${}_{i}\widetilde{\mathcal{R}}_{i}\to[\mathfrak{b}/B]$

sending $(g,s)$ to $gs$ mod $t$. We also have the cartesian square

(15)
${{}_{i}\widetilde{\mathcal{R}}_{i}}$${\left[\widetilde{\mathfrak{g}}/G\right]=[\mathfrak{b}/B]}$${{}_{i}\mathcal{R}_{i}}$${\left[\mathfrak{g}/G\right]}$$\scriptstyle{\varphi^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\varphi}$

similar to (7). Now, note that by finite-dimensional Springer theory

$H_{*}^{{\widetilde{\mathbf{I}}}\rtimes{\mathbb{C}}^{\times}}({}_{i}\mathcal{R}_{i})={}_{i}\widetilde{\mathcal{A}}_{i}\mathbf{e}$

and

$H_{*}^{{\widetilde{G}}_{\mathcal{O}}\rtimes{\mathbb{C}}^{\times}}({}_{i}\mathcal{R}_{i})=\mathbf{e}{}_{i}\widetilde{\mathcal{A}}_{i}\mathbf{e}$

Finally
${}_{i+1}\mathcal{A}_{i}\cong\mathbf{e}{}_{i+1}\widetilde{\mathcal{A}}_{i}\mathbf{e}$
and

$\mathbf{e}_{-}{}_{i}\widetilde{\mathcal{A}}_{i}\mathbf{e}[2\dim
G/B]\cong\mathbf{e}{}_{i+1}\widetilde{\mathcal{A}}_{i}\mathbf{e}$

again similarly to the proof of Lemma 2.2.

∎

###  6.2. Localization of the spherical algebra in the adjoint case

We now analyze the ${\mathbb{Z}}$-algebra introduced in the previous section
via localization to fixed points. In particular, we may deduce results about
the associated graded of the Bruhat filtration for the convolution algebras,
using an ”abelianization” procedure appearing e.g. in [14]. We should note
that similar fixed-point analysis does not apply to the Springer action itself
unless we are in a situation similar to [72, 83, 26], but we are still able to
deduce many results about the convolution action on general grounds in Section
7.

We let $G$ and $N$ be arbitrary for now. Suppose $\mathbf{P}=G_{\mathcal{O}}$.
The spaces ${}_{i}\mathcal{R}_{j}$ have natural closed embeddings to
${}_{i}\mathcal{R}_{j}\hookrightarrow
G_{\mathcal{K}}\times^{G_{\mathcal{O}}}t^{j}N_{\mathcal{O}}$. Moreover, there
is the embedding of the zero-section

$z:\operatorname{Gr}_{G}\hookrightarrow
G_{\mathcal{K}}\times^{G_{\mathcal{O}}}t^{j}N_{\mathcal{O}}$

and an inclusion of ”the fibers above fixed points”

$\iota:{}_{i}\mathcal{R}_{j}|_{\operatorname{Gr}_{T}}\hookrightarrow{}_{i}\mathcal{R}_{j}.$

The latter map gives rise to an equivariant pushforward $\iota_{*}$ (see [9,
Remark 5.23]). The map $z$ for $G=T$ gives the maps ${}_{i}z_{j}^{*}$ from
Theorem 5.13. Similarly to [9, 10] we then have

######  Proposition 6.6.

We have an embedding

${}_{i}z_{j}^{*}(\iota_{*})^{-1}:{}_{i}\mathcal{A}_{j}^{\hbar}\hookrightarrow\mathcal{A}_{T,0}^{\hbar}[\hbar^{-1},(\text{generalized
roots}+m\hbar+nc)^{-1}|m,n\in{\mathbb{Z}}].$

Note that this is not a ring homomorphism unless $i=j$, but a bimodule
homomorphism, as in Theorem 5.13.

Let $\pi:{}_{i}\mathcal{R}_{j}\to\operatorname{Gr}_{G}$ be the projection. We
use the Cartan decomposition of the affine Grassmannian into $G_{\mathcal{O}}$
orbits:

$\operatorname{Gr}_{G}=\bigsqcup_{\lambda\in
X_{*}^{+}}\operatorname{Gr}_{G}^{\lambda},\quad\operatorname{Gr}_{G}^{\lambda}=G_{\mathcal{O}}t^{\lambda}G_{\mathcal{O}}/G_{\mathcal{O}}$

The closures of these orbits will be denoted by
$\operatorname{Gr}^{\leq\lambda}_{G}=\overline{\operatorname{Gr}_{G}^{\lambda}}$.
Then the subvariety
${}_{i}\mathcal{R}_{j}^{\leq\lambda}:=\pi^{-1}(\operatorname{Gr}^{\leq\lambda}_{G})$
gives rise to a class in equivariant Borel-Moore homology as in [9, Section
2].

In particular, we have the following localization formula.

######  Lemma 6.7.

For a minuscule cocharacter $\lambda$, we have

(16)
${}_{i}z_{j}^{*}(\iota_{*})^{-1}f\cap[{}_{i}\mathcal{R}_{j}^{\leq\lambda}]=\sum_{\lambda^{\prime}=w\lambda\in
W\lambda}\frac{wf\times
e(t^{\lambda^{\prime}}t^{j}N_{\mathcal{O}}/t^{\lambda^{\prime}}t^{j}N_{\mathcal{O}}\cap
t^{i}N_{\mathcal{O}})}{e(T_{\lambda^{\prime}}\operatorname{Gr}_{G}^{\lambda})}u_{\lambda^{\prime}}$

###### Proof.

We are using Borel-Moore homology, so results of Brion [13] apply and the
formula follows from Theorem 5.13. For the case $i=j=0$, see [9, Proposition
6.6].

∎

If $\lambda$ is not minuscule, the corresponding Schubert variety is not
smooth and there is no nice formula for
$[{}_{i}\mathcal{R}_{j}^{\leq\lambda}]$. Still, the right hand side of the
equation (6.7) yields the formula for the associated graded with respect to
the Bruhat filtration, see e.g. [10, Eq. (6.3)]. We first consider the case
$G=GL_{n}$.

######  Theorem 6.8.

Let $G=GL_{n}$, then the following hold:

(a) For $c=\hbar=0$, arbitrary cocharacter $\lambda$ and a function $f(y)$
which is symmetric under the stabilizer of $\lambda$ we have the following:

${}_{j}z_{i}^{*}\iota_{*}^{-1}\operatorname{gr}[{}_{j}\mathcal{R}_{i}^{\lambda}][f]=\sum_{\lambda^{\prime}\in
W\lambda}f^{\prime}\prod_{s\neq
r}\frac{(y_{r}-y_{s})^{\max(\lambda^{\prime}_{r}-\lambda^{\prime}_{s}+i,j)}}{(y_{r}-y_{s})^{\lambda^{\prime}_{r}-\lambda^{\prime}_{s}+i}(y_{r}-y_{s})^{\max(\lambda^{\prime}_{r}-\lambda^{\prime}_{s},0)}}u^{\lambda^{\prime}}$

Here $f^{\prime}$ is the image of $f$ under any permutation in $W$ which sends
$\lambda$ to $\lambda^{\prime}$. If $\lambda$ is minuscule, the formula is
exact without taking associated graded.

