We prove that Lusztig’s semi-infinite Deligne–Lusztig variety for \(\textrm{GSp}\) (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 \(X^0_{w_r}(b)_{\mathcal {L}}\) for \(\textrm{GSp}\) 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 \(X_h\) defined by Chan and Ivanov, and show that \(X_h\) 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 \(X^0_{w_r}(b)_{\mathcal {L}}\) above, even in the \(\textrm{GSp}\) case. This reinterprets previous constructions of representations from \(X_h\) as instances of Lusztig’s conjectural picture.
