2.1.1 Root data

Let G be a connected complex quasi-simple algebraic group over $\mathbb {C}$ , with maximal torus T, Borel subgroup B, opposite Borel subgroup $B^-$ and root datum $(\mathrm {X}^\ast (T), \mathrm {X}_\ast (T), \Phi , {\check \Phi })$ . Let N and $N^-$ be the unipotent radicals of B and $B^-$ , respectively. Let r be the rank of G and h its Coxeter number. Let $G_{\mathrm {sc}}$ be the simply connected cover of G and $T_{\mathrm {sc}}$ its maximal torus. Let ${\check G}$ be the Langlands dual group of G, defined over the field $\mathbb {C}$ . Let ${\check T}$ be the dual torus and ${\check B}$ the corresponding Borel subgroup. Let $\mathfrak {g}$ , $\mathfrak {b}$ , $\mathfrak {t}$ be the Lie algebras of G, B, T. Let $\Phi _{\mathrm {af}}$ and $\Phi _{\mathrm {re}}=\Phi \oplus \mathbb {Z}\delta $ be the sets of affine roots and affine real roots, where $\delta $ is the minimal positive imaginary root. Let $\Sigma $ and $\Sigma _{\mathrm {af}}=\Sigma \sqcup \{\alpha _0\}$ be the sets of simple finite and affine roots. Let ${\check \alpha }$ denote the affine coroot associated with the affine root $\alpha $ . We abbreviate ${\check {\Lambda }}=\mathrm {X}_\ast (T)$ , $\Lambda =\mathrm {X}^\ast (T)$ and $Q=\mathbb {Z}\Phi $ , ${\check Q}=\mathbb {Z}{\check \Phi }$ . Let $\langle -,-\rangle $ be the canonical pairing on $\Lambda \times {\check {\Lambda }}$ . Set also $P={\check Q}^*$ and ${\check P}=Q^*$ . Recall that $Q\subset \Lambda \subset P$ and ${\check Q}\subset {\check {\Lambda }}\subset {\check P}$ . Let e be the order of the group ${\check {\Lambda }}/{\check Q}$ . Let $Z=Z(G)$ , ${\check Z}=Z({\check G})$ be the centres and $\pi _1=\pi _1(G)$ , ${\check \pi }_1=\pi _1({\check G})$ the fundamental groups. We have

$$ \begin{align*}\pi_1=\mathrm{X}^\ast({\check Z})={\check{\Lambda}}/{\check Q}=P/\Lambda,\quad{\check\pi}_1=\mathrm{X}^\ast(Z)=\Lambda/Q={\check P}/{\check{\Lambda}}.\end{align*} $$

For any commutative ring R, we abbreviate

$$ \begin{align*}\mathfrak{t}_R={\check{\Lambda}}\otimes_{\mathbb{Z}} R ,\quad \mathfrak{t}^*_R=\Lambda\otimes_{\mathbb{Z}} R.\end{align*} $$

Let $\mathfrak {t}$ and $\mathfrak {t}^*$ be the $\mathbb {Z}$ -schemes whose sets of R-points are $\mathfrak {t}_R$ and $\mathfrak {t}^*_R$ . We will usually omit the subscript R, hoping it will not create any confusion. Let W be the Weyl group. We fix a W-invariant positive definite symmetric bilinear form $(-,-)$ on $\mathfrak {t}_{\mathbb {R}}$ such that $({\check \alpha },{\check \alpha })=2$ for each short coroot. We have $({\check {\Lambda }},{\check {\Lambda }})\subset \frac {1}{e}\mathbb {Z}$ and $({\check {\Lambda }},{\check Q})\subset \mathbb {Z}$ . We may identify $\alpha =2{\check \alpha }/({\check \alpha },{\check \alpha })$ for each root $\alpha $ . We write $\Sigma =\{\alpha _i\}$ . Let $a_{ij}$ be the $(i,j)$ -th entry of the Cartan matrix of G. For any root $\alpha $ we set $d_{\alpha }=(\alpha ,\alpha )/2$ . We have $\alpha _i=d_{\alpha _i}{\check \alpha }_i$ and $(\alpha _i,\alpha _j)=d_{\alpha _i}a_{ij}$ . Let ${\check \rho }$ be half the sum of the positive coroots and $\theta $ the highest root.

Let $W_{\mathrm {af}}=W\ltimes {\check Q}$ and $W_{\mathrm {ex}}=W\ltimes {\check {\Lambda }}$ be the affine Weyl group and the extended one. For each $\mu \in {\check {\Lambda }}$ , we write $\tau _\mu =1\ltimes \mu $ . The group $W_{\mathrm {ex}}$ acts on the lattice ${\check {\Lambda }}\oplus \mathbb {Z} \partial $ by

(2.1) $$ \begin{align}w(\lambda+m \partial)=w(\lambda)+m \partial,\quad \tau_\mu(\lambda+m \partial)=\lambda+m\mu+m\partial.\end{align} $$

for $w\in W$ and $\lambda ,\mu \in {\check {\Lambda }}$ . The dual action on $\Lambda \oplus \mathbb {Z}\delta $ is such that

(2.2) $$ \begin{align}w(\beta+k\delta)=w(\beta)+k\delta,\quad \tau_\mu(\beta+k\delta)=\beta+k\delta-\langle\beta,\mu\rangle\delta, \end{align} $$

for $w\in W$ , $\beta \in \Lambda $ and $\mu \in {\check {\Lambda }}$ .

The group W acts on the torus T, hence on $\mathbb {C}[T]$ . When viewed as a function on T, we will denote the character $\beta \in \Lambda $ by the symbol $e^\beta $ . Similarly, for each cocharacter $\lambda \in {\check {\Lambda }}$ and each $z\in \mathbb {C}^{\times }$ we write $z^\lambda $ for the element $\lambda (z)$ in T. We have

$$ \begin{align*}\mathbb{C}[T\times\mathbb{G}_m]=\mathbb{C}[q^{\pm 1}][e^{\beta+m\delta}\,;\,\beta\in\Lambda\,,\,m\in\mathbb{Z}]\,/\,(q-e^{\delta}), \quad \mathrm{X}^*(\mathbb{G}_m)=\mathbb{Z}\delta.\end{align*} $$

The $W_{\mathrm {ex}}$ -action on $\Lambda \oplus \mathbb {Z}\delta $ yields a $W_{\mathrm {ex}}$ -action on $T\times \mathbb {G}_m$ such that

(2.3) $$ \begin{align}w(t,z)=(w(t),z),\quad \tau_\mu(t,z)=(tz^\mu,z)\end{align} $$

for $w\in W$ , $\mu \in {\check {\Lambda }}$ , $t\in T$ and $z\in \mathbb {C}^{\times }$ . Let

(2.4) $$ \begin{align} (x,f)\mapsto {}^x\!f \end{align} $$

be the induced $W_{\mathrm {ex}}$ -action on the algebra $\mathbb {C}[T\times \mathbb {G}_m]$ .

Under the obvious identification ${\check {\Lambda }}\cong {\check {\Lambda }}+\partial $ , the group $W_{\mathrm {ex}}$ acts on ${\check {\Lambda }}$ by

(2.5) $$ \begin{align}w(\lambda)=w(\lambda),\quad \tau_\mu(\lambda)=\lambda+\mu\end{align} $$

for $w\in W$ , $\lambda ,\mu \in {\check {\Lambda }}$ . For each affine real root $\alpha +m\delta $ let $s_{\alpha +m\delta }$ be the reflection in $W_{\mathrm {af}}$ relatively to the affine hyperplane

$$ \begin{align*}H_{\alpha+m\delta}=\{\lambda\in \mathfrak{t}_{\mathbb{R}}\,;\, \langle\alpha,\lambda\rangle+m=0\}.\end{align*} $$

We have $s_{\alpha +m\delta }=s_{\alpha }\tau _{m{\check \alpha }}$ and $s_{\alpha +m\delta }(\lambda )=\lambda -(\langle \alpha ,\lambda \rangle +m){\check \alpha }$ for each $\lambda \in {\check {\Lambda }}$ . The $\bullet $ -action of $W_{\mathrm {ex}}$ on ${\check {\Lambda }}$ is given by $x\bullet \lambda =x(\lambda +{\check \rho })-{\check \rho }$ for each $x\in W_{\mathrm {ex}}$ .

The sets of dominant coweights and restricted dominant coweights are

$$ \begin{align*} &{\check{\Lambda}}^+=\{\lambda\in{\check{\Lambda}}\,;\,0\leqslant\langle\alpha,\lambda\rangle,\,\forall\alpha\in\Sigma\},\\ &{\check{\Lambda}}_\ell=\{\mu\in{\check{\Lambda}}\,;\,0\leqslant\langle\alpha,\mu\rangle<\ell,\,\forall\alpha\in\Sigma\}. \end{align*} $$

For each $x\in W_{\mathrm {af}}$ the coweight $x\bullet 0$ is dominant if and only if x is minimal in its coset in $W\setminus W_{\mathrm {af}}$ . For each ${\check \alpha }\in {\check Q}$ let $w_{\check \alpha }$ be the minimal element in the coset $W\tau _{\check \alpha }$ . We get a bijection

$$ \begin{align*}{\check Q}\to{\check{\Lambda}}^+\cap W_{\mathrm{af}}\bullet 0,\quad {\check\alpha}\mapsto w_{\check\alpha}\bullet 0=w\bullet{\check\alpha},\end{align*} $$

where $w\in W$ is such that $w\bullet {\check \alpha }$ is dominant.

For each affine real root $\alpha +m\delta $ let $U_{\alpha +m\delta }$ be the unipotent root subgroup in $G(\mathcal {K})$ associated with the affine real root $\alpha +m\delta $ . It is the image of the group homomorphism

$$ \begin{align*}\mathbb{G}_a\to G(\mathcal{K}),\quad z\mapsto n_{\alpha+m\delta}(z):=n_{\alpha}(zt^m),\end{align*} $$

where $n_{\alpha }:\mathbb {G}_a\to U_{\alpha }$ is a fixed group isomorphism. Let $\mathfrak {u}_{\alpha }$ be the Lie algebra of $U_{\alpha }$ and $e_{\alpha }\in \mathfrak {u}_{\alpha }\otimes \mathcal {K}$ be the differential of $n_{\alpha }$ at 1.

2.1.2 Affine flag manifolds

Let ${{\mathcal {F}}\ell }= G(\mathcal {K} )/\mathcal {I}$ denote the affine flag manifold, with $\mathcal {I}$ the usual Iwahori subgroup. For any subset $J\subset \Sigma _{\mathrm {af}}$ let $P^J\subset G(\mathcal {K})$ be the standard parahoric subgroup of type J and $\mathfrak {p}^J$ its Lie algebra. Let ${}^0P^J$ be the pro-unipotent radical of $P^J$ and ${}^0\mathfrak {p}^J$ its Lie algebra. Let $G_J$ be the Levi component and $\mathfrak {g}_J$ its Lie algebra. Let $\Phi _{\mathrm {re}}^J$ and $\Phi _{{\mathrm {re}},J}$ be the sets of all real affine roots that occur in ${}^0\mathfrak {p}^J$ and in $\mathfrak {g}_J$ respectively. Let $W_J\subset W_{\mathrm {af}}$ be the subgroup generated by J and $W^J_{\mathrm {af}}=W_{\mathrm {af}}/W_J$ , $W^J_{\mathrm {ex}}=W_{\mathrm {ex}}/W_J$ . The partial affine flag manifold of type J is the fpqc quotient ${{\mathcal {F}}\ell }^J=G(\mathcal {K} )/P^J$ . It is an ind-scheme. The connected components of ${{\mathcal {F}}\ell }^J$ are parametrized by the algebraic fundamental group $\pi _1$ of G. We have canonical identifications

$$ \begin{align*}\pi_1=N_{G(\mathcal{K})}(\mathcal{I})/\mathcal{I}, \quad W_{\mathrm{ex}}=W_{\mathrm{af}}\rtimes\pi_1\end{align*} $$

where $\pi _1$ acts on $W_{\mathrm {af}}$ by Dynkin diagram automorphisms. Let ${{\mathcal {F}}\ell }^{J,\circ }$ be the connected component of the base point $P^J/P^J$ . The set of fixed points for the left T-action on ${{\mathcal {F}}\ell }^J$ is

$$ \begin{align*}({{\mathcal {F}}\ell}^J)^T=\{\delta_x^J\,;\,x\in W^J_{\mathrm{ex}}\},\quad \delta^J_x=xP^J/P^J.\end{align*} $$

If no confusion is possible we abbreviate $\delta _x=\delta ^J_x$ . We may also abbreviate $\delta _\mu =\delta _{\tau _\mu }$ for each weight $\mu \in {\check {\Lambda }}$ . If $J=\Sigma $ then ${{\mathcal {F}}\ell }^J={{\mathcal {G}}{\mathfrak {r}} }$ is the affine Grassmannian and the set $W^J_{\mathrm {ex}}$ is identified with the lattice ${\check {\Lambda }}$ . The set of T-fixed points in ${{\mathcal {G}}{\mathfrak {r}} }$ is

$$ \begin{align*}{{\mathcal{G}}{\mathfrak{r}} }^T=\{\delta_\mu\,;\, \mu\in{\check{\Lambda}}\},\quad \delta_\mu=t^\mu G(\mathcal{O})/G(\mathcal{O}).\end{align*} $$