(b) For general $c,\hbar$ we have

${}_{j}z_{i}^{*}\iota_{*}^{-1}\operatorname{gr}[{}_{j}\mathcal{R}_{i}^{\lambda}][f]=\sum_{\lambda^{\prime}\in
W\lambda}f^{\prime}\frac{\prod_{\lambda^{\prime}_{r}-\lambda^{\prime}_{s}+i<j}\prod_{\ell=0}^{j-(\lambda^{\prime}_{r}-\lambda^{\prime}_{s}+i)-1}(y_{r}-y_{s}+(\lambda^{\prime}_{r}-\lambda^{\prime}_{s}+i+\ell)+c)}{\prod_{s\neq
r}\prod_{\ell=0}^{\max(\lambda^{\prime}_{r}-\lambda^{\prime}_{s},0)}(y_{s}-y_{r}+\ell\hbar)}u^{\lambda^{\prime}}$

where the notations are as above.

###### Proof.

We compute the right hand side in the equation (16). If
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in X_{*}(T)\subset X_{*}(GL_{n})$ we
get

$t^{\lambda}.t^{i}N_{\mathcal{O}}=\begin{pmatrix}t^{i}N_{\mathcal{O}}&t^{\lambda_{1}-\lambda_{2}+i}N_{\mathcal{O}}&t^{\lambda_{1}-\lambda_{3}+i}N_{\mathcal{O}}&\cdots&t^{\lambda_{1}-\lambda_{n}+i}N_{\mathcal{O}}\\\
t^{\lambda_{2}-\lambda_{1}+i}N_{\mathcal{O}}&t^{i}N_{\mathcal{O}}&t^{\lambda_{2}-\lambda_{3}+i}N_{\mathcal{O}}&\cdots&t^{\lambda_{2}-\lambda_{n}+i}N_{\mathcal{O}}\\\
t^{\lambda_{3}-\lambda_{1}+i}N_{\mathcal{O}}&t^{\lambda_{3}-\lambda_{2}+i}N_{\mathcal{O}}&t^{i}N_{\mathcal{O}}&\cdots&t^{\lambda_{3}-\lambda_{n}+i}N_{\mathcal{O}}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
t^{\lambda_{n}-\lambda_{1}+i}N_{\mathcal{O}}&t^{\lambda_{n}-\lambda_{2}+i}N_{\mathcal{O}}&t^{\lambda_{n}-\lambda_{3}+i}N_{\mathcal{O}}&\cdots&t^{i}N_{\mathcal{O}}\end{pmatrix}$

Whence we compute the Euler class at $c=\hbar=0$:

$e(t^{\lambda}t^{i}N_{\mathcal{O}}/t^{\lambda}t^{i}N_{\mathcal{O}}\cap
t^{j}N_{\mathcal{O}})=\prod_{s\neq
r}\frac{(y_{r}-y_{s})^{\max(\lambda_{r}-\lambda_{s}+i,j)}}{(y_{r}-y_{s})^{\lambda_{r}-\lambda_{s}+i}}$

For general $c,\hbar$ the factors with
$\lambda^{\prime}_{r}-\lambda^{\prime}_{s}+i\geq j$ still contribute 1, and
the formula for the Euler class reads as

$\prod_{\lambda^{\prime}_{r}-\lambda^{\prime}_{s}+i<j}\prod_{\ell=0}^{j-(\lambda^{\prime}_{r}-\lambda^{\prime}_{s}+i)-1}(y_{r}-y_{s}+(\lambda^{\prime}_{r}-\lambda^{\prime}_{s}+i+\ell)+c).$

It is well known that the tangent space
$T_{\lambda}\operatorname{Gr}^{\lambda}$ is naturally identified with
$\frac{N_{\mathcal{O}}}{N_{\mathcal{O}}\cap t^{\lambda}.N_{\mathcal{O}}}$,
from which we get

$e(T_{\lambda}\operatorname{Gr}_{G}^{\lambda})=\prod_{s\neq
r}\prod_{\ell=0}^{\max(\lambda_{r}-\lambda_{s},0)}(y_{s}-y_{r}+\ell\hbar).$

∎

######  Remark 6.9.

The above formula makes sense even if $f$ is not symmetric with respect to the
stabilizer of $\lambda$. In this case, we first symmetrize with respect to the
stabilizer of $\lambda$ and then symmetrize with respect to the whole group
$W=S_{n}$.

For a general group $G$ and $N=\operatorname{Ad}$, we write can write the
formula as follows.

######  Proposition 6.10.

For arbitrary $G$ and arbitrary coweight $\lambda$ we have:

${}_{j}z_{i}^{*}\iota_{*}^{-1}\operatorname{gr}[{}_{j}\mathcal{R}_{i}^{\leq\lambda}]=\sum_{\lambda^{\prime}\in
W\lambda}\frac{\prod_{\alpha(\lambda^{\prime})+i<j}\prod_{\ell=0}^{i-\alpha(\lambda^{\prime})-j-1}(y_{\alpha}+(\alpha(\lambda^{\prime})+j+\ell)\hbar+c)}{\prod_{\alpha\in\Phi}\prod_{\ell=0}^{\max(0,\alpha(\lambda^{\prime}))-1}(y_{\alpha}+\ell\hbar)}u_{\lambda^{\prime}}$

###### Proof.

The proof of Theorem 6.8 is naturally adopted to arbitrary root systems. We
will not need it below and leave the details for the reader. ∎

######  Lemma 6.11.

For $G=GL_{n}$ and $N=\operatorname{Ad}$ and the minuscule coweight
$\omega_{m}=(1,\ldots,1,0,\ldots,0)$ and $i\geq j$, we have

(17) $\displaystyle
z^{*}\iota_{*}^{-1}[{}_{i}\mathcal{R}_{i}^{\leq\omega_{m}}]=$
$\displaystyle\sum_{I\subset[n],|I|=m}\prod_{r\in I,s\notin
I}\frac{y_{s}-y_{r}+(i-1)\hbar+c}{y_{r}-y_{s}}u^{I}$ (18) $\displaystyle
z^{*}\iota_{*}^{-1}[{}_{i+1}\mathcal{R}_{i}^{\leq\omega_{m}}]=$
$\displaystyle\sum_{I\subset[n],|I|=m}\frac{(\prod_{r\in I,s\notin
I}(y_{s}-y_{r}+(i-1)\hbar+c)(y_{s}-y_{r}+i\hbar+c))(\prod_{r\in I,s\in I\text{
or }r\notin I,s\notin I}y_{r}-y_{s}+i\hbar+c)}{\prod_{r\in I,s\notin
I}y_{r}-y_{s}}u^{I}$ (19) $\displaystyle
z^{*}\iota_{*}^{-1}[{}_{j}\mathcal{R}_{i}^{\leq\omega_{m}}]=$
$\displaystyle\prod_{k=i+1}^{j-1}\prod_{r,s}(y_{r}-y_{s}+k\hbar+c)\cdot
z^{*}\iota_{*}^{-1}[{}_{i+1}\mathcal{R}_{i}^{\leq\omega_{m}}]$

for $j\geq i+2$.

