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 $$K^0(\mathbb{P}^{n-1}) = K^0(\mathrm{pt})[t, t^{-1}]/(t-\chi\_1)(t-\chi\_2) \cdots (t - \chi_n)$$ 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.

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.