On the semi-infinite Deligne–Lusztig varieties for GSp

Abstract We prove that Lusztig’s semi-infinite Deligne–Lusztig variety for GSp GSp (and its inner form) is isomorphic, as a set with action, to an affine Deligne–Lusztig variety at infinite level, generalizing a result of Chan–Ivanov. Furthermore, we show that a component of some affine Deligne–Lusztig variety X⁰ₖ㶂 (b) ₋ X w r 0 (b) L for GSp GSp can be written, up to perfection, as a direct product of a classical Deligne–Lusztig variety with an affine space. We also study the varieties Xₕ X h defined by Chan and Ivanov, and show that Xₕ X h at infinite level can be realized as a subset of semi-infinite Deligne–Lusztig varieties defined using components of affine Deligne–Lusztig varieties such as X⁰ₖ㶂 (b) ₋ X w r 0 (b) L above, even in the GSp GSp case. This reinterprets previous constructions of representations from Xₕ X h as instances of Lusztig’s conjectural picture.