###### Proof.

This is a direct application of Theorem 6.8, recall that since $\omega_{m}$ is
minuscule we do not need to take associated graded. A symmetrization of
$\omega_{m}$ leads to a weight

$\lambda^{\prime}=(\lambda^{\prime}_{1},\ldots,\lambda^{\prime}_{n}),\
\lambda^{\prime}_{r}=\begin{cases}1&\text{if}\ r\in I\\\
0&\text{otherwise}\end{cases}$

for some $m$-element subset $I$. In this case
$\max(\lambda^{\prime}_{r}-\lambda^{\prime}_{s},0)$ equals 1 if $r\in
I,s\notin I$ and 0 otherwise which gives the denominator. For the numerator,
we observe

$\lambda^{\prime}_{r}-\lambda^{\prime}_{s}+i=\begin{cases}i-1&\text{if}\
r\notin I,s\in I\\\ i&\text{if}\ r\in I,s\in I\ \text{or}\ r\notin I,s\notin
I\\\ i+1&\text{if}\ r\in I,s\notin I\\\ \end{cases}$

and the result follows. ∎

More generally, we have the formula for arbitrary $G$ and minuscule $\lambda$.

######  Lemma 6.12.

For $N=\operatorname{Ad}$ and $\lambda$ minuscule,

(20) $\displaystyle
z^{*}\iota_{*}^{-1}[{}_{i}\mathcal{R}_{i}^{\leq\lambda}][f]=$
$\displaystyle\sum_{\lambda^{\prime}=w\lambda\in
W\lambda}wf\times\prod_{\alpha(\lambda^{\prime})=1}\frac{-y_{\alpha}+(i-1)\hbar+c}{y_{\alpha}}u^{\lambda^{\prime}}$
(21) $\displaystyle
z^{*}\iota_{*}^{-1}[{}_{i+1}\mathcal{R}_{i}^{\leq\lambda}]=$
$\displaystyle\sum_{\lambda^{\prime}=w\lambda\in
W\lambda}wf\times\frac{\left(\prod_{\begin{subarray}{c}\alpha(\lambda^{\prime})=1\end{subarray}}(y_{\alpha}+(i-1)\hbar+c)(y_{\alpha}+i\hbar+c)\right)\left(\prod_{\begin{subarray}{c}\alpha(\lambda^{\prime})=0\end{subarray}}y_{\alpha}+i\hbar+c\right)}{y_{\alpha}}u^{\lambda^{\prime}}$
(22) $\displaystyle z^{*}\iota_{*}^{-1}[{}_{j}\mathcal{R}_{i}^{\leq\lambda}]=$
$\displaystyle\prod_{k=i+1}^{j-1}\prod_{\alpha\in\Phi}(y_{\alpha}+k\hbar+c)\cdot
z^{*}\iota_{*}^{-1}[{}_{i+1}\mathcal{R}_{i}^{\leq\lambda}]$

######  Lemma 6.13.

Let $\varepsilon(x)=\max(x+i,j)-(x+i)-\max(x,0)$, then

$\varepsilon(x)+\varepsilon(-x)=\begin{cases}j-i&\text{if}\ |x|\geq|j-i|,\\\
(j-i)+|j-i|-|x|&\text{if}\ |x|\leq|j-i|.\end{cases}$

###### Proof.

Let us first prove that for arbitrary $x,d$ one has

(23) $\max(x,d)+\max(-x,d)=d+\max(|x|,|d|)=\begin{cases}d+|x|&\text{if}\
|x|\geq|d|,\\\ d+|d|&\text{if}\ |x|\leq|d|.\\\ \end{cases}$

Clearly, $\max(x,d)+\max(-x,d)=\max(|x|,d)+\max(-|x|,d)$. For $d\geq 0$ we get
$\max(-|x|,d)=d$ and (23) is clear. For $d<0$ we can rewrite

$\max(|x|,d)+\max(-|x|,d)=|x|-\min(|x|,|d|)=d+\max(|x|,|d|).$

Now we can prove lemma, by letting $d=j-i$. Note that
$\max(x+i,j)=i+\max(x,j-i)$, therefore

$\varepsilon(x)+\varepsilon(-x)=\max(x+i,j)-(x+i)-\max(x,0)+\max(-x+i,j)-(-x+i)-\max(-x,0)=$
$\max(x,j-i)-\max(x,0)+\max(-x,j-i)-\max(-x,0).$

Now we can use (23) with $d=j-i$. ∎

######  Corollary 6.14.

Let $G=GL_{n}$. At $c=\hbar=0$ we get

${}_{j}z_{i}^{*}\operatorname{gr}\iota_{*}^{-1}[{}_{j}\mathcal{R}_{i}^{\lambda}][f]=\pm{\mathrm{Sym}}\left(f\cdot\Delta^{j-i}\prod_{r<s,|\lambda_{r}-\lambda_{s}|<|j-i|}(y_{r}-y_{s})^{|j-i|-|\lambda_{r}-\lambda_{s}|}u^{\lambda}\right)$

###### Proof.

Consider a pair $r<s$. In the right hand side of Theorem 6.8 we get

$(y_{r}-y_{s})^{\varepsilon(\lambda^{\prime}_{r}-\lambda^{\prime}_{s})}(y_{s}-y_{r})^{\varepsilon(\lambda^{\prime}_{s}-\lambda^{\prime}_{r})}=\pm(y_{r}-y_{s})^{\varepsilon(\lambda^{\prime}_{r}-\lambda^{\prime}_{s})+\varepsilon(\lambda^{\prime}_{s}-\lambda^{\prime}_{r})}.$

By Lemma 6.13, the result follows. ∎

######  Example 6.15.

Again for $G=GL_{n}$, assume that $j=i+1$, then at $c=\hbar=0$ we get

${}_{i+1}z_{i}^{*}\operatorname{gr}\iota_{*}^{-1}[{}_{i+1}\mathcal{R}_{i}^{\lambda}][f]=\pm\Delta\mathrm{Alt}\left(f\cdot\prod_{r<s,\lambda_{r}=\lambda_{s}}(y_{r}-y_{s})u^{\lambda}\right)$

For arbitrary groups and $\hbar=c=0$ we get a similar formula.

######  Corollary 6.16.

For arbitrary $G$ and $\lambda$ we have

${}_{j}z_{i}\iota_{*}^{-1}\operatorname{gr}[{}_{j}\mathcal{R}_{i}^{\lambda}][f]={\mathrm{Sym}}_{W}\left(f\Delta^{j-i}\prod_{\alpha\in\Phi^{+},|\alpha(\lambda)|<|j-i|}(y_{\alpha})^{|j-i|-|\alpha(\lambda)|}u^{\lambda}\right)$

###### Proof.

The proof follows from setting $\hbar=c=0$ in Theorem 6.10 in exactly the same
way as Theorem 6.8 and Corollary 6.14.∎

###  6.3. Factorization of bimodules

######  Lemma 6.17.

Suppose that $\lambda$ is an arbitrary integral coweight for $GL_{n}$ and
$d>0$. Then there exist $d$ coweights $\mu^{(0)},\ldots,\mu^{(d-1)}$ such that
$\mu^{(0)}+\ldots+\mu^{(d-1)}=\lambda$ and for all $i$ and $j$ the following
holds:

1) If $|\lambda_{i}-\lambda_{j}|<d$ then

$d-|\lambda_{i}-\lambda_{j}|=\sum_{k,\mu^{(k)}_{i}=\mu^{(k)}_{j}}1.$

2) If $|\lambda_{i}-\lambda_{j}|>d$ then $\mu^{(k)}_{i}\neq\mu^{(k)}_{j}$ for
all $k$.

