Quantum cohomology and the Satake isomorphism

Abstract

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.

References in corpus (2)

• Perverse sheaves on a Loop group and Langlands' duality
• Schubert calculus and singularity theory

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• Cyclic stratum of Frobenius manifolds, Borel-Laplace $(\mathbf{\alpha },\mathbf{\beta })$-multitransforms, and integral representations of solutions of Quantum Differential Equations
• An update of quantum cohomology of homogeneous varieties