2 The formal Peterson subalgebra

In this section, we recall the definition of a small torus FADA and the formal Peterson algebra following [Reference Calmès, Zainoulline and Zhong2, Reference Zhong18].

Given an oriented algebraic cohomology theory $h(\text {-}),$ in the sense of Levine–Morel (see [Reference Levine and Morel15]) there is an associated formal group law F over a commutative ring R with characteristic 0. Here, $R=h(pt)$ is the coefficient ring, and F is defined from the Quillen formula for the characteristic class of a tensor product of line bundles. For example, for connective K-theory (see, e.g., [Reference Richmond and Zainoulline17]) we have $F_\beta (x,y)=x+y-\beta xy$ over the polynomial ring $R=\mathbb {Z}[\beta ]$ . Specializing to $\beta =1$ (resp. $\beta =0$ ), one obtains the usual K-theory (resp. cohomology). In these notes, by usual cohomology, we always mean its algebraic part: the Chow ring (modulo rational equivalence) with rational coefficients.

Given a lattice $\Lambda $ (free abelian group of finite rank) and a formal group law F, consider the associated formal group algebra S of [Reference Calmés, Petrov and Zainoulline1] that is the quotient of the power series ring

$$\begin{align*}S=R[\![\Lambda]\!]_F=R[\![x_\lambda \colon \lambda\in \Lambda]\!]/\mathcal{J}_F, \end{align*}$$

where $\mathcal {J}_F$ is the closure of the ideal of relations

$$\begin{align*}(x_0, x_{\lambda_1+\lambda_2}-F(x_{\lambda_1}, x_{\lambda_2})\colon \lambda_1,\lambda_2\in \Lambda). \end{align*}$$

In the case $F=F_\beta ,$ we set S to be the quotient of the polynomial ring

$$\begin{align*}S:=R[x_\lambda\colon \lambda\in \Lambda]/\langle x_0,x_{\lambda_1+\lambda_2}-F_\beta(x_{\lambda_1}, x_{\lambda_2})\rangle. \end{align*}$$

Let $\Phi $ be a finite irreducible root system with a fixed subset $I=\{\alpha _1,\ldots ,\alpha _n\}$ of simple roots. Let Q and $Q^{\vee }$ denote the root and the coroot lattice, respectively. Let W denote the Weyl group, generated by simple reflections $s_{\alpha _i}$ , $\alpha _i\in I$ . Consider an affine root system corresponding to the extended Dynkin diagram for $\Phi $ with the extra simple root

$$\begin{align*}\alpha_0=-\theta+\delta\in Q\oplus \mathbb{Z}\delta, \end{align*}$$

where $\theta \in \Phi $ is the highest root and $\delta $ is the so called null root so that $s_{\alpha _0}= t_{\theta ^{\vee }} s_\theta $ is an extra generator of the respective affine Weyl group $W_{\mathrm {a}}= Q^{\vee } \rtimes W$ . Recall that the latter is generated by reflections $s_{\alpha +k\delta }= t_{-k\alpha ^{\vee }} s_\alpha $ , where $s_\alpha \in W$ is a reflection and $t_\lambda $ , $\lambda \in Q^{\vee }$ is a translation. The affine Weyl group $W_{\mathrm {a}}$ acts on the lattice Q via W that is

$$\begin{align*}t_\lambda w(\mu)=w(\mu), ~\mu\in Q, ~w\in W, ~\lambda\in Q^{\vee}. \end{align*}$$

Therefore, it also acts on the formal group algebra $S=R[\![Q]\!]_F$ .

Suppose $x_\alpha $ is a regular (not a zero-divisor) element in S for each $\alpha \in \Phi $ . In particular, this holds if $2$ is not a zero-divisor in R (see [Reference Calmès, Zainoulline and Zhong4, Lemma 2.2]). Let $\mathfrak {Q}=S[\tfrac {1}{x_\alpha }\colon \alpha \in \Phi ]$ be the localization of S at $x_\alpha $ s. Consider the twisted group algebra $\mathfrak {Q}_{W_{\mathrm {a}}}$ associated with the affine Weyl group $W_{\mathrm {a}}$ . By definition, it is a free left $\mathfrak {Q}$ -module $\mathfrak {Q}_{W_{\mathrm {a}}}=\mathfrak {Q}\otimes _RR[W_{\mathrm {a}}]$ with basis $\{\eta _u\}_{u\in W_{\mathrm {a}}}$ and the product given by

$$\begin{align*}c\eta_u\cdot c'\eta_{u'}=cu(c')\eta_{uu'}, ~c,c'\in \mathfrak{Q}, ~u,u'\in W_{\mathrm{a}}. \end{align*}$$

For each $\alpha \in \Phi $ , define elements

$$ \begin{align*} \kappa_\alpha &=\tfrac{1}{x_\alpha}+\tfrac{1}{x_{-\alpha}}\; \text{(which is an element of }S),\\ X_\alpha &=\tfrac{1}{x_\alpha}(1-\eta_{s_\alpha})\; \text{called the Demazure element},\\ Y_\alpha &=\kappa_\alpha-X_\alpha \;\text{called the push-pull element}, \\ X_{\alpha_0} &=\tfrac{1}{x_{-\theta}}(1-\eta_{s_0}),\; \text{and} \\ Y_{\alpha_0} &=\kappa_\theta-X_{\alpha_0}. \end{align*} $$

All these elements satisfy the quadratic relations (e.g., $X_\alpha ^2=\kappa _\alpha X_\alpha $ ) and the twisted braid relations (see, e.g., [Reference Calmès, Zainoulline and Zhong2]). For simplicity of notation, we will omit $\alpha $ or s in the indices, i.e., we will write $x_i=x_{\alpha _i}$ , $s_i=s_{\alpha _i}$ , $\eta _i=\eta _{s_{i}}$ , $X_i=X_{\alpha _i}$ , and $Y_i=Y_{\alpha _i}$ .

Similarly, consider the twisted group algebra $\mathfrak {Q}_{Q^{\vee }}=\mathfrak {Q}\otimes _RR[Q^{\vee }]$ . It is a free $\mathfrak {Q}$ -module with basis $\{\eta _{t_\lambda }\}_{\lambda \in Q^{\vee }}$ . Observe that $\mathfrak {Q}_{Q^{\vee }}$ is commutative since $t_\lambda (c)=c$ , $c\in \mathfrak {Q}$ . Consider two homomorphisms of left $\mathfrak {Q}$ -modules

$$ \begin{align*} \mathrm{pr}\colon&\ \mathfrak{Q}_{W_{\mathrm{a}}} \to \mathfrak{Q}_{Q^{\vee}}, ~c\eta_{t_\lambda w}\mapsto c\eta_{t_\lambda}, ~c\in \mathfrak{Q}, w\in W,\; \text{and}\\ \imath\colon&\ \mathfrak{Q}_{Q^{\vee}} \to \mathfrak{Q}_{W_{\mathrm{a}}}, ~c\eta_{t_\lambda}\mapsto c\eta_{t_\lambda}. \end{align*} $$

By definition, $\imath $ is a section of $\mathrm {pr}$ , and it is a ring homomorphism. Set $\psi =\imath \circ \mathrm {pr}$ , so $\psi |_{\imath (\mathfrak {Q}_{Q^{\vee }})}=\mathrm {id}$ . Define elements

$$\begin{align*}Z_\alpha=\tfrac{1}{x_{-\alpha}}(1-\eta_{t_{\alpha^{\vee}}}), ~\alpha\in \Phi. \end{align*}$$

Proof For $z=c\eta _u$ , $c\in \mathfrak {Q}$ , $u\in W_{\mathrm {a}}$ , and $\alpha _i\in I,$ we have

$$ \begin{align*} \mathrm{pr} (c\eta_u X_i)&=\mathrm{pr}\big(c\eta_u\tfrac{1}{x_{\alpha_i}}(1-\eta_{s_i})\big)\\ &=\mathrm{pr}\big(\tfrac{c}{u(x_{\alpha_i})} (\eta_u-\eta_{us_i})\big)=0. \end{align*} $$

As for the $X_0$ , we obtain

$$ \begin{align*} \psi(X_0)&=\imath\circ \mathrm{pr}(\tfrac{1}{x_{-\theta}}(1-\eta_{s_0}))\\ &=\tfrac{1}{x_{-\theta}}\imath\circ \mathrm{pr}((1-\eta_{t_{\theta^{\vee}}s_\theta}))\\ &=\tfrac{1}{x_{-\theta}}(1-\eta_{t_{\theta^{\vee}}})=Z_{\theta}, \end{align*} $$

and the proof is finished.

Denote $I_{\mathrm {a}}=\{\alpha _0,\ldots ,\alpha _n\}$ . Following [Reference Calmès, Zainoulline and Zhong2] define the small torus FADA $\mathbf {D}_{W_{\mathrm {a}}}$ to be the subring of $\mathfrak {Q}_{W_{\mathrm {a}}}$ generated by S and the elements $X_i$ , $\alpha _i\in I_{\mathrm {a}}$ . Set $\mathbf {D}_{W_{\mathrm {a}}/W}=\mathrm {pr}(\mathbf {D}_{W_{\mathrm {a}}})$ to be its image in $\mathfrak {Q}_{Q^{\vee }}$ . We called it the relative FADA. Assuming $\mathbb {Q} \subset R$ and using the small torus GKM descripition it is proven in [Reference Zhong18, Lemma 5.1] that the map $\imath $ induces a map $\mathbf {D}_{W_{\mathrm {a}}/W}\to \mathbf {D}_{W_{\mathrm {a}}}$ . We then define the formal Peterson subalgebra to be the image of the relative FADA

$$\begin{align*}\mathbf{D}_{Q^{\vee}}:=\imath(\mathbf{D}_{W_{\mathrm{a}}/W}).\end{align*}$$

According to [Reference Zhong18, Theorem 5.7] the formal Peterson subalgebra $\mathbf {D}_{Q^{\vee }}$ is a Hopf subalgebra in $\mathfrak {Q}_{Q^{\vee }}$ . Moreover, $\mathbf {D}_{Q^{\vee }}$ coincides with the centralizer $C_{\mathbf {D}_{W_{\mathrm {a}}}}(S)$ of the formal group algebra S in the FADA $\mathbf {D}_{W_{\mathrm {a}}}$ .

3 Properties of the FADA and the formal Peterson subalgebra

In the present section, we establish several properties of the FADA and the formal Peterson subalgebra. For the K-theory, some of these properties were proven in [Reference Kato7] using different arguments. We start with the following version of the projection formula.

Observe that for the K-theory the property (ii) played a key role in [Reference Kato7, Theorem 1.7].

