  We prove that the geometric Satake correspondence admits quantum corrections
for minuscule Grassmannians of Dynkin types $A$ and $D$. We find, as a
corollary, that the quantum connection of a spinor variety $OG(n,2n)$ can be
obtained as the half-spinorial representation of that of the quadric
$Q_{2n-2}$. We view the (quantum) cohomology of these Grassmannians as endowed
simultaneously with two structures, one of a module over the algebra of
symmetric functions, and the other, of a module over the Langlands dual Lie
algebra, and investigate the interaction between the two. In particular, we
study primitive classes $y$ in the cohomology of a minuscule Grassmannian $G/P$
that are characterized by the condition that the operator of cup product by $y$
is in the image of the Lie algebra action. Our main result states that quantum
correction preserves primitivity. We provide a quantum counterpart to a result
obtained by V. Ginzburg in the classical setting by giving explicit formulas
for the quantum corrections to homogeneous primitive elements.