###### Proof.

We define $\mu^{(k)}$ by “dividing $\lambda$ by $d$ with remainder”. More
precisely, let $\lambda_{i}=dq_{i}+r_{i}$ where $0\leq r_{i}<d$. We define

$\mu^{(k)}_{i}=\begin{cases}q_{i}+1&\text{for}\ k<r_{i}\\\ q_{i}&\text{for}\
k\geq r_{i}\\\ \end{cases}$

Clearly, $\mu^{(0)}_{i}+\ldots+\mu^{(d-1)}_{i}=\lambda_{i}$. Without loss of
generality, we can assume that $\lambda_{j}\geq\lambda_{i}$. We have the
following cases:

1) $\lambda_{j}=dq_{i}+r_{j},r_{j}\geq r_{i}$. In this case
$\mu^{(k)}_{i}=\mu^{(k)}_{j}$ for $k<r_{i}$ and $k\geq r_{j}$, so

$\sum_{k,\mu^{(k)}_{i}=\mu^{(k)}_{j}}1=d-(r_{j}-r_{i})=d-(\lambda_{j}-\lambda_{i}).$

2) $\lambda_{j}=d(q_{i}+1)+r_{j},r_{j}<r_{i}$. In this case
$\mu^{(k)}_{i}=\mu^{(k)}_{j}$ for $r_{j}\leq k<r_{i}$ and

$\sum_{k,\mu^{(k)}_{i}=\mu^{(k)}_{j}}1=r_{i}-r_{j}=d-(\lambda_{j}-\lambda_{i}).$

3) If $\lambda_{j}>d(q_{i}+1)+r_{j}$ then $\mu^{(k)}_{j}\geq q_{i}+2$ for
$k<r_{i}$ and $\mu^{(k)}_{j}\geq q_{i}+1$ for $k\geq r_{i}$, so
$\mu^{(k)}_{i}\neq\mu^{(k)}_{j}$ for all $k$.

∎

######  Example 6.18.

Suppose that $d=2$, then we split $\lambda=\mu^{(0)}+\mu^{(1)}$ as follows. If
$\lambda_{i}=2k$ is even, we set $\mu_{i}^{(0)}=\mu_{i}^{(1)}=k$; if
$\lambda_{i}=2k+1$ is odd, we set $\mu_{i}^{(0)}=k+1$ and $\mu_{i}^{(1)}=k$.
Clearly, if $\lambda_{i}=\lambda_{j}$ then both $\mu_{i}^{(0)}=\mu_{j}^{(0)}$
and $\mu_{i}^{(1)}=\mu_{j}^{(1)}$. If $|\lambda_{i}-\lambda_{j}|=1$, it is not
hard to see that exactly one of equations $\mu_{i}^{(0)}=\mu_{j}^{(0)}$ and
$\mu_{i}^{(1)}=\mu_{j}^{(1)}$ holds.

######  Corollary 6.19.

Suppose that $G=GL_{n}$, $j-i=d$ and $c=\hbar=0$. For an arbitrary coweight
$\lambda$ and $\mu^{(k)}$ as in Lemma 6.17 we have

(24)
${}_{j}z_{i}^{*}\iota_{*}^{-1}\operatorname{gr}[{}_{j}\mathcal{R}_{i}^{\lambda}]=\pm\prod_{k=0}^{d-1}{}_{i+k+1}z_{i+k}^{*}\iota_{*}^{-1}\operatorname{gr}\left[{}_{i+k+1}\mathcal{R}_{i+k}^{\mu^{(k)}}\right]+\mathrm{lower\
order\ terms}.$

###### Proof.

By Lemma 6.17 we get

$\prod_{r<s,|\lambda_{r}-\lambda_{s}|<d}(y_{r}-y_{s})^{d-|\lambda_{r}-\lambda_{s}|}=\prod_{k}\prod_{r<s,\mu^{(k)}_{r}=\mu^{(k)}_{s}}(y_{r}-y_{s}).$

By Corollary 6.14, the left hand side of Eq. (24) is a symmetric polynomial
with leading term (in the dominance order on the $u_{\lambda}$)

$\pm\Delta^{d}\prod_{r<s,|\lambda_{r}-\lambda_{s}|<d}(y_{r}-y_{s})^{d-|\lambda_{r}-\lambda_{s}|}u^{\mathrm{sort}(\lambda)}$

while the right hand side is a product of $d$ symmetric polynomials with
leading terms

$\pm\Delta\prod_{r<s,\mu^{(k)}_{r}=\mu^{(k)}_{s}}(y_{r}-y_{s})u^{\mathrm{sort}(\mu^{(k)})}$

It is easy to see that in the above construction
$\mathrm{sort}(\lambda)=\mathrm{sort}(\mu^{(0)})+\ldots+\mathrm{sort}(\mu^{(d-1)})$,
so the result follows. ∎

######  Remark 6.20.

It seems reasonable to conjecture analogs of Lemma 6.17 and Corollary 6.19 for
other groups, at least for simply laced groups. This would have the
consequence that the isomorphism constructed in Theorem 6.21 would hold for
other groups, showing for instance that the global sections of the line bundle
we construct equal $A_{G}^{d}$. Since the line bundle is not expected to be
ample outside the simply laced case (see [53]) we do not expect the result to
hold in general.

We also note that the combinatorics appearing in the Lemma are closely related
to the root-system chip-firing of [24]. It would be interesting to make the
connection more precise. The second author thanks Pavel Galashin for
correspondence regarding this point.

###  6.4. The geometric ${\mathbb{Z}}$-algebra for the adjoint representation

We are ready to prove the main result of this section.

######  Theorem 6.21.

When $G=GL_{n}$, the ${\mathbb{Z}}$-algebras $\mathcal{A}$ and $\mathcal{B}$
are isomorphic. For general $G$, there is an injection
$\mathcal{A}\hookrightarrow\mathcal{B}$ inducing
${}_{j-1}\mathcal{A}_{j}\cong{}_{j-1}\mathcal{B}_{j}$ and
${}_{j}\mathcal{A}_{j}\cong{}_{j}\mathcal{B}_{j}$.

###### Proof.

We need to prove the following facts:

  * (a)

${}_{i}\mathcal{A}_{i}\cong{}_{i}\mathcal{B}_{i}$ as algebras

  * (b)

${}_{i}\mathcal{A}_{i+1}\cong{}_{i}\mathcal{B}_{i+1}$ as bimodules over
${}_{i}\mathcal{A}_{i}$ (resp. ${}_{i}\mathcal{B}_{i}$) and
${}_{i+1}\mathcal{A}_{i+1}$ (resp. ${}_{i+1}\mathcal{B}_{i+1}$)

  * (c)

${}_{i}\mathcal{A}_{i+1}\cdots{}_{j-1}\mathcal{A}_{j}\hookrightarrow{}_{i}\mathcal{A}_{j}$
and this is an isomorphism for $G=GL_{n}$. Note that
${}_{i}\mathcal{B}_{j}\cong{}_{i}\mathcal{B}_{i+1}\cdots{}_{j-1}\mathcal{B}_{j}$
by definition.

Part (a) follows from Theorem 6.1. Part (b) follows from Theorem 6.5.

