The Harder–Borel quasi-isomorphism conjecture for generalized automorphic complexes

Let $Y$ be the locally symmetric space and $E(\pi\otimes\varphi)$ the coefficient bundle in the supplied setting. Write $\Omega^*(Y,E(\pi\otimes\varphi))$ for its de Rham complex, let ${\cal A}_F^*(Y,E(\pi\otimes\varphi))$ be the automorphic-form subcomplex with generalized infinitesimal character $\chi_F$, and let ${\cal A}_{F,ch}^*(Y,E(\pi\otimes\varphi))$ be the coclosed harmonic automorphic subcomplex.

Harder–Borel conjecture. The inclusions

$${\cal A}_{F,ch}^*(Y,E(\pi\otimes\varphi))\subset {\cal A}_F^*(Y,E(\pi\otimes\varphi))\subset \Omega^*(Y,E(\pi\otimes\varphi))$$

are quasi-isomorphisms.

This is the precise complex-level formulation of the Harder–Borel assertion: the automorphic and coclosed harmonic subcomplexes should compute the same cohomology as the full de Rham complex. The supplied text does not state whether it has been resolved.