For any $Z \in Pr^L$, the assignment

is a sheaf of spaces because taking limit commutes with taking mapping spaces (in the second variable).

Note that sheaves of spaces on a manifold are hypercomplete, because it has finite homotopy dimension, which in turn comes from the fact that it has finite covering dimension, see HTT 7.2.

Therefore, $h_X(Z)$ satifisfies hyperdescent. By Yoneda lemma, so does $F$.

Note that this argument carries actually to sheaves with coefficients in any $\infty$-category. Thus the topos of sheaves of spaces is universal in this sense.

The computation there is however extremely confusing to me, largely due to the rather involved definition of convolution product and their approach of exposition (and my incompetence of course!)

I will try to dot the i's and cross the t's.

For simplicity, assume our group is $G=T=\C^\times$, and the representation $N = \C_\xi$ is given by a single character $\xi$.

A quick recap on the definition of $\Rc$. Define $\Tc = G_\Kc\times_{G_\Oc}N_\Oc$ and consider the embedding $\Tc \hookrightarrow \Gr_G \times N_\Kc$ given by quotient on the first factor and multiplying on the second. The variety of triples are then given by $\Rc := \Tc \cap \Gr_G \times N_\Oc$.

Recall that when $G$ is the torus $\C^\times$, the affine grassmannian is just the coweight lattice $Y = \mathbb{Z}$.

We then have,

It follows that as a vector space,

Here $r^\lambda$ is the fundamental class of the component corresponding to $\lambda$.

So in order to get the ring structure, it suffices to compute $r^\lambda r^\mu$.

Now we look at the following convolution diagram.

The difficulty of analyzing this convolution product lies in the fact that it is defined via the rather mysterious restriction with support morphism giving a map $H_\bullet^{G_\Oc}(\Rc) \otimes H_\bullet^{G_\Oc}(\Rc) \rightarrow H_\bullet^{G_\Oc \times G_\Oc}(p^{-1}(\Rc\times\Rc))$ , which is the usual restriction unless everything is smooth. Good news is everything is indeeed smooth in the abelian case.

Now identify $T_\Kc \times_{T_\Oc} \Rc$ with

with the map given by $[g_1, [g_2, s]] \mapsto ([g_1], [g_1g_2], [g_1g_2s])$.

To calculate $r^\lambda r^\mu$ it is then enough to look at the component given by $(\lambda, \nu = \lambda + \mu)$.

Let's first have a look at the map $m$. Under the identification $\Tc = \coprod {\lambda} \times z^\lambda N_\Oc$, it is simply given by the projection.

The trick is to define a map $p':T_\Kc \times_{T_\Oc} \Rc \rightarrow \Tc \times \Rc$ by sending $(\lambda,\nu,s)$ to $(\lambda, s, \nu - \lambda, z^{-\lambda}s)$.

It enjoys the property that $ p^\prime_{\Tc} q = p_\Tc, p^\prime_\Rc q a = p_\Rc $,

where $ a $ is an automorphism of $ T_\Kc\times \Rc $ inducing identity map on the quotient $ T_\Kc\times_{T_\Oc} \Rc $.

Therefore, the convolution can be computed as

Recall that if $\iota: V \hookrightarrow X$ is a $T$-invariant smooth closed subvariety, we have $\iota^\ast\iota_\ast([V]) = e(N_{V/X})[V]$ in equivariant (Borel-Moore/co-) homology.

Thus we have

So

We know $r^\lambda r^\mu$ is supported on $R_{\nu}$, so it suffices to divide by the normal bundle of $\Rc_\nu$ in $\Tc_\nu$ to get the coefficient.

Therefore,

Counting dimensions case-by-case shows that the above is equal to

Here $d(k, l):=[kl < 0]\min(\lvert k \rvert,\lvert l \rvert)$.

This computation generalizes trivially to higher-dimensional tori and representations, giving

if the representation $N$ is given by characters $\xi_i$.

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.

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

, 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,

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

Note that we have the following identity,

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: Birkhäuser, 1997.