The Aubert–Baum–Plymen–Solleveld conjecture for Bernstein components

Let $G$ be a split connected reductive $p$-adic group and let $\mathfrak{s} \in \mathcal{B}(G)$ be an inertial pair. Write $T_{\mathfrak{s}}$ for the associated complex torus, $W_{\mathfrak{s}}$ for its finite Weyl group, $\operatorname{Irr}(G)_{\mathfrak{s}}$ for the corresponding Bernstein component, $\Phi(G)_{\mathfrak{s}}$ for the corresponding set of Langlands parameters, and $K_{\mathfrak{s}}$ for the compact part of $T_{\mathfrak{s}}$. The spectral extended quotient is $T_{\mathfrak{s}} \sslash \widehat{W_{\mathfrak{s}}}$. Aubert–Baum–Plymen–Solleveld conjecture. The cuspidal support map

$$\operatorname{Sc}:\operatorname{Irr}(G)_{\mathfrak{s}}\longrightarrow T_{\mathfrak{s}}/W_{\mathfrak{s}}$$

is one-to-one if and only if the action of $W_{\mathfrak{s}}$ on $T_{\mathfrak{s}}$ is free. There is a canonically defined bijection $\mu_{\mathfrak{s}}:T_{\mathfrak{s}}\sslash\widehat{W_{\mathfrak{s}}}\longrightarrow\operatorname{Irr}(G)_{\mathfrak{s}}$ fitting into a commutative triangle with $\Phi(G)_{\mathfrak{s}}$. Moreover, $\mu_{\mathfrak{s}}$ maps $K_{\mathfrak{s}}\sslash\widehat{W_{\mathfrak{s}}}$ onto the tempered representations, there is an algebraic family of finite morphisms $\theta_z:T_{\mathfrak{s}}\sslash\widehat{W_{\mathfrak{s}}}\to T_{\mathfrak{s}}/W_{\mathfrak{s}}$ with $\theta_1$ equal to the natural projection and $\theta_{\sqrt q}=\operatorname{Sc}\circ\mu_{\mathfrak{s}}$, and the stated cocharacter, component, and labeling properties hold. The conjecture is intended to describe the representation-theoretic and Langlands-parameter structure of each Bernstein component; the source does not provide a resolution status.