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Abstract We study the supersingular locus of a reduction at an inert prime of the Shimura variety attached to$$\textrm{GU}(2,n-2)$$ $\text{GU}(2,n-2)$. More concretely, we realize irreducible components of the supersingular locus as closed subschemes of flag schemes over Deligne–Lusztig varieties defined by explicit conditions after taking perfections. Moreover, we study the intersections of the irreducible components. Stratifications of Deligne–Lusztig varieties defined using powers of Frobenius action appear in the description of the intersections. 

Abstract We construct a moduli space $$\textsf {LP}_{G}$$ of $$\operatorname {SL}_{2}$$-parameters over $${\mathbb {Q}}$$, and show that it has good geometric properties (e.g., explicitly parametrized geometric connected components and smoothness). We construct a Jacobson–Morozov morphism$$\textsf {JM}\colon \textsf {LP}_{G}\to \textsf {WDP}_{G}$$ (where $$\textsf {WDP}_{G}$$ is the moduli space of Weil–Deligne parameters considered by several other authors). We show that $$\textsf {JM}$$ is an isomorphism over a dense open of $$\textsf {WDP}_{G}$$, that it induces an isomorphism between the discrete loci $$\textsf {LP}^{\textrm {disc}}_{G}\to \textsf {WDP}_{G}^{\textrm {disc}}$$, and that for any $${\mathbb {Q}}$$-algebra $$A$$ it induces a bijection between Frobenius semi-simple equivalence classes in $$\textsf {LP}_{G}(A)$$ and Frobenius semi-simple equivalence classes in $$\textsf {WDP}_{G}(A)$$ with constant (up to conjugacy) monodromy operator.