Pinned Jordan Decomposition of Characters and Depth-Zero Hecke Algebras

Abstract

We construct a pinned canonical Jordan decomposition of characters for finite reductive groups in cases where the relevant dual centralizers may be disconnected. For a connected reductive group \(G\) over a finite field, with a fixed pinning, and for a semisimple element \(s\in G^*\), we construct a canonical bijection between the Lusztig series \(\mathcal E(G,s)\) and the unipotent characters of \(C_{G^*}(s)^{F^*}\). This refines Lusztig's orbit-valued Jordan decomposition for groups with disconnected centre, and is characterized by compatibility with Deligne--Lusztig character formulae and Harish--Chandra series. We also treat a class of possibly disconnected reductive groups with abelian component group whose rational components admit pinning-preserving representatives. In this setting the natural result is an enriched disconnected Jordan decomposition: the target records the connected unipotent Jordan datum, the source Clifford class, and the corresponding projective Clifford label. When the transported Clifford classes agree with the ordinary Clifford classes on the dual-centralizer side, this enriched target recovers the usual unipotent characters of the corresponding disconnected dual centralizer. The construction uses pinned-normalized preferred extensions of cuspidal unipotent characters, Clifford theory, relative Weyl group comparison, Malle's matching, and connected and disconnected Howlett--Lehrer theory. As an application, we give a pinned canonical form of the finite-field input in Ohara's comparison of depth-zero Hecke algebra parameters with the unipotent case.