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

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 PGLn of Finkelberg and Rybnikov.

quantization, affine Grassmannian, quantum groups, Yangian
Mathematical Subject Classification 2010
Primary: 20G42
Secondary: 53D55, 14M15, 14D24
Received: 10 January 2013
Revised: 2 August 2013
Accepted: 31 August 2013
Published: 10 August 2014
Joel Kamnitzer
Department of Mathematics
University of Toronto
40 St. George Street
Toronto, ON M5S 2E4
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
Oded Yacobi
School of Mathematics and Statistics
University of Sydney
NSW 2006