In type $A$, it is instructive to review what part (b) says in order to prove
part (c). We can compute the bases in the associated graded spaces on both
sides: $\operatorname{gr}{}_{i}\mathcal{B}_{i+1}=A$ is the space of
antisymmetric polynomials in
${\mathbb{C}}[x_{1}^{\pm},\ldots,x_{n}^{\pm},y_{1},\ldots,y_{n}]$ and by Lemma
3.25(a) it has a vector space basis $\Delta_{S}$ parametrized by all
$n$-element subsets of ${\mathbb{Z}}_{\geq 0}\times{\mathbb{Z}}$. On the other
hand, in $\operatorname{gr}{}_{i}\mathcal{A}_{i+1}$ we have a basis
$\operatorname{gr}[{}_{j}\mathcal{R}_{i}^{\lambda}][f]$ parametrized by a
weight $\lambda$ and a function $f$. By Example 6.15 and Lemma 3.25(c) these
can be explicitly identified by setting $f$ to be the product of Schur
polynomials. Finally, having a filtered homomorphism inducing an isomorphism
on associated graded spaces gives an isomorphism.

Let us prove part (c) for $G$ arbitrary. By Corollary 5.7 the convolution
product gives a natural map

(25)
${}_{i}\mathcal{A}_{i+1}\bigotimes_{{}_{i+1}\mathcal{A}_{i+1}}\cdots\bigotimes_{{}_{j-1}\mathcal{A}_{j-1}}{}_{j-1}\mathcal{A}_{j}\to{}_{i}\mathcal{A}_{j},$

To check that (25) is injective, it is sufficient to check that it becomes an
isomorphism after localization in the multiplicative set generated by
$\\{y_{\alpha}+n\hbar+mc|\alpha\in\Phi,m,n\in{\mathbb{Z}}\\}$ which we get
from [9, Remark 3.24]. Finally, we need to prove that it is an isomorphism for
$G=GL_{n}$. To prove that it is surjective, we first consider the commutative
limit $\hbar=c=0$ and take the associated graded with respect to the Bruhat
filtration. Then surjectivity follows from Corollary 6.19.

Next, we use parts (a) and (b) of the theorem to rewrite the left hand side of
(25) as

${}_{i}\mathcal{B}_{i+1}\bigotimes_{{}_{i+1}\mathcal{B}_{i+1}}\cdots\bigotimes_{{}_{j-1}\mathcal{B}_{j-1}}{}_{j-1}\mathcal{B}_{j}={}_{i}\mathcal{B}_{j}.$

By Theorem 4.18 this is free over ${\mathbb{C}}[y_{1},\ldots,y_{n}]^{S_{n}}$.
Since the space ${}_{i}\mathcal{R}_{j+1}$ is equivariantly formal, the
bimodule ${}_{i}\mathcal{A}_{j+1}$ is free over
${\mathbb{C}}[\hbar]\otimes{\mathbb{C}}[y_{1},\ldots,y_{n}]^{S_{n}}$ as well.
Therefore (25) is surjective for general $c,\hbar$ .

∎

######  Remark 6.22.

For $G=GL_{n}$, Simental [78] classified Harish-Chandra bimodules for the
rational Cherednik algebra and proved that the shift bimodule is the unique
Harish-Chandra bimodule which sends polynomial representation to the
polynomial representation. In particular, this implies an analogue of Theorem
6.21 for the rational Cherednik algebra.

It would be interesting to know if the methods of [78] can be generalized to
the trigonometric case to give an alternate proof of Theorem 6.21.

Combining the above result with the $\operatorname{\mathrm{Proj}}$
construction we get

######  Corollary 6.23.

When $G=GL_{n}$, the graded algebra
$\bigoplus_{i}{}_{j-i}\mathcal{A}_{j}^{\hbar=c=0}$ is naturally isomorphic to
the homogeneous coordinate ring of
$\operatorname{Hilb}^{n}({\mathbb{C}}\times{\mathbb{C}}^{*})$ for any $j$.

###### Proof.

Specialize the above theorem for $c=\hbar=0$ and use Theorem 4.18. ∎

######  Remark 6.24.

One should also compare this to the results in [11] which essentially show
${}_{i}\mathcal{A}_{j}\cong\mathcal{O}(j-i)$ in the case
$G=GL_{n},N=\operatorname{Ad}\oplus V^{\ell}$ for $\ell\geq 1$, using
factorization and results about the Hilbert schemes on
$A_{\ell-1}$-resolutions.

###  6.5. A flag ${\mathbb{Z}}$-algebra

In this section, we sketch to what extent the construction of
${}_{i}\mathcal{A}_{j}$ extends to the flag level, i.e. when we replace
$\mathfrak{g}(\mathcal{O})$ by the standard Iwahori subalgebra and
$G(\mathcal{O})$ by the Iwahori subgroup $\mathbf{I}$. This gives a Springer-
theoretic construction of the ”one-step” shift bimodule
${}_{i-1}\mathcal{A}_{i}$. On the level of affine Springer fibers, the
analogous geometry is discussed in Section 7.4.

Let $ev_{0}^{-1}(\mathfrak{b})={\mathfrak{i}}$ be the standard Iwahori
subalgebra. Then $ev_{0}^{-1}(0)\subset{\mathfrak{i}}$. Consider the sequence
of subalgebras

$\mathfrak{g}_{0}:=\mathfrak{g}(\mathcal{O})\supset\mathfrak{g}_{1/2}:={\mathfrak{i}}\supset\mathfrak{g}_{1}:=t\mathfrak{g}(\mathcal{O})\supset
t{\mathfrak{i}}\supset t^{2}\mathfrak{g}(\mathcal{O})\supset
t^{2}{\mathfrak{i}}\supset\cdots$

Then, as $k$-vector spaces (but importantly, not as Lie algebras) we have the
subquotients

$\mathfrak{g}_{i}/\mathfrak{g}_{i+1/2}\cong\begin{cases}\mathfrak{b},&i\in{\mathbb{Z}}+1/2\\\
{\mathfrak{n}}_{-},&i\in{\mathbb{Z}}\end{cases}$

######  Example 6.25.

For $G=SL_{2}$ we have

$\begin{pmatrix}\mathcal{O}&\mathcal{O}\\\
{\mathfrak{m}}&\mathcal{O}\end{pmatrix}\supset\begin{pmatrix}{\mathfrak{m}}&{\mathfrak{m}}\\\
{\mathfrak{m}}&{\mathfrak{m}}\end{pmatrix}\supset\begin{pmatrix}{\mathfrak{m}}&{\mathfrak{m}}\\\
{\mathfrak{m}}^{2}&{\mathfrak{m}}\end{pmatrix}\supset\cdots$

so that
$\mathfrak{g}_{1/2}/\mathfrak{g}_{1}\cong\mathfrak{b},\;\mathfrak{g}_{1}/\mathfrak{g}_{3/2}\cong{\mathfrak{n}}_{-}$.

