On the semi-infinite Deligne–Lusztig varieties for $$\textrm{GSp}$$

Abstract We prove that Lusztig’s semi-infinite Deligne–Lusztig variety for $$\textrm{GSp}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>GSp</mml:mtext> </mml:math> (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 $$X^0_{w_r}(b)_{\mathcal {L}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>X</mml:mi> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mn>0</mml:mn> </mml:msubsup> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>L</mml:mi> </mml:msub> </mml:mrow> </mml:math> for $$\textrm{GSp}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>GSp</mml:mtext> </mml:math> 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 $$X_h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:math> defined by Chan and Ivanov, and show that $$X_h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:math> 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 $$X^0_{w_r}(b)_{\mathcal {L}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>X</mml:mi> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mn>0</mml:mn> </mml:msubsup> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>L</mml:mi> </mml:msub> </mml:mrow> </mml:math> above, even in the $$\textrm{GSp}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>GSp</mml:mtext> </mml:math> case. This reinterprets previous constructions of representations from $$X_h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:math> as instances of Lusztig’s conjectural picture.