2.1.3 The $\ell $ -affine Weyl group

Recall that e is the order of the group ${\check {\Lambda }}/{\check Q}$ , see §2.1.1. From now on we will assume that $\ell \geqslant h$ is prime to e and to $3$ if G contains a component of type $G_2$ . Let $\mu _\ell \subset \mathbb {C}^{\times }$ be the set of $\ell $ -th roots of unity and $\zeta \in \mu _\ell $ to be a primitive element. Let $W_{\ell ,{\mathrm {af}}}=W\ltimes \ell {\check Q}$ and $W_{\ell ,{\mathrm {ex}}}=W\ltimes \ell {\check {\Lambda }}$ be the $\ell $ -affine Weyl group and the $\ell $ -extended Weyl group. Let $\Sigma _{\ell ,{\mathrm {af}}}=\{s_{\alpha +\ell m\delta }\,;\, s_{\alpha +m\delta }\in \Sigma _{\mathrm {af}}\}$ . The fundamental $\ell $ -alcove in $\mathfrak {t}_{\mathbb {R}}$ is

$$ \begin{align*}A_{\ell,1}=\{\mu\in\mathfrak{t}_{\mathbb{R}}\,;\, 0< \langle\alpha,\mu\rangle< \ell\,,\,\forall\alpha\in\Phi^+\}.\end{align*} $$

Let $\bar A_{\ell ,1}$ be its closure. It is a fundamental domain for the action of $W_{\ell ,{\mathrm {af}}}$ on $\mathfrak {t}_{\mathbb {R}}$ . The facet of type J in $\bar A_{\ell ,1}$ is the subset

$$ \begin{align*}F_J=\bar A_{\ell,1}\cap\bigcap_{s_{\alpha+m\delta}\in J}H_{\alpha+\ell m\delta}.\end{align*} $$

Let $W_{\ell ,J}$ be the subgroup of $W_{\ell ,{\mathrm {af}}}$ generated by J. We write $W_{\ell ,{\mathrm {af}}}^{J}=W_{\ell ,{\mathrm {af}}}/W_{\ell ,J}$ and $W_{\ell ,{\mathrm {ex}}}^{J}=W_{\ell ,{\mathrm {ex}}}/W_{\ell ,J}$ . We will abbreviate $\ell {\check {\Lambda }}=W_{\ell ,{\mathrm {ex}}}^\Sigma .$ We consider the $\bullet $ -action of $W_{\ell ,{\mathrm {ex}}}$ on ${\check {\Lambda }}$ . Let

$$ \begin{align*}\Xi_{\mathrm{sc}}={\check{\Lambda}}/W_{\ell,{\mathrm{af}}},\quad\Xi={\check{\Lambda}}/W_{\ell,{\mathrm{ex}}}.\end{align*} $$

Since $\bar A_{\ell ,1}-{\check \rho }$ is a fundamental domain for the $\bullet $ -action of $W_{\ell ,{\mathrm {af}}}$ on $\mathfrak {t}_{\mathbb {R}}$ , each coset in $\Xi _{\mathrm {sc}}$ is uniquely represented by an element in the set $(\bar A_{\ell ,1}-{\check \rho })\cap {\check {\Lambda }}$ . Further, we have $W_{\ell ,{\mathrm {ex}}}=W_{\ell ,{\mathrm {af}}}\rtimes \pi _{\ell ,1}$ where $\pi _{\ell ,1}\subset W_{\ell ,{\mathrm {ex}}}$ is the subgroup stabilizing $(\bar A_{\ell ,1}-{\check \rho })\cap {\check {\Lambda }}$ . The group $\pi _{\ell ,1}$ is canonically identified with $\pi _1$ , yielding a $\pi _1$ -action on $\Xi _{\mathrm {sc}}$ such that $\Xi =\Xi _{\mathrm {sc}}/\pi _1$ . Since the integer $\ell $ is prime to e, an easy computation yields

(2.6) $$ \begin{align}\ell{\check Q}={\check Q}\cap\ell{\check{\Lambda}}.\end{align} $$

Hence, the $\pi _1$ -action on $\Xi _{\mathrm {sc}}$ is free. Given $\omega $ in $\Xi _{\mathrm {sc}}$ , let

$$ \begin{align*}J_{\omega}=\{s\in\Sigma_{\ell,{\mathrm{af}}}\,;\,s\bullet\omega=\omega\},\end{align*} $$

where $\omega $ is viewed as an element in ${\check {\Lambda }}$ . We abbreviate $W_{\ell ,\omega }=W_{\ell ,J_{\omega }}$ and $W_{\ell ,{\mathrm {af}}}^{\omega }=W_{\ell ,{\mathrm {af}}}^{J_{\omega }}$ . We have

$$ \begin{align*}W_{\ell,\omega}=\{x\in W_{\ell,{\mathrm{af}}}\,;\,x\bullet\omega=\omega\}.\end{align*} $$

We define

$$ \begin{align*}W_{\zeta^{\omega}}=\{w\in W\,;\,w\bullet\zeta^{\omega}=\zeta^{\omega}\}.\end{align*} $$

Using (2.6) and the obvious group homomorphism $W_{\ell ,{\mathrm {ex}}}\to W$ , we get

(2.7) $$ \begin{align} W_{\ell,\omega}=\{x\in W_{\ell,{\mathrm{ex}}}\,;\,x\bullet\omega=\omega\}\cong W_{\zeta^{\omega}}. \end{align} $$

2.1.4 Fixed points

For any subset $J\subset \Sigma _{\mathrm {af}}$ , let ${{\mathcal {F}}\ell }^J_\ell $ be the partial affine flag manifold with t replaced by $t^\ell $ . We have ${{\mathcal {F}}\ell }^J_\ell =G(\mathcal {K}_\ell )/P^J_\ell $ where $P^J_\ell \subset G(\mathcal {K}_\ell )$ is the parahoric subgroup of type J. Note that

(2.8) $$ \begin{align}U_{\alpha+\ell m\delta}\subset P^J_\ell\iff F_J\subset H_{\alpha+\ell m\delta}.\end{align} $$

For $\omega \in \Xi _{\mathrm {sc}}$ we abbreviate

$$ \begin{align*}{{\mathcal {F}}\ell}^{\omega}_\ell={{\mathcal {F}}\ell}^{J_{\omega}}_\ell,\quad P^{\omega}_\ell=P^{J_{\omega}}_\ell,\quad \mathfrak{p}^{\omega}_\ell=\mathfrak{p}^{J_{\omega}}_\ell,\quad\text{etc.} \end{align*} $$

Note that $J_{\omega \gamma }=\gamma (J_{\omega })$ for any element $\gamma \in \pi _1$ , where $\gamma $ acts on $\Sigma _{\mathrm {af}}$ through an automorphism of the affine Dynkin diagram. Hence, the partial affine flag manifold ${{\mathcal {F}}\ell }^{\omega }_\ell $ depends only on the class of $\omega $ in the set $\Xi =\Xi _{\mathrm {sc}}\,/\,\pi _1$ . Let ${{\mathcal {G}}{\mathfrak {r}} }^\zeta $ be the set of fixed points of the $\mu _\ell $ -action on ${{\mathcal {G}}{\mathfrak {r}} }$ by loop rotation. By [Reference Riche and Williamson36, §4], we have the following.

Proof. Let $\omega _{\mathrm {sc}}$ be a lifting of $\omega $ in $\Xi _{\mathrm {sc}}$ . By loc. cit., we have an isomorphism

(2.9) $$ \begin{align} \bigsqcup_{\omega\in \Xi} {{\mathcal {F}}\ell}^{\omega}_\ell\to{{\mathcal{G}}{\mathfrak{r}} }^\zeta,\quad g\delta^{\omega}_1\mapsto g\delta_{\omega_{\mathrm{sc}}},\quad g\in G(\mathcal{K}_\ell). \end{align} $$

The formula for $\pi _0({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ follows from (2.9) and the isomorphism $\pi _0({{\mathcal {F}}\ell }^{\omega }_\ell )\cong \pi _1$ .

2.2.1 The deformation to the normal cone

Let A be a commutative $\mathbf {k}$ -algebra and let I be an ideal in A. Let $\hbar $ be a formal variable. We define

$$ \begin{align*} &\widetilde{\mathfrak{R}}_I(A)=\sum_{n\geqslant 0}A[\hbar](\hbar^{-1}I)^n\subset A[\hbar^{\pm 1}], \\ &\mathfrak{R}_I(A)=\widetilde{\mathfrak{R}}_I(A)/\hbar\,\widetilde{\mathfrak{R}}_I(A)=\sum_{n\geqslant 0}I^n/I^{n+1} \ \text{ with }\ I^0=A. \end{align*} $$

For $X=\mathrm {Spec}(A)$ and $Y=\mathrm {Spec}(A/I)\subset X$ , we write

$$ \begin{align*}N_Y(X)=\mathrm{Spec}(\mathfrak{R}_I(A)),\quad \widetilde{N}_Y(X)=\mathrm{Spec}(\widetilde{\mathfrak{R}}_I(A)).\end{align*} $$

The scheme $\widetilde {N}_Y(X)$ is the deformation of X to the normal cone of Y. The scheme $N_Y(X)$ is the normal cone. Note that $\widetilde {N}_Y(X)$ is a scheme over $X\times \mathbb {G}_a$ and $N_Y(X)$ is its fibre at $0\in \mathbb {G}_a$ .

2.2.2 The schemes $\Omega $ and $\Gamma $

Consider the $\mathbf {k}$ -scheme homomorphism

$$ \begin{align*}[\ell]:T/W\to T/W,\quad t\,\text{mod}\,W\mapsto t^\ell\,\text{mod}\,W,\quad t\in T.\end{align*} $$

Let $\Omega $ be the scheme-theoretic preimage of the closed point $1\in T/W$ under the map $[\ell ]$ . Let $\Omega _{\mathrm {sc}}=\Omega \times _{T/W}T_{\mathrm {sc}}/W$ , where $T_{\mathrm {sc}}\to T=T_{\mathrm {sc}}/\pi _1$ is the obvious projection. We have $\Omega \cong \Omega _{\mathrm {sc}}\,/\,\pi _1$ . Let $\mathbf {C}=\mathbf {k}[\mathfrak {t}]/(\mathbf {k}[\mathfrak {t}]^W_+)$ be the coinvariant ring. By (2.7) the group $W_{\ell ,\omega }$ is identified with the subgroup $W_{\zeta ^{\omega }}$ of W. Hence, it acts on the ring $\mathbf {C}$ . We define $\Omega _{\omega }= \mathrm {Spec}(\mathbf {C}^{W_{\ell ,\omega }})$ . For each $\omega $ we consider the closed point $t_{\omega }=\zeta ^{2(\omega +{\check \rho })}$ of the torus T. We have the following; see, e.g., [Reference Lachowska29, thm. 4.1].

Let $\Gamma $ be the fibre product of schemes

(2.10) $$ \begin{align}\Gamma=T\times_{T/W}T/W.\end{align} $$

The map $T\to T/W$ is $t\mapsto t^\ell \,\text {mod}\, W$ . The map $T/W\to T/W$ is $[\ell ]$ . We have

$$ \begin{align*}\Gamma=\mathrm{Spec}\Big(\mathbf{k}[{\check{\Lambda}}]\otimes_{\mathbf{k}[\ell{\check{\Lambda}}]^W}\mathbf{k}[{\check{\Lambda}}]^W\Big).\end{align*} $$

We can describe $\Gamma $ as follows. Consider the union of the graphs of the elements in W in $T^2$ . Let $\widetilde \Gamma $ be its preimage under the map $T^2\to T^2$ given by $(z_1,z_2)\mapsto (z_1^\ell , z_2^\ell )$ . Then $\Gamma $ is the image of $\widetilde \Gamma $ in $T\times (T/W)$ . The formal neighbourhood $\Gamma _{\hat 1}$ of $\Gamma $ along the fibre at $1\in T$ is a disjoint union

(2.11) $$ \begin{align}\Gamma_{\hat 1}=\bigsqcup_{\omega\in\Xi}\Gamma_{\hat\omega}. \end{align} $$

Each connected component $\Gamma _{\hat \omega }$ is isomorphic to $\mathfrak {t}_{\hat 0}\times \Omega _{\omega }$ . Let

$$ \begin{align*}I_{{\Omega}}\subset \mathbf{k}[T/W],\quad I_{\Gamma}\subset\mathbf{k}[T\times T/W],\end{align*} $$

be the vanishing ideals of $\Omega $ and $\Gamma $ . Let

$$ \begin{align*}\mathbf{k}[\widetilde{N}_{\Gamma}(T\times T/W)]_{\hat 0},\quad \mathbf{k}[\widetilde{N}_{\Gamma}(T\times T/W)]_{\widehat{1,0}},\quad \mathbf{k}[N_{\Gamma}(T\times T/W)]_{\hat 1},\end{align*} $$

be respectively the completions of the $\mathcal {A}[T]$ -algebra $\mathbf {k}[\widetilde {N}_{\Gamma }(T\times T/W)]$ at the point $0\in \mathbb {G}_a$ , its completion at $(1,0)\in T\times \mathbb {G}_a$ , and the completion of the $\mathbf {k}[T]$ -algebra $\mathbf {k}[N_{\Gamma }(T\times T/W)]$ at the point $1\in T$ .

