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


1. 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
2. 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...



Equivariant characteristic classes on $\mathbb{P}^n$
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
$$ \text{Td}(\mathbb{P}^n\_{\mathbb{C}}) = \prod\_{i=0}^n \frac{\zeta-\lambda\_i}{1-e^{\lambda\_i-\zeta}}.$$