Proof (i) Let $\xi =c_1\eta _{t_{\lambda _1}}$ and $z=c_2\eta _{t_{\lambda _2}w}$ , where $w\in W$ , $c_i\in \mathfrak {Q}$ , $\lambda _i\in Q^{\vee }$ . Then, we obtain

$$ \begin{align*} \mathrm{pr}(\imath(\xi)z)&=\mathrm{pr}(c_1\eta_{t_{\lambda_1}}c_2\eta_{t_{\lambda_2}w})=\mathrm{pr}(c_1t_{\lambda_1}(c_2)\eta_{t_{\lambda_1+\lambda_2}w})\\&=c_1t_{\lambda_1}(c_2)\eta_{t_{\lambda_1+\lambda_2}}=c_1\eta_{t_{\lambda_1}}c_2\eta_{t_{\lambda_2}}=\xi \mathrm{pr} (z). \end{align*} $$

(ii) Let $z=c\eta _{t_{\lambda }v}$ and $z'=c'\eta _{t_{\lambda '}v'}$ , $v,v'\in W$ . Then, we get

$$ \begin{align*} \mathrm{pr}(z\sigma z')&=\mathrm{pr}\big(c\eta_{t_{\lambda}v}\sum_{w\in W}\eta_w c'\eta_{t_{\lambda'}v'}\big)\\ &=\mathrm{pr}\big(\sum_{w\in W}ct_\lambda vw(c')\eta_{t_\lambda vwt_{\lambda'}v'}\big). \end{align*} $$

Since $t_\lambda vw t_{\lambda '} v'=t_\lambda (vwt_{\lambda '}(vw)^{-1}) vw v'=t_\lambda t_{vw(\lambda ')}vwv'$ , reindexing the sum by $w'=vw$ we obtain

$$ \begin{align*} \mathrm{pr}(z\sigma z') &=\mathrm{pr}(\sum_{w'\in W}ct_\lambda w'(c')\eta_{t_{\lambda}t_{w'(\lambda')}w'v'})\\ &=\sum_{w'\in W}ct_\lambda w'(c')\eta_{t_{\lambda}t_{w'(\lambda')}}. \end{align*} $$

On the other side, $\mathrm {pr}(z)=c\eta _{t_\lambda }$ and

$$ \begin{align*} \mathrm{pr}(\sigma z')&=\mathrm{pr}\big(\sum_w \eta_w c'\eta_{t_{\lambda'}v'}\big)=\mathrm{pr}\big(\sum_w w(c') \eta_{wt_{\lambda'}v'}\big)\\&=\mathrm{pr}\big(\sum_w w(c') \eta_{t_{w(\lambda')}wv'}\big)=\sum_w w(c') \eta_{t_{w(\lambda')}}. \end{align*} $$

The result then follows.

We now extend the Hecke action on the Peterson algebra for the K-theory introduced in [Reference Kato7, Section 2] to the action on the formal Peterson algebra.

We define an action of $\mathfrak {Q}_{W_{\mathrm {a}}}$ on $\mathfrak {Q}_{Q^{\vee }}$ by

$$\begin{align*}z\diamond \xi:=\mathrm{pr}(z\imath(\xi)), ~z\in \mathfrak{Q}_{W_{\mathrm{a}}}, ~\xi\in \mathfrak{Q}_{Q^{\vee}}. \end{align*}$$

More explicitly, we have

(3.1) $$ \begin{align} c\eta_{t_\lambda w}\diamond c'\eta_{t_{\lambda'}}=cw(c')\eta_{t_{\lambda+w(\lambda')}}, ~c,c'\in \mathfrak{Q}, ~w\in W. \end{align} $$

Direct computation shows that $W_{\mathrm {a}}$ is an action.

For $w\in W$ , $\xi \in \mathfrak {Q}_{Q^{\vee },}$ and $\alpha \in \Phi $ define

$$ \begin{align*} w(\xi)&=\eta_w\diamond \xi, \;\text{and}\\ \Delta_\alpha(\xi)&=X_\alpha\diamond \xi=\tfrac{1}{x_\alpha}(\xi-s_\alpha( \xi)). \end{align*} $$

We then have the following.

Proof (i) Let $z=c\eta _{t_{\lambda }w}$ and $z'=c'\eta _{t_{\lambda '}w'}$ . Then, we obtain

$$ \begin{align*} \mathrm{pr}(zz')&=\mathrm{pr}(c\eta_{t_{\lambda}w}c'\eta_{t_{\lambda'}w'})= \mathrm{pr}(ct_{\lambda}w(c')\eta_{t_{\lambda+w(\lambda')}ww'})\\ &=ct_{\lambda}w(c')\eta_{t_{\lambda+w(\lambda')}}=c\eta_{t_{\lambda}w}\diamond c'\eta_{t_{\lambda'}}. \end{align*} $$

(ii) For $\xi =c\eta _{t_\lambda ,}$ we get

$$ \begin{align*} X_0\diamond (c\eta_{t_\lambda})&=\tfrac{c}{x_{-\theta}}\eta_{t_\lambda}-\tfrac{s_0(c)}{x_{-\theta}}\eta_{t_{s_\theta(\lambda)}t_{\theta^{\vee}}}\\ &=\tfrac{c}{x_{-\theta}}\eta_{t_\lambda}-\tfrac{s_\theta(c)}{x_{-\theta}}\eta_{t_{s_\theta(\lambda)}t_{\theta^{\vee}}}-\tfrac{s_\theta(c)}{x_{-\theta}}\eta_{t_{s_\theta(\lambda)}}+\tfrac{s_\theta(c)}{x_{-\theta}}\eta_{t_{s_\theta(\lambda)}}\\ &=\Delta_{-\theta}(c\eta_{t_\lambda})+Z_{\theta}s_\theta(c\eta_{t_\lambda}) \end{align*} $$

and the result follows.

Proof Let $z,z'\in \mathbf {D}_{W_{\mathrm {a}}}$ , and let $\xi =\mathrm {pr}(z')$ . Then, we have

$$\begin{align*}z\diamond \xi=z\diamond \mathrm{pr} (z')=\mathrm{pr} (zz')\in \mathrm{pr}(\mathbf{D}_{W_{\mathrm{a}}})=\mathbf{D}_{W_{\mathrm{a}}/W} \end{align*}$$

and the lemma follows.

Identifying the formal Peterson algebra $\mathbf {D}_{Q^{\vee }}$ (resp. $\imath (\mathfrak {Q}_{Q^{\vee }})$ ) with $\mathbf {D}_{W_{\mathrm {a}}/W}$ (resp. $\mathfrak {Q}_{Q^{\vee }}$ ) via the ring homomorphism $\imath $ we obtain an action of $\mathbf {D}_{W_{\mathrm {a}}}$ on $\mathbf {D}_{Q^{\vee }}$ and an action of $\mathfrak {Q}_{W_{\mathrm {a}}}$ on $\imath (\mathfrak {Q}_{Q^{\vee }})$ . From this point on, we write $\xi $ as both an element in $\mathbf {D}_{W_{\mathrm {a}}/W}$ (resp. $\mathfrak {Q}_{Q^{\vee }}$ ) and in $\mathbf {D}_{Q^{\vee }}=\imath (\mathbf {D}_{W_{\mathrm {a}}/W})$ (resp. $\imath (\mathfrak {Q}_{Q^{\vee }})$ ). If we consider a product $\xi _1\xi _2$ with $\xi _i\in \mathfrak {Q}_{Q^{\vee }}$ , we may assume it is in $\mathfrak {Q}_{W_{\mathrm {a}}}$ . However, for the product $z\xi $ with $z\in \mathfrak {Q}_{W_{\mathrm {a}}}$ and $\xi \in \mathfrak {Q}_{Q^{\vee }}$ , we view $\xi $ as an element in $\mathfrak {Q}_{W_{\mathrm {a}}}$ via the map $\imath $ . Following these identifications we obtain

(3.2) $$ \begin{align} z\diamond \xi=\psi(z\xi),~ \text{where}\; \xi\in \mathbf{D}_{Q^{\vee}}\subset \imath(\mathfrak{Q}_{Q^{\vee}}),\, z\in \mathbf{D}_{W_{\mathrm{a}}}\subset \mathfrak{Q}_{W_{\mathrm{a}}}, \end{align} $$

and Lemma 3.1 gives

(3.3) $$ \begin{align} \psi(\xi z)=\xi \psi(z).\end{align} $$

Example 3.4 Consider the affine root system of extended Dynkin type $\hat A_2$ . It has three simple roots $\alpha _0,\alpha _1,\alpha _2$ and the highest root $\theta =\alpha _1+\alpha _2$ . Denote $X_{ij}=X_{i}X_j$ for simplicity. Direct computations then give:

$$ \begin{align*} \psi(X_{10}) &= X_1\diamond X_0=\tfrac{1}{x_1}Z_{\alpha_1+\alpha_2}-\tfrac{1}{x_1}Z_{\alpha_2},\\ \psi(X_{20}) &=\tfrac{1}{x_2}Z_{\alpha_1+\alpha_2}-\tfrac{1}{x_2}Z_{\alpha_1},\\ \psi(X_{210}) &=X_2\diamond \psi(X_{10}). \end{align*} $$

Finally, we describe the centre of FADA.

Lemma 3.5 (i) For any $\xi \in \mathfrak {Q}_{Q^{\vee }}$ and $\alpha _i\in I$ , we have

$$\begin{align*}\eta_i\diamond \xi=\xi~\Longleftrightarrow~\eta_i\xi=\xi \eta_i. \end{align*}$$

Moreover, if this condition is satisfied, we have

$$\begin{align*}c\eta_i\diamond(\xi\xi')= \xi(c\eta_i\diamond\xi'). \end{align*}$$

(ii) The centres of $\mathfrak {Q}_{W_{\mathrm {a}}}$ and $\mathbf {D}_{W_{\mathrm {a}}}$ can be described as follows:

$$ \begin{align*} Z(\mathfrak{Q}_{W_{\mathrm{a}}}) &=\{\xi\in \mathfrak{Q}_{Q^{\vee}}\colon\ \eta_w\xi=\xi \eta_w, \forall w\in W\}=(\mathfrak{Q}_{Q^{\vee}})^W,\\ Z(\mathbf{D}_{W_{\mathrm{a}}}) &=\{\xi\in \mathbf{D}_{Q^{\vee}}\colon\ \eta_w\xi=\xi \eta_w, \forall w\in W\}=(\mathbf{D}_{Q^{\vee}})^W. \end{align*} $$

(iii) There are ring homomorphisms

$$\begin{align*}\mathfrak{Q}_{W_{\mathrm{a}}}\to \mathrm{End}_{(\mathfrak{Q}_{Q^{\vee}})^W}(\mathfrak{Q}_{Q^{\vee}}), ~ \mathbf{D}_{W_{\mathrm{a}}}\to \mathrm{End}_{(\mathbf{D}_{Q^{\vee}})^W}(\mathbf{D}_{Q^{\vee}}), ~ z\mapsto z\diamond \text{--}. \end{align*}$$

Proof (i) For a given $\xi =\sum _\lambda c_{\lambda } \eta _{t_\lambda }$ , $c_\lambda \in \mathfrak {Q,}$ we get

$$ \begin{align*} \eta_i\xi&=\sum_{\lambda}s_i(c_{\lambda})\eta_i\eta_{t_\lambda}=\sum_\lambda s_i(c_{\lambda})\eta_{t_{s_i(\lambda)}}\eta_{i}\\&=\sum_{\lambda'} s_i(c_{s_i(\lambda')})\eta_{t_{\lambda'}}\eta_i, \text{ where }\lambda'=s_i(\lambda). \end{align*} $$

On the other side, we have

$$\begin{align*}\eta_i\diamond\xi =\psi(\eta_i\xi)=\psi(\eta_i\sum_\lambda c_\lambda \eta_{t_\lambda})=\sum_\lambda s_i(c_{s_i(\lambda)})\eta_{t_\lambda}. \end{align*}$$

Therefore, $\eta _i\diamond \xi =\xi $ if and only if $s_i(c_{s_i(\lambda )})=c_\lambda $ for any $\lambda \in Q^{\vee }$ , which is equivalent to say that $\eta _i\xi =\xi \eta _i$ .

