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In our previous paper we suggested a conjecture relating the structure of thesmall quantum cohomology ring of a smooth Fano variety to the structure of itsderived category of coherent sheaves. Here we generalize this conjecture, makeit more precise, and support by the examples of (co)adjoint homogeneousvarieties of simple algebraic groups of Dynkin types $A_n$ and $D_n$, i.e.,flag varieties $Fl(1,n;n+1)$ and isotropic orthogonal Grassmannians $OG(2,2n)$;in particular we construct on each of those an exceptional collection invariantwith respect to the entire automorphism group. For $OG(2,2n)$ this is the firstexceptional collection proved to be full.

Residual categories for (co)adjoint Grassmannians in classical types