The irreducible-component conjecture for Kottwitz–Viehmann varieties

Let $G$ be the reductive group defining the Kottwitz–Viehmann variety, let $F$ be its base field, let $T$ be a maximal torus, and let $G(F)^{\mathrm{rs}}$ denote the regular semisimple locus. For $\gamma\in G(F)^{\mathrm{rs}}$ and $\lambda\in X_*(T)_+$, let $\nu_\gamma\in X_*(T)_{\mathbb{Q},+}$ be the Newton point and let $\mu\in X_*(T)_+$ be the minimal dominant integral coweight dominating $\nu_\gamma$. Write $X_\gamma^\lambda$ for the corresponding Kottwitz–Viehmann variety, $G_\gamma(F)$ for the loop-group centralizer, and $\operatorname{Irr}$ for the set of irreducible components. Let $m_{\lambda\mu}$ denote the dimension of the $\mu$-weight space in the irreducible representation of the Langlands dual group $\hat{G}$ with highest weight $\lambda$. Irreducible-component conjecture. The number of irreducible components of the quotient stack $[X_\gamma^\lambda/G_\gamma(F)]$, equivalently the number of $G_\gamma(F)$-orbits on $\operatorname{Irr}(X_\gamma^\lambda)$, equals $m_{\lambda\mu}$:

$$\#\operatorname{Irr}[X_\gamma^\lambda/G_\gamma(F)]=m_{\lambda\mu}.$$

Although $X_\gamma^\lambda$ may have infinitely many irreducible components, the quotient stack always has finitely many. The conjecture proposes a representation-theoretic description of this finite number in terms of a weight multiplicity for the Langlands dual group; no resolution is supplied in the source.