Strongly Regular Elements and Harish-Chandra Theory

Definition

Let $G$ be a connected reductive group. A semisimple element $\gamma \in G$ is strongly regular if its centraliser $C_G(\gamma)$ is a maximal torus (i.e.\ connected and abelian of rank $r$): $$C_G(\gamma) = T \quad \text{(a maximal torus)}.$$

The set of strongly regular elements is denoted $G^{\mathrm{sr}}$.

Comparison with regular elements

For simply connected $G$ (or adjoint $G$): regular semisimple $\iff$ strongly regular (since $C_G(\gamma)$ is automatically connected).

For general reductive $G$: strongly regular $\Rightarrow$ regular semisimple, with the difference measured by $\pi_0(C_G(\gamma))$.

Open dense subset

$G^{\mathrm{sr}}$ is open dense in $G$ (and in particular in the set of semisimple elements). Its complement has codimension $\ge 1$.

Role in harmonic analysis

Harish-Chandra's theorem: For a regular semisimple element $\gamma \in G(F)$ ($F$ a local field), the orbital integral $$O_\gamma(f) = \int_{G(F)/C_G(\gamma)(F)} f(g \gamma g^{-1})\, dg$$ is absolutely convergent for any $f \in C_c^\infty(G(F))$.

The function $\gamma \mapsto O_\gamma(f)$ (as $\gamma$ varies over $G^{\mathrm{sr}}(F)$) extends to all regular semisimple elements and satisfies the germ expansion near the identity: $$O_\gamma(f) \sim \sum_{\mathcal{O}} c_{\mathcal{O}}(\gamma) \cdot O_{\mathcal{O}}(f)$$ where $\mathcal{O}$ ranges over nilpotent orbits and $c_\mathcal{O}(\gamma)$ are the Shalika germs.

Lie algebra version

On $\mathfrak{g} = \mathrm{Lie}(G)$, the strongly regular elements of $\mathfrak{g}$ are the regular semisimple elements $\mathfrak{g}^{\mathrm{rs}} = \{X \in \mathfrak{g} : C_G(X) \text{ is a max torus}\}$.

The invariant polynomials $\mathfrak{g} \to \mathfrak{t}/W$ restrict to a smooth fibration over the strongly regular locus, with fibres $G/T$ (flag variety).

Stable orbital integrals

For strongly regular $\gamma \in G(F)$, the stable orbital integral is $$SO_\gamma(f) = \sum_{\gamma' \sim_{\mathrm{st}} \gamma} O_{\gamma'}(f),$$ where the sum is over $G(F)$-conjugacy classes within the stable class of $\gamma$. These are the building blocks of the geometric side of the stable trace formula.