On the Satake Correspondence for the Equivariant Quantum Differential Equations and qKZ Difference Equations of Grassmannians

Abstract We consider the joint system of equivariant quantum differential equations (qDE) and qKZ difference equations for the Grassmannian G ( k , n ), which parametrizes k -dimensional subspaces of $${{\mathbb {C}}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . First, we establish a connection between this joint system for G ( k , n ) and the corresponding system for the projective space $${{\mathbb {P}}}^{n-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . Specifically, we show that, under suitable Satake identifications of the equivariant cohomologies of G ( k , n ) and $${{\mathbb {P}}}^{n-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> , the joint system for G ( k , n ) is gauge equivalent to a differential-difference system on the k -th exterior power of the cohomology of $${{\mathbb {P}}}^{n-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . Secondly, we demonstrate that the Б-theorem for Grassmannians, as stated in Cotti and Varchenko (Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem, Integrability, quantization, and geometry: dedicated to the memory of Boris Dubrovin, 1950, 2019, 2021), Tarasov and Varchenko (J Geom Phys 184:104711, 2023), is compatible with the Satake identification. This implies that the Б-theorem for $${{\mathbb {P}}}^{n-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> extends to G ( k , n ) through the Satake identification. As a consequence, we derive determinantal formulas and new integral representations for multi-dimensional hypergeometric solutions of the joint qDE and qKZ system for G ( k , n ). Finally, we analyze the Stokes phenomenon for the joint system of qDE and qKZ equations associated with G ( k , n ). We prove that the Stokes bases of solutions correspond to explicit K -theoretical classes of full exceptional collections in the derived category of equivariant coherent sheaves on G ( k , n ). Furthermore, we show that the Stokes matrices equal the Gram matrices of the equivariant Euler–Poincaré–Grothendieck pairing with respect to these exceptional K -theoretical bases.