Springer correspondence

introduction

• In his paper [Spr76], Springer defined a certain representation of the Weyl group of a reductive group on the cohomology of a set of fixed points in the flag variety by a nilpotent element, which we now call the Springer representation.
• In particular, for various sets of such fixed points, called Springer fibers, the top degree cohomology gives all the irreducible representations of the corresponding Weyl group.
• The Springer correspondence makes a link between the characters of a Weyl group and the geometry of the nilpotent cone of the corresponding semisimple Lie algebra
• extend this to an equivalence between the triangulated category generated by the Springer perverse sheaves and the derived category of differential graded modules over a dg-ring related to the Weyl group

nilpotent variety

• Given a semisimple algebraic group $G$ with Lie algebra $\mathfrak{g}$, we call the closure of a $G$-orbit in the nilpotent elements of $\mathfrak{g}$ a nilpotent variety.

related items

• 틀:수학노트

expositions

• Achar, Pramod N., Anthony Henderson, Daniel Juteau, and Simon Riche. “Modular Generalized Springer Correspondence: An Overview.” arXiv:1510.08962 [math], October 29, 2015. http://arxiv.org/abs/1510.08962.
• Clausen, http://www.math.harvard.edu/theses/senior/clausen/clausen.pdf

encyclopedia

• http://en.wikipedia.org/wiki/Nilpotent_cone
• http://en.wikipedia.org/wiki/Springer_correspondence

articles

• Julianna Tymoczko, The geometry and combinatorics of Springer fibers, arXiv:1606.02760 [math.AG], June 08 2016, http://arxiv.org/abs/1606.02760
• Aubert, Anne-Marie, Ahmed Moussaoui, and Maarten Solleveld. “Generalizations of the Springer Correspondence and Cuspidal Langlands Parameters.” arXiv:1511.05335 [math], November 17, 2015. http://arxiv.org/abs/1511.05335.
• Chen, Tsao-Hsien, Kari Vilonen, and Ting Xue. “Hessenberg Varieties, Intersections of Quadrics, and the Springer Correspondence.” arXiv:1511.00617 [math], November 2, 2015. http://arxiv.org/abs/1511.00617.
• Achar, Pramod N., Anthony Henderson, Daniel Juteau, and Simon Riche. “Modular Generalized Springer Correspondence III: Exceptional Groups.” arXiv:1507.00401 [math], July 1, 2015. http://arxiv.org/abs/1507.00401.
• Juteau, Daniel. “Modular Springer Correspondence, Decomposition Matrices and Basic Sets.” arXiv:1410.1471 [math], October 6, 2014. http://arxiv.org/abs/1410.1471.
• Rider, Laura, and Amber Russell. “Perverse Sheaves on the Nilpotent Cone and Lusztig’s Generalized Springer Correspondence.” arXiv:1409.7132 [math], September 24, 2014. http://arxiv.org/abs/1409.7132.
• Rider, Laura. “Formality for the Nilpotent Cone and a Derived Springer Correspondence.” arXiv:1206.4343 [math], June 19, 2012. http://arxiv.org/abs/1206.4343.
• Ciubotaru, Dan. “Spin Representations of Weyl Groups and the Springer Correspondence.” Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal) 2012, no. 671 (2011): 199–222. doi:10.1515/CRELLE.2011.160.
• Springer, T.A.. "Trigonometric Sums, Green Functions on Finite Groups and Representation of Weyl Groups.." Inventiones mathematicae 36 (1976): 173-208
  • http://ams.mpim-bonn.mpg.de/mathscinet-getitem?mr=MR0442103