####  6.5.1. Bimodules

Consider now the spaces

${}_{j}\widetilde{\mathcal{R}}_{i}:=\\{[g,s]\in
G_{\mathcal{K}}\times^{\mathbf{I}}\mathfrak{g}_{i}|gs\in\mathfrak{g}_{j}\\},\;i\in\frac{1}{2}{\mathbb{Z}}$

And

${}_{j}\mathcal{R}_{i}:=\\{[g,s]\in
G_{\mathcal{K}}\times^{G_{\mathcal{O}}}\mathfrak{g}_{i}|gs\in\mathfrak{g}_{j}\\},\;i\in{\mathbb{Z}}.$

######  Proposition 6.26.

Let $[\mathfrak{b}/B]\xrightarrow{r}[\mathfrak{g}/G]$ be the Grothendieck-
Springer resolution. Then for $i\in{\mathbb{Z}}$ we have the cartesian
diagrams

${{{}_{i+1/2}\widetilde{\mathcal{R}}_{i+1/2}}}$${\left[\mathfrak{b}/B\right]}$${{{}_{i}\mathcal{R}_{i}}}$${\left[\mathfrak{g}/G\right]}$$\scriptstyle{\psi}$$\scriptstyle{\phi}$

In particular,

${}_{i}\mathcal{A}_{i}\cong\mathbf{e}\;{}_{i+1/2}\mathcal{A}_{i+1/2}\;\mathbf{e}$

by Springer theory. On the other hand, it is easy to see that

$\phi^{-1}(0)=\\{[g,s]\in
G_{\mathcal{K}}\times^{G_{\mathcal{O}}}\mathfrak{g}_{i}|gs\in
t\mathfrak{g}_{i}\\}={}_{i+1}\mathcal{R}_{i}$

and

$(r\circ\psi)^{-1}(0)=\\{[g,s]\in
G_{\mathcal{K}}\times^{\mathbf{I}}\mathfrak{g}_{i+1/2}|gs\in\mathfrak{g}_{i+1}\\}={}_{i+1}\widetilde{\mathcal{R}}_{i+1/2}.$

In particular,

${}_{i+1}\mathcal{A}_{i}=\mathbf{e}_{-}\;{}_{i+1/2}\widetilde{\mathcal{A}}_{i+1/2}\Delta\;\mathbf{e}=\mathbf{e}\;{}_{i+1}\widetilde{\mathcal{A}}_{i+1/2}\;\mathbf{e}.$

From what we have before,

${}_{i}\widetilde{\mathcal{A}}_{i}$

is the trigonometric Cherednik algebra when $i\in 1/2+{\mathbb{Z}}$. The
algebra for $i\in{\mathbb{Z}}$ is not a Cherednik algebra, but indeed a matrix
algebra over the spherical Cherednik algebra, as in [86, 85].

##  7\. Generalized affine Springer theory

###  7.1. Generalized affine Springer fibers

In this section we generalize the Springer action from [26, 42] to the line
operators discussed above. Let $\mathbf{P}$ be a parahoric subgroup,
$N_{\mathbf{P}}$ be a lattice in $N_{\mathcal{K}}$ stable under $\mathbf{P}$.
Given this data, denote $\eta=(\mathbf{P},N_{\mathbf{P}})$. Further, suppose
that

$1\to G\to{\widetilde{G}}\to G_{F}\to 1$

is an extension of algebraic groups and that $\widetilde{\mathbf{P}}$ is a
parahoric subgroup of ${\widetilde{G}}_{\mathcal{K}}$ which fits into an
extension

$1\to\mathbf{P}\to{\widetilde{\mathbf{P}}}\to(G_{F})_{\mathcal{O}}\to 1$

so that ${\widetilde{\mathbf{P}}}\cap G_{\mathcal{K}}=\mathbf{P}$. Let
${\widetilde{G}}^{\mathcal{O}}_{\mathcal{K}}$ be the preimage in
${\widetilde{G}}_{\mathcal{K}}$ of $(G_{F})_{\mathcal{O}}$.

######  Definition 7.1.

Let $v\in N_{\mathcal{K}}$. The generalized affine Springer fiber of $v$ is
the ind-subscheme of $\operatorname{Fl}_{\mathbf{P}}$ defined by

${}_{\eta}M_{v}:=\\{g\in\operatorname{Fl}_{\mathbf{P}}|g^{-1}.v\in
N_{\mathbf{P}}\\}.$

######  Remark 7.2.

Recall that if $N=\operatorname{Ad}$, $\mathbf{P}$ is a fixed parahoric
subgroup, $N_{\mathbf{P}}=\operatorname{Lie}(\mathbf{P})$,
${}_{\eta}M_{\gamma}={\mathrm{Sp}}_{\gamma}^{\mathbf{P}}$, the classical
affine Springer fiber for $\mathbf{P}$.

######  Definition 7.3.

The orbital variety of $\gamma\in N_{\mathcal{K}}$ and
$\eta=(\mathbf{P},N_{\mathbf{P}})$ is

${}_{\eta}\mathbb{O}_{\gamma}:={\widetilde{G}}_{\mathcal{K}}.\gamma\cap
N_{\mathbf{P}}.$

######  Remark 7.4.

Note that the orbital variety only depends on the lattice $N_{\mathbf{P}}$.
However, we always use it in conjunction with $\mathbf{P}$, explaining the
slightly redundant notation with $\eta$.

In particular, we have

######  Lemma 7.5 ([26, 42]).

We have an isomorphism of stacks
$[{}_{\eta}\mathbb{O}_{\gamma}/\mathbf{P}]\cong[L_{\gamma}\backslash{}_{\eta}M_{\gamma}]$,
where $L_{\gamma}$ is the stabilizer of $\gamma$ in $G_{\mathcal{K}}$.

######  Lemma 7.6.

Suppose ${}_{\eta}M_{\gamma}$ is finite-dimensional over ${\mathbb{C}}$. Then
${}_{\eta}M_{\gamma}$ admits a $L_{\gamma}$-equivariant dualizing complex
$\omega_{{}_{\eta}M_{\gamma}}$, and the equivariant Borel-Moore homology
$H_{*}^{H_{\gamma}}({}_{\eta}M_{\gamma})=:H^{\mathbf{P}}_{*}(H_{\gamma}\backslash
X_{\gamma})$ is well-defined for any algebraic subgroup $H_{\gamma}\subseteq
L_{\gamma}$. Here
$X_{\gamma}=\\{g\in{\widetilde{G}}_{\mathcal{K}}^{\mathcal{O}}\rtimes{\mathbb{C}}^{\times}|g^{-1}.\gamma\in
N_{\mathbf{P}}\\}$.

###### Proof.

We can approximate $M_{\gamma}$ by finite-type $H_{\gamma}$-stable varieties,
and then take the colimit. ∎

######  Example 7.7.

Suppose $N=\operatorname{Ad}$, and $\gamma$ is split regular semisimple. Then
$L_{\gamma}$ is a split maximal torus in $G_{\mathcal{K}}$, in particular the
loop group of a split maximal torus $T\subset G$. Let $H_{\gamma}\subset
L_{\gamma}$ be the subgroup of ”constant loops” of this torus.
$H_{*}^{H_{\gamma}}({\mathrm{Sp}}_{\gamma})$ is studied e.g. in [51, 30, 1,
15].

######  Remark 7.8.

Note that generically, $L_{\gamma}$ is a torus in $G_{\mathcal{K}}$. We may
extend the setup to the flavor-deformed equivariant version by considering
${}_{\eta}\widetilde{\mathbb{O}}_{\gamma}:={\widetilde{G}}_{\mathcal{K}}^{\mathcal{O}}\rtimes{\mathbb{C}}^{\times}.\gamma$
and and its quotient by ${\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}$
instead. We leave constructions of these extended notions to the reader or
refer to [26].

