Yangians and quantizations of slices in the affine Grassmannian

Abstract

Abstract. We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians -these are subalgebras of the Yangian we introduce which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type. Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. This conjecture also implies the conjectural quantization of the Zastava spaces for P GL n of Finkelberg-Rybnykov.

On category O for affine Grassmannian slices and categorified tensor products

“…When is finite type, the definition of first appeared in [, Section 4C] in the case when is dominant, and in [, Appendix B] for all . Our present definition of is a straightforward generalization to arbitrary simply‐laced Kac–Moody type.…”

“…In , 80% of the authors proved that is an affine Poisson variety. We also introduced the truncated shifted Yangian , an algebra which quantizes .…”

“…Let be a generic set of parameters (cf. [, Section 2.5]). In this case, we have that , and by Theorem , the quadratic dual of the category of is the quadratic dual of .…”

Categorical cell decomposition of quantized symplectic algebraic varieties

“…It is a finite-dimensional affine conic symplectic singularity. If is a sum of miniscule coweights, then it is shown in [37,Theorem 2.9] that Gr admits a symplectic resolution Gr , given by closed convolution of Schubert varieties associated to miniscule weights. If T G is a maximal torus, then T acts Hamiltonian on both…”

“…Conjecturally, any such quantization is given by a quotient Y .c/ of a shifted Yangian Y ; see [37,Conjecture 4.11].…”