Given a representation $V$ of a group $G$, we can think of $V$ as a vector bundle over the classifying stack $BG$, and we can define its index $\chi(BG; V)$ to be the dimension of the $G$-invariant part $V^G$ of the representation.
Now let $G = GL(1)$ ($\mathbb{G}_m$ or $\mathbb{G}_m$ or $\mathbb{C}^\ast$ if you like). Then let $\phi_n : \mathbb{P}^n \to BG$ be the map corresponding to the bundle $\mathcal{O}(1)$ (or maybe we should take map corresponding to the bundle $\mathcal{O}(-1)$, I always mix it up).
Let $V$ be a representation of $G$. Then we can compute the index (i.e. sheaf cohomology Euler characteristic) of $\phi_n^\ast V$ over $\mathbb{P}^n$. We can show, just by doing the calculation, that for sufficiently large $n$ (how large depends on the representation $V$), this index is a polynomial in $n$. Then, plugging in $n=-1$ into this polynomial, we get one can show that the result is equal to $\chi(BG; V)$, defined as above.