Suppose that ${\widetilde{G}}=G\times{\mathbb{C}}^{*}$, so
$G_{F}={\mathbb{C}}^{*}$ is the flavor group above. The group
${\widetilde{G}}$ acts on $N$ via $v\mapsto hg^{-1}\gamma g$. We denote the
resulting GASF

$\underline{M}_{\gamma}={\widetilde{G}}_{\mathcal{K}}.\gamma\cap
N_{\mathbf{P}}/{\widetilde{\mathbf{P}}}$

Since $[h]=[t^{d}]\in X^{*}({\mathbb{C}}^{*})={\mathbb{Z}}$ we see that
$\underline{M}_{\gamma}$ splits into components

$M_{t^{-d}\gamma}\cong\\{g\in\operatorname{Gr}_{G}|t^{d}g^{-1}\gamma
g=g^{-1}t^{d}\gamma
g\in\mathfrak{g}(\mathcal{O})\\}={\mathrm{Sp}}_{t^{-d}\gamma}.$

We recognize this to be the affine Springer fiber of $t^{-d}\gamma$ in
$\operatorname{Fl}_{\mathbf{P}}$, or in other words that

$\underline{M}_{\gamma}=\bigsqcup_{d\in{\mathbb{Z}}}M_{t^{d}\gamma}.$

###  7.2. Springer action from the Coulomb perspective

We now define the Coulomb branch version of the Springer action, and in
particular the geometric action of our ${\mathbb{Z}}$-algebra.

######  Theorem 7.9.

The following convolution diagram defines naturally associative maps

$H_{*}^{{\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}}({}_{\eta}\mathcal{R}_{\eta^{\prime}})\otimes
H_{*}^{L_{\gamma}}({}_{\eta^{\prime}}M_{\gamma})\to
H_{*}^{L_{\gamma}}({}_{\eta}M_{\gamma}).$
${{}_{\eta}\mathcal{R}_{\eta^{\prime}}\times{}_{\eta^{\prime}}\mathbb{O}_{\gamma}}$${p^{-1}({}_{\eta}\mathcal{R}_{\eta^{\prime}}\times{}_{\eta^{\prime}}\mathbb{O}_{\gamma})}$${q(p^{-1}({}_{\eta}\mathcal{R}_{\eta^{\prime}}\times{}_{\eta^{\prime}}\mathbb{O}_{\gamma}))}$${{}_{\eta}\mathbb{O}_{\gamma}}$${G_{\mathcal{K}}\times_{\mathbf{P}^{\prime}}N_{\mathbf{P}^{\prime}}\times{}_{\eta^{\prime}}\mathbb{O}_{\gamma}}$${G_{\mathcal{K}}\times{}_{\eta^{\prime}}\mathbb{O}_{\gamma}}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{q}$$\scriptstyle{m}$

Here $p:(g,s)\mapsto([g,s],s)$, $q$ is the quotient by the diagonal action of
$\mathbf{P}^{\prime}$ and $m$ is the map sending $[g,s]\mapsto g.s$.

###### Proof.

We explain the maps $p,q,m$ induce in BM homology. Consider the space

${}_{\eta}\mathcal{P}_{\eta^{\prime}}:=\\{(g,s)\in{\widetilde{G}}^{\mathcal{O}}_{\mathcal{K}}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}\times
N_{\mathbf{P}^{\prime}}|g^{-1}.s\in N_{\mathbf{P}}\\}$

and note there are maps $\pi_{1}:{}_{\eta}\mathcal{P}_{\eta^{\prime}}\to
N_{\mathbf{P}^{\prime}}$ and $(g,s)\mapsto s$ and
$\pi_{2}:{}_{\eta}\mathcal{P}_{\eta^{\prime}}\to N_{\mathbf{P}}$ given by
$(g,s)\mapsto g^{-1}.s$. Then consider
$\mathcal{F}^{v}_{\mathbf{P},N_{\mathbf{P}}}:=\omega_{{}_{\eta^{\prime}}\mathbb{O}_{\gamma}}[-2\dim{\widetilde{\mathbf{P}}}]$,
which is an object in the
${\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}$-equivariant
derived category of $N_{\mathbf{P}}$.

First of all, we have the ”pull-back with support” map $p^{*}$ (see [9,
Section 3(ii)])

$\displaystyle p^{*}:\,$ $\displaystyle
H_{{\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}\times{\widetilde{\mathbf{P}}}^{\prime}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}}^{-*}({}_{\eta}\mathcal{R}_{\eta^{\prime}}\times
N_{\mathbf{P}},(\omega_{{}_{\eta}\mathcal{R}_{\eta^{\prime}}})\boxtimes({}_{\eta^{\prime}}\mathcal{F}_{\gamma}))$
(26)
$\displaystyle=H_{*}^{{\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}}({}_{\eta}\mathcal{R}_{\eta^{\prime}})\otimes
H_{*}^{{\widetilde{\mathbf{P}}}^{\prime}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}}({}_{\eta^{\prime}}\mathbb{O}_{\gamma})\to
H^{*}_{{\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}\times{\widetilde{\mathbf{P}}}^{\prime}\rtimes{\mathbb{C}}^{\times}_{rot}}({}_{\eta}\mathcal{P}_{\eta^{\prime}},\pi_{1}^{!}({}_{\eta^{\prime}}\mathcal{F}_{\gamma})).$

Further, we have a map
$\pi_{1}^{!}{}_{\eta^{\prime}}\mathcal{F}_{\gamma}\to\pi^{!}_{2}{}_{\eta}\mathcal{F}_{\gamma}$
and since $\pi_{2}=m\circ q$, we get

$q_{*}:H^{*}_{{\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}\times{\widetilde{\mathbf{P}}}^{\prime}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}}({}_{\eta}\mathcal{P}_{\eta^{\prime}},\pi_{1}^{!}{}_{\eta^{\prime}}\mathcal{F}_{\gamma})\to
H^{*}_{{\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}}(q({}_{\eta}\mathcal{P}_{\eta^{\prime}}),m^{!}{}_{\eta}\mathcal{F}_{\gamma})$

Finally, $m$ is (ind-)proper because its fibers are closed subvarieties of a
partial affine flag variety, so that using the adjunction
$m_{!}m^{!}\to\operatorname{\mathrm{id}}$ we get a map

$(m\circ
q)_{*}:H_{*}^{{\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}}(q({}_{\eta}\mathcal{P}_{\eta^{\prime}}),m^{!}\mathcal{F}^{v}_{\mathbf{P},N_{\mathbf{P}}})\to
H_{*}^{{\widetilde{\mathbf{P}}}\rtimes{\mathbb{C}}^{\times}_{\mathrm{rot}}}({}_{\eta}\mathbb{O}_{\gamma})=H_{*}^{L_{\gamma}}({}_{\eta}M_{\gamma})$

See [26, Theorem 4.5] and [42] for more details, for example the proof of
associativity of the maps.