Now if this condition is satisfied, then

$$\begin{align*}c\eta_i\diamond (\xi \xi')=\psi(c\eta_i\xi \xi')=\psi(\xi c\eta_i\xi')\overset{({3.3})} =\xi \psi(c\eta_i\xi')\overset{({3.2})} =\xi(c \eta_i\diamond \xi'). \end{align*}$$

(ii) Since $\mathfrak {Q}_{Q^{\vee }}=\imath (\mathfrak {Q}_{Q^{\vee }})=C_{\mathfrak {Q}_{W_{\mathrm {a}}}}(\mathfrak {Q})$ , we have $Z(\mathfrak {Q}_{W_{\mathrm {a}}})\subset \mathfrak {Q}_{Q^{\vee }}$ , and the first identity then follows. By part (i), we know that $\eta _i\diamond \xi =\xi $ , $\forall \alpha _i\in I$ is equivalent to $\eta _w \xi =\xi \eta _w$ , $\forall w\in W$ that is equivalent to $z\xi =\xi z$ , $\forall z\in \mathfrak {Q}_{W}$ . Since $\xi $ already commutes with $\eta _{t_\lambda }$ , $\lambda \in Q^{\vee }$ , $\xi $ belongs to the centre $Z(\mathfrak {Q}_{W_{\mathrm {a}}})$ . Conversely, if $\xi \in Z(\mathfrak {Q}_{W_{\mathrm {a}}})\cap \mathfrak {Q}_{Q^{\vee }}$ , then it is invariant under all $\eta _i$ , $\alpha _i\in I$ . The description of the centre $Z(\mathbf {D}_{W_{\mathrm {a}}})$ follows similarly.

(iii) Follows from parts (i) and (ii).

4 Borel isomorphisms

In this section, we study Borel isomorphisms involving the FADA and the formal Peterson subalgebra. We assume $\mathbb {Q} \subset R$ throughout this section.

Consider the left S-linear dual $\mathbf {D}_W^{*}$ embedded into $\mathfrak {Q}_W^{*}$ . The latter has a $\mathfrak {Q}$ -basis $\{f_w\}_{w\in W}$ . Following [Reference Calmès, Zainoulline and Zhong4, Section 11] there is an (equivariant) characteristic map

(4.1) $$ \begin{align} \mathfrak{c}\colon S\to \mathbf{D}_W^{*},\quad a\mapsto \sum_{w\in W}w(a)f_w \end{align} $$

which induces the Borel isomorphism (see [Reference Calmès, Zainoulline and Zhong4, Theorem 11.4])

(4.2) $$ \begin{align} \rho\colon S\otimes_{S^W}S\to \mathbf{D}_W^{*}, \quad a\otimes b\mapsto a\mathfrak{c}(b)=\sum_{w\in W}aw(b)f_w. \end{align} $$

Recall that $\sigma = \sum _{w\in W}\eta _{w}\in \mathbf {D}_{W_{\mathrm {a}}}$ . Denote $\mathbf {x}=\prod _{\alpha \in \Phi ^+}x_{-\alpha }$ and $Y = \sigma \tfrac {1}{\mathbf {x}}$ .

By [Reference Calmès, Zainoulline and Zhong3, Lemma 10.12] ( $Y=Y_\Pi $ ) we have $Y\in \mathbf {D}_W$ . We denote by $X_{I_u}$ , $Y_{I_u}$ products corresponding to a reduced sequence $I_u$ of $u\in W_{\mathrm {a}}$ .

Proof Observe that $Y=\tfrac {1}{|W|} \sigma Y$ , so $Y S\subset Y\mathbf {D}_{W}\subset \sigma Y \mathbf {D}_{W}\subset \sigma \mathbf {D}_W$ . Conversely,

$$\begin{align*}\sigma \mathbf{D}_W=Y\mathbf{x} \mathbf{D}_W\subset Y \mathbf{D}_W. \end{align*}$$

Note that $\mathbf {D}_W$ is also a right S module with basis $X_{I_v}, v\in W$ , and $Y X_{I_v}=\delta _{v,e} Y$ . So given $X_{I_v}b\in \mathbf {D}_{W}$ with $v\in W$ , $b\in S$ , we have

$$\begin{align*}Y X_{I_v} b=\delta_{v,e}Y b\in Y S. \end{align*}$$

So $\sigma \mathbf {D}_W\subset Y S$ , and the result follows.

Proof According to [Reference Calmès, Zainoulline and Zhong3, Lemma 10.3] $\mathbf {x} f_e\in \mathbf {D}_W^{*}$ . Let $\sum _i a_i\otimes b_i\in S\otimes _{S^W}S$ so that $\rho (\sum _i a_i\otimes b_i)=\mathbf {x} f_e$ . Then,

$$\begin{align*}\sum_i a_iw(b_i) = \begin{cases} \mathbf{x}, & w=e,\\ 0, & \text{otherwise}. \end{cases} \end{align*}$$

Therefore,

$$\begin{align*}\sum_i a_iY b_i = \sum_{w\in W} \sum_i a_i w(b_i)\eta_{w}\tfrac{1}{\mathbf{x}}=1. \end{align*}$$

Finally, by Lemma 4.1, for any $z\in \mathbf {D}_W,$ we can write $z=\sum _i a_iY b_iz=\sum _i a_iY b^{\prime }_i$ for some $b^{\prime }_i\in S$ .

Proof Let $z=c\eta _{t_{\lambda }u}$ , where $c\in \mathfrak {Q}$ , $u\in W$ . We have

$$ \begin{align*} \psi(\sigma z) & =\psi\Big( \sum_{w\in W} \eta_w c\eta_{t_{\lambda}u}\Big) = \psi\Big( \sum_{w\in W} w(c)\eta_{t_{w(\lambda)}}\eta_{wu}\Big) \\ &= \sum_{w\in W} w(c)\eta_{t_{w(\lambda)}}=\sum_{w\in W}\eta_w ct_\lambda \eta_{w^{-1}}, \end{align*} $$

so it obviously belongs to $Z(\mathbf {D}_{W_{\mathrm {a}}})$ .

As for the opposite inclusion, take $z\in Z(\mathbf {D}_{W_a})$ . Since $Z(\mathbf {D}_{W_{\mathrm {a}}})\subset C_{\mathbf {D}_{W_{\mathrm {a}}}}(S)=\mathbf {D}_{Q^{\vee }}$ , we get $\psi (z)=z$ . Observe that $\mathrm {pr}(z'\sigma )=|W|\mathrm {pr}(z')$ for any $z'\in \mathbf {D}_{W_{\mathrm {a}}}$ . So we obtain

$$ \begin{align*} \psi\big(\sigma z\tfrac{1}{|W|}\big)& = \psi\big(z\sigma \tfrac{1}{|W|}\big) =\psi(z) = z. \end{align*} $$

Thus, $z\in \psi (\sigma \mathbf {D}_{W_{\mathrm {a}}})$ , and the proof is finished.

Consider two ring homomorphisms induced by the usual mutiplication:

(4.3) $$ \begin{align} \Theta\colon S\otimes_{S^W} Z(\mathbf{D}_{W_{\mathrm{a}}}) & \longrightarrow \mathbf{D}_{Q^{\vee}}. \end{align} $$

(4.4) $$ \begin{align} \Xi\colon \mathbf{D}_{W}\otimes_{S^W} Z(\mathbf{D}_{W_{\mathrm{a}}}) & \longrightarrow \mathbf{D}_{W_{\mathrm{a}}}. \end{align} $$

Note that in the definition of $\Theta $ and $\Xi ,$ one can switch the tensor factors. Moreover, both homomorphisms are left S-linear. The following is our first main result.

Remark 4.5 The geometric interpretation of this theorem is well-known for equivariant homology and equivariant K-homology. As explained in [Reference Lam, Lee and Shimozono9, Reference Lam and Shimozono12], we have the following isomorphisms of algebras

$$\begin{align*}\mathbf{D}_W \simeq H^T_*(G/B),\qquad \mathbf{D}_{W_{\mathrm{a}}} \simeq H_*^{T}(Fl_G),\qquad \mathbf{D}_{Q^{\vee}} \simeq H_*^{T}(Gr_G),\end{align*}$$

where $G/B$ is the flag variety, $Fl_G$ is the affine flag variety, and $Gr_G$ is the affine Grassmannian. By (ii) of Lemma 3.5, we could identify

$$ \begin{align*}Z(\mathbf{D}_{W_{\mathrm{a}}})=(\mathbf{D}_{Q^{\vee}})^W = H_*^T(Gr_G)^W = H^G_*(Gr_G).\end{align*} $$

As a result, the morphisms $\Theta $ and $\Xi $ can be viewed as the isomorphisms of algebras

$$ \begin{align*} \Theta\colon H_T^{*}(pt)\otimes_{H_G^{*}(pt)}H^G_*(Gr_G)& \simeq H_*^T(Gr_G), \\ \Xi\colon H^T_*(G/B)\otimes_{H_G^{*}(pt)}H^G_*(Gr_G)& \simeq H_*^T(Fl_G). \end{align*} $$

The isomorphism $\Xi $ can be also rewritten in a more familiar form

$$\begin{align*}H^T_*(G/B)\otimes_{H_T^{*}(pt)}H^T_*(Gr_G) \simeq H_*^T(Fl_G) ,\end{align*}$$

which is what Corollary 4.8 implies.

For a general equivariant oriented cohomology theory h, it follows from [Reference Calmès, Zainoulline and Zhong5] that $\mathbf {D}_W^{*}$ is isomorphic to the equivariant oriented cohomology $h_T(G/B)$ . However, the respective results for $Fl_G$ and $Gr_G$ are not known since they are not varieties of finite type. Therefore, our isomorphism in this case serves as the algebraic analogs of the potentially-correct geometric result. Observe also that in general, the isomorphisms in Corollary 4.8 are only isomorphisms of modules.