2.2.3 The equivariant K-theory of the affine Grassmannian

We first recall a few facts about the equivariant K-theory of the affine Grassmannian. All results here are standard and can be found in [Reference Lam, Schilling and Shimozono33], for instance. For $H=T\times \mathbb {G}_m$ , T or $\mathbb {G}_m$ , let $K^0_H({{\mathcal {G}}{\mathfrak {r}} })$ be the H-equivariant $0$ -th K-cohomology group of the affine Grassmannian tensored with $\mathbf {k}$ . It can be realized as the limit in the category of $R_H$ -modules of the $\mathbf {k}$ -linear Grothendieck groups $K_0^H(U)$ of H-equivariant coherent sheaves over U, where U runs over the set of all finite unions of $G(\mathcal {O})$ -orbits in the thick affine Grassmannian of G. As a module over the ring $R_H$ , we have

(2.12) $$ \begin{align} K^0_H({{\mathcal{G}}{\mathfrak{r}} })=\prod_{x\in {\check{\Lambda}}}R_H\cdot[\mathscr{O}^x], \end{align} $$

where $\mathscr {O}^x$ is the structure sheaf of the finite-codimensional Schubert variety labelled by x. Recall that ${\check {\Lambda }}=W_{\mathrm {ex}}/W$ . For future use, we allow e-th roots of the element q in $\mathbb {C}[T\times \mathbb {G}_m]$ . Recall the rings $\mathbf {A}$ and $\mathcal {A}$ introduced in (1.1). Set

(2.13) $$ \begin{align}\begin{aligned} K_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} })&=\bigoplus_{x\in {\check{\Lambda}}}\mathbf{A}[T]\cdot[\mathscr{O}^x]\subset K^0_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} })\otimes_{R_{T\times\mathbb{G}_m}}\mathbf{A}[T] ,\\ K_{T}({{\mathcal{G}}{\mathfrak{r}} })&=\bigoplus_{x\in {\check{\Lambda}}}\mathbf{k}[T]\cdot[\mathscr{O}^x]\subset K^0_{T}({{\mathcal{G}}{\mathfrak{r}} })\otimes_{R_T}\mathbf{k}[T] ,\\ K_{\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} })&=\bigoplus_{x\in {\check{\Lambda}}}\mathbf{A}\cdot[\mathscr{O}^x]\subset K^0_{\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} })\otimes_{R_{\mathbb{G}_m}}\mathbf{A} \end{aligned} \end{align} $$

The $W_{\mathrm {ex}}$ -action on the ring $\mathbb {C}[T\times \mathbb {G}_m]$ in (2.3) extends to a $W_{\mathrm {ex}}$ -action on $\mathbf {A}[T]$ . The $G(\mathcal {K})$ -action on the affine Grassmannian by left multiplication yields a $W_{\mathrm {ex}}$ -action on the space $K_{T\times \mathbb {G}_m}({{\mathcal {G}}{\mathfrak {r}} })$ . This action is called the left action. It is compatible with the $W_{\mathrm {ex}}$ -action on the ring $\mathbf {A}[T]$ . The restriction to the $T\times \mathbb {G}_m$ -fixed point $\delta _x\in {{\mathcal {G}}{\mathfrak {r}} }$ yields a $W_{\mathrm {ex}}$ -equivariant map

$$ \begin{align*}\mathrm{res}_x:K_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} })\to\mathbf{A}[T],\quad \forall\ x\in{\check{\Lambda}}.\end{align*} $$

In the rest of this section, except in Proposition 2.5, we will assume that the group G is simply connected. By [Reference Kostant and Kumar27], taking the product over all x yields an $\mathbf {A}[T]$ -algebra embedding

(2.14) $$ \begin{align}K_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} })\to\mathrm{Fun}({\check{\Lambda}}\,,\,\mathbf{A}[T]),\end{align} $$

which takes the class $[\mathscr {O}^0]$ to the constant function $1$ . We view the set of maps $\mathrm {Fun}({\check {\Lambda }}\,,\,\mathbf {A}[T])$ as an $(\mathbf {A}[T]\,,\,\mathbf {A}[T/W])$ -bimodule such that

$$ \begin{align*}(f\cdot \psi)(x)=\psi(x)f,\quad (\psi\cdot g)(x)=\psi(x)\,{}^x\! g,\end{align*} $$

for $x\in {\check {\Lambda }}$ , $f\in \mathbf {A}[T]$ , $g\in \mathbf {A}[T/W]$ , $\psi \in \mathrm {Fun}({\check {\Lambda }}\,,\,\mathbf {A}[T])$ . Here ${}^x\! g$ is the twist of the function g by the left action of x on $T/W\times \mathbb {G}_m$ as in (2.4). The image of the map (2.14) is an $(\mathbf {A}[T]\,,\,\mathbf {A}[T/W])$ -subbimodule containing the constant function 1, see [Reference Kostant and Kumar27]. We consider the algebra homomorphism

(2.15) $$ \begin{align} \mathbf{A}[T\times T/W]\to K_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} }),\quad f\otimes g\mapsto f\cdot 1\cdot g. \end{align} $$

Forgetting the T-action we also have a restriction map $K_{\mathbb {G}_m}({{\mathcal {G}}{\mathfrak {r}} })\to \mathrm {Fun}({\check {\Lambda }}\,,\,\mathbf {A})$ and an algebra homomorphism $\mathbf {A}[T/W]\to K_{\mathbb {G}_m}({{\mathcal {G}}{\mathfrak {r}} })$ . Recall that $K_{T\times \mathbb {G}_m}({{\mathcal {G}}{\mathfrak {r}} })_{\hat \zeta }$ and $K_{\mathbb {G}_m}({{\mathcal {G}}{\mathfrak {r}} })_{\hat \zeta }$ are the completions at the point $q_e=\zeta _e$ . Recall also that we have identified $\mathbf {A}_{\hat \zeta }$ and the completion $\mathcal {A}_{\hat 0}$ of $\mathcal {A}$ at $\hbar =0$ , i.e., we have $\mathbf {A}_{\hat \zeta }=\mathcal {A}_{\hat 0}=\mathbf {k}[[\hbar ]].$ Let the closed subset $\Gamma \subset T\times T/W$ be as in (2.10). See §1.5 for details.

Proof. Let us concentrate on (a) because the proof of (b) is similar. We must check that the map (2.15) extends through the canonical embedding $\mathbf {A}[T\times T/W]\to \mathbf {k}[\widetilde {N}_{\Gamma }(T\times T/W)]_{\hat 0}$ to an algebra homomorphism $\mathbf {k}[\widetilde {N}_{\Gamma }(T\times T/W)]_{\hat 0}\to K_{T\times \mathbb {G}_m}({{\mathcal {G}}{\mathfrak {r}} })_{\hat \zeta }$ . To do that, we first note that the map (2.15) yields an algebra homomorphism

(2.16) $$ \begin{align}\mathbf{A}[T\times T/W]_{\hat\zeta} \to K_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} })_{\hat\zeta}.\end{align} $$

Recall that we write $q_e=\zeta _e+\hbar $ , see §1.5. Taking the specialization at $\hbar =0$ , we get a map

$$ \begin{align*}\mathbf{k}[T\times T/W]\to K_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} })_\zeta.\end{align*} $$

We claim that this map takes $I_{\Gamma }$ to $\{0\}$ . Further, the $\mathbf {A}$ -module $K_{T\times \mathbb {G}_m}({{\mathcal {G}}{\mathfrak {r}} })$ is free by (2.13).

Now, recall the following general fact. Let $I\subset A$ be an ideal and $\widetilde I$ be an A-module with a fixed isomorphism of A-modules $I\to \widetilde I$ such that $x\mapsto \tilde x$ . Let $S_A(\widetilde I)$ be the symmetric algebra of the A-module $\widetilde I$ . Let $J\subset S_A(\widetilde I)[\hbar ]$ be the ideal generated by $\{\hbar \tilde x-x\,;\,x\in I\}$ and $J_{\hat 0}\subset S_A(\widetilde I)[[\hbar ]]$ its completion. There is an exact sequence

$$ \begin{align*}0\to J\to S_A(\widetilde I)[\hbar]\to\widetilde{\mathfrak{R}}_I(A)\to 0.\end{align*} $$

Recall that if $0\to K\to M\to N\to 0$ is a short exact sequence of modules over a commutative ring R and N is flat, then there is a short exact sequence $0\to K^\wedge \to M^\wedge \to N^\wedge \to 0$ of completed modules with respect to any ideal of R. Since $\widetilde {\mathfrak {R}}_I(A)$ is a flat $\mathbf {k}[\hbar ]$ -module, we have an exact sequence of completed modules

$$ \begin{align*}0\to J_{\hat 0}\to S_A(\widetilde I)[[\hbar]]\to\widetilde{\mathfrak{R}}_I(A)_{\hat 0}\to 0.\end{align*} $$

Now, let B be a $\mathbf {k}[[\hbar ]]$ -algebra that is torsion free over $\mathbf {k}[[\hbar ]]$ . Let $f:A[[\hbar ]]\to B$ be a morphism of commutative $\mathbf {k}[[\hbar ]]$ -algebras such that $f(I)\subset \hbar B$ . Then $f(x)=\hbar g(x)$ for some unique map $g:I\to B$ such that $g(ax)=f(a)g(x)$ for each $a\in A$ , $x\in I$ . The maps f and g yield an algebra homomorphism

$$ \begin{align*}\tilde f:S_A(\widetilde I)[[\hbar]]\to B ,\quad a\mapsto f(a) ,\quad \tilde x\mapsto g(x)\end{align*} $$

which factorizes through $S_A(\widetilde I)[[\hbar ]]\,/\,J_{\hat 0}\to B$ . Hence it yields a map $\widetilde {\mathfrak {R}}_I(A)_{\hat 0}\to B$ .

We apply this general construction to the map (2.16). The claim yields the desired morphism

(2.17) $$ \begin{align} \mathbf{k}[\widetilde{N}_{\Gamma}(T\times T/W)]_{\hat 0}\to K_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} })_{\hat\zeta}. \end{align} $$

Its composition with (2.14) is a map

(2.18) $$ \begin{align}\mathbf{k}[\widetilde{N}_{\Gamma}(T\times T/W)]_{\hat 0} \to\text{Fun}({\check{\Lambda}}\,,\,\mathbf{k}[T][[\hbar]]).\end{align} $$

Since the first projection $\Gamma \to T$ is flat, the left-hand side of (2.18) is flat over $\mathbf {k}[T][[\hbar ]]$ . Hence, to prove that (2.17) is injective, it suffices to prove that the localized morphism below is injective

$$ \begin{align*}\mathbf{k}[\widetilde{N}_{\Gamma}(T\times T/W)]\otimes_{\mathcal{A}[T]}\mathrm{Frac}(\mathcal{A}[T]) \to\text{Fun}({\check{\Lambda}}\,,\,\mathrm{Frac}(\mathcal{A}[T])).\end{align*} $$

Note that the left-hand side is

$$ \begin{align*}\mathbf{k}[T\times T/W][\hbar]\otimes_{\mathcal{A}[T]}\mathrm{Frac}(\mathcal{A}[T])= \mathrm{Frac}(\mathcal{A}[T])\otimes_{\mathbf{k}}\mathbf{k}[T/W]\end{align*} $$

The injectivity follows.

To prove the claim, note that the ideal $I_\Gamma $ of $\mathbf {k}[T\times T/W]$ is generated by the set $\{f\otimes 1-1\otimes f\,;\,f\in [\ell ]^*\mathbf {k}[T/W]\}$ and that $\ell {\check {\Lambda }}$ acts trivially on $\mathbf {k}[T]$ by (2.2).

2.2.4 The equivariant cohomology of ${{\mathcal {G}}{\mathfrak {r}} }$ and ${{\mathcal {G}}{\mathfrak {r}} }^\zeta $

Now, we consider the equivariant cohomology of partial affine flag manifolds. All results here can be found in [Reference Lam, Lapointe, Morse, Schilling, Shimozono and Zabrocki32, §4], for instance. Recall that

$$ \begin{align*}H^{\bullet}_{T\times{\mathbb{G}}_m}=\mathbf{k}[\mathfrak{t}\times{\mathbb{G}}_a]=\mathcal{A}[\mathfrak{t}], \quad H^{\bullet}_T=\mathbf{k}[\mathfrak{t}].\end{align*} $$

For each subset $J\subset \Sigma _{\mathrm {af}}$ the equivariant cohomology group $H^{\bullet }_{T\times \mathbb {G}_m}({{\mathcal {F}}\ell }^J)$ is a limit in the category of graded $H^{\bullet }_{T\times \mathbb {G}_m}$ -modules in the same way as the equivariant K-cohomology group considered above. Let $[{{\mathcal {F}}\ell }^{J,x}]$ be the fundamental class of the finite-codimensional Schubert variety labelled by the coset x in $W^J$ . We have

$$ \begin{align*}H_{T\times\mathbb{G}_m}^{\bullet}({{\mathcal {F}}\ell}^J)=\bigoplus_{x\in W^J}H_{T\times\mathbb{G}_m}^{\bullet}\cdot[{{\mathcal {F}}\ell}^{J,x}].\end{align*} $$

The $G(\mathcal {K})$ -action on ${{\mathcal {F}}\ell }^J$ by left multiplication yields a $W_{\mathrm {ex}}$ -action on the cohomology space $H^{\bullet }_{T\times \mathbb {G}_m}({{\mathcal {F}}\ell }^J)$ which will be referred to as the left action. The group $W_{\mathrm {ex}}$ acts also on $H^{\bullet }_{T\times \mathbb {G}_m}({{\mathcal {F}}\ell })$ via the affine Springer action, see §2.3.3 for details. We will refer to this action as the right action. The group $W_{\mathrm {ex}}$ acts on

$$ \begin{align*}H^{\bullet}_{T\times\mathbb{G}_m}=\mathbf{k}[\mathfrak{t}\times\mathbb{G}_a]=\mathrm{Sym}(\Lambda\oplus\mathbb{Z}\delta)\otimes\mathbf{k}\end{align*} $$

as in (2.2). The induced $W_{\mathrm {ex}}$ -action on $\mathfrak {t}\times \mathbb {G}_a$ is given by

(2.19) $$ \begin{align} w(\lambda,\hbar)=(w(\lambda),\hbar),\quad \tau_\mu(\lambda,\hbar)=(\lambda+\hbar\mu,\hbar),\quad \lambda,\mu\in{\check{\Lambda}},\quad w\in W,\quad\hbar\in\mathbf{k}. \end{align} $$

The right action is $H^{\bullet }_{T\times \mathbb {G}_m}$ -linear and the left one is $H^{\bullet }_{T\times \mathbb {G}_m}$ -skew-linear.

