Equivariant characteristic classes on $\mathbb{P}^n$ Let $T=(\mathbb{C}^*)^n$ act on $\mathbb{P}^n$ torically by
$$t.[x_0:\dots:x_n]=[x_0\;:\;t_1x_1\;:\;\ldots \;:\;t_nx_n]$$
I would like to know an expression for


*

*the equivariant Chern character $\mathrm{ch}^T(\chi\cdot\mathcal{O}(d))$, where $\chi\cdot\mathcal{O}(d)$ is the equivariant line bundle isomorphic to $\mathcal{O}(d)$ and given by the character $\chi:T\to \mathbb{C}^*$, and

*the equivariant Todd genus $\mathrm{Td}^T(\mathbb{P}^n)$ .


In alternative to 1., I'd be happy with $\mathrm{ch}^T(\mathcal{O}/\mathcal{I}_Z)$ where $\mathcal{I}_Z$ is the ideal sheaf of a closed invariant subscheme $Z\subset\mathbb{P}^n$. Also, take $n=2$ if you want.
The more concrete and explicit, the better...
 A: I am just posting my comments as an answer.  Denote by $A$ the equivariant Chow ring $\text{CH}^*_T(\text{Spec}(\mathbb{C}))$.  For $i=1,\dots,n$, denote by $\lambda_i\in A^1$ the first Chern class of the character $\chi_i$ with $\chi_i(t) = t_i$.  Denote by $\lambda_0$ the class $0$, i.e., the first Chern class of the constant character.  Then $A$ is the polynomial ring $\mathbb{Z}[\lambda_1,\dots,\lambda_n]$.  
Denote by $B$ the equivariant Chow ring $\text{CH}^*_T(\mathbb{P}^n_{\mathbb{C}})$ as an $A$-algebra.
Give $\mathcal{O}(1)$ on $\mathbb{P}^n_{\mathbb{C}}$ the unique $T$-linearization so that the associated action of $T$ on $H^0(\mathbb{P}^n_{\mathbb{C}},\mathcal{O}(1)) = \text{span}(x_0,x_1,\dots,x_n)$ is the specified action.  Denote by $\zeta\in B^1$ the first Chern class of $\mathcal{O}(1)$ with this $T$-linearization.  Then $B$ equals $A[\zeta]/\langle (\zeta-\lambda_0)(\zeta-\lambda_1)\cdots (\zeta-\lambda_n) \rangle$.  
The given $T$-linearization of $\mathcal{O}(1)$ induces a $T$-linearization of $\mathcal{O}(d)$ that has first Chern class $d\zeta$.  Thus, the first Chern class of $\chi\cdot \mathcal{O}(d)$ equals $\chi + d\zeta$.  Therefore the Chern character of $\chi\cdot \mathcal{O}(d)$ equals $e^{\chi + d\zeta}$.  
Similarly, the Todd class of the tangent sheaf of $\mathbb{P}^n_{\mathbb{C}}$ with its intrinsic $T$-linearization equals
$$ \text{Td}(\mathbb{P}^n_{\mathbb{C}}) = \prod_{i=0}^n \frac{\zeta-\lambda_i}{1-e^{\lambda_i-\zeta}}.$$