The proof of the theorem will occupy the rest of this section. We start proving the surjectivity first.

Proof Consider the following diagram

Since $\psi $ is an S-module homomorphism, this diagram commutes.

By Lemma 4.2, we can write $1=\sum _i a_iY b_i$ for some $a_i$ , $b_i\in S$ . For any $z\in \mathbf {D}_{W_{\mathrm {a}}}$ , we then have $z=\sum _i a_i Y b_i z$ . This shows that elements of $Y \mathbf {D}_{W_{\mathrm {a}}}$ generate $\mathbf {D}_{W_{\mathrm {a}}}$ as a left S-module. Similarly to the proof of Lemma 4.1, we obtain that $Y \mathbf {D}_{W_{\mathrm {a}}}=\sigma \mathbf {D}_{W_{\mathrm {a}}}$ . So the top horizontal map is surjective, and the result follows.

Proof Since the elements of $\mathbf {D}_W$ and of $Z(\mathbf {D}_{W_{\mathrm {a}}})$ commute with each others, the image of $\Xi $ is the subalgebra generated by $\mathbf {D}_W$ and $Z(\mathbf {D}_{W_{\mathrm {a}}})$ . It contains S and $X_i$ for $\alpha _i\in I$ by definition, so it suffices to show that it contains $X_0$ as well. Observe that

$$ \begin{align*} X_0 &= \tfrac{1}{x_{\theta}}(1-\eta_{s_{\theta}}\eta_{t_{-\theta^{\vee}}})\\ &= \tfrac{1}{x_{\theta}}(1-\eta_{s_{\theta}}) +\eta_{s_{\theta}}\tfrac{1}{x_{-\theta}}(1-\eta_{t_{-\theta^{\vee}}})\\ &=X_{\theta} + \eta_{s_{\theta}}\tfrac{x_{\theta}}{x_{-\theta}}Z_{\theta}. \end{align*} $$

Since $Z_{\theta }\in \mathbf {D}_{Q^{\vee }}$ by [Reference Zhong18, Lemma 4.1], we have $X_\theta \in \mathbf {D}_W$ and $\eta _{s_\theta }\in \mathbf {D}_W$ . So $X_0$ belongs to the subalgebra generated by $\mathbf {D}_W$ and $Z(\mathbf {D}_{W_{\mathrm {a}}})$ .

Proof Since $\mathbf {D}_{Q^{\vee }}\supset Z(\mathbf {D}_{W_{\mathrm {a}}})$ , by Lemma 4.7, both maps are surjective. To prove the injectivity, we change the base to $\mathfrak {Q}$ -modules by applying the exact functors $\text {--}\otimes _S\mathfrak {Q}$ and $\mathfrak {Q}\otimes \text {--}$ . It then suffices to show that the induced maps

$$\begin{align*}\mathbf{D}_W\otimes_{S}\mathbf{D}_{Q^{\vee}}\otimes_S\mathfrak{Q} = \mathfrak{Q}_W\otimes_{\mathfrak{Q}} \mathfrak{Q}_{Q^{\vee}} \longrightarrow \mathfrak{Q}_{W_{\mathrm{a}}} =\mathbf{D}_{W_{\mathrm{a}}}\otimes_{S} {\mathfrak{Q}} \end{align*}$$

$$\begin{align*}\mathfrak{Q}\otimes_S\mathbf{D}_{Q^{\vee}}\otimes_{S}\mathbf{D}_W = \mathfrak{Q}_{Q^{\vee}}\otimes_{\mathfrak{Q}} \mathfrak{Q}_{W} \longrightarrow \mathfrak{Q}_{W_{\mathrm{a}}} ={\mathfrak{Q}}\otimes_{S} \mathbf{D}_{W_{\mathrm{a}}} \end{align*}$$

are injective. But these are even isomorphisms. So, the conclusion follows.

We now discuss injectivity of the maps in the theorem.

For any parabolic subgroup $W_P$ of $W,$ we denote by $W^P$ the subset of minimal length left coset representatives. Consider the $\mathfrak {Q}$ -linear dual $\mathfrak {Q}^{*}_{W^P}=\mathrm {Hom}(W^P,\mathfrak {Q})$ with a basis $\{f_{w}\}_{w\in W^P}$ . One can also identify it with the invariants $(\mathfrak {Q}_W^{*})^{W_P}$ by identifying each $f_w$ , $w\in W^P$ with $\sum _{v\in W_P}f_{wv}\in (\mathfrak {Q}_W^{*})^{W_P}$ (see [Reference Calmès, Zainoulline and Zhong3, Section 11] for more details).

Lemma 4.9 The map $\rho _{P,\mathfrak {Q}}\colon S\otimes _{S^W}\mathfrak {Q}^{W_P}\to \mathfrak {Q}_{W^P}^{*}$ defined by

$$\begin{align*}\rho_{P,\mathfrak{Q}}(c_1\otimes c_2):=\sum_{w\in W^P}c_1w(c_2)f_w\end{align*}$$

is an isomorphism.

Proof Assume first that $P=B$ (the Borel case). Then, the map $\rho _{P,\mathfrak {Q}}$ is obtained from the isomorphism $\rho $ by the base change with the functor $\text {--}\otimes _S\mathfrak {Q}$ . So $\rho _{B,\mathfrak {Q}}$ is an isomorphism.

For a general parabolic $W_P,$ there is a commutative diagram

Both vertical maps identify the top with the $W_P$ -invariant subsets of the bottom, so the top horizontal map is an isomorphism.

Proof Let $z=\sum _{\lambda \in Q^{\vee }} c_{\lambda }\eta _{t_{\lambda }}\in Z(\mathbf {D}_{W_{\mathrm {a}}})\subset \mathbf {D}_{Q^{\vee }}$ with $c_\lambda \in \mathfrak {Q}$ . Since $\eta _u z=z\eta _u$ for any $u\in W$ , we have

(*) $$ \begin{align} \forall u\in W,\quad uc_{\lambda} =c_{u(\lambda)}. \end{align} $$

These properties give us an injective map:

$$\begin{align*}\phi: Z(\mathbf{D}_{W_{\mathrm{a}}})\to \bigoplus_{\lambda\in Q^{\vee}_{\geq0}}\mathfrak{Q}^{W_{\lambda}},\qquad \sum_{\lambda\in Q^{\vee}} c_{\lambda}\eta_{t_{\lambda}} \longmapsto (c_{\lambda})_{\lambda\in Q^{\vee}_{\geq0}}, \end{align*}$$

where $Q^{\vee }_{\ge 0}$ is the set of dominant coroots and $W_{\lambda }$ is the stabilizer of $\lambda $ , which is a parabolic subgroup of W.

Let $W^\lambda $ denote the set of minimal length representatives of the cosets $W/W_\lambda $ . Consider the following diagram

where $\Theta '$ is the direct sum of maps

$$ \begin{align*}S\otimes_{S^W} \mathfrak{Q}^{W_{\lambda}}\longrightarrow \bigoplus_{w\in W^{\lambda}} \mathfrak{Q}\eta_{t_{w(\lambda)}},\quad c_1\otimes c_2\longmapsto \sum_{w\in W^{\lambda}} c_1 w(c_2)\eta_{t_{w(\lambda)}},\end{align*} $$

for all $\lambda \in Q_{\geq 0}^{\vee }$ . Since by Lemma 4.9, each such component map is injective, so is $\Theta '$ .

By direct computations and by the property (*), the diagram is commutative. Since both maps $\operatorname {id}\otimes \phi $ and $\Theta '$ are injective, so is $\Theta $ .

Proof It follows from the combination of previous results:

$$ \begin{align*} \mathbf{D}_W\otimes_{S^W} Z(\mathbf{D}_{W_{\mathrm{a}}}) & \simeq \mathbf{D}_W\otimes_{S}S\otimes_{S^W} Z(\mathbf{D}_{W_{\mathrm{a}}})\\ & \simeq \mathbf{D}_W\otimes_S \mathbf{D}_{Q^{\vee}} & (\Theta \text{ is an isomorphism)}\\ & \simeq \mathbf{D}_{W_{\mathrm{a}}} & \text{(Corollary}~{4.8}). &\\[-34pt] \end{align*} $$

5 The $\hat A_1$ -case

In this section, we discuss an example of the formal Peterson subalgebra for the affine root system of type $\hat A_1$ . We show that it provides a natural model for “quantum” oriented cohomology of $\mathbb {P}^1$ .

Recall that a root system of type $\hat A_1$ has two simple roots, $\alpha _1=\theta =\alpha $ and $\alpha _0=-\alpha +\delta $ , and each $w\in W_{\mathrm {a}}$ has a unique reduced decomposition. We follow the notation of [Reference Lam, Schilling and Shimozono11, Section 4.3] and define for $i\ge 1$ :

$$ \begin{align*} \begin{array}{lcr@{\,=\,}lcr@{\,=\,}l} \sigma_0=e, & & \sigma_{2i} & (s_1s_0)^i=t_{-i\alpha^{\vee}}, & & \sigma_{2i+1} &s_0\sigma_{2i}, \\ & &\sigma_{-2i} &(s_0s_1)^i=t_{i\alpha^{\vee}}, & &\sigma_{-(2i+1)} &s_1\sigma_{-2i}. \end{array} \end{align*} $$

The set of minimal length coset representatives of $W_{\mathrm {a}}/W$ is then $W_{\mathrm {a}}^{-}=\{\sigma _i\colon i\ge 0\}$ .

The root lattice is $Q=\mathbb {Z} \alpha $ , and in the formal group algebra $S=R[\![Q]\!]_F,$ we have $x_n=x_{n\alpha }=n\cdot _F x_{\alpha }$ , where $n\cdot _F x$ , $n\in \mathbb {Z}$ is the n-fold formal sum (inverse) of x.

As for the Demazure elements, we have

$$ \begin{align*} X_j^2 &=\kappa_{\alpha_j} X_j, \; j=0,1,\;\text{ and }\\ \kappa_{\alpha_j} &=\kappa_\alpha=\tfrac{1}{x_\alpha}+\tfrac{1}{x_{-\alpha}}\text{ is }W_{\mathrm{a}}\text{-invariant}. \end{align*} $$

Set $\mu =-\frac {x_{-1}}{x_1}$ . Observe that if F is of the form $F(x,y)=\frac {x+y-\beta xy}{g(x,y)}$ for some power series $g(x,y)$ , we have $x_{-1}=\tfrac {x_1}{\beta x_1-1}$ , hence, $\mu =\tfrac {1}{1-\beta x_1}$ . Given a reduced expression $w=s_is_j\ldots $ , we will use the notation $Y_{ij\cdots }$ for $Y_w=Y_{I_w}$ . Denote $\mathfrak {X}_w=\mathrm {pr}(X_w)$ and $\mathfrak {Y}_w=\mathrm {pr}(Y_w)$ .