The cup product by elements of $H^{\bullet }_{T\times \mathbb {G}_m}$ yields a graded ring homomorphism $l:H^{\bullet }_{T\times \mathbb {G}_m}\to H_{T\times \mathbb {G}_m}^{\bullet }({{\mathcal {F}}\ell }).$ The first $T\times \mathbb {G}_m$ -equivariant Chern classes of the line bundles $\mathcal {L}(\lambda )=G(\mathcal {K})\times _{\mathcal {I}}\lambda $ with $\lambda \in \Lambda \oplus \mathbb {Z}\delta $ yield a graded ring homomorphism $r:H^{\bullet }_{T\times \mathbb {G}_m}\to H_{T\times \mathbb {G}_m}^{\bullet }({{\mathcal {F}}\ell }).$ Note that $\mathcal {L}(\delta )$ is a trivial line bundle on ${{\mathcal {F}}\ell }$ with first $T\times \mathbb {G}_m$ -equivariant Chern class $\text {c}_1(\mathcal {L}(\delta ))=l(\hbar )$ . Thus the tensor product $l\otimes r$ is a graded ring homomorphism that fits into the commutative diagram

The identification on the right-hand side is proved as in [Reference Bezrukavnikov and Finkelberg10, thm. 1], where a similar statement is given for the affine Grassmannian ${{\mathcal {G}}{\mathfrak {r}} }$ . The lower map is the obvious one; the elements $f_1,\dots ,f_r$ are the homogeneous generators of the ring $\mathbf {k}[\mathfrak {t}]^W$ , and $g_i$ is a polynomial of degree $\deg (g_i)=\deg (f_i)-1$ . Let

(2.20) $$ \begin{align} \Delta\subset\mathfrak{t}\times\mathfrak{t} \end{align} $$

be the pull-back of the diagonal of $\mathfrak {t}/W$ by the obvious projection $\mathfrak {t}\times \mathfrak {t}\to \mathfrak {t}/W\times \mathfrak {t}/W$ . The map $l\otimes r$ extends to a graded ring isomorphism

$$ \begin{align*}\mathbf{k}[\widetilde N_\Delta(\mathfrak{t}\times\mathfrak{t})]=\mathbf{k}[\mathfrak{t}\times\mathfrak{t}\times\mathbb{G}_a][g_i]/(l(f_i)-r(f_i)-\hbar g_i)\to H_{T\times\mathbb{G}_m}^{\bullet}({{\mathcal {F}}\ell}).\end{align*} $$

Under this identification, the left and right $W_{\mathrm {ex}}$ -actions on the right-hand side correspond to the left and right actions on $\mathfrak {t}\times \mathfrak {t}\times \mathbb {G}_a$ given by (2.19) fixing respectively the second and the first factor $\mathfrak {t}$ in the Cartesian product. The pull-back by the projection ${{\mathcal {F}}\ell }\to {{\mathcal {F}}\ell }^J$ yields an isomorphism

$$ \begin{align*}H_{T\times\mathbb{G}_m}^{\bullet}({{\mathcal {F}}\ell}^J)= H_{T\times\mathbb{G}_m}^{\bullet}({{\mathcal {F}}\ell})^{W_J},\end{align*} $$

see, e.g., [Reference Kumar28, cor. 11.3.14]. The invariants on the right-hand side are relative to the right action of $W_J$ on $H_{T\times \mathbb {G}_m}^{\bullet }({{\mathcal {F}}\ell })$ . The right $W_{\mathrm {ex}}$ -action on $\mathfrak {t}\times \mathfrak {t}\times \mathbb {G}_a$ preserves the closed subset $\Delta \times \{0\}$ . Hence $W_{\mathrm {ex}}$ acts on the scheme $\widetilde N_\Delta (\mathfrak {t}\times \mathfrak {t})$ . We deduce that there is a graded ring isomorphism

(2.21) $$ \begin{align} \mathbf{k}[\widetilde N_{\Delta}(\mathfrak{t}\times\mathfrak{t})/W_J]\to H_{T\times\mathbb{G}_m}^{\bullet}({{\mathcal {F}}\ell}^J). \end{align} $$

Note that if $W_J\subset W$ then $\widetilde N_{\Delta }(\mathfrak {t}\times \mathfrak {t})/W_J=\widetilde N_{\Delta _J}(\mathfrak {t}\times \mathfrak {t}/W_J)$ where $\Delta _J\subset \mathfrak {t}\times (\mathfrak {t}/W_J)$ is the image of $\Delta $ . For instance, in the special case $W_J=W$ we recover [Reference Bezrukavnikov and Finkelberg10, thm. 1].

Now, we consider the equivariant cohomology of ${{\mathcal {G}}{\mathfrak {r}} }^\zeta $ . For each coset $x\in W^J$ , the restriction to the fixed point $\delta _x$ in $({{\mathcal {F}}\ell }^J)^{T\times \mathbb {G}_m}$ yields a map

$$ \begin{align*}\mathrm{res}_x:H^{\bullet}_{T\times\mathbb{G}_m}({{\mathcal {F}}\ell}^J)\to \mathcal{A}[\mathfrak{t}].\end{align*} $$

Thus, for $x\in {\check {\Lambda }}$ , the restriction to $\delta _x$ yields maps

$$ \begin{align*}\mathrm{res}_x:H^{\bullet}_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} }^\zeta)\to\mathcal{A}[\mathfrak{t}],\quad \mathrm{res}_x:H^{\bullet}_{\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} }^\zeta)\to\mathcal{A}.\end{align*} $$

Let $H_T^{\bullet }({{\mathcal {G}}{\mathfrak {r}} }^\zeta )_{\hat 0}$ be the completion of the $H_T^{\bullet }$ -algebra $H^{\bullet }_T({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ at the point $0\in \mathfrak {t}$ , and $H^{\bullet }_{T\times \mathbb {G}_m}({{\mathcal {G}}{\mathfrak {r}} }^\zeta )_{\widehat {0,0}}$ be the completion of the $H^{\bullet }_{T\times \mathbb {G}_m}$ -algebra $H^{\bullet }_{T\times \mathbb {G}_m}({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ at the point $(0,0)$ in $\mathfrak {t}\times \mathbf {k}$ .

Lemma 2.4. Let G be simply connected. There are algebra isomorphisms

$$ \begin{align*} \mathbf{k}[\widetilde{N}_{\Gamma}(T\times T/W)]_{\widehat{1,0}}\to H^{\bullet}_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} }^\zeta)_{\widehat{0,0}},\quad \mathbf{k}[N_{\Gamma}(T\times T/W)]_{\hat 1}\to H^{\bullet}_T({{\mathcal{G}}{\mathfrak{r}} }^\zeta)_{\hat 0}. \end{align*} $$

Proof. The deformation to the normal cone commutes with flat base change. We have

$$ \begin{align*}\widetilde{N}_{\Delta}(\mathfrak{t}\times \mathfrak{t})_{\widehat{0,0}}/W_{\ell,\omega}= \widetilde{N}_{\Delta/W_{\ell,\omega}}(\mathfrak{t}\times \mathfrak{t}/W_{\ell,\omega})_{\widehat{0,0}}.\end{align*} $$

We have the following Cartesian square whose horizontal maps are scheme isomorphisms

Here $\Delta $ is as in (2.20) and the group $W_{\ell ,\omega }$ acts on it through the right action. We deduce that

$$ \begin{align*}\widetilde{N}_{\Gamma_{\hat\omega}}(T\times T/W)_{\widehat{1,0}}= \widetilde{N}_{\Delta}(\mathfrak{t}\times \mathfrak{t})_{\widehat{0,0}}/W_{\ell,\omega}. \end{align*} $$

Thus, the partition (2.11) implies that

(2.22) $$ \begin{align} \mathbf{k}[\widetilde{N}_{\Gamma}(T\times T/W)]_{\widehat{1,0}}= \bigoplus_{\omega\in \Xi}\mathbf{k}[\widetilde{N}_{\Delta}(\mathfrak{t}\times \mathfrak{t})/W_{\ell,\omega}]_{\widehat{0,0}}. \end{align} $$

Using (2.21), we get a canonical ring isomorphism

(2.23) $$ \begin{align} \mathbf{k}[\widetilde{N}_{\Gamma}(T\times T/W)]_{\widehat{1,0}}= H^{\bullet}_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} }^\zeta)_{\widehat{0,0}}. \end{align} $$

The ind-scheme ${{\mathcal {G}}{\mathfrak {r}} }^\zeta $ is equivariantly formal by (2.9). Hence, we have

$$ \begin{align*}H^{\bullet}_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} }^\zeta)\otimes_{\mathcal{A}}\mathbf{k}=H^{\bullet}_T({{\mathcal{G}}{\mathfrak{r}} }^\zeta).\end{align*} $$

Since $H^{\bullet }_T({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ is flat over $H^{\bullet }_T$ , we have

$$ \begin{align*}H^{\bullet}_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} }^\zeta)_{\hat 0}\otimes_{\mathcal{A}}\mathbf{k}=H^{\bullet}_T({{\mathcal{G}}{\mathfrak{r}} }^\zeta)_{\hat 0},\end{align*} $$

where both completions are with respect to the maximal ideal of the ring $H^{\bullet }_T$ , hence

(2.24) $$ \begin{align} H^{\bullet}_{T\times\mathbb{G}_m}({{\mathcal{G}}{\mathfrak{r}} }^\zeta)_{\widehat{0,0}}\otimes_{\mathcal{A}}\mathbf{k}=H^{\bullet}_T({{\mathcal{G}}{\mathfrak{r}} }^\zeta)_{\hat 0}. \end{align} $$

Similarly, since $\mathbf {k}[N_{\Gamma }(T\times T/W)]$ is flat over $\mathbf {k}[T]$ , we have

$$ \begin{align*}\mathbf{k}[\widetilde{N}_{\Gamma}(T\times T/W)]_{\widehat{1}}\otimes_{\mathcal{A}}\mathbf{k}= \mathbf{k}[N_{\Gamma}(T\times T/W)]_{\hat 1},\end{align*} $$

hence

(2.25) $$ \begin{align} \mathbf{k}[\widetilde{N}_{\Gamma}(T\times T/W)]_{\widehat{1,0}}\otimes_{\mathcal{A}}\mathbf{k}= \mathbf{k}[N_{\Gamma}(T\times T/W)]_{\hat 1}. \end{align} $$

Thus, comparing (2.23), (2.24) and (2.25), we deduce that specializing (2.23) to the point $\hbar =0$ yields

$$ \begin{align*}\mathbf{k}[N_{\Gamma}(T\times T/W)]_{\hat 1} =H^{\bullet}_{T}({{\mathcal{G}}{\mathfrak{r}} }^\zeta)_{\hat 0}.\\[-32pt]\end{align*} $$

In the next proposition, we remove the assumption that G is simply connected. Recall that

(2.26) $$ \begin{align} {{\mathcal{G}}{\mathfrak{r}} }^\zeta =\bigsqcup_{\omega\in \Xi} {{\mathcal {F}}\ell}^{\omega}_\ell,\quad \Gamma_{\hat 1}=\bigsqcup_{\omega\in \Xi}\Gamma_{\hat\omega},\quad \Omega=\bigsqcup_{\omega\in \Xi}\Omega_{\omega}. \end{align} $$

The following proposition is the main result of this section. It summarizes and completes the results above.

2.3.1 Affine Springer fibres

Let $({{\mathcal {F}}\ell }^J)^T$ denote the fixed point set of the action of the torus T on the flag manifold ${{\mathcal {F}}\ell }^J$ by left translations. The stabilizer in T of any point of an arbitrary 1-dimensional T-orbit E in any T-variety is the kernel of a character of T, which is uniquely determined up to integer multiples. We pick such a character $\alpha _E$ with minimal norm. It is unique up to a sign.

Fix an element $\gamma $ in $\mathfrak {g}\otimes \mathcal {K}$ . Let ${}^0\!{{\mathcal {F}}\ell }^J_\gamma $ be the closed set of all $x\in {{\mathcal {F}}\ell }^J$ such that $\gamma $ belongs to the Lie algebra of the pro-unipotent radical of the stabilizer of x in $G(\mathcal {K})$ . Let ${{\mathcal {F}}\ell }^J_\gamma $ be the closed set of all $x\in {{\mathcal {F}}\ell }^J$ such that $\gamma $ belongs to the Lie algebra of the stabilizer of x. Following the terminology of Borho-MacPherson [Reference Borho and MacPherson13], we may call ${}^0\!{{\mathcal {F}}\ell }^J_\gamma $ an affine Spaltenstein fibre, ${{\mathcal {F}}\ell }^J_\gamma $ an affine Steinberg fibre and ${{\mathcal {F}}\ell }_\gamma $ an affine Springer fibre. Note that if $\gamma $ is topologically nilpotent then ${{\mathcal {F}}\ell }_\gamma ={}^0\!{{\mathcal {F}}\ell }^\emptyset _\gamma ={{\mathcal {F}}\ell }^\emptyset _\gamma $ . From now on let $\gamma _\ell =s\otimes t^\ell $ , where $s\in \mathfrak {t}$ is regular semi-simple, and set $\gamma =\gamma _1$ . The torus T centralizes $\gamma $ ; thus, it acts on $ {}^0\!{{\mathcal {F}}\ell }^J_\gamma $ . Recall that $({{\mathcal {F}}\ell }^J)^T=\{\delta _x\,;\,x\in W^J_{\mathrm {ex}}\}$ .

