I spent the last few days1 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.
Short papers with lots of nice examples
1: Julianna Tymoczko's introduction to equivariant cohomology, An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson
2: Early parts of Knutson-Tao's paperpaper Puzzles and (equivariant) cohomology of Grassmannians, on the equivariant cohomology of the Grassmannian
More detailed references on equivariant cohomology:
3: Fulton's notesnotes (since superceded by the book Equivariant Cohomology in Algebraic Geometry). 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 paperpaper, Equivariant de Rham Theory and Graphs, explaining which parts of the story are pure combinatorics.
References specifically for K-theory
5: Chapters 5 and 6 of Complex Geometry and Representation TheoryComplex Geometry and Representation Theory, 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 paperpaper G-actions on graphs. This is similar to the previous paper of theirs that I cite, but terser and for K-theory.
7: A paperpaper of Nielsen, Diagonalizably linearized coherent sheaves, which seems to have anticipated a lot of the results in this field, and has some nice examples at the end. (Note: The MathSciNet reviewMathSciNet review at the link is in French, but the paperpaper is in English.)
8: Knutson and Rosu's paperpaper Equivariant K-theory and Equivariant Cohomology, 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.
1 For those who are curious, I am not learning this material for the first time. Rather, I originally learned it by skimming 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.
