| Abstract: |
This review surveys and meta-analyses five decades of work at the confluence of three programmes: character sheaves on commutative group schemes and group ind-schemes, the local Langlands correspondence for an algebraic torus T over a non-archimedean local field F, and the inertial refinement classifying smooth characters by the restriction of their parameter to the inertia subgroup I_F. Writing LT for the loop ind-scheme colim_n T_n attached to T and CS(LT) for the Picard groupoid of multiplicative rank-one local systems on it, the central object of study is the comparison square asserting that the sheaf-function dictionary applied to the Contou-Carrère Fourier transform agrees with Langlands' isomorphism composed with dualisation. We formalise thirty contributions through a coding of formal invariants: object class, ambient triangulated category, dualising functor, hypotheses on residue characteristic and on the splitting field, and the type of inertial output. On that basis we reconstruct the logical dependency skeleton of the field, quantify its thematic and chronological structure, and audit which asserted compatibilities carry explicit proofs. Our principal mathematical synthesis is the statement that for a split torus in equal characteristic the depth filtration on CS(LT) matches the upper-numbering ramification filtration on H^1 (W_F,T-dual), so that the inertial correspondence is realised by a filtered equivalence of Picard groupoids rather than by an imported bijection. Four gaps are isolated: wild ramification in mixed characteristic, non-standardised normalisations, the absent geometry of norm maps for non-split tori, and the missing comparison functor to categorical local Langlands on the Fargues-Fontaine curve. |
