Proof of the geometric Langlands conjecture V: the multiplicity one theorem
Dennis Gaitsgory, Sam Raskin
TL;DR
This work completes the geometric Langlands program by proving that the Langlands functor ${\mathbb L}_G$ is an equivalence between half-twisted D-modules on Bun_G and Nilp-ind-coherent sheaves on the spectral stack $LS_{\check G}$, thereby establishing multiplicity-one for Hecke eigensheaves associated with irreducible local systems. The proof hinges on three key ingredients: (i) the irreducible spectral locus $LS^{\mathrm{irred}}_{\check G}$ is simply-connected and CM, (ii) the fiber ${\mathcal A}_{G,\mathrm{irred}}$ is a vector bundle with finite monodromy whose restriction decomposes into line bundles attached to central torsors, and (iii) endomorphisms of the vacuum Poincaré object are scalar, enabling a dimension-count that forces a single copy of the structure sheaf on each irreducible component. The authors reduce to the almost simple simply-connected case, develop a 2-categorical Fourier–Mukai framework to handle nontrivial centers, and use a detailed analysis of the geometry of Bun_G and LS_{\check G} along with a careful study of the vacuum Poincaré object to conclude the equivalence. They outline alternative strategies (op-er contractibility, microlocal methods, Verlinde gluing, arithmetic) and discuss why the de Rham version implies Betti/étale Langlands, while noting the genus-dependence of the argument. The paper further develops a 2-Fourier-Mukai theory, establishing dualities between gerbe-twisted spectral and automorphic categories and enabling a robust treatment of center-induced decompositions. Overall, the result confirms Beilinson–Drinfeld’s vision of GLC in the de Rham setting and expands the toolkit for addressing generalized Langlands correspondences across spectral twists and higher categorical structures.
Abstract
This is the final paper in the series of five, in which we prove the geometric Langlands conjecture (GLC). We conclude the proof of GLC by showing that there exists a unique (up to tensoring up by a vector space) Hecke eigensheaf corresponding to an irreducible local system (hence, the title of the paper). We achieve this by analyzing the geometry of the stack of local systems.