Example 5.1 Direct computations give:

$$ \begin{align*} \imath(\mathfrak{X}_0)&=\imath(\mathfrak{X}_{\sigma_1})=X_0+X_1-x_{-1}X_{01} \\ \imath(\mathfrak{X}_{10})&=\imath(\mathfrak{X}_{\sigma _2})=X_{10}+\mu X_{01} \\ \imath(\mathfrak{X}_{010})&=\imath(\mathfrak{X}_{\sigma _3})=X_{010}+X_{101}-x_{-1}X_{1010} \end{align*} $$

and

$$ \begin{align*} \mathfrak{Y}_0 &=\mathfrak{Y}_{\sigma_1}=\tfrac{1}{x_1}+\tfrac{1}{x_{-1}}\eta_{t_{\alpha^{\vee}}} \\ \mathfrak{Y}_{10} &=\mathfrak{Y}_{\sigma_2}=Y_1\diamond \mathfrak{Y}_0=\tfrac{2}{x_1x_{-1}}+\tfrac{1}{x_{-1}^2}\eta_{t_{\alpha^{\vee}}}+\tfrac{1}{x_1^2}\eta_{t_{-\alpha^{\vee}}} \\ \mathfrak{Y}_{010} &=\mathfrak{Y}_{\sigma_3}=\mathfrak{Y}_0\mathfrak{Y}_{10}=\tfrac{3}{x_1^2x_{-1}}+\tfrac{3}{x_1x_{-1}^2}\eta_{t_{\alpha^{\vee}}}+\tfrac{1}{x_1^3}\eta_{t_{-\alpha^{\vee}}}+\tfrac{1}{x_{-1}^3}\eta_{t_{2\alpha^{\vee}}}. \end{align*} $$

These computations also show that $\mathfrak {X}_{\sigma _{i}}$ , $i=1,2,3$ satisfy identities similar to those of [Reference Lam, Li, Mihalcea and Shimozono10, Lemma 3].

We now look at various products of elements $\mathfrak {Y}_w\in \mathbf {D}_{Q^{\vee }}$ .

Proof Observe that $\sigma _{2i}=s_1s_0s_1\cdots s_0$ , so $Y_{\sigma _{2i}}=Y_1Y_{\sigma _{2i-1}}=(1+\eta _1)\frac {1}{x_{-\alpha }}Y_{\sigma _{2i-1}}$ . By Lemma 3.1, we obtain

$$\begin{align*}\mathfrak{Y}_{w\sigma_{2i}}=\mathrm{pr}(Y_{w}Y_{\sigma_{2i}})= \mathrm{pr}(Y_w)\mathrm{pr}(Y_{\sigma_{2i}})=\mathfrak{Y}_w\mathfrak{Y}_{\sigma_{2i}} \end{align*}$$

and the result follows.

Lemma 5.3 For each $i\ge 1$ we have $\mathfrak {Y}_{\sigma _{2i}}\in (\mathbf {D}_{Q^{\vee }})^W$ , and for $j=0,1$

$$\begin{align*}Y_j\diamond \mathfrak{Y}_{w}= \begin{cases} \mathfrak{Y}_{s_iw}, &\text{if }s_jw>w,\\ \kappa_\alpha\mathfrak{Y}_{w}, &\text{ if }s_j<s_jw. \end{cases} \end{align*}$$

In particular, we have

(5.1) $$ \begin{align} Y_{\sigma_i}\diamond \mathfrak{Y}_0=Y_{\sigma_i}\diamond \mathfrak{Y}_{\sigma_1}= \begin{cases} \mathfrak{Y}_{\sigma_1}, &\text{if }i=0, \\ \kappa_\alpha\mathfrak{Y}_{\sigma_i}, &\text{ if }i>0,\\ \mathfrak{Y}_{\sigma_{-i+1}}, &\text{ if }i<0. \end{cases} \end{align} $$

Proof Since $\sigma _{2i}=s_1\sigma _{2i-1}$ , we get $Y_{\sigma _{2i}}=(1+\eta _1)\tfrac {1}{x_{-\alpha }}Y_{\sigma _{2i-1}}$ , which implies that $\eta _1Y_{\sigma _{2i}}=Y_{\sigma _{2i}}$ . By Lemma 3.1, we then obtain

$$\begin{align*}\eta_1\diamond \mathfrak{Y}_{\sigma_{2i}}=\eta_1\diamond\mathrm{pr}(Y_{\sigma_{2i}})= \mathrm{pr}(\eta_1Y_{\sigma_{2i}})=\mathrm{pr}(Y_{\sigma_{2i}})=\mathfrak{Y}_{\sigma_{2i}}, \end{align*}$$

therefore, $\mathfrak {Y}_{\sigma _{2i}}\in (\mathbf {D}_{Q^{\vee }})^W$ . Similarly, we obtain $Y_j\diamond \mathfrak {Y}_{w}=\mathrm {pr}(Y_jY_{w})$ , and the formula for the action follows.

Proof The first part follows from (5.1). For the second part, we have

$$\begin{align*}X_0\diamond \mathfrak{Y}_{\sigma_1}=(\kappa_\alpha-Y_0)\diamond\mathfrak{Y}_{\sigma_1}= \kappa_\alpha\mathfrak{Y}_{\sigma_1}-\kappa_\alpha\mathfrak{Y}_{\sigma_1}=0, \end{align*}$$

so $X_0\in \ker \pi $ .

Conversely, let $z=\sum _{i\ge 1}a_iY_{\sigma _i}+\sum _{j\ge 0}b_jY_{\sigma _{-j}}\in \ker \pi $ . We then obtain

$$ \begin{align*} z\diamond \mathfrak{Y}_{\sigma_1}&=(\sum_{i\ge 1}a_iY_{\sigma_i}+\sum_{j\ge 0}b_jY_{\sigma_{-j}})\diamond \mathfrak{Y}_{\sigma_1}\\& =\sum_{i\ge 1}a_i\kappa_\alpha \mathfrak{Y}_{\sigma_{i}}+\sum_{j\ge 0}b_j\mathfrak{Y}_{\sigma_{j+1}}\\ &=\sum_{i\ge 1}a_i\kappa_\alpha \mathfrak{Y}_{\sigma_i}+\sum_{k\ge 1}b_{k-1}\mathfrak{Y}_{\sigma_k}\\ &=\sum_{i\ge 1}(a_i\kappa_\alpha +b_{i-1})\mathfrak{Y}_{\sigma_i}. \end{align*} $$

Therefore, if $z\diamond \mathfrak {Y}_{\sigma _1}=0$ , then $b_{i-1}=-a_i\kappa _\alpha $ for all $i\ge 1$ , and we obtain

$$ \begin{align*} z&=\sum_{i\ge 1}a_iY_{\sigma_i}-\sum_{j\ge 0}\kappa_\alpha a_{j+1}Y_{\sigma_{-j}}\\&=\sum_{i\ge 1}a_iY_{\sigma_{1-i}}Y_0-\sum_{k\ge 1}\kappa_\alpha a_{k}Y_{\sigma_{1-k}}\\ &=\sum_{i\ge 1}a_iY_{1-i}(Y_0-\kappa_\alpha)\\ &=(\sum_{i\ge 1}a_iY_{1-i})(-X_0)\in \mathbf{D}_{W_{\mathrm{a}}} X_0. \end{align*} $$

From the identities of Example 5.1 it follows that

$$\begin{align*}\mathfrak{Y}_0^2=x_{-1}\mathfrak{Y}_{010}+\mu\mathfrak{Y}_{10}, \end{align*}$$

and we obtain the following presentation of the formal Peterson algebra in terms of generators and relations:

(5.2) $$ \begin{align} \mathbf{D}_{Q^{\vee}}\simeq S[\mathfrak{s},\mathfrak{t}]/(\mathfrak{s}^2-x_{-1}\mathfrak{s}\mathfrak{t}-\mu \mathfrak{t}), ~\mathfrak{Y}_0\mapsto \mathfrak{s}, \mathfrak{Y}_{10}\mapsto \mathfrak{t}. \end{align} $$

According to Lemma 5.2, we may define the localization

$$\begin{align*}\mathbf{D}_{Q^{\vee}\!,\mathrm{loc}}=\mathbf{D}_{Q^{\vee}}[\tfrac{1}{\mathfrak{Y}_{2i}}, i\ge 1]. \end{align*}$$

From (5.2), we then obtain our second main result.

Theorem 5.6 We have the following presentation

$$\begin{align*}\mathbf{D}_{Q^{\vee}\!,\mathrm{loc}}\simeq S[\mathfrak{t},{\mathfrak{t}}^{-1}][\mathfrak{s}]/(\mathfrak{s}^2-x_{-1}\mathfrak{s}\mathfrak{t}-\mu \mathfrak{t}).\end{align*}$$

In particular, the action of $\mathbf {D}_{W_{\mathrm {a}}}$ on $\mathbf {D}_{Q^{\vee }}$ extends to an action on the localization $\mathbf {D}_{Q^{\vee }\!,\mathrm {loc}}$ by

$$\begin{align*}z\diamond (\frac{\xi}{\mathfrak{Y}_{2i}})=\frac{z\diamond \xi}{z\diamond \mathfrak{Y}_{2i}},\quad i\ge 1,\, \xi\in \mathbf{D}_{Q^{\vee}}. \end{align*}$$

Observe that it is well-defined since for any $i,j\ge 1,$ we have

$$\begin{align*}z\diamond (\frac{\xi \mathfrak{Y}_{\sigma_{2j}}}{\mathfrak{Y}_{\sigma_{2{(i+j)}}}})=\frac{z\diamond (\xi\mathfrak{Y}_{\sigma_{2j}})}{z\diamond (\mathfrak{Y}_{\sigma_{2i}}\mathfrak{Y}_{\sigma_{2j}})}\overset{{5.3}}=\frac{(z\diamond \xi)\mathfrak{Y}_{\sigma_{2j}}}{(z\diamond \mathfrak{Y}_{\sigma_{2i}})\mathfrak{Y}_{\sigma_{2j}}}=\frac{z\diamond\xi}{z\diamond \mathfrak{Y}_{\sigma_{2i}}}=z\diamond \frac{\xi}{\mathfrak{Y}_{\sigma_{2i}}}. \end{align*}$$

It then follows from Corollary 5.4 that

Finally, by the result of [Reference Zhong18] $\mathbf {D}_{Q^{\vee }}$ is a Hopf algebra with coproduct defined by

$$\begin{align*}\triangle(a\eta_{t_\lambda})=a\eta_{t_\lambda}\otimes \eta_{t_\lambda}=\eta_{t_\lambda}\otimes a\eta_{t_\lambda}. \end{align*}$$

