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In this paper we study higher level Deligne--Lusztig representations ofreductive groups over discrete valuation rings, with finite residue field$\mathbb{F}_q$. In previous work we proved that, at even levels, thesegeometrically constructed representations are isomorphic to certainalgebraically constructed representations (referred to as the algebraisationtheorem 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 thesign being explicitly determined for $q\geq7$). As consequences, we obtain (2)the regular semisimplicity of orbits of generic higher level Deligne--Lusztigrepresentations, and the dimension formula; in the course of the proof, we give(3) an induction formula of higher level Deligne--Lusztig representations, anda new proof of the character formula at regular semisimple elements.

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