Conjugacy Classes in the Weyl Group

Setup

Let $G$ be a connected reductive group with Weyl group $W = N_G(T)/T$. As an abstract group, $W$ is a finite Coxeter group generated by simple reflections $\{s_\alpha : \alpha \in \Delta\}$.

Classification (Carter)

The conjugacy classes of $W$ are classified by admissible diagrams: certain subgraphs $\Gamma$ of the extended Dynkin diagram $\tilde{\Delta}$, subject to the condition that the corresponding sub-root-system decomposes as a product of root systems of types $A$, $D$, or $E$ only (no $B$, $C$, $F$, $G$).

Each admissible diagram $\Gamma$ gives a conjugacy class $C_\Gamma \subset W$; its representative is the product of simple reflections for the nodes of $\Gamma$.

Elliptic elements

An element $w \in W$ is elliptic if it has no fixed vector in the real reflection representation, i.e.\ $V^w = 0$ where $V = X^*(T) \otimes \mathbb{R}$. Equivalently, $w$ is not contained in any proper parabolic subgroup of $W$.

Elliptic elements play a key role in:

• The elliptic representation theory of $p$-adic groups (depth-zero supercuspidal representations).
• The elliptic pairing on class functions: $(f, g)_{\mathrm{ell}} = |W|^{-1}\sum_{w \text{ ell}} f(w)g(w^{-1})$.

Connection to Springer correspondence

Under the Springer correspondence: $$\bigsqcup_{\mathcal{O}} \mathrm{Irr}(A(\mathcal{O})) \xleftrightarrow{\;\sim\;} \widehat{W},$$ each irreducible $W$-representation $\phi_w$ (attached to a conjugacy class $C_w \subset W$) corresponds to a pair $(\mathcal{O}, \mathcal{L})$. The Springer character $\chi_w : W \to \mathbb{Z}$ encodes which unipotent orbit $\mathcal{O}$ is attached to $C_w$.

Connection to Kazhdan–Lusztig theory

Conjugacy classes in $W$ parametrise cells in the Kazhdan–Lusztig cell decomposition of $W$: two-sided cells $\mathbf{c}$ of $W$ biject with special unipotent orbits, and the number of left cells within $\mathbf{c}$ equals the number of irreducible representations of $W$ attached to $\mathcal{O}_\mathbf{c}$.

Number of conjugacy classes by type