Proof. Any element $\omega \in \Xi _{\mathrm {sc}}$ is identified with an element in ${\check {\Lambda }}$ . So the adjoint action by $t^{\omega }$ on $\mathfrak {g}\otimes \mathcal {O}$ yields a well-defined Lie subalgebra in $\mathfrak {g}\otimes \mathcal {K}$ . To prove (a), observe first that

$$ \begin{align*} \mathfrak{g}\otimes\mathcal{K}_\ell\cap\mathrm{Ad}_{t^{-\omega}}(\mathfrak{g}\otimes\mathcal{O})=\mathfrak{p}^{\omega}_\ell. \end{align*} $$

Since $\mathfrak {g}\otimes \mathcal {K}_\ell \cap \mathrm {Ad}_{t^{-\omega }}(\mathfrak {g})$ is reductive, we deduce that

$$ \begin{align*}\mathfrak{g}\otimes\mathcal{K}_\ell\cap\mathrm{Ad}_{t^{-\omega}}(\mathfrak{g}\otimes t\mathcal{O})={}^0\mathfrak{p}^{\omega}_\ell\end{align*} $$

is the pro-unipotent radical of $\mathfrak {p}^{\omega }_\ell $ . Let $\omega _{\mathrm {sc}}\in \Xi _{\mathrm {sc}}$ be a lifting of $\omega \in \Xi $ . By (2.9), we have

$$ \begin{align*} {}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell} &=\bigsqcup_{\omega\in \Xi}\{g\delta_{\omega_{\mathrm{sc}}}\,;\, g\in G(\mathcal{K}_\ell)\,,\, \mathrm{Ad}_g^{-1}(\gamma_\ell)\in\mathfrak{g}\otimes\mathcal{K}_\ell\cap\mathrm{Ad}_{t^{-\omega}}(\mathfrak{g}\otimes t\mathcal{O})\}\\ &=\bigsqcup_{\omega\in \Xi}\{g\delta_{\omega_{\mathrm{sc}}}\,;\, g\in G(\mathcal{K}_\ell)\,,\, \mathrm{Ad}_g^{-1}(\gamma_\ell)\in {}^0\mathfrak{p}^{\omega}_\ell\}\\ &=\bigsqcup_{\omega\in \Xi} {}^0\!{{\mathcal {F}}\ell}^{\omega}_{\ell,\gamma_\ell}. \end{align*} $$

Part (b) is obvious. Now, we concentrate on part (d). The Iwasawa decomposition gives

$$ \begin{align*}{{\mathcal {F}}\ell}^J=\bigsqcup_{x\in W^J_{\mathrm{ex}}}N(\mathcal{K})\delta_x.\end{align*} $$

Intersecting this decomposition with ${}^0\!{{\mathcal {F}}\ell }_\gamma ^J$ provides an affine paving of ${}^0\!{{\mathcal {F}}\ell }_\gamma ^J$ . More precisely, choose an element $n\in N(\mathcal {K})$ . It is of the following form

$$ \begin{align*}n=\prod_{\alpha\in\Phi^+}n_{\alpha}(a_{\alpha} t^{k_{\alpha}}),\quad k_{\alpha}\in\mathbb{Z},\quad a_{\alpha}\in\mathbb{C}.\end{align*} $$

Let $x\in W_{\mathrm {ex}}$ with $x=\tau _\mu w$ and $w\in W$ , $\mu \in {\check {\Lambda }}$ . We have

(2.30) $$ \begin{align} \mathrm{Ad}_{nx}^{-1}(\gamma)=\mathrm{Ad}_x^{-1}(\gamma)+ \sum_{\alpha\in\Phi^+}(\langle\alpha,s\rangle\, a_{\alpha}\, t^{1+k_{\alpha}-\langle\alpha,\mu\rangle}+f_{\alpha}(t)) e_{w^{-1}(\alpha)} \end{align} $$

where $f_{\alpha }(t) $ is a polynomial of degree $>1+k_{\alpha }-\langle \alpha ,\mu \rangle $ . Since $\mathrm {Ad}_x^{-1}(\gamma )\in {}^0\mathfrak {p}^J$ , we deduce that

$$ \begin{align*} n\delta_x\in {}^0\!{{\mathcal {F}}\ell}^J_\gamma&\iff w^{-1}(\alpha)+(1+k_{\alpha}-\langle\alpha,\mu\rangle)\delta\in\Phi_{\mathrm{re}}^J,\ \forall \alpha\in\Phi^+,\\ n\delta_x\neq\delta_x&\iff w^{-1}(\alpha)+(k_{\alpha}-\langle\alpha,\mu\rangle)\delta\notin\Phi_{{\mathrm{re}},J}\cup\Phi_{\mathrm{re}}^J,\ \forall \alpha\in\Phi^+. \end{align*} $$

Hence, there is an isomorphism

(2.31) $$ \begin{align} {}^0\!{{\mathcal {F}}\ell}^J_\gamma\cap N(\mathcal{K})\delta_x\cong \prod_{\alpha}U_{\alpha+m_{\alpha}\delta}, \end{align} $$

where the product is over all roots $\alpha \in \Phi ^+$ such that

(2.32) $$ \begin{align} (w^{-1}(\alpha)+\mathbb{Z}\delta)\cap\Phi_{{\mathrm{re}},J}=\emptyset. \end{align} $$

The integer $m_{\alpha }$ is given by

$$ \begin{align*}m_{\alpha}=\max\{m\in\mathbb{Z}\,;\, w^{-1}(\alpha)+(m-\langle\alpha,\mu\rangle)\delta\notin\Phi^J_{\mathrm{re}}\}.\end{align*} $$

This proves (d). To prove part (e), note that (2.2) yields

$$ \begin{align*}x^{-1}(\alpha+m\delta)=w^{-1}(\alpha)+(m+\langle \alpha,\mu\rangle)\delta.\end{align*} $$

Hence the condition (2.32) is equivalent to $x^{-1}s_{\alpha +m\delta }x\notin W_J$ for all $m\in \mathbb {Z}$ . Now, the closure relation of the one-dimensional orbits in ${{\mathcal {F}}\ell }^J$ is well-known, and ${}^0\!{{\mathcal {F}}\ell }^J_\gamma $ is a union of T-orbits. Thus, we are reduced to describing which one-dimensional orbits in ${{\mathcal {F}}\ell }^J$ are contained in ${}^0\!{{\mathcal {F}}\ell }^J_\gamma $ . For each one-dimensional T-orbit $L\subset {}^0\!{{\mathcal {F}}\ell }^J_\gamma $ , there is a unique affine cell $C_x={}^0\!{{\mathcal {F}}\ell }^J_\gamma \cap N(\mathcal {K})\delta _x$ such that $L=L\cap C_x\cong \mathbb {A}^1$ . By (2.31), the intersection $L\cap C_x$ is of the form $U_{\alpha +m_{\alpha }\delta }\cdot \delta _x$ . Finally, the closure of $U_{\alpha +m_{\alpha }\delta }\cdot \delta _x$ in ${}^0\!{{\mathcal {F}}\ell }^J_\gamma $ is the same as its closure in ${{\mathcal {F}}\ell }^J$ . It is equal to $U_{\alpha +m_{\alpha }\delta }\cdot x\cup \{\delta _{s_{\alpha +m_{\alpha }\delta }x}\}$ .

Finally, let us consider part (c). Fix an order

$$ \begin{align*}\{\alpha_1+m_1\delta,\dots,\alpha_r+m_r\delta\}=\Phi_{\mathrm{re}}^+\setminus x(\Phi^J_{\mathrm{re}}\sqcup\Phi_{{\mathrm{re}},J}),\end{align*} $$

such that $\alpha _i-\alpha _j+(m_i-m_j)\delta \notin \Phi _{\mathrm {re}}^+\ \text {if}\ i<j.$ We have an isomorphism of affine spaces

$$ \begin{align*}\mathbb{A}^r\to \mathcal{I} \delta_x,\quad (z_1,z_2,\dots,z_r)\mapsto n_{\alpha_1+m_1\delta}(z_1)n_{\alpha_2+m_2\delta}(z_2)\cdots n_{\alpha_r+m_r\delta}(z_r)\delta_x.\end{align*} $$

For $nx\in \mathcal {I} \delta _x$ given by the image of $(z_1,\dots ,z_r)$ under the above map, we have

$$ \begin{align*}\mathrm{Ad}_{nx}^{-1}(\gamma)=\mathrm{Ad}_x^{-1}(\gamma)+ \sum_{k=1}^r(\langle\alpha_k,s\rangle z_k+f_k(z_1,\cdots,z_{k-1}))e_{x^{-1}(\alpha_k)+(m_k+1)\delta}, \end{align*} $$

where $f_k(z_1,\dots ,z_{k-1})$ is a polynomial in $z_1,\dots ,z_{k-1}$ . Thus

$$ \begin{align*}\mathrm{Ad}_{nx}^{-1}(\gamma)\in {}^0\mathfrak{p}^J\iff \langle\alpha_k,s\rangle z_k+f_k(z_1,\cdots,z_{k-1})=0\end{align*} $$

whenever $x^{-1}(\alpha _k)+(m_k+1)\delta \notin \Phi ^J_{\mathrm {re}}$ . Since s is regular, we have $\langle \alpha _k,s\rangle \neq 0$ . Hence this equality determines $z_k$ uniquely. If $x^{-1}(\alpha _k)+(m_k+1)\delta \in \Phi ^J_{\mathrm {re}}$ , then $z_k$ can take any value. Thus ${}^0\!{{\mathcal {F}}\ell }^J_\gamma \cap \mathcal {I}\delta _x$ is an affine space of dimension

$$ \begin{align*}\sharp\{\gamma\in\Phi^+_{\mathrm{re}}\,;\,x^{-1}(\gamma)\notin\Phi^J_{\mathrm{re}}\sqcup\Phi_{{\mathrm{re}},J}\,,\, x^{-1}(\gamma)\in(\Phi^J_{\mathrm{re}}-\delta)\}.\\[-32pt]\end{align*} $$

2.3.2 The GKM presentation of $H^{\bullet }_T({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })$

Let i be the obvious inclusion of ${}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell }$ in ${{\mathcal {G}}{\mathfrak {r}} }^\zeta $ . The ind-varieties ${}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell }$ and ${{\mathcal {G}}{\mathfrak {r}} }^\zeta $ are both equivariantly formal. Hence, using the localization theorem as in [Reference Goresky, Kottwitz and MacPherson23], the restrictions to the fixed-point subsets yield the following commutative diagram of algebra embeddings

(2.33)

We cannot apply the GKM method to get a presentation of the cohomology group $H^{\bullet }_T({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ . To do so, we must consider the $T\times \mathbb {G}_m$ -action. However we can apply it to the cohomology group $H^{\bullet }_T({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })$ . To do that, we first consider the connected components of ${}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell }$ separately. Recall the integer $m_{\alpha }$ in (2.29).

Lemma 2.7. For each element $\omega \in \Xi $ there is an $H^{\bullet }_T$ -algebra isomorphism

$$ \begin{align*}H^{\bullet}_T({}^0\!{{\mathcal {F}}\ell}^{\omega}_{\ell,\gamma_\ell})= \left\{ (a_x)\in \mathrm{Fun}(W_{\ell,{\mathrm{ex}}}^{\omega}\,,H^{\bullet}_T)\,;\, x^{-1}s_{\alpha+\ell n\delta}x\notin W_{\ell,\omega} \ \forall n\in\mathbb{Z} \Rightarrow a_x\equiv a_{s_{\alpha+\ell m\delta}x}\mathrm{mod} \alpha\ \forall m\in\mathbb{Z} \right\}\end{align*} $$

and an $H^{\bullet }_{T\times \mathbb {C}^{\times }}$ -algebra isomorphism

$$ \begin{align*}H^{\bullet}_{T\times\mathbb{C}^{\times}}( {}^0\!{{\mathcal {F}}\ell}^{\omega}_{\ell,\gamma_\ell})&= \{ (a_x)\in \mathrm{Fun}(W_{\ell,{\mathrm{ex}}}^{\omega}\,,H^{\bullet}_{T\times\mathbb{C}^{\times}})\,;\, x^{-1}s_{\alpha+\ell n\delta}x\notin W_{\ell,\omega} \\ &\qquad \forall n\in\mathbb{Z} \Rightarrow a_x\equiv a_{s_{\alpha+\ell m_{\alpha}\delta}x}\mathrm{mod} \alpha+\ell m_{\alpha}\delta \}\end{align*} $$

From Lemmas 2.6 and 2.7 we get the following.

