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

\[h_X(Z): U \mapsto \operatorname{map}(Z, F(U))\]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,

\[\Rc = \coprod_\lambda \{\lambda\} \times (z^\lambda N_\Oc \cap N_\Oc).\]It follows that as a vector space,

\[H_*^{G_\Oc}(\Rc) = \oplus_\lambda \C[\tf] r^\lambda.\]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.

\[\begin{CD} \Rc \times \Rc @<<< p^{-1}(\Rc\times\Rc) @>>> q(p^{-1}(\Rc\times\Rc)) @>>> \Rc \\ @VVV @VVV @VVV @VVV \\ \Tc \times \Rc @<p = (p_\Tc, p_\Rc)<< G_\Kc \times \Rc @>q>> G_\Kc \times_{G_\Oc} \Rc @>m>> \Tc \\ \end{CD}\]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

\[\coprod_{\lambda, \nu} \{\lambda\} \times \{\nu\} \times z^\lambda N_\Oc \cap z^\nu N_\Oc\]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.

\[\lambda \times \nu \times z^\lambda N_\Oc \cap z^\nu N_\Oc \rightarrow \nu \times z^\nu N_\Oc\]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

\[r^\lambda r^\mu = m_*p'^*(r^\lambda \boxtimes r^\mu) = m_* p'^*([z^\lambda N_\Oc \times (N_\Oc \cap z^\mu N_\Oc)]\]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

\[p'^*_\Tc([N_\Oc \cap z^\lambda N_\Oc]) = e(z^\lambda N_\Oc/N_\Oc\cap z^\lambda N_\Oc) [z^\lambda N_\Oc \cap z^\nu N_\Oc],\] \[p'^*_\Rc([N_\Oc \cap z^\mu N_\Oc]) = [z^\lambda N_\Oc \cap z^\nu N_\Oc]\]So

\[p'^*([(N_\Oc \cap z^\lambda N_\Oc) \times (N_\Oc \cap z^\mu N_\Oc)]) = e(z^\lambda N_\Oc/N_\Oc\cap z^\lambda N_\Oc)[z^\lambda N_\Oc \cap z^\nu N_\Oc]\] \[m_*p'^*([(N_\Oc \cap z^\lambda N_\Oc) \times (N_\Oc \cap z^\mu N_\Oc)]) = e(z^\nu N_\Oc/z^\lambda N_\Oc\cap z^\nu N_\Oc) e(z^\lambda N_\Oc/N_\Oc\cap z^\lambda N_\Oc) [z^\nu N_\Oc]\]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,

\[r^\lambda r^\mu = m_*p'^*([(N_\Oc \cap z^\lambda N_\Oc) \times (N_\Oc \cap z^\mu N_\Oc)]) = \frac { e(z^\lambda N_\Oc/N_\Oc\cap z^\lambda N_\Oc) e(z^{\nu}N_\Oc/z^{\lambda}N_\Oc\cap z^\nu N_\Oc) } { e(z^\nu N_\Oc/z^\nu N_\Oc \cap N_\Oc) } r^{\lambda + \mu}.\]Counting dimensions case-by-case shows that the above is equal to

\[r^\lambda r^\mu = \xi^{d(\langle\lambda,\xi\rangle, \langle\mu, \xi\rangle)} r^{\lambda + \mu}.\]Here $d(k, l):=[kl < 0]\min(\lvert k \rvert,\lvert l \rvert)$.

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

\[r^\lambda r^\mu = \prod_i \xi_i^{d(\langle\lambda,\xi_i\rangle, \langle\mu, \xi_i\rangle)} r^{\lambda + \mu}\]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.

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.

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!

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

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