group

TL;DR

The paper provides a geometric realization of the Weyl group multiplication by studying $B$-double cosets through the flag variety $\mathcal{B}$ and the orbits $O(w)$ in $\mathcal{B}\times\mathcal{B}$. It defines the abstract Weyl group $\mathbf W$ and proves that the convolution $O(w_1)*O(w_2)$ contains a unique closed orbit $O(w_1w_2)$, thereby encoding the group law geometrically and matching the Coxeter data $(W(G,T),S(G,T))$ under a pinning. The authors also show that $\mathbf W$ acts on the abstract Cartan $\mathbf T$ in a way compatible with the pinning, and they provide a combinatorial model for convolution via a $*$ product on $W$, reducing to a unique Bruhat-minimal element $x_1x_2$. The work concludes with open questions about a complete orbit-by-orbit description of $O(w_1)*O(w_2)$ and a concrete GL$_3$ example illustrating limits of the Bruhat-interval description, highlighting gaps between geometry and combinatorics.

Abstract

Let $G$ be a reductive group with Borel $B$ and Weyl group $W$. Then $B$-double cosets in $G$ are indexed by the Weyl group, say $O(w)$ for $w\in W$. Then we prove the minimal $B$-double coset in the convolution $O(w_1)*O(w_2)$ is $O(w_1w_2)$, which gives a geometric characterization of multiplication in $W$. This defines the abstract Weyl group $\mathbf W$ which is a Coxeter group acting on the abstract Cartan $\mathbf T$.

Theorems & Definitions (15)

• Theorem 1.1
• Definition 1
• Theorem 1.2
• Definition 2
• Lemma 2.1
• Theorem 2.2
• Definition 3: humphreys
• Lemma 3.1: humphreys
• Lemma 3.2
• proof