Proposition 2.8. There is an $H^{\bullet }_T$ -algebra isomorphism

$$ \begin{align*}H^{\bullet}_T({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell})= \left\{ (a_\lambda)\in \mathrm{Fun}({\check{\Lambda}}\,,H^{\bullet}_T)\,\middle |\, \langle\alpha\,,\,\lambda+{\check\rho}\rangle\notin\ell\mathbb{Z}\Rightarrow a_\lambda\equiv a_{s_{\alpha+\ell m\delta}\bullet\lambda}\mathrm{mod}\alpha\ \forall m\in\mathbb{Z} \right\}\end{align*} $$

Proof. Use the following identifications of $W_{\ell ,{\mathrm {ex}}}$ -sets

$$ \begin{align*} {\check{\Lambda}}=\bigsqcup_{\omega\in \Xi} W_{\ell,{\mathrm{ex}}}\bullet\omega= \bigsqcup_{\omega\in \Xi} W_{\ell,{\mathrm{ex}}}/W_{\ell,\omega} =\bigsqcup_{\omega\in \Xi} W_{\ell,{\mathrm{ex}}}^{\omega},\end{align*} $$

the decomposition in Lemma 2.6 and the following equivalences

$$ \begin{align*}s_{\alpha+\ell n\delta}\bullet\lambda\neq\lambda,\ \forall n\in\mathbb{Z} \iff s_{\alpha} \bullet\zeta^\lambda\neq\zeta^\lambda\iff \langle\alpha\,,\,\lambda+{\check\rho}\rangle\notin\ell\mathbb{Z}.\\[-32pt]\end{align*} $$

2.3.3 Affine Weyl group symmetries of $H^{\bullet }_T({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })$

The $G(\mathcal {K}_\ell )$ -action on ${{\mathcal {G}}{\mathfrak {r}} }^\zeta $ gives rise to the left $W_{\ell ,{\mathrm {ex}}}$ -action on $H^{\bullet }_T({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ . Similarly, the $T(\mathcal {K}_\ell )$ -action on ${}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell }$ yields an $\ell {\check {\Lambda }}$ -action on $H^{\bullet }_T({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })$ . We claim that this action extends to a $W_{\ell ,{\mathrm {ex}}}$ -action on $H^{\bullet }_T({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })$ again called the left action. Indeed, under the restriction to fixed points

$$ \begin{align*}\mathrm{res}:H^{\bullet}_T({{\mathcal{G}}{\mathfrak{r}} }^\zeta)\to\mathrm{Fun}({\check{\Lambda}},H^{\bullet}_T)\end{align*} $$

the left $W_{\ell ,{\mathrm {ex}}}$ -action on $H^{\bullet }_T({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ is given by the following rule

(2.34) $$ \begin{align} x(a_\lambda)={}^xa_{x\bullet\lambda},\quad \lambda\in {\check{\Lambda}},\quad x\in W_{\ell,{\mathrm{ex}}},\quad (a_{\lambda})\in \mathrm{Fun}({\check{\Lambda}},H^{\bullet}_T). \end{align} $$

Here the map $f\mapsto {}^xf$ is the $W_{\ell ,{\mathrm {ex}}}$ -action on $H^{\bullet }_T=\mathbf {k}[\mathfrak {t}]$ dual to (2.5). Similarly, under the restriction

$$ \begin{align*}\mathrm{res}:H^{\bullet}_T({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell})\to\mathrm{Fun}({\check{\Lambda}},H^{\bullet}_T)\end{align*} $$

the left $\ell {\check {\Lambda }}$ -action on $H^{\bullet }_T({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })$ is given by

$$ \begin{align*}\mu(a_\lambda)=a_{\tau_\mu\bullet\lambda},\quad \lambda,\mu\in {\check{\Lambda}},\quad (a_\lambda)\in \mathrm{Fun}({\check{\Lambda}},H^{\bullet}_T).\end{align*} $$

By Proposition 2.8, this $\ell {\check {\Lambda }}$ -action extends uniquely to a $W_{\ell ,{\mathrm {ex}}}$ -action on $H^{\bullet }_T({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })$ such that (2.34) holds, proving the claim. The element $\gamma _\ell $ is homogeneous. Hence, both ind-schemes ${{\mathcal {G}}{\mathfrak {r}} }^\zeta $ and ${}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell }$ are cohomologically pure by [Reference Goresky, Kottwitz and MacPherson22, cor. 1.3]. Note that loc. cit. deals with homology, but the result is the same in cohomology. We deduce that their T-equivariant cohomology is T-equivariantly formal. Thus, we have

$$ \begin{align*}H^{\bullet}_T({{\mathcal{G}}{\mathfrak{r}} }^\zeta)\otimes_{H^{\bullet}_T}\mathbf{k}=H^{\bullet}({{\mathcal{G}}{\mathfrak{r}} }^\zeta),\quad H^{\bullet}_T({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell})\otimes_{H^{\bullet}_T}\mathbf{k}=H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell}).\end{align*} $$

The left $W_{\ell ,{\mathrm {ex}}}$ -actions on $H^{\bullet }_T({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ and $H^{\bullet }_T({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })$ are $H^{\bullet }_T$ -skew-linear. Thus, they specialize to $W_{\ell ,{\mathrm {ex}}}$ -actions on the cohomology spaces $H^{\bullet }({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ and $H^{\bullet }({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })$ . The restriction

$$ \begin{align*}i^*:H^{\bullet}_T({{\mathcal{G}}{\mathfrak{r}} }^\zeta)\to H^{\bullet}_T({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell})\end{align*} $$

commutes with the left $W_{\ell ,{\mathrm {ex}}}$ -action. By formality, it specializes to the restriction

$$ \begin{align*}i^*:H^{\bullet}({{\mathcal{G}}{\mathfrak{r}} }^\zeta)\to H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell}),\end{align*} $$

which is also $W_{\ell ,{\mathrm {ex}}}$ -equivariant. For each subset $S\subset W_{\ell ,{\mathrm {ex}}}$ , we write $H^{\bullet }({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })^S$ for the space of S-invariants with respect to the left action. The left $W_{\ell ,{\mathrm {af}}}$ -action on $H^{\bullet }({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ is trivial, because it comes from the left action of the neutral connected component of $G(\mathcal {K}_\ell )$ . Hence $i^*$ maps into the space $H^{\bullet }({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })^{W_{\ell ,{\mathrm {af}}}}$ . The connected components of ${{\mathcal {G}}{\mathfrak {r}} }$ are parametrized by the group $\pi _1$ , hence the algebras $H^{\bullet }({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ and $H^{\bullet }({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })$ are both $\pi _1$ -graded. The canonical inclusion $\pi _1\subset W_{\ell ,{\mathrm {ex}}}$ recalled in §2.1.3 yields an action of the group $\pi _1$ by algebra automorphisms on $H^{\bullet }({{\mathcal {G}}{\mathfrak {r}} }^\zeta )$ and $H^{\bullet }({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })$ . Since $W_{\ell ,{\mathrm {ex}}}=W_{\ell ,{\mathrm {af}}}\rtimes \pi _1$ , the algebra $H^{\bullet }({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })^{W_{\ell ,{\mathrm {af}}}}$ is preserved by the $\pi _1$ -action, and the map $i^*$ is a $\pi _1$ -equivariant algebra homomorphism

(2.35) $$ \begin{align} i^*:H^{\bullet}({{\mathcal{G}}{\mathfrak{r}} }^\zeta)\to H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell})^{W_{\ell,{\mathrm{af}}}}. \end{align} $$

In particular, it takes $H^{\bullet }({{\mathcal {G}}{\mathfrak {r}} }^\zeta )^{\pi _1}$ into $H^{\bullet }({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })^{W_{\ell ,{\mathrm {ex}}}}$ .

The group $W_{\mathrm {ex}}$ also acts on $H^{\bullet }({{\mathcal {F}}\ell }_\gamma )$ via the affine Springer action defined by G. Lusztig in [Reference Lusztig34]. We will use the description of the Springer action given in [Reference Goresky, Kottwitz and MacPherson23, §14.4]. More precisely, under restriction to the fixed points, this action is given by

$$ \begin{align*}y(a_x)=(a_{xy^{-1}}),\quad x,y\in W_{\mathrm{ex}}.\end{align*} $$

We will refer to the Springer action as the right action. To avoid confusion, we will write $(a_x)y^{-1}$ instead of $y(a_x)$ . The left and right actions commute with each other. They both preserve the cohomological grading. The right action is $H^{\bullet }_T$ -linear and the left one is $H^{\bullet }_T$ -skew-linear.

From now on and until the end of Proposition 2.11, we focus on the component $J=\emptyset $ , and safely remove $\ell $ in the discussion.

Proof. We first concentrate on part (a). We will prove the equivalent statement

$$ \begin{align*}\dim H_{\bullet}({{\mathcal {F}}\ell}_\gamma)_{W_{\mathrm{ex}}}\leqslant (h+1)^r\end{align*} $$

where the left-hand side is the dimension of the space of coinvariants in the Borel-Moore homology group for the left action. We may assume that the group G is simply connected, because the terms of the inequality do not depend on the fundamental group. Then ${\check {\Lambda }}$ is the coroot lattice ${\check Q}$ . For each affine element $x\in W_{\mathrm {ex}}$ we consider the affine space $C_x={{\mathcal {F}}\ell }_\gamma \cap \mathcal {I}\delta _x$ . From the proof of Lemma 2.6 we deduce that

$$ \begin{align*} &C_x=x\left(\prod_{\alpha\in E_x}U_{\alpha}\right)\delta_1,\\ &E_x=x^{-1}(\Phi_{\mathrm{re}}^+)\cap\Phi^-_{\mathrm{re}}\cap(\Phi^+_{\mathrm{re}}-\delta)=x^{-1}(\Phi_{\mathrm{re}}^+)\cap((\Phi^+-\delta)\sqcup\Phi^-). \end{align*} $$

Hence, for each affine reflection $s_i$ we have

$$ \begin{align*} s_iC_x= C_{s_ix} &\iff E_x=E_{s_ix}\\ &\iff \Phi_{\mathrm{re}}^+\cap(x(\Phi^+-\delta)\sqcup x(\Phi^-))=s_i(\Phi_{\mathrm{re}}^+)\cap(x(\Phi^+-\delta)\sqcup x(\Phi^-))\\ &\iff\pm x^{-1}(\alpha_i)\notin (\Phi^+-\delta)\sqcup\Phi^-\\ &\iff x^{-1}(\alpha_i)\notin E_x\cup(-E_{s_ix}). \end{align*} $$

We consider the equivalence relation on $W_{\mathrm {ex}}$ such that

$$ \begin{align*}y\sim x\iff y^{-1}C_y=x^{-1}C_x.\end{align*} $$

We have

(2.36) $$ \begin{align}x\sim s_ix\iff x^{-1}(\alpha_i)\notin E_x\cup(-E_{s_ix}). \end{align} $$

Let $\overline {C_x}$ be the closure of $C_x$ in ${{\mathcal {F}}\ell }_\gamma $ , and $b_x=[\overline {C_x}]$ be its fundamental class in $H_{\bullet }({{\mathcal {F}}\ell }_\gamma )$ . Then $\{b_x\,;\, x\in W_{\mathrm {ex}}\}$ is a basis of $H_{\bullet }({{\mathcal {F}}\ell }_\gamma )$ such that $b_y=b_x$ in $H_{\bullet }({{\mathcal {F}}\ell }_\gamma )_{W_{\mathrm {ex}}}$ whenever $y\sim x$ . Let $M\subset W_{\mathrm {ex}}$ be the set of all elements that are minimal in their $\sim $ -class. We deduce that

$$ \begin{align*}\dim H_{\bullet}({{\mathcal {F}}\ell}_\gamma)_{W_{\mathrm{ex}}}\leqslant \sharp(W_{\mathrm{ex}}/\sim)\leqslant \sharp M.\end{align*} $$

We are reduced to giving an upper bound for the integer $\sharp M$ . For each $x\in W_{\mathrm {ex}}$ let $A_x=x^{-1}(A_1)$ where

$$ \begin{align*}A_1=\{\mu\in\mathfrak{t}_{\mathbb{R}}\,;\, 0< \langle\alpha,\mu\rangle< 1\,,\,\forall\alpha\in\Phi^+\}.\end{align*} $$

We write $\langle \alpha ,A_{x^{-1}}\rangle =\langle x^{-1}(\alpha ),{\check \rho }\rangle /h$ for each $\alpha \in \Lambda $ . We use the same convention as in (2.5):

(2.37) $$ \begin{align}\langle\delta,\mu\rangle=1,\quad\forall\mu\in\mathfrak{t}_{\mathbb{R}}.\end{align} $$

Note that

$$ \begin{align*}|\langle\Phi,A_1\rangle|\subset(0,1),\quad \langle\alpha_i,A_{x^{-1}}\rangle<0\iff s_ix<x.\end{align*} $$

Pick an element $x\in M$ . From (2.36) we deduce that

$$ \begin{align*}\langle\alpha_i,A_{x^{-1}}\rangle<0\Rightarrow x^{-1}(\alpha_i)\in E_x\cup(-E_{s_ix}).\end{align*} $$

Write $x^{-1}=w^{-1}\tau _\mu $ with $w\in W$ and $\mu \in {\check {\Lambda }}$ . By (2.2) and (2.37) we have

(2.38) $$ \begin{align} x^{-1}(\alpha_i)=w^{-1}(\alpha_i)-\langle\alpha_i,\mu\rangle\delta,\quad \langle\alpha_i,A_{x^{-1}}\rangle=\langle \alpha_i,A_{w^{-1}}\rangle-\langle\alpha_i,\mu\rangle. \end{align} $$

First, assume that $i\neq 0$ . If $x^{-1}(\alpha _i)\in E_x$ then $x^{-1}(\alpha _i)\in (\Phi ^+-\delta )\sqcup \Phi ^-$ . Hence, either $w^{-1}(\alpha _i)\in \Phi ^+$ and $\langle \alpha _i,\mu \rangle =1$ , or $w^{-1}(\alpha _i)\in \Phi ^-$ and $\langle \alpha _i,\mu \rangle =0$ . Hence $\langle \alpha _i,A_{x^{-1}}\rangle <0$ in both cases. Similarly, if $x^{-1}(\alpha _i)\in -E_{s_ix}$ , then $\langle \alpha _i,A_{x^{-1}}\rangle>0$ . Therefore, the computation above and (2.38) yield

$$ \begin{align*}\langle\alpha_i,A_{x^{-1}}\rangle<0\Rightarrow x^{-1}(\alpha_i)\in E_x \Rightarrow \langle\alpha_i,A_{x^{-1}}\rangle>-1.\end{align*} $$

On the other hand, if $\langle \alpha _i,A_{x^{-1}}\rangle \geqslant 0$ then (2.38) yields either $w^{-1}(\alpha _i)\in \Phi ^+$ and $\langle \alpha _i,\mu \rangle \leqslant 0$ , or $w^{-1}(\alpha _i)\in \Phi ^-$ and $\langle \alpha _i,\mu \rangle \leqslant -1$ . Hence, in both cases we have $\langle \alpha _i,A_{x^{-1}}\rangle>-1$ .

