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 situations where the dual centralizers may be disconnected. For a connected reductive group (\bG) over a finite field, equipped with a pinning, and for a semisimple element (s\in G^*), we construct a uniquely determined bijection [ \J_s:\cE(G,s)\xrightarrow{\sim}\Uch\bigl(C_{\bG^}(s)^{F^}\bigr). ] This refines Lusztig's orbit-valued Jordan decomposition for groups with disconnected centre, and is characterized by compatibility with the Deligne--Lusztig character formula and with Harish--Chandra series. We then extend the construction to possibly disconnected reductive groups with abelian component group, obtaining a canonical bijection between disconnected Lusztig series and unipotent characters of the corresponding disconnected dual centralizers. The main technical input is a canonical choice of preferred extensions of cuspidal unipotent characters to their inertia groups. The construction uses Lusztig's preferred extensions, Clifford theory, relative Weyl group comparison, and connected and disconnected forms of Howlett--Lehrer theory. These tools allow the cuspidal Jordan decomposition to be extended functorially to all Harish--Chandra series. As an application, we prove a pinned canonical reduction from depth-zero Bernstein blocks of tame (p)-adic reductive groups to unipotent blocks. More precisely, for a depth-zero Bernstein type ((K_{x_0},ρ_{x_0})), the associated Hecke algebra is canonically isomorphic, after fixing the same pinning, to a unipotent Hecke algebra. This refines an earlier result of Ohara. This isomorphism preserves the standard anti-involutions.