# Weyl character formula is torus localization

There is a geometric proof from [1] of the following celebrated Weyl character formula that I find extremely exciting.

\[\operatorname{ch}(V_\lambda) = \frac { \sum_{w\in W}(-1)^{\ell(w)} e^{w(\lambda + \rho)} } { \sum_{\alpha\in R^+}(e^{\alpha/2} - e^{-\alpha/2}) }\]It is the thing that for the first time convinced me we should care about flag varieties and their K-theories and cohomologies.

## K-theory

Let's start with some definitions. Given a rational character $\lambda \in \hom(T,\mathbb{C}^\times)$, we can define a line bundle $L_\lambda$ on the flag variety $\mathcal{B} = G/B$ by the formula $L_\lambda=G\times_B\mathbb{C}_\lambda$ where $B$ acts on $\mathbb{C}_\lambda$ by $B \rightarrow B/[B,B] = T\xrightarrow{\lambda}\mathbb{C}^\times$.

Now assume $w_0(\lambda)$ is an anti-dominant weight, in which case $L_{w_0(\lambda)}$ is ample. It turns out, by Borel-Weil-Bott, all higher cohomologies of $L_{w_0(\lambda)}$ vanish and $\Gamma(\mathcal{B},{w_0(L_\lambda)})$ is the simple $G$-module with highest weight $w_0^2(\lambda)=\lambda$.

To make things easier, we introduce the geometric choice of positive roots $R^+_{g} = w_0(R^+)$. With this convention, $L_\lambda$ is ample if and only if $\lambda$ is dominant.

Under the above identifications, we can rewrite Weyl character formula in terms of $K$-theory as follows, where $p$ is the projection to a point.

\[p_*L_{w_0(\lambda)} = \Delta^{-1}\sum_{w\in W}(-1)^{\ell(w)}e^{w(\lambda + \rho)}.\]Note that RHS is equal to $ w_0(\Delta^{-1}) \sum_{w\in W} (-1)^{\ell(w)}e^{w(w_0\lambda + w_0\rho)} $. So it suffices to prove

\[p_*L_\lambda = \Delta^{-1}\sum_{w\in W}(-1)^{\ell(w)}e^{w(\lambda + \rho)}\], provided $\Delta$ and $\rho$ are defined using the geometric choice of positive roots, which we will assume from now on.

## Torus localization

Now let's do torus localization. (finally!)

Let's fix a borel subalgebra $\mathfrak{b}$, then the torus fixed $\mathcal{B}^T$ are nothing but the borel subalgebras containing $\operatorname{Lie}T$, and are given by $w(\mathfrak{b})$, indexed by the Weyl group.

Now by torus localization,

\[[L_\lambda] = \sum_{w\in W} \frac { [L_\lambda|_{w(\mathfrak{b})}] } { \sum(-1)^i[\Lambda^iN^\vee_{w(\mathfrak{b})}\mathcal{B}] }.\]Essentially by definition, \(L_\lambda|_{w(\mathfrak{b})} = \mathbb{C}_{w(\lambda)}\). On the other hand, a closer look at the of the flag variety shows that $T_{w(\mathfrak{b})}\mathcal{B} \simeq \mathfrak{g}/w(\mathfrak{b})$. So $N^\vee_{w(\mathfrak{b})}\mathcal{B}\simeq w(\mathfrak{n})$. Recall that with the geometric choic of positive roots, $\mathfrak{n}$ are the negative weights.

Pushing-forward to a point, we have

\[p_*[L_\lambda] = \sum_w \frac { [\mathbb{C}_{w(\lambda)}] } { \sum (-1)^i[\Lambda ^i w(\mathfrak n)] } = \sum_w \frac { e^w(\lambda) } { \prod_{\alpha\in R^+} (1 - e^{-w(\alpha)}) }.\]Note that we have the following identity,

\[\prod_{\alpha \in R^+}(1 - e^{-w(\alpha)})=e^{-w(\rho)}w(\Delta)=e^{-w(\rho)}(-1)^{\ell(w)}\Delta.\]Putting this back we get the Weyl character formula. Voila!

## references

[1] Chriss, Neil, and Victor Ginzburg. Representation theory and complex geometry. Vol. 42. Boston: BirkhĂ¤user, 1997.