In our case, we obtain

$$ \begin{align*} \triangle (\mathfrak{Y}_0)&=\tfrac{1}{x_1}(1-\mu)+\mu(\mathfrak{Y}_0\otimes 1+1\otimes \mathfrak{Y}_0)+x_{-1}\mathfrak{Y}_0\otimes \mathfrak{Y}_0,\\ \triangle (\mathfrak{Y}_{10})&=\kappa_\alpha^2+(\tfrac{x_1}{x^2_{-1}}-\tfrac{1}{x_1})(1\otimes \mathfrak{Y}_0+\mathfrak{Y}_0\otimes 1)+(1+\tfrac{x_1^2}{x_{-1}^2})\mathfrak{Y}_0\otimes \mathfrak{Y}_0\\ &\quad +\tfrac{1}{\mu}(\mathfrak{Y}_{10}\otimes 1+1\otimes \mathfrak{Y}_{10})-\tfrac{x_1^2}{x_{-1}}(\mathfrak{Y}_0\otimes \mathfrak{Y}_{10}+\mathfrak{Y}_{10}\otimes \mathfrak{Y}_0)+x_1^2\mathfrak{Y}_{10}\otimes \mathfrak{Y}_{10}. \end{align*} $$

Example 5.9 In particular, for the cohomology we get

$$ \begin{align*} \triangle (\mathfrak{Y}_0)&= 1\otimes \mathfrak{Y}_0+\mathfrak{Y}_0\otimes 1-x_\alpha\mathfrak{Y}_0\otimes \mathfrak{Y}_0,\\ \triangle (\mathfrak{Y}_{10})&= 2\mathfrak{Y}_0\otimes \mathfrak{Y}_0+(1\otimes \mathfrak{Y}_{10}+\mathfrak{Y}_{10}\otimes 1)\\&\quad +x_\alpha (\mathfrak{Y}_0\otimes \mathfrak{Y}_{10}+\mathfrak{Y}_{10}\otimes \mathfrak{Y}_0)+x_\alpha^2 \mathfrak{Y}_{10}\otimes \mathfrak{Y}_{10}, \end{align*} $$

and for the K-theory (identifying $x_\alpha =1-e^{-\alpha }$ ) we get

$$ \begin{align*} \triangle (\mathfrak{Y}_0) &= -e^\alpha+e^\alpha(\mathfrak{Y}_0\otimes 1+1\otimes \mathfrak{Y}_0)+(1-e^\alpha)\mathfrak{Y}_0\otimes \mathfrak{Y}_0,\\ \triangle (\mathfrak{Y}_{10}) &= 1-(1+e^\alpha)(\mathfrak{Y}_0\otimes 1+1\otimes \mathfrak{Y}_0)+(1+e^{-2\alpha})\mathfrak{Y}_0\otimes \mathfrak{Y}_0\\ &\quad + e^{-\alpha}(\mathfrak{Y}_{10}\otimes 1+1\otimes \mathfrak{Y}_{10}) + (e^{-\alpha}-e^{-2\alpha})(\mathfrak{Y}_0\otimes \mathfrak{Y}_{10}+\mathfrak{Y}_{10}\otimes \mathfrak{Y}_0)\\ &\quad +(1-e^{-\alpha})^2\mathfrak{Y}_{10}\otimes \mathfrak{Y}_{10}. \end{align*} $$

6 The dual of the formal Peterson subalgebra

In this section, we study the dual of the formal Peterson subalgebra.

Consider the $\mathfrak {Q}$ -linear dual of the twisted group algebra $\mathfrak {Q}_{W_{\mathrm {a}}}^{*}=\mathrm {Hom}_{\mathfrak {Q}}(\mathfrak {Q}_{W_{\mathrm {a}}}, \mathfrak {Q})$ . It is generated by $f_w$ , $w\in W_{\mathrm {a}}$ . Following [Reference Lenart, Zainoulline and Zhong14] there are two actions of $\mathfrak {Q}_{W_{\mathrm {a}}}$ on the dual $\mathfrak {Q}_{W_{\mathrm {a}}}^{*}$ defined as follows:

$$\begin{align*}a\eta_w\bullet bf_v=bvw^{-1}(a)f_{vw^{-1}}\;\text{ and }\; a\eta_w \odot bf_v=aw(b)f_{wv}. \end{align*}$$

Indeed, the $\odot $ -action comes from left multiplication in $\mathfrak {Q}_{W_{\mathrm {a}}}$ , and the $\bullet $ -action comes from right multiplication. Observe that these two actions commute, which makes $\mathfrak {Q}_{W_{\mathrm {a}}}^{*}$ into a $\mathfrak {Q}$ - $\mathfrak {Q}$ -bimodule. Moreover, $(z\bullet f)(z')=f(zz')$ , $z,z'\in \mathfrak {Q}_{W_{\mathrm {a}}}$ , $f\in \mathfrak {Q}_{W_{\mathrm {a}}}^{*}$ .

We now define two tensor products.

The first one is the tensor product $\mathfrak {Q}_{W_{\mathrm {a}}}\otimes \mathfrak {Q}_{W_{\mathrm {a}}}$ of left $\mathfrak {Q}$ -modules that is

$$\begin{align*}az\otimes z'=z\otimes az', ~a\in \mathfrak{Q}, ~z,z'\in \mathfrak{Q}_{W_{\mathrm{a}}}.\end{align*}$$

There is a canonical map $\Delta \colon \mathfrak {Q}_{W_{\mathrm {a}}}\to \mathfrak {Q}_{W_{\mathrm {a}}}\otimes \mathfrak {Q}_{W_{\mathrm {a}}}$ given by $a\eta _w\mapsto a\eta _w\otimes \eta _w$ . This map defines a co-commutative coproduct structure on $\mathfrak {Q}_{W_{\mathrm {a}}}$ with the co-unit $\mathfrak {Q}\to \mathfrak {Q}_{W_{\mathrm {a}}}$ , $a\mapsto a\eta _e$ .

The second tensor product $\hat \otimes $ was introduced in [Reference Lanini, Xiong and Zainoulline13]. Here, we provide a different but equivalent definition:

$$\begin{align*}\mathfrak{Q}_{W_{\mathrm{a}}}\hat\otimes \mathfrak{Q}_{W_{\mathrm{a}}}:=\mathfrak{Q}_{W_{\mathrm{a}}}\times \mathfrak{Q}_{W_{\mathrm{a}}}/\langle (a\eta_w, b\eta_v)-(aw(b)\eta_w , \eta_v)\rangle. \end{align*}$$

Observe that $\mathfrak {Q}_{W_{\mathrm {a}}}\hat \otimes \mathfrak {Q}_{W_{\mathrm {a}}}$ is also a left $\mathfrak {Q}$ -module.

Similarly, we define:

$$\begin{align*}\mathfrak{Q}_{W_{\mathrm{a}}}^{*}\hat\otimes \mathfrak{Q}^{*}_{W_{\mathrm{a}}}=\mathfrak{Q}^{*}_{W_{\mathrm{a}}}\times \mathfrak{Q}_{W_{\mathrm{a}}}^{*}/\langle (af_w,bf_v)-(aw(b)f_w,f_v)\rangle. \end{align*}$$

By definition, there is an isomorphism of $\mathfrak {Q}$ -modules:

$$\begin{align*}\mathfrak{Q}_{W_{\mathrm{a}}}^{*}\hat\otimes \mathfrak{Q}_{W_{\mathrm{a}}}^{*}\simeq (\mathfrak{Q}_{W_{\mathrm{a}}}\hat\otimes \mathfrak{Q}_{W_{\mathrm{a}}})^{*}, \quad (af_w\hat\otimes bf_v)(c\eta_x\hat\otimes d\eta_y):=acw(bd)\delta_{w,x}\delta_{v,y}. \end{align*}$$

There is a left $\mathfrak {Q}$ -module homomorphism defined by the product structure of $\mathfrak {Q}_{W_{\mathrm {a}}}$ :

$$\begin{align*}m\colon \mathfrak{Q}_{W_{\mathrm{a}}}\hat\otimes\mathfrak{Q}_{W_{\mathrm{a}}}\longrightarrow \mathfrak{Q}_{W_{\mathrm{a}}},\quad z_1\hat\otimes z_2\mapsto z_1z_2, \end{align*}$$

whose dual is given by

$$\begin{align*}m^{*}\colon\mathfrak{Q}_{W_{\mathrm{a}}}^{*}\longrightarrow (\mathfrak{Q}_{W_{\mathrm{a}}}\hat\otimes \mathfrak{Q}_{W_{\mathrm{a}}})^{*}\simeq \mathfrak{Q}_{W_{\mathrm{a}}}^{*}\hat\otimes \mathfrak{Q}_{W_{\mathrm{a}}}^{*}, \quad m^{*}(cf_w)=\prod_{u}cf_u\hat\otimes f_{u^{-1}w}. \end{align*}$$

Indeed, given any element $a\eta _u\hat \otimes b\eta _v=au(b)\eta _u\hat \otimes \eta _v$ , we have

$$ \begin{align*} m^{*}(cf_w)(a\eta_u\hat\otimes b\eta_v)&=cf_w(au(b)\eta_u\eta_v)=cf_w(au(b)\eta_{uv})\\ & =\eta_{uv,w}cau(b)=(\prod_{u}cf_u\hat\otimes f_{u^{-1}w})(a\eta_u\hat\otimes b\eta_v). \end{align*} $$

Recall (see also [Reference Zhong18, Section 1.7]) that there is the Borel map defined via the characteristic map

$$\begin{align*}\rho\colon \mathfrak{Q}\otimes_{\mathfrak{Q}^{W_{\mathrm{a}}}}\mathfrak{Q}\longrightarrow \mathfrak{Q}_{W_{\mathrm{a}}}^{*},\quad a\otimes b\mapsto a \mathbf{c}(b)=\prod_{w\in W_{\mathrm{a}}}aw(b)f_w. \end{align*}$$

Similar to [Reference Lanini, Xiong and Zainoulline13] one obtains the following commutative diagram:

Definition 6.1 We define the $0$ -th Hochschild homology of the bimodule $\mathfrak {Q}_{W_{\mathrm {a}}}^{*}$ to be the quotient

$$\begin{align*}\mathop{\textrm{HH}_0}(\mathfrak{Q}_{W_{\mathrm{a}}}^{*}):=\mathfrak{Q}_{W_{\mathrm{a}}}^{*}/\langle a\bullet f-a\odot f\colon a\in \mathfrak{Q}, f\in \mathfrak{Q}_{W_{\mathrm{a}}}\rangle. \end{align*}$$

Consider the dual of the map $\imath \colon \mathfrak {Q}_{Q^{\vee }}\to \mathfrak {Q}_{W_{\mathrm {a}}}$ , $\eta _{t_\lambda }\mapsto \eta _{t_\lambda } $ . It gives a surjection

$$\begin{align*}\imath^{*}\colon \mathfrak{Q}_{W_{\mathrm{a}}}^{*}\twoheadrightarrow \mathfrak{Q}_{Q^{\vee}}^{*},\quad af_{t_\lambda w}\mapsto a\delta_{w,e}f_{t_\lambda}. \end{align*}$$

We have

$$\begin{align*}\imath^{*}(a\bullet bf_{t_\lambda v}-a\odot bf_{t_\lambda v})=\imath^{*}(v(a)bf_{t_\lambda v}-abf_{t_{\lambda v}})=(v(a)b-ab)\delta_{v,e}f_{t_\lambda v}=0. \end{align*}$$

So it induces a surjection

$$\begin{align*}\imath^{*}\colon \mathop{\textrm{HH}_0}(\mathfrak{Q}_{W_{\mathrm{a}}}^{*})\twoheadrightarrow \mathfrak{Q}_{Q^{\vee}}^{*}. \end{align*}$$

On the other hand, $\ker \imath ^{*}=\prod _{w\neq e, \lambda \in Q^{\vee }}\mathfrak {Q} f_{t_\lambda w}$ . Now for any w, let $x_\mu \in S$ so that $w(\mu )\neq \mu $ , then we obtain

$$\begin{align*}f_{t_\lambda w}=\tfrac{1}{x_\mu-w(x_\mu)}(x_\mu\odot f_{t_\lambda w}-x_\mu \bullet f_{t_\lambda w})\in \ker \imath^{*}. \end{align*}$$

Therefore, we have proven the following lemma.

