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11 - Knizhnik–Zamolodchikov-Type Equations, Selberg Integrals and Related Special Functions
- By V. Tarasov, A. Varchenko
Edited by Tom H. Koornwinder, Universiteit van Amsterdam, Jasper V. Stokman, Universiteit van Amsterdam
Book:

Encyclopedia of Special Functions: The Askey-Bateman Project

Published online:

30 September 2020

Print publication:

15 October 2020, pp 368-401
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Summary

The KZ equations are a fundamental mathematical structure related to hypergeometric functions. Solutions of all versions of KZ equations are given by multidimensional hypergeometric integrals. The semi-classical limit of KZ equations leads to basic quantum chain models of mathematical physics and representation theory. In this chapter we describe the main examples of the KZ equations (rational, trigonometric, elliptic, differential or difference) with integral hypergeometric solutions. We also describe the semi-classical limit of KZ equations and associated Bethe ansatz method as the semi-classical limit of the hypergeometric solutions.

Geometric Langlands correspondence for $\text{SL}(2)$, $\text{PGL}(2)$ over the pair of pants
Part of
- Families, fibrations
- Lie groups
David Nadler, Zhiwei Yun
Journal:

Compositio Mathematica / Volume 155 / Issue 2 / February 2019

Published online by Cambridge University Press:

07 February 2019, pp. 324-371

Print publication:

February 2019
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We establish the geometric Langlands correspondence for rank-one groups over the projective line with three points of tame ramification.

Nearby cycles of Whittaker sheaves on Drinfeld’s compactification
Part of
- Lie groups
Justin Campbell
Journal:

Compositio Mathematica / Volume 154 / Issue 8 / August 2018

Published online by Cambridge University Press:

18 July 2018, pp. 1775-1800

Print publication:

August 2018
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In this article we give a geometric construction of a tilting perverse sheaf on Drinfeld’s compactification, by applying the nearby cycles functor to a family of nondegenerate Whittaker sheaves. Its restrictions along the defect stratification are shown to be certain perverse sheaves attached to the nilpotent radical of the Langlands dual Lie algebra. We also describe the subquotients of the monodromy filtration using the Picard–Lefschetz oscillators introduced by Schieder. We give an argument that the subquotients are semisimple based on the action, constructed by Feigin, Finkelberg, Kuznetsov, and Mirković, of the Langlands dual Lie algebra on the global intersection cohomology of quasimaps into flag varieties.