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5 - The Springer Correspondence
Edited by David Jordan, University of Edinburgh, Nadia Mazza, Lancaster University, Sibylle Schroll, Universität zu Köln
Book:

Modern Trends in Algebra and Representation Theory

Published online:

25 November 2023

Print publication:

17 August 2023, pp 169-211
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Summary

We present some key concepts and tools in the field of geometric representation theory. We review the background necessary to state the Springer correspondence for an arbitrary semisimple Lie algebra. We then study the notion of convolution in Borel–Moore homology and see how to apply it to the Springer correspondence. Finally, we reframe these ideas in the language of perverse sheaves and intersection homology.

13 - Green Functions
François Digne, Université de Picardie Jules Verne, Amiens, Jean Michel, Centre National de la Recherche Scientifique (CNRS), Paris
Book:

Representations of Finite Groups of Lie Type

Published online:

14 February 2020

Print publication:

05 March 2020, pp 225-241
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Summary

We review invariants of reflection groups and reflection cosets up to giving a formula for the order of a finite reductive group. We then review the Springer correspondence between local systems on unipotent classes and characters of the Weyl group, and use it to describe the Lusztig–Shoji algorithm to compute Green functions.

2nd edition
François Digne, Jean Michel
Published online:

14 February 2020

Print publication:

05 March 2020
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On its original publication, this book provided the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to reflect the continuous evolution of the subject, including entirely new chapters on Hecke algebras, Green functions and Lusztig families. The authors cover the basic theory of representations of finite groups of Lie type, such as linear, unitary, orthogonal and symplectic groups. They emphasise the Curtis–Alvis duality map and Mackey's theorem and the results that can be deduced from it, before moving on to a discussion of Deligne–Lusztig induction and Lusztig's Jordan decomposition theorem for characters. The book contains the background information needed to make it a useful resource for beginning graduate students in algebra as well as seasoned researchers. It includes exercises and explicit examples.

Springer correspondence for the split symmetric pair in type $A$
Part of
- Linear algebraic groups and related topics
- Algebraic groups
Tsao-Hsien Chen, Kari Vilonen, Ting Xue
Journal:

Compositio Mathematica / Volume 154 / Issue 11 / November 2018

Published online by Cambridge University Press:

11 October 2018, pp. 2403-2425

Print publication:

November 2018
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In this paper we establish Springer correspondence for the symmetric pair $(\text{SL}(N),\text{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at $q=-1$. Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at $q=-1$ arise in geometry.

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