∎

While the convolution diagram in Theorem 7.9 is rather abstract and the maps
in Borel-Moore homology involved are defined sheaf-theoretically, in easy
cases it is possible to analyze the action as follows. Similar to [26, Section
4.2], we define the Hecke stack for $\gamma,\eta$ which has
${\mathbb{C}}$-points

${}_{\eta}\mathcal{R}_{\eta^{\prime}}^{\gamma}({\mathbb{C}})=\\{(s_{2},g,s_{1})\in{}_{\eta}\mathbb{O}_{\gamma}\times
G_{\mathcal{K}}\times{}_{\eta^{\prime}}\mathbb{O}_{\gamma}|g.s_{1}=s_{2}\\}/\mathbf{P}.$

Here the quotient is by the action $h.(s_{2},g,s_{1})=(s_{2},gh^{-1},hs_{1})$.
There is a natural Schubert stratification of
${}_{\eta}\mathcal{R}^{\gamma}_{\eta^{\prime}}$ inherited from
${}_{\eta}\mathcal{R}_{\eta}$, where

${}_{\eta}\mathcal{R}^{\gamma}_{\eta^{\prime}}\hookrightarrow{}_{\eta}\mathcal{R}_{\eta^{\prime}}$

via $[s_{2},g,s_{1}]\mapsto[g,s_{1}]$. Similarly we have maps

(27)
${{}_{\eta}\mathcal{R}^{\gamma}_{\eta^{\prime}}}$${{}_{\eta}M_{\gamma}}$${{}_{\eta^{\prime}}\mathbb{O}_{\gamma}}$

We will use this diagram later on in our computation of certain shift maps.

In the adjoint case, the name ”Springer action” is warranted, as it coincides
with the action defined by Yun, Oblomkov-Yun [72] (and Varagnolo-Vasserot
[82]):

######  Theorem 7.10.

Let $N=\operatorname{Ad}$ and
$\eta=(\mathbf{I},N_{\mathbf{I}})=\eta^{\prime}$. Then the action of the
algebra $\widetilde{\mathcal{A}}_{G,\mathbf{I}}$ on
$H_{*}^{L_{\gamma}}(M_{\gamma})$ defined by Theorem 7.9 coincides with the one
defined in [72] on the equivariant homology of affine Springer fibers, under
the isomorphism of Theorem 6.1

###### Proof.

Theorem 6.1 shows that the Springer action of simple reflections in the affine
Weyl group is the same. The equivariant parameters act by Chern classes of
line bundles on the affine flag variety, and that the relations are the same
follows from Theorem 6.1. ∎

The novel feature in allowing arbitary $\eta,\eta^{\prime}$ shows the
following.

######  Corollary 7.11.

The convolution product in Theorem 7.9 gives maps

${}_{j}\mathcal{A}_{i}^{\hbar}\otimes H_{*}^{L_{\gamma}}(M_{t^{i}\gamma})\to
H_{*}^{L_{\gamma}}(M_{t^{j}\gamma})$

that naturally assemble into an action of the ${\mathbb{Z}}$-algebra
$B^{\hbar}=\bigoplus_{i\leq j}{}_{j}\mathcal{A}_{i}^{\hbar}$ Moreover, the
action in Theorem 5.6 not including loop rotation, i.e. setting $\hbar=0$,
defines maps

$H_{*}^{G_{\mathcal{O}}}(\mathcal{R}_{{\widetilde{G}},N}^{d})\times
H_{*}^{L}(M_{\gamma}^{d^{\prime}})\to H_{*}^{L}(M_{\gamma}^{d+d^{\prime}}).$

In particular, the above corollary gives a geometric construction of ”column
vector” modules for our geometric ${\mathbb{Z}}$-algebra $B=\bigoplus_{i\leq
j\leq 0}{}_{i}\mathcal{A}_{j}$.

###  7.3. The adjoint case

In the case $N=\operatorname{Ad}$, the construction of these affine Springer
theoretic modules is also closely related to the construction of a commutative
(partial) resolution as in the previous sections, in the following way. For
$G=GL_{n}$, by the results of [11], the commutative limit
${}_{i+d}\mathcal{A}_{i}^{\hbar=0}$ is identified with the sections of
$\mathcal{O}(d)$ on the Hilbert scheme of points
$\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}})$. In
particular,

$\operatorname{\mathrm{Proj}}\bigoplus_{d\geq
0}{}_{i+d}\mathcal{A}_{i}^{\hbar=0}\cong\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}}).$

In general, we have the following proposition.

######  Proposition 7.12.

Let $\hslash=0$. Then every $\gamma\in N_{\mathcal{K}}$ as in Theorem 7.9
gives a quasicoherent sheaf $\mathcal{F}_{\gamma}$ on the partial resolution
of the Coulomb branch given by

$\operatorname{\mathrm{Proj}}\bigoplus_{d\geq
0}{}_{i+d}\mathcal{A}_{i}^{\hbar=0}.$

######  Corollary 7.13.

When $G=GL_{n}$, the above construction gives a quasicoherent sheaf on
$\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}})$ associated
to $\gamma\in N_{\mathcal{K}}$.

It is in general hard to compute which sheaf this is. In all examples we have
checked, this should be a coherent sheaf for regular semisimple elements. This
is a conjecture that we discuss in Section 8.

Recall that Lemma 2.13 tells us that the support of $\mathcal{F}_{\gamma}$ is
determined by $H_{\gamma}$, i.e. the equivariance we consider, as well as the
splitting type of $\gamma$.

######  Example 7.14.

When $\gamma=zt^{d}$ as above, we get twists of the ”Procesi bundle” as shown
in [51] which are supported everywhere. See Proposition 8.11 for the precise
statement.

When $\gamma$ is elliptic, these sheaves are supported on the punctual Hilbert
scheme over $(1,0)\in{\mathbb{C}}^{\times}\times{\mathbb{C}}$.

###  7.4. Action on representations

Let ${\widetilde{\mathrm{Sp}}}_{\gamma}^{\prime}=\\{g\mathbf{I}|g\gamma
g^{-1}\in t\mathfrak{g}(\mathcal{O})\\}$. Consider the Springer module
$M_{\gamma}=H_{*}^{L_{\gamma}}({\mathrm{Sp}}_{\gamma})$. Then we have a
natural map $M_{\gamma}\to M_{t\gamma}$ given by inclusion. There are also
maps

$\widetilde{M}_{\gamma}\to\widetilde{M}_{t\gamma}^{\prime}\to\widetilde{M}_{t\gamma}$

given by inclusion, or in other words convolving with the identity or point
class in ${}_{1/2}\widetilde{\mathcal{A}}_{1}$ and then by the point class in
${}_{1}\widetilde{\mathcal{A}}_{3/2}$.

There are also natural Gysin maps
$\widetilde{M}_{t\gamma}\to\widetilde{M}_{t\gamma}^{\prime}\to\widetilde{M}_{\gamma}$.
The first one is codimension zero, and the second one codimension $\dim G/B$.
Composed, on the level of equivariant parameters, these look like the square
of the Vandermonde determinant $\Delta$. The issue arises from the
normalization in the embedding to difference-reflection operators, in which
the point class of ${}_{1}\widetilde{\mathcal{A}}_{3/2}$ is naturally
identified with $\Delta$ (and a Cartan part), but acts as the identity on
components (effectively, it cuts down the tangent spaces of the components by
$\Delta$).

Note also we have two projections

${\widetilde{\mathrm{Sp}}}_{t\gