Could you suggest me a basic reading list on the Springer resolution? Is there a textbook I can refer to? Or do I need to start with the original paper?

Unfortunately googling for "Springer" and "resolution" was not very helpful so far, due to the existence of a certain publisher called Springer which, from some strange reasons, produces many books discussing resolution of singularities ... :p

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    $\begingroup$ This doesn't answer your question, but why not Google for "Springer resolution"? $\endgroup$ – Yemon Choi Jun 13 '11 at 2:40
  • $\begingroup$ That didn't give me any expository article either, alas. google.com/search?q=%22springer+resolution%22 $\endgroup$ – Yuji Tachikawa Jun 13 '11 at 2:46
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    $\begingroup$ How about searching MathSciNet for "Springer resolution" anywhere? $\endgroup$ – Gerald Edgar Jun 13 '11 at 3:15
  • $\begingroup$ Also, I added some tags $\endgroup$ – David White Sep 13 '11 at 12:30

I would highly recommend chapter 3 from Chriss and Ginzburg's textbook "Representation Theory and Complex Geometry" (really, the entire book is worthy of recommendation).

In a very similar vein, I would also recommend Ginzburg's article "Geometric methods in the representation theory of Hecke algebras and quantum groups", which can be found on the arxiv at: http://arxiv.org/abs/math/9802004. In some ways, I actually prefer these notes to the aforementioned textbook.

While I'm not quite as familiar with it, the textbook "Nilpotent orbits, primitive ideals, and characteristic classes" by Borho, Brylinski, and MacPherson is a very nice reference. In my opinion, however, it serves as more of a secondary reference if you're just interested in the basics of the Springer resolution.

This article and two textbooks, together with the abundant references found within, should be more than enough to get you started. There are, of course, numerous papers on the subject. In particular, I would recommend the 1983 paper by Borho and MacPherson.

  • $\begingroup$ I agree that Ginzburg provides the best entry point for the geometric representation theory viewpoint; his own work develops creative new analogues of Springer theory. Two minor corrections: the spelling is MacPherson; and the research-level monograph by Borho, Brylinski, MacPherson isn't at all a "textbook" but coupled with their related papers is an important older source. (By now Springer-Verlag owns most of the books mentioned, including this one and Chriss-Ginzburg ... ) $\endgroup$ – Jim Humphreys Jun 13 '11 at 13:26
  • $\begingroup$ Thank you for the corrections. Given the influence that MacPherson's work has had on me, it's a bit embarrassing that I misspelled his name. I suppose I referred to Borho, Brylinski, and MacPherson as a "textbook" because I tend to incorrectly think of anything published in a hardcover as a "textbook". $\endgroup$ – Mike Skirvin Jun 13 '11 at 16:41
  • $\begingroup$ If you have access to springerlink, you can get the whole book here: springerlink.com/content/978-0-8176-4937-1 $\endgroup$ – Chuck Hague Jun 16 '11 at 18:20

Springer developed his resolution of singularities for the unipotent (or nilpotent) variety of a semsimple algebraic group (or its Lie algebra) as part of a bigger program to construct Weyl group representations in a new way by making the Weyl group act on cohomology groups of fibers of the resolution. This is a multifaceted subject by now and can be approached in somewhat different ways. As Mike indicates in his answer, Ginzburg and others provide at least a partial pathway (mainly in characteristic 0) using modern algebraic geometry and emphasizing the best-behaved example $SL_n$. Some of the machinery used is necessarily rather sophisticated, as in Springer's original papers.

Papers by Steinberg and Spaltenstein developed a lot of detail about the Springer resolution and its fibers, including dimension formulas. Steinberg's 1974 Springer Lecture Notes 366 volume is based on Vinay Deodhar's write-up of Steinberg's Tata lectures covering some of this material. In my 1995 AMS monograph Conjugacy Classes in Semisimple Algebraic Groups I gave a more comprehensive treatment in the framework of the Borel-Chevalley structure theory of semisimple (or reductive) groups over arbitrary algebraically closed fields. See especially Chapter 6 for the Springer resolution in this setting, as well as the brief survey in Chapter 9 of Springer's construction of Weyl group representations (which Ginzburg outlines in his Chapter 3).

The Springer resolution taken in isolation is well enough presented in these sources, though in different styles and with different assumptions on the field. But there is no standard textbook treatment of the related Weyl group story, which is much harder to approach straightforwardly. Carter's 1985 book Finite Groups of Lie Type does contain a lot of concrete details in special cases, however. Which source to consult depends a lot on what you want to do next.

  • $\begingroup$ Thank you very much for the comment. Unfortunately your book in the library was borrowed by somebody else... $\endgroup$ – Yuji Tachikawa Jun 14 '11 at 1:28

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