#### Vol. 8, No. 4, 2014

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Yangians and quantizations of slices in the affine Grassmannian

### Joel Kamnitzer, Ben Webster, Alex Weekes and Oded Yacobi

Vol. 8 (2014), No. 4, 857–893
##### 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 and 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 PGL${}_{n}$ of Finkelberg and Rybnikov.

##### Keywords
quantization, affine Grassmannian, quantum groups, Yangian
##### Mathematical Subject Classification 2010
Primary: 20G42
Secondary: 53D55, 14M15, 14D24
##### Milestones
Received: 10 January 2013
Revised: 2 August 2013
Accepted: 31 August 2013
Published: 10 August 2014
##### Authors
 Joel Kamnitzer Department of Mathematics University of Toronto 40 St. George Street Toronto, ON M5S 2E4 Canada Ben Webster Department of Mathematics University of Virginia 141 Cabell Drive Charlottesville, VA 22904 United States Alex Weekes Department of Mathematics University of Toronto 40 St. George Street Toronto, ON M5S 2E4 Canada Oded Yacobi School of Mathematics and Statistics University of Sydney NSW 2006 Australia