Now, assume that $i=0$ . If $x^{-1}(\alpha _0)\in E_x$ then $x^{-1}(\alpha _0)\in (\Phi ^+-\delta )\sqcup \Phi ^-$ . Hence, either $w^{-1}(\alpha _0)\in \Phi ^++\delta $ and $\langle \alpha _i,\mu \rangle =2$ , or $w^{-1}(\alpha _0)\in \Phi ^-+\delta $ and $\langle \alpha _i,\mu \rangle =1$ . Hence $\langle \alpha _0,A_{x^{-1}}\rangle \in (-1,0)$ in both cases. Similarly, if $x^{-1}(\alpha _0)\in -E_{s_0x}$ , then $\langle \alpha _0,A_{x^{-1}}\rangle \in (0,1)$ . Therefore, the computation above and (2.38) yield

$$ \begin{align*}\langle\alpha_0,A_{x^{-1}}\rangle<0\Rightarrow x^{-1}(\alpha_0)\in E_x \Rightarrow \langle\alpha_0,A_{x^{-1}}\rangle>-1.\end{align*} $$

On the other hand, if $\langle \alpha _0,A_{x^{-1}}\rangle \geqslant 0$ then (2.38) yields either $w^{-1}(\alpha _0)\in \Phi ^++\delta $ and $\langle \alpha _0,\mu \rangle \leqslant 2$ , or $w^{-1}(\alpha _i)\in \Phi ^-+\delta $ and $\langle \alpha _0,\mu \rangle \leqslant 1$ . Hence, in both cases we have $\langle \alpha _0,A_{x^{-1}}\rangle>-1$ .

Therefore, the case-by-case computations above imply that

$$ \begin{align*} x\in M &\Rightarrow\langle\alpha_i,A_{x^{-1}}\rangle>-1\,,\,\forall \alpha_i\in\Sigma_{\mathrm{af}} \end{align*} $$

Hence, from (2.37) we deduce that

$$ \begin{align*} x\in M&\Rightarrow\langle\theta,A_{x^{-1}}\rangle<2\,,\,\langle\alpha_i,A_{x^{-1}}\rangle>-1\,,\,\forall \alpha_i\in\Sigma,\\ &\Rightarrow\langle\theta,A_{x^{-1}}+{\check\rho}\rangle<h+1\,,\,\langle\alpha_i,A_{x^{-1}}+{\check\rho}\rangle>0\,,\,\forall \alpha_i\in\Sigma,\\ &\Rightarrow A_{x^{-1}}+{\check\rho}\subset(h+1)A_1. \end{align*} $$

Recall that $(h+1) A_1$ is a fundamental domain for the action of $W_{h+1,{\mathrm {af}}}$ on $\mathfrak {t}_{\mathbb {R}}$ . Hence, we have

$$ \begin{align*}W_{h+1,{\mathrm{af}}}\cdot((h+1)A_1)=W_{{\mathrm{af}}}\cdot A_1.\end{align*} $$

We deduce that

$$ \begin{align*}\sharp\{x\in W_{{\mathrm{af}}}\,;\, A_x\subset (h+1) A_1\}=\sharp(W_{{\mathrm{af}}}/W_{h+1,{\mathrm{af}}})=\sharp( {\check Q}/(h+1){\check Q})=(h+1)^r.\end{align*} $$

Thus, we have $\sharp M \leqslant (h+1)^r$ .

Now we prove part (b). Since this dimension formula was obtained independently by another argument in [Reference Boixeda Alvarez and Losev12], see Proposition 2.11, we will be brief. The diagonal coinvariant ring is $DR=\mathbf {k}[\mathfrak {t}\times \mathfrak {t}]/I_+$ , where $I_+$ is the ideal generated by homogeneous diagonal W-invariants of positive degrees. In [Reference Haiman24, prop. 3.6], Haiman proved that this ring has dimension $(h+1)^r$ . We now use the graded Double Affine Hecke algebra (GDAHA) action on the equivariant Borel-Moore homology of ${{\mathcal {F}}\ell }_\gamma $ and ${{\mathcal {F}}\ell }$ considered in [Reference Varagnolo and Vasserot41] and [Reference Oblomkov and Yun35]. By definition, it commutes with the pushforward by the closed immersion $i:{{\mathcal {F}}\ell }_\gamma \to {{\mathcal {F}}\ell }$ .

We claim that, using this representation, the construction of Carlsson and Oblomkov defined in [Reference Carlsson and Oblomkov16, thm. 7.1] yields a representation of $DR$ on $H_{\bullet }({{\mathcal {F}}\ell }_\gamma )$ and $H_{\bullet }({{\mathcal {F}}\ell })$ that commutes with $i_\ast $ . Hence, the image of $i_\ast $ is a $DR$ -submodule. Let $[G/B]$ be the fundamental class corresponding to $G/B$ in $H_{\bullet }({{\mathcal {F}}\ell })$ . Since $G/B\subset {{\mathcal {F}}\ell }_\gamma $ , the class $[G/B]$ belongs to the image of $i_\ast $ . Further, by [Reference Carlsson and Oblomkov16, thm. 7.1], the map

(2.39) $$ \begin{align}DR\to H_{\bullet}({{\mathcal {F}}\ell}),\quad x\mapsto x\cdot [G/B]\end{align} $$

is injective. Hence, the $DR$ -submodule in $H_{\bullet }({{\mathcal {F}}\ell })$ generated by $[G/B]$ has dimension $(h+1)^r$ . We deduce that $\dim i_*(H_{\bullet }({{\mathcal {F}}\ell }_\gamma ))\geqslant (h+1)^r$ . By taking the duals and identifying the dual of the Borel-Moore homology groups $H_{\bullet }({{\mathcal {F}}\ell }_\gamma )$ , $H_{\bullet }({{\mathcal {F}}\ell })$ with the cohomology groups $H^{\bullet }({{\mathcal {F}}\ell }_\gamma )$ , $H^{\bullet }({{\mathcal {F}}\ell })$ , we deduce that

$$ \begin{align*}\dim i^*(H^{\bullet}({{\mathcal {F}}\ell}))\geqslant (h+1)^r.\end{align*} $$

Since the left $W_{\mathrm {ex}}$ -action on $H^{\bullet }({{\mathcal {F}}\ell })$ is trivial, we have

$$ \begin{align*}i^*(H^{\bullet}({{\mathcal {F}}\ell}))\subset H^{\bullet}({{\mathcal {F}}\ell}_\gamma)^{W_{\mathrm{ex}}}.\end{align*} $$

But by part (a), we know that $\dim H^{\bullet }({{\mathcal {F}}\ell }_\gamma )^{W_{\mathrm {ex}}}\leqslant (h+1)^r$ . Thus, we have

$$ \begin{align*}\dim H^{\bullet}({{\mathcal {F}}\ell}_\gamma)^{W_{\mathrm{ex}}}= (h+1)^r.\end{align*} $$

To prove our claim, by taking the duals, it is enough to prove that the formulas in [Reference Carlsson and Oblomkov16] yield a $DR$ -action that commutes with the restriction $H^{\bullet }({{\mathcal {F}}\ell })\to H^{\bullet }({{\mathcal {F}}\ell }_\gamma )$ . To do this, using the equivariant formality, we relate the $DR$ -action to the GDAHA-action on the equivariant cohomology spaces $H^{\bullet }_{U}({{\mathcal {F}}\ell })$ and $H^{\bullet }_{U}({{\mathcal {F}}\ell }_\gamma )$ , where $U\subset T\times \mathbb {C}^{\times }$ is a one-parameter subgroup that fixes $\gamma $ up to a scalar multiple and $\mathbb {C}^{\times }$ is the loop rotation.

The GDAHA-action commutes with the restriction map $H^{\bullet }_{U}({{\mathcal {F}}\ell })\to H^{\bullet }_{U}({{\mathcal {F}}\ell }_\gamma )$ , see, e.g., [Reference Oblomkov and Yun35]. To compare the $DR$ -action with the GDAHA-action, we must explicitly identify the latter. The GDAHA-action is generated by the operators $x_i$ of multiplication by first Chern classes of line bundles associated with affine simple roots $\alpha _i\in \Sigma _{\mathrm {af}}$ , by a modified version of the Lusztig $W_{\mathrm {af}}$ -action and by a further symmetry $\rho $ that comes from the action of the extended affine Weyl group which is not contained in $W_{\mathrm {af}}$ .

The action of a simple reflection $s_i$ in $W_{\mathrm {af}}$ , viewed as an element of the GDAHA, is related to the standard action $\bar s_i$ that permutes the $x_j$ ’s as in [Reference Carlsson and Oblomkov16, (5.12), (6.11)], i.e., we have

(2.40) $$ \begin{align}s_i=\bar s_i+\nu\frac{1}{x_i-x_{i+1}}(1-\bar s_i)\end{align} $$

where $\nu $ is the action of the central parameter of the GDAHA. The $DR$ -action is defined in [Reference Carlsson and Oblomkov16, §7.1] using the GDAHA-action of elements $\psi _i$ , $\rho $ in the extended affine Weyl group defined in [Reference Carlsson and Oblomkov16, §5.1]. In [Reference Carlsson and Oblomkov16] the element $\gamma $ is elliptic and the central parameter is $\nu =m/n$ where m, n are relatively prime integers depending on the choice of $\gamma $ . In our case $\gamma $ is split, corresponding to $\nu =1$ and $m=n$ . In both cases, the formula (2.40) holds. Checking that the $DR$ -action on $H^{\bullet }({{\mathcal {F}}\ell })$ descends to $H^{\bullet }({{\mathcal {F}}\ell }_\gamma )$ reduces to checking that the operators $s_i$ and $\rho $ descend. This follows from the definition of the GDAHA-action, as mentioned above. This can also be checked by a direct computation using the GKM presentation of both sides of $H^{\bullet }_{U}({{\mathcal {F}}\ell })\to H^{\bullet }_{U}({{\mathcal {F}}\ell }_\gamma )$ using Lemma 2.7. For instance, the action of $\rho $ on the GKM presentation is given by the conjugation by an element in the extended affine Weyl group and it also preserves the relations in Lemma 2.7.

The proof above shows that in type A, we have $i^*(H^{\bullet }({{\mathcal {F}}\ell }))=H^{\bullet }({{\mathcal {F}}\ell }_\gamma )^{W_{\mathrm {ex}}}$ . We obtain the following

Using Proposition 2.9, Boixeda Alvarez and Losev proved the following stronger result.

Let $e_1,\dots ,e_r$ be the exponents of the group W. We will prove the following.

Theorem 2.12. Assume that ${\check G}$ is simply connected. We have

(2.41) $$ \begin{align} \dim H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell})^{W_{\ell,{\mathrm{ex}}}}= \frac{1}{|W|}\prod_{i=1}^r\big((h+1)\ell-h+e_i\big). \end{align} $$

Remark 2.13. In type $SL_{r+1}$ it is conjectured in [Reference Lachowska and Qi31] that

$$ \begin{align*}\dim Z(\mathrm{u}_\zeta)=\frac{1}{(h+1)\ell}\begin{pmatrix}(h+1)\ell\\h\end{pmatrix}.\end{align*} $$

For $G=PSL_{r+1}$ the formula (2.41) gives

$$ \begin{align*}\dim H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell})^{W_{\ell,{\mathrm{ex}}}}= \frac{1}{(r+1)!}\prod_{i=1}^r\big((h+1)\ell-i\big) =\frac{1}{(h+1)\ell}\begin{pmatrix}(h+1)\ell\\h\end{pmatrix}. \end{align*} $$

2.3.4 Relation with the Lusztig elliptic Springer fibres

Let $\gamma \in {\mathfrak g}\otimes \mathcal {K}$ be an arbitrary regular semi-simple element. Fix a proper subset $J\subset \Sigma _{\mathrm {af}}$ . Let ${{\mathcal {F}}\ell }^J_\gamma $ and ${}^0\!{{\mathcal {F}}\ell }^J_\gamma $ be the affine Steinberg and Spaltenstein fibres at $\gamma $ as in §2.3.1. Let $1_J$ , $\varepsilon _J$ be the trivial and signature idempotents in $\mathbf {k} W_J$ . The cohomology of ${}^0\!{{\mathcal {F}}\ell }^J_\gamma $ is given by the following proposition. Set $d_J=\dim {{\mathcal {F}}\ell }-\dim {{\mathcal {F}}\ell }^J$ .