Lemma 6.1 There is an isomorphism $\mathop {\textrm {HH}_0}(\mathfrak {Q}_{W_{\mathrm {a}}}^{*})\simeq \mathfrak {Q}_{Q^{\vee }}^{*}$ which fits into the following commutative diagram

By definition, we have the commutative diagram of left S-modules:

It is also easy to see that the surjection $\imath ^{*}\colon \mathfrak {Q}_{W_{\mathrm {a}}}^{*}\twoheadrightarrow \mathfrak {Q}_{Q^{\vee }}^{*}$ induces a surjective map

$$\begin{align*}\imath^{*\prime}\colon \mathop{\textrm{HH}_0}(\mathbf{D}_{W_{\mathrm{a}}}^{*})\twoheadrightarrow \mathbf{D}_{Q^{\vee}}^{*}. \end{align*}$$

Our goal is to show that the isomorphism and the diagram of Lemma 6.1 can be restricted to the formal Peterson subalgebra $\mathbf {D}_{Q^{\vee }}$ . Namely, we want to prove the following.

Since the product structure on both the domain and the codomain is induced by the coproduct structure

$$\begin{align*}\mathfrak{Q}_{W_{\mathrm{a}}}\longrightarrow \mathfrak{Q}_{W_{\mathrm{a}}}\otimes\mathfrak{Q}_{W_{\mathrm{a}}}, \quad a\eta_w\mapsto a\eta_w\otimes \eta_w, \end{align*}$$

where the codomain is the tensor product of left $\mathfrak {Q}$ -modules, the map $\imath ^{*\prime }$ is a ring homomorphism.

Moreover, since the coproduct structure on both the domain and the codomain is induced by the product structure in $\mathfrak {Q}_{W_{\mathrm {a}}}$ and $\mathfrak {Q}_{Q^{\vee }}$ , the map $\imath ^{*\prime }$ is a coalgebra homomorphism. Therefore, it only suffices to prove the injectivity of $\imath ^{*\prime }$ .

To prove the latter, we introduce the following filtration on the dual $\mathbf {D}_{W_{\mathrm {a}}}^{*}$ of the FADA.

Definition 6.2 Let $w_\lambda $ be as defined in the appendix. Set $F_i=\bigcup _{\ell (w_{\lambda })\geq i} t_\lambda W$ . For any $f\in \mathfrak {Q}_{W_{\mathrm {a}}}^{*}=\mathrm {Hom}(W_{\mathrm {a}}, \mathfrak {Q})$ set $\mathrm {supp} f=\{w\in W_{\mathrm {a}}\colon f(w)\neq 0\}$ . Define the i-th stratum of the filtration to be

$$\begin{align*}\mathcal{Z}_i:=\{f\in \mathbf{D}_{W_{\mathrm{a}}}^{*}\colon \mathrm{supp} f\subseteq F_i\}. \end{align*}$$

We have $\mathcal {Z}_i=\prod _{w\in F_i} S\cdot Y_{I_{w}}^{*}$ . Each $\mathcal {Z}_i$ is a S-bimodule, since $x\bullet af_u=u(x)af_u$ and $x\odot af_u=xa f_u$ , $x,a\in S$ , $u\in W_{\mathrm {a}}$ .

Consider the following two conditions:

(6.1) $$ \begin{align} f(\eta_{t_{\lambda}u})\in x_\alpha^{\ell_{\alpha}(w_\lambda)} S \quad \text{ for any } u\in W, \end{align} $$

(6.2) $$ \begin{align} f(\eta_{t_{\lambda}u}-\eta_{t_{\lambda}s_{\alpha}u}) \in x_\alpha^{\ell_{\alpha}(w_\lambda)+1} S \quad \text{ for any } u\in W \text{ and root } \alpha\in \Phi. \end{align} $$

Proof Let $f\in \mathcal {Z}_i$ and $\ell (w_\lambda )=i$ . Assume $\left <\lambda ,\alpha ^{\vee }\right>\leq 0$ . For $k\in [1, \ell _{\alpha }(w_{\lambda })]$ by Lemma 7.2 of the appendix, we conclude that $w_{\lambda +k\alpha ^{\vee }}<w_{\lambda }$ and, in particular, $\ell (w_{\lambda +k\alpha ^{\vee }})<\ell (w_{\lambda })$ . So $f(\eta _{\lambda +k\alpha ^{\vee }})=0$ , and we have for any $u\in W$ ,

$$ \begin{align*} (Z_\alpha^{\ell_\alpha(w_\lambda)}\eta_{t_\lambda}\odot f)(\eta_u)&=\frac{1}{x_{-\alpha}^{\ell_\alpha(w_\lambda)}}f\big((1-\eta_{t_\alpha^{\vee}})^{\ell_{\alpha}(w_{\lambda})}\eta_{t_{\lambda}u}\big) \\ &=\frac{1}{x_{-\alpha}^{\ell_\alpha(w_\lambda)}}f(\eta_{t_{\lambda}u})\in S,\; \text{ and}\\ (Z_\alpha^{\ell_\alpha(w_\lambda)}\eta_{t_\lambda}X_\alpha\odot f)(\eta_u)&=\frac{1}{x_{-\alpha}^{\ell_\alpha(w_\lambda)}x_{\alpha}}f\big((1-\eta_{t_\alpha^{\vee}})^{\ell_{\alpha}(w_{\lambda})}(\eta_{t_{\lambda}u}-\eta_{t_{\lambda}s_{\alpha}u})\big) \\ &=\frac{1}{x_{-\alpha}^{\ell_\alpha(w_\lambda)}x_{\alpha}}f(\eta_{t_{\lambda}u}-\eta_{t_{\lambda}s_{\alpha}u}) \in S. \end{align*} $$

Note that $\frac {x_\alpha }{x_{-\alpha }}$ is invertible in S, so we can replace $x_\alpha $ by $x_{-\alpha }$ whenever needed.

Similarly, if $\left <\lambda ,\alpha ^{\vee }\right> <0$ , then $\ell (w_{\lambda -k\delta })<\ell (w_{\lambda })$ for $k\in [1,\ell _{\alpha }(w_{\lambda })]$ . Thus,

$$ \begin{align*}(Z_{-\alpha}^{\ell_{\alpha}(w_\lambda)}\odot f)(\eta_u)&=\frac{1}{x_\alpha^{\ell_\alpha(w_\lambda)}}f\big((1-\eta_{t_{-\alpha^{\vee}}})^{\ell_{\alpha}(w_{\lambda})}\eta_{t_{\lambda}u}\big) \\ &=\frac{1}{x_{\alpha}^{\ell_\alpha(w_\lambda)}}f(\eta_{t_{\lambda}w})\in S,\;\text{ and}\\ (Z_{-\alpha}^{\ell_\alpha(w_\lambda)}\eta_{t_\lambda}X_{\alpha}\odot f)(\eta_u)&=\frac{1}{x_\alpha^{\ell_\alpha(w_\lambda)+1}}f \big((1-\eta_{t_{-\alpha^{\vee}}})^{\ell_{\alpha}(w_{\lambda})}(\eta_{t_{\lambda}w}- \eta_{t_{\lambda}s_{\alpha}u})\big) \\ &=\frac{1}{x_\alpha^{\ell_\alpha(w_\lambda)+1}}f(\eta_{t_{\lambda}u}-\eta_{t_{\lambda}s_{\alpha}u}) \in S. \end{align*} $$

The result then follows.

Define

$$ \begin{align*}\mathcal{Y}_{(i)}=\left\{f\in \mathrm{Hom}(F_i\backslash F_{i+1},S)\colon f \text{ satisfies the conditions (6.1)} \text{ and (6.2)} \right\}. \end{align*} $$

Here, $f(\eta _{v_1}-\eta _{v_2}):=f(v_1)-f(v_2)$ . Note that $\mathcal {Y}_{(i)}$ is a S-bimodule in the usual sense, that is $(a\odot f)(v)=af(v)$ and $(a\bullet f)(v)=v(a)f(v)$ . By Lemma 6.3, we have a natural projection $\mathcal {Z}_i\to \mathcal {Y}_{(i)}$ , which induces an injective S-bimodule map

$$ \begin{align*}\mathrm{res}\colon \mathcal{Z}_i/\mathcal{Z}_{i+1}\longrightarrow \mathcal{Y}_{(i)}.\end{align*} $$

Proof We only need to prove that $\mathrm {res}$ is surjective. Let $f\in \mathcal {Y}_{(i)}$ . We pick a minimal element $w\in \mathrm {supp}(f)$ . We first show that

$$ \begin{align*}f(\eta_{w})\in \prod_{\alpha\in \Phi^+} x_{\alpha}^{\ell_{\alpha}(w)}S.\end{align*} $$

As $x_{\alpha }$ ’s are relatively prime, it reduces to show

(6.3) $$ \begin{align} f(\eta_w)\in x_{\alpha}^{\ell_{\alpha}(w)}S \end{align} $$

for each root $\alpha $ .

Let $w=t_{\lambda }u$ for $\lambda \in Q^{\vee }$ , $u\in W$ and $\ell _\alpha (w_\lambda )=i$ . By Lemma 7.1 of the appendix, $\ell _{\alpha }(w)\in \{\ell _{\alpha }(w_{\lambda }),\ell _{\alpha }(w_{\lambda })+1\}$ .

