The algebraisation of higher level Deligne--Lusztig representations II: odd levels

In this paper we study higher level Deligne–Lusztig representations of reductive groups over discrete valuation rings, with finite residue field ${\mathbb{F}}_q$. In previous work we proved that, at even levels, these geometrically constructed representations are isomorphic to certain algebraically constructed representations (referred to as the algebraisation theorem at even levels). In this paper we work with an arbitrary level $>1$. Our main result is (1) the algebraisation theorem at all levels $>1$ (with the sign being explicitly determined for $q\ge 7$). As consequences, we obtain (2) the regular semisimplicity of orbits of generic higher level Deligne–Lusztig representations, and the dimension formula; in the course of the proof, we give (3) an induction formula of higher level Deligne–Lusztig representations, and a new proof of the character formula at regular semisimple elements.