References for equivariant K-theory

Question

I want a good introduction to localization in equivariant $K$-theory. This introduction can be simple in several ways:

1. I only care about torus actions.
2. I only care about $K^0$.
3. I only care about very nice spaces. I would be fine if the only spaces considered were $G/P$'s and smooth projective toric varieties.

However, I want this exposition to include the following:

1. How to compute a $K$-class from a Hilbert series.
2. Given $X \to Y$ nice spaces, how describe push back and pull forward in terms of the map $X^T \to Y^T$.
3. Ideally, this reference would also give the generators and relations presentation of $K$-theory for the sort of examples mentioned above. I mean formulas like $$K^0(\mathbb{P}^{n-1}) = K^0(\mathrm{pt})[t, t^{-1}]/(t-\chi\_1)(t-\chi\_2) ... (t - \chi_n)$$ where $\chi_i$ are the characters of the torus action. But I can find other references for this sort of thing.

The best reference I currently know is the appendix to Knutson-Rosu.

Answer 1

I like the book by Chriss and Ginzburg (Representation Theory and Complex Geometry, https://doi.org/10.1007/978-0-8176-4938-8) very much, and I think it fits many of your requirements.

Answer 2

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 paper Puzzles and (equivariant) cohomology of Grassmannians, on the equivariant cohomology of the Grassmannian

More detailed references on equivariant cohomology:

3: Fulton's notes (since superceded by the book Equivariant Cohomology in Algebraic Geometry).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 paper, 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 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 paper G-actions on graphs. This is similar to the previous paper of theirs that I cite, but terser and for K-theory.

7: A paper 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 review at the link is in French, but the paper is in English.)

8: Knutson and Rosu's paper 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.

Answer 3

My understanding is that Atiyah and Segal did a lot of this to deal with equivariant index theory, so that's the place that I'd start: Atiyah and Segal's original papers, or any decent book on index theory (I seem to remember a question about that some time ago). I don't know if that would answer your specific queries, though.