If $\ell _{\alpha }(w)=\ell _{\alpha }(w_{\lambda })$ , then (6.3) follows from (6.1) directly. If $\ell _{\alpha }(w)=\ell _{\alpha }(w_{\lambda })+1$ , by (7.2) and (7.3), we have

$$ \begin{align*} \ell_{\alpha}(t_{\lambda}s_{\alpha}u)=\ell_{\alpha}(w_{\lambda})<\ell_{\alpha}(w). \end{align*} $$

It implies $t_{\lambda }s_{\alpha }u<w$ (note that $t_{\lambda }s_{\alpha }u$ and w are always comparable under the Bruhat order). By (6.2), we have

$$ \begin{align*}f(\eta_{t_{\lambda}u}-\eta_{t_{\lambda}s_{\alpha}u}) =f(\eta_{w}) \in x_\alpha^{\ell_{\alpha}(w_\lambda)+1} S. \end{align*} $$

Observe that the images of $Y_{I_w}^{*}$ for $w\in Z_i\setminus Z_{i+1}$ form a basis of $\mathcal {Z}_i/\mathcal {Z}_{i+1}$ , and we have $Y_{I_{w}}^{*}(\eta _{w})=\prod _{\alpha>0} x_{\alpha }^{\ell _{\alpha }(w)}$ .

The conclusion then follows after replacing f by $f-\frac {f(\eta _w)}{\prod _{\alpha \in \Phi ^+}x_\alpha ^{\ell _\alpha (w_\lambda )}}Y_{I_w}^{*}$ .

Denote for each $\lambda \in Q^{\vee }$ , $\Delta _{\lambda }= \prod _{\alpha>0} x_{\alpha }^{\ell _{\alpha }(w_{\lambda })}\in S$ . It is clear that we have an S-bimodule isomorphism

$$ \begin{align*}\mathcal{Y}_{(i)}\simeq \bigoplus_{\ell(w_{\lambda})=i} \Delta_\lambda \cdot \mathbf{D}_W.\end{align*} $$

To finish the proof of Theorem 6.2, we define a filtration on $ \mathbf {D}_{Q^{\vee }}^{*}$ by

$$\begin{align*}\mathcal{X}_i=\{f\in \mathbf{D}_{Q^{\vee}}^{*}\colon \mathrm{supp}(f)\subset F_i\}. \end{align*}$$

Then, $Y_{I_w}^{*}$ with $\ell (w_\lambda )=i$ is a S-basis of $\mathcal {X}_i/\mathcal {X}_{i+1}$ . So the rank of $\mathcal {X}_i/\mathcal {X}_{i+1}$ is $|F_i\backslash F_{i+1}|$ . Moreover, by definition, we know that $\imath ^{*\prime }$ induces a map on each associated graded piece:

$$\begin{align*}\imath^{*\prime}\colon \mathcal{Z}_i/\mathcal{Z}_{i+1}\to \mathcal{X}_{i}/\mathcal{X}_{i+1}. \end{align*}$$

From Lemma 6.4, the rank of $\mathcal {Z}_i/\mathcal {Z}_{i+1}$ is $|F_i\backslash F_{i+1}|$ , therefore, $\imath ^{*\prime }$ is an isomorphism.

7 Appendix

Here, we prove several combinatorial properties of the affine Weyl group that are used in the proof of Theorem 6.2.

For $w\in W_{\mathrm {a}}$ , denote

$$ \begin{align*}\ell_{\alpha}(w)=|\{\beta=\pm \alpha+k\delta>0 \colon w^{-1}(\beta)<0\}|.\end{align*} $$

It is clear that $\ell (w)=\sum _{\alpha>0} \ell _\alpha (w)$ . Also denote by $w_{\lambda }\in W_a^{-}$ the minimal representative of $t_{\lambda }W$ . Then, $w_{\lambda }\leq w_{\mu }$ if and only if there exists $w\in t_{\lambda }W$ and $y\in t_{\mu }W$ such that $w\leq y$ . Note that $w\leq y$ implies $\ell _{\alpha }(w)\leq \ell _{\alpha }(y)$ for all $\alpha \in \Phi _+$ , so after fixing $\alpha $ $\ell _\alpha (w_\lambda )$ becomes minimal for elements w from $t_\lambda W$ .

Lemma 7.1 We have the following property:

$$ \begin{align*}\ell_{\alpha}(w_{\lambda})= \begin{cases} -\left<\lambda,\alpha\right>, & \text{ if }\left<\lambda,\alpha\right>\leq 0,\\ \left<\lambda,\alpha\right>-1, & \text{ if } \left<\lambda,\alpha\right>> 0. \end{cases}\end{align*} $$

Proof For $w=t_{\lambda }u\in t_{\lambda }W$ , $\beta =\pm \alpha +k\delta>0$ (so $k\ge 0$ ), we have

(7.1) $$ \begin{align} w^{-1}(\beta)=w^{-1}(\pm \alpha+k\delta) = \pm u^{-1}\alpha+(k\pm \left<\lambda,\alpha\right>)\delta. \end{align} $$

If $\langle \lambda ,\alpha \rangle \le 0$ , then $w^{-1}(\beta )<0$ implies $\beta =\alpha +k\delta $ , and moreover, $k\in [0,\ell _\alpha (w)-1]$ (since $\ell _\alpha (w)=|Inv_\alpha (w)|$ ). So we have

(7.2) $$ \begin{align}\ell_{\alpha}(w)=-\left<\lambda,\alpha\right>+ \begin{cases} 1, & u^{-1}(\alpha)<0,\\ 0, & u^{-1}(\alpha)>0, \end{cases} \end{align} $$

and the minimal value is $-\langle \lambda ,\alpha \rangle $ .

If $\left <\lambda ,\alpha \right>> 0$ , then $w^{-1}(\beta )<0$ if and only if $\beta =-\alpha +k\delta $ and $k\in [1,\ell _{\alpha }(w)]$ , in which case we have

(7.3) $$ \begin{align}\ell_{\alpha}(w)=\left<\lambda,\alpha\right>- \begin{cases} 1, & u^{-1}(\alpha)<0,\\ 0, & u^{-1}(\alpha)>0. \end{cases} \end{align} $$

The minimal value is $\langle \lambda ,\alpha \rangle -1$ .

Proof If $\left <\lambda ,\alpha \right>\leq 0$ , consider $w\in t_\lambda W$ such that $w=t_{\lambda }u$ with $u^{-1}(\alpha )<0$ . We have

$$\begin{align*}w^{-1}(\alpha)=u^{-1}(\alpha)+\langle\lambda,\alpha\rangle\delta<0,\end{align*}$$

so $w>s_\alpha w$ . From (7.2), we get $\ell _{\alpha }(w)=\ell _{\alpha }(w_{\lambda })+1=-\left <\lambda ,\alpha \right>+1$ .

Since $1\leq k\leq \ell _{\alpha }(w_{\lambda })=-\left <\lambda ,\alpha \right>$ , we get

$$\begin{align*}(s_{\alpha}w)^{-1}(-\alpha+k\delta)=u^{-1}\alpha+(k+\left<\lambda,\alpha\right>)\delta <0,\end{align*}$$

which implies

$$\begin{align*}s_\alpha w>s_{-\alpha+k\delta}s_{\alpha}w=t_{k\alpha^{\vee}}w.\end{align*}$$

Therefore, $w>t_{k\alpha ^{\vee } }w$ .

Since $w\in t_{\lambda }W$ and $t_{k\alpha ^{\vee }}w\in t_{\lambda +k\alpha ^{\vee }}W$ , we get $w_{\lambda }>w_{\lambda +k\alpha ^{\vee }}$ .

If $\left <\lambda ,\alpha \right>> 0$ , consider $w=t_{\lambda }u$ with $u^{-1}(\alpha )>0$ . We have

$$\begin{align*}w^{-1}(-\alpha+\delta) =u^{-1}(\alpha)+(1-\left<\lambda,\alpha\right>)\delta <0,\end{align*}$$

so $w>s_{-\alpha +\delta }w$ . From (7.3), we have $\ell _{\alpha }(w)=\langle \lambda ,\alpha \rangle =\ell _{\alpha }(w_{\lambda })+1. $

Since $1\leq k\leq \ell _{\alpha }(w)-1=\ell _{\alpha }(w_{\lambda })=\left <\lambda ,\alpha \right>-1$ , we get

$$ \begin{align*}s_{-\alpha+\delta} w>s_{\alpha+(k-1)\delta}s_{-\alpha+\delta}x=t_{-k\alpha^{\vee}}w.\end{align*} $$

So $w>t_{-k\alpha ^{\vee }}w$ . Since $w\in t_{\lambda }W$ and $t_{k\alpha ^{\vee }}w\in t_{\lambda +k\alpha ^{\vee }}W$ , we have $w_{\lambda }>w_{\lambda +k\alpha ^{\vee }}$ .

Lemma 7.3 If $\langle \lambda ,\alpha \rangle =0$ , then we have the following sequence:

$$\begin{align*}w_{\lambda}<w_{\lambda+\alpha^{\vee}}<w_{\lambda-\alpha^{\vee}}<w_{\lambda+2\alpha^{\vee}}<w_{\lambda-2\alpha^{\vee}}<\cdots. \end{align*}$$

If $\langle \lambda ,\alpha \rangle =1$ , then we have the following sequence:

$$\begin{align*}w_{\lambda}<w_{\lambda-\alpha^{\vee}}<w_{\lambda+\alpha^{\vee}}<w_{\lambda-2\alpha^{\vee}}<w_{\lambda+2\alpha^{\vee}}<\cdots. \end{align*}$$

The lengths $\ell _\alpha $ are given by $(0,1,2,3,4,\ldots )$ .

Proof We only prove the case when $\langle \lambda ,\alpha \rangle =0$ . Consider $\lambda +k\alpha ^{\vee }$ and $\lambda -k\alpha ^{\vee }$ with $k> 0$ , then $\langle \lambda -k\alpha ^{\vee },\alpha \rangle =-2k<0$ , and $2k=\ell _\alpha (w_{\lambda -k\alpha ^{\vee }})$ , so by Lemma 7.2,

$$\begin{align*}w_{\lambda-k\alpha^{\vee}}>w_{\lambda-k\alpha^{\vee}+2k\alpha^{\vee}}=w_{\lambda+k\alpha^{\vee}}. \end{align*}$$

Finally, consider ${\lambda -k\alpha ^{\vee }}$ and $\lambda +(k+1)\alpha ^{\vee }$ , $k\ge 0$ , then $\langle \lambda +(k+1)\alpha ^{\vee },\alpha \rangle =2(k+1)\ge 2$ , and $\ell _\alpha (w_{\lambda +(k+1)\alpha ^{\vee }})=2k+1$ , so by Lemma 7.2,

$$\begin{align*}w_{\lambda+(k+1)\alpha^{\vee}}\ge w_{\lambda+(k+1)\alpha^{\vee}-(2k+1)\alpha^{\vee}}=w_{\lambda-k\alpha^{\vee}}.\\[-34pt]\end{align*}$$