I spent the last few days<sup>1</sup> reading references on equivariant K-theory. I just pulled together the following bibliography for my collaborators and I can't see a reason not to make it public. This list tilts heavily towards combinatorics and classical algebraic geometry.

My general conclusion is that there are several good references on equivariant K-theory localization, but they are all too dense for someone who has never seen this stuff before. Fortunately, there are good, slow, introductions to equivariant cohomology, and the two fields are similar enough that you can get the motivation and examples from the cohomology papers and then dive into the K-theory tomes.

This post is Community Wiki, so feel free to add your own favorites.

<h2>Short papers with lots of nice examples</h2>

1:      Julianna Tymoczko's [introduction][1] to equivariant cohomology

2:      Early parts of Knutson-Tao's [paper][2] on the equivariant cohomology of the Grassmannian

<h2>More detailed references on equivariant cohomology:</h2>

3:      Fulton's [notes][3]. See lectures 4 and 5 for localization, lectures 6 - 10 for special features of homogeneous spaces.

4:      Section 2 of Guillemin and Zara's first [paper][4], explaining which parts of the story are pure combinatorics.

<h2>References specifically for K-theory</h2>

5:      Chapters 5 and 6 of <em>Complex Geometry and Representation Theory</em>, by Chriss and Ginzburg. Lots and lots of detail, and focuses on flag varieties in particular. One of the only places I've found that describes how to pushforward to something other than a point.

6:      Guillemin and Zara's K-theory [paper][5]. This is similar to the previous paper of theirs that I cite; it is terser but more relevant. 

7: A [paper][6] of Nielsen, which seems to have anticipated a lot of the results in this field, and has some nice examples at the end. (Note: The MathSciNet review is in French, but the paper is in English.)

8:      Knutson and Rosu's [paper][7], establishing localization in equivariant K-theory and explaining equivariant Grothendieck-Riemann-Roch.

William Fulton also tells me that he and Graham are working on an expository article, which I expect will be excellent.

<sup>1</sup> For those who are curious, I am not learning this material for the first time. Rather, I originally learned it by reading a lot of papers, going to talks and asking Allen Knutson about anything that confused me. I am now playing the role of Allen to others, so I need to learn the subject better.


  [1]: http://arxiv.org/abs/math/0503369
  [2]: http://front.math.ucdavis.edu/0112.5150
  [3]:  http://www.math.lsa.umich.edu/~dandersn/eilenberg/index.html
  [4]: http://front.math.ucdavis.edu/9808.5135
  [5]: http://front.math.ucdavis.edu/0007.5165
  [6]: http://www.ams.org/mathscinet-getitem?mr=366928
  [7]: http://front.math.ucdavis.edu/9912.5088