# Ext groups in the equivariant derived category

I apologize in advance that this question is probably too basic for MO, but I reckoned I would not get an answer on Math.Stackexchange.

I am starting to learn about perverse sheaves, the decomposition theorem, the equivariant derived category and applications in representation theory (as in the book of Chriss and Ginzburg, for example). What I don't quite understand at the moment is the difference between taking Ext in the derived category and the equivariant derived category. For example, if I let $$\mu : G/B \to \{pt\}$$ be the projection of a flag variety onto a point and let $C$ denote the constant sheaf on $G/B$ then $$Ext^*(\mu_*C,\mu_*C) \cong H_*(G/B \times G/B) \cong End_k(H_*(G/B))$$ and $$Ext_G^*(\mu_*C,\mu_*C) \cong H_*(G/B \times G/B) \otimes H_*^G(\{pt\})$$ is the NilHecke ring. So let us consider the simplest possible example. Let $C$ now denote the constant sheaf on a point, $G$ a one-dimensional torus $\mathbb{C}^*$. Then, if I take Ext in the derived category, I get $$Ext^*(C,C) = k.$$ This means that $$Ext^m(C,C) = Hom(C,C[m]) = 0$$ for $m \neq 0$. If I now regard $C$ as an object in the equivariant derived category (i.e. more precisely a triple consisting of the constant sheaf on a point, the constant sheaf on the classifying space $BG$ and an isomorphism between the respective pullbacks to $EG$) and take Ext in the equivariant derived category I get $$Ext_G^*(C,C) = k[x],$$ i.e. $$Ext^m_G(C,C) = Hom_G(C,C[m]) = k$$ for $m \geq 0$. My question is: why do we have non-trivial morphisms between $C$ and its shifts $C[m]$ in the equivariant derived category? What are they explicitly?

First of all, you made a mistake: $x$ has degree 2, so your equation is only correct for $m$ even.
As for what the classes are, they're cohomology classes on $\mathbb{CP}^\infty\cong B\mathbb{C}^*$. If you like cellular homology, you can decompose $\mathbb{CP}^\infty$ into cells: there will be one of each even dimension. The cocycle with value 1 on this cell is the class you want.
More generally, you should think of taking an equivariant sheaf on a $G$-variety $X$, and looking at the induced sheaf on $(X\times EG)/G$. So, Ext of the constant sheaf of $X$ with itself is the equivariant cohomology of $X$. That, together with the general nonsense of the 6-functor formalism is the main thing you need to know about equivariant derived categories.