Proposition 2.14. There is a commutative diagram

where $i^*$ is the restriction and $p_*$ is the Gysin map. The map $q_*$ restricts to a graded vector space isomorphism from the space $H^{\bullet }({{\mathcal {F}}\ell }_\gamma )\cdot \varepsilon _J$ of anti-invariants for the right $W_J$ -action to $H^{\bullet }({}^0\!{{\mathcal {F}}\ell }^J_\gamma )[-2d_J].$

Proof. The proof uses a sheaf-theoretic interpretation of the affine Springer representation, which goes back to the work of G. Lusztig. We will use Yun’s approach, see, e.g., [Reference Yun42, §2.3]. We have the following commutative diagram of stacks with Cartesian squares

where $B_J=G_J\cap \mathcal {I}$ and $\mathfrak {n}_J$ is the nilpotent radical of its Lie algebra. The map $q_2$ maps $xP^J/P^J$ to the image of $\mathrm {Ad}_x^{-1}(\gamma )$ in $\mathfrak {g}_J$ for each $x\in G(\mathcal {K})$ . The map $q_1$ is defined similarly. By base change, we have

$$ \begin{align*}H^{\bullet}({{\mathcal {F}}\ell}_\gamma)=H^{\bullet}({{\mathcal {F}}\ell}^J_\gamma,(\pi_2)_!q_1^*(\underline{\mathbf{k}})) =H^{\bullet}({{\mathcal {F}}\ell}^J_\gamma,q_2^*(\pi_1)_!(\underline{\mathbf{k}})).\end{align*} $$

The map $\pi _1$ is semi-small. Thus the complex $(\pi _1)_!(\underline {\mathbf {k}})$ is semi-simple and contains the direct summand $(i_1)_!(\underline {\mathbf {k}})[-2d_J]$ . Thus, by base change, the projection onto this direct summand yields a map

$$ \begin{align*}q_*:H^{\bullet}({{\mathcal {F}}\ell}_\gamma)\to H^{\bullet}({{\mathcal {F}}\ell}^J_\gamma,q_2^*(i_1)_!(\underline{\mathbf{k}}))[-2d_J]\end{align*} $$

Note that the right-hand side is equal to $H^{\bullet }({{\mathcal {F}}\ell }^J_\gamma ,(i_2)_!(\underline {\mathbf {k}}))[-2d_J]$ , which is the same as $H^{\bullet }({}^0\!{{\mathcal {F}}\ell }^J_\gamma )[-2d_J].$ Further, the Springer theory for the group $G_J$ yields an algebra homomorphism

$$ \begin{align*}\mathbf{k}[W_J]\to\mathrm{End}_{D^b({{\mathcal {F}}\ell}^J)}((\pi_1)_!(\underline{\mathbf{k}}))\end{align*} $$

which maps the sign idempotent to the projection to the direct summand

$$ \begin{align*}(i_1)_!(\underline{\mathbf{k}})[-2d_J]\subset(\pi_1)_!(\underline{\mathbf{k}}).\end{align*} $$

So the map $q_*$ satisfies the properties in the proposition.

Let $\text {ht}(\alpha )$ be the height of a root $\alpha $ . Let $e_{\alpha }\in \mathfrak {g}$ be a non-zero root vector of weight $\alpha \in \Phi $ . Assume that $(h,n)=1$ . G. Lusztig has defined for each positive integer n a regular semi-simple element $l_n$ such that

$$ \begin{align*}l_n=\sum_{hk-\text{ht}(\alpha)=n}t^ke_{\alpha},\end{align*} $$

where the sum is over all pairs $(k,\alpha )\in \mathbb {Z}\times \Phi $ as above. Assume that ${\check G}$ is simply connected. Then $l_n$ is known to be elliptic and, by [Reference Sommers38, thm. 7.3], we have

(2.42) $$ \begin{align} H^{\bullet}({{\mathcal {F}}\ell}_{l_n})=\mathbf{k}[{\check{\Lambda}}/n{\check{\Lambda}}] \end{align} $$

as $W_J$ -modules for the right action. We will abbreviate $\gamma '=\gamma ^{\prime }_1=l_{h+1}$ . Let $\gamma =ts$ as before. From Propositions 2.11, 2.14 and (2.42) we deduce the following.

We can now complete the proof of Theorem 2.12.

Proof of Theorem 2.12

We set

$$ \begin{align*}\gamma^{\prime}_\ell=\sum_{hk-\text{ht}(\alpha)=h+1}t^{\ell k}e_{\alpha}.\end{align*} $$

First, assume that $(h,\ell )=1$ . A regular semi-simple element $a\in \mathfrak {g}\otimes \mathcal {K}$ is said to be homogeneous of slope $d/m\in \mathbb {Q}$ if $s^da(t)$ and $a(s^mt)$ are $G(\mathcal {K})$ -conjugate for any $s\in \mathbb {C}^{\times }$ . By [Reference Oblomkov and Yun35, cor. 5.4.5], if a, $a'$ have the same slope, then $H^{\bullet }({{\mathcal {G}}{\mathfrak {r}} }_a)\cong H^{\bullet }({{\mathcal {G}}{\mathfrak {r}} }_{a'})$ . The regular semi-simple elements $t^{-1}\gamma ^{\prime }_\ell $ and $l_{(h+1)\ell -h}$ are elliptic and homogeneous of slope $\ell (h+1)/h-1$ . Hence, we have $H^{\bullet }({{\mathcal {G}}{\mathfrak {r}} }_{t^{-1}\gamma ^{\prime }_\ell })\cong H^{\bullet }({{\mathcal {G}}{\mathfrak {r}} }_{l_{(h+1)\ell -h}})$ . Let $e_1^J,\dots ,e_j^J$ be the exponents of the group $W_J$ . By [Reference Sommers38, prop. 6.1], we have

$$ \begin{align*} \dim H^{\bullet}({{\mathcal {F}}\ell}^J_{l_n})=(n+e^J_1)\dots(n+e^J_j)n^{r-j}\,/\,|W_J|. \end{align*} $$

We deduce that

(2.43) $$ \begin{align} \begin{aligned} \dim H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }_{\gamma^{\prime}_\ell})&= \dim H^{\bullet}({{\mathcal{G}}{\mathfrak{r}} }_{t^{-1}\gamma^{\prime}_\ell})\\ &=\dim H^{\bullet}({{\mathcal{G}}{\mathfrak{r}} }_{l_{(h+1)\ell-h}})\\ &=\frac{1}{|W|}\prod_{i=1}^r\big((h+1)\ell-h+e_i\big). \end{aligned} \end{align} $$

Since ${\check G}$ is simply connected, we have ${\check \rho }\in {\check {\Lambda }}$ . Further, the action of the element $(z^{-\ell {\check \rho }},z^h)$ in $T\times \mathbb {G}_m$ takes $\gamma ^{\prime }_\ell $ to $z^{(h+1)\ell }\gamma ^{\prime }_\ell $ . Hence $\gamma ^{\prime }_\ell $ is fixed by the $\mu _\ell $ -action by loop rotation, and we may consider both projective varieties ${}^0{{\mathcal {G}}{\mathfrak {r}} }_{\gamma ^{\prime }_\ell }$ and ${}^0{{\mathcal {G}}{\mathfrak {r}} }_{\gamma ^{\prime }_\ell }^\zeta $ . Since $(h,\ell )=1$ , the $\mu _\ell $ -action on ${}^0{{\mathcal {G}}{\mathfrak {r}} }$ by loop rotation extends to a $\mathbb {G}_m$ -action which preserves ${}^0{{\mathcal {G}}{\mathfrak {r}} }_{\gamma ^{\prime }_\ell }$ . By [Reference Goresky, Kottwitz and MacPherson22, thm. 1.2], the variety ${}^0{{\mathcal {G}}{\mathfrak {r}} }_{\gamma ^{\prime }_\ell }$ has a $\mathbb {G}_m$ -equivariant affine paving. Therefore, the Grothendieck group $K_{\mathbb {G}_m}({}^0{{\mathcal {G}}{\mathfrak {r}} }_{\gamma ^{\prime }_\ell })$ is free as an $R_{\mathbb {G}_m}$ -module, the Chern character yields isomorphisms of $\mathbf {k}$ -vector spaces

$$ \begin{align*}H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma^{\prime}_\ell})=K({}^0{{\mathcal{G}}{\mathfrak{r}} }_{\gamma^{\prime}_\ell}^\zeta), \quad H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }_{\gamma^{\prime}_\ell})=K({}^0{{\mathcal{G}}{\mathfrak{r}} }_{\gamma^{\prime}_\ell}),\end{align*} $$

and for any closed subgroup $H\subset \mathbb {G}_m$ , we have

$$ \begin{align*}K_H({}^0{{\mathcal{G}}{\mathfrak{r}} }_{\gamma^{\prime}_\ell})= K_{\mathbb{G}_m}({}^0{{\mathcal{G}}{\mathfrak{r}} }_{\gamma^{\prime}_\ell})\otimes_{R_{\mathbb{G}_m}}R_H.\end{align*} $$

Thus, the localization theorem in K-theory implies that

(2.44) $$ \begin{align}\dim H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma^{\prime}_\ell})= \dim K({}^0{{\mathcal{G}}{\mathfrak{r}} }_{\gamma^{\prime}_\ell}^\zeta)=\dim K({}^0{{\mathcal{G}}{\mathfrak{r}} }_{\gamma^{\prime}_\ell})=\dim H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }_{\gamma^{\prime}_\ell}). \end{align} $$

Next, from Lemma 2.6(a) we deduce that

$$ \begin{align*} H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell})^{W_{\ell,{\mathrm{ex}}}} &=\bigoplus_{\omega\in \Xi}H^{\bullet}({}^0{{\mathcal {F}}\ell}^{\omega}_{\ell,\gamma_\ell})^{W_{\ell,{\mathrm{ex}}}} =\bigoplus_{\omega\in \Xi}H^{\bullet}({}^0{{\mathcal {F}}\ell}^{\omega}_{\gamma})^{W_{\mathrm{ex}}} \end{align*} $$

The same proof with $\gamma _\ell $ replaced by $\gamma ^{\prime }_\ell $ implies that

$$ \begin{align*} H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma^{\prime}_\ell}) &=\bigoplus_{\omega\in \Xi}H^{\bullet}({}^0{{\mathcal {F}}\ell}^{\omega}_{\ell,\gamma^{\prime}_\ell}) =\bigoplus_{\omega\in \Xi}H^{\bullet}({}^0{{\mathcal {F}}\ell}^{\omega}_{\gamma'}). \end{align*} $$

Thus, Corollary 2.15 implies that

(2.45) $$ \begin{align} H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell})^{W_{\ell,{\mathrm{ex}}}}=H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma^{\prime}_\ell}). \end{align} $$

From (2.43), (2.44) and (2.45) we deduce that if $(h,\ell )=1$ then

(2.46) $$ \begin{align} \dim H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell})^{W_{\ell,{\mathrm{ex}}}}= \frac{1}{|W|}\prod_{i=1}^r\big((h+1)\ell-h+e_i\big). \end{align} $$

Now, we claim that the integer $\dim H^{\bullet }({}^0{{\mathcal {G}}{\mathfrak {r}} }^\zeta _{\gamma _\ell })^{W_{\ell ,{\mathrm {ex}}}}$ is a polynomial in $\ell $ . Thus the equality (2.46) holds for any $\ell $ . To prove the claim, recall first that

$$ \begin{align*}\dim H^{\bullet}({}^0{{\mathcal{G}}{\mathfrak{r}} }^\zeta_{\gamma_\ell})^{W_{\ell,{\mathrm{ex}}}}= \sum_{\omega\in\Xi}\dim H^{\bullet}({}^0{{\mathcal {F}}\ell}^{\omega}_{\ell,\gamma_\ell})^{W_{\ell,{\mathrm{ex}}}}.\end{align*} $$

Since the dimension of $H^{\bullet }({}^0{{\mathcal {F}}\ell }^{\omega }_{\ell ,\gamma _\ell })^{W_{\ell ,{\mathrm {ex}}}}$ does not depend on $\ell $ , we must check that for each subset $J\subset \Sigma _{\mathrm {af}}$ , the cardinality of the set $\{\omega \in \Xi \,;\,J_{\omega }=J\}$ is a polynomial in $\ell $ . We have $\Xi _{\mathrm {sc}}\cong \bar A_{\ell ,1}\cap {\check {\Lambda }}$ and $\pi _1$ acts freely on $\Xi _{\mathrm {sc}}$ with the quotient equal to $\Xi $ , see §2.1.3. Hence, it is enough to check that the cardinality of the set

$$ \begin{align*}\{\mu\in\mathfrak{t}_{\mathbb{R}}\,;\, 0\leqslant \langle\alpha,\mu\rangle\leqslant \ell\,,\,\forall\alpha\in\Phi^+\,,\, \mu\ \text{of\ type}\ J\}\cap{\check{\Lambda}}\end{align*} $$

is a polynomial in $\ell $ . This number is given by the Ehrhart polynomial of the interior of the face of type J of the alcove (which is an integral polytope), proving the claim.

3.1.1 Quantum groups

Let q be a formal variable. We abbreviate $q_i=q^{d_i}$ for each i. Let the field $\mathbf {F}$ be as in (1.1), with the integer e equal to the order of the group ${\check {\Lambda }}/{\check Q}$ , see §2.1.1. The quantum group $\mathscr {U}_q$ associated with ${\check G}$ is the $\mathbf {F}$ -algebra generated by $E_i$ , $F_i$ , $K_\lambda $ for $i\in I$ , $\lambda \in {\check {\Lambda }}$ , subject to the quantum Chevalley relations and

$$ \begin{align*} &K_\lambda K_\mu=K_{\lambda+\mu}, \quad K_0=1,\quad K_i=K_{{\check\alpha}_i},\\ &K_\lambda E_i K_{-\lambda}=q_i^{(\lambda,\alpha_i)} E_i=q^{(\lambda,{\check\alpha}_i)} E_i,\quad K_\lambda F_i K_{-\lambda}=q_i^{-(\lambda,\alpha_i)} F_i=q^{-(\lambda,{\check\alpha}_i)} F_i,\\ &[E_i, F_j]=\delta_{i,j}\frac{K_i-K_i^{-1}}{q_i-q^{-1}_i}. \end{align*} $$

Let $\mathscr {U}_q^+$ , $\mathscr {U}_q^0$ , $\mathscr {U}_q^-$ be respectively subalgebras generated by $E_i$ ’s, $K_\lambda $ ’s, $F_i$ ’s. Multiplication yields a canonical isomorphism of vector spaces $\mathscr {U}_q^-\otimes \mathscr {U}_q^0\otimes \mathscr {U}_q^+= \mathscr {U}_q.$ Given any weight $\mu \in {\check {\Lambda }}$ , let

$$ \begin{align*}\varepsilon_\mu:\mathscr{U}_q^0\to\mathbf{F}\end{align*} $$

be the algebra homomorphism taking $K_\lambda $ to $q^{(\lambda ,\mu )}$ for each $\lambda \in {\check {\Lambda }}$ .

3.1.2 Integral forms

For each $m\in \mathbb {Z}$ and $n\in \mathbb {N}$ we introduce the following elements in $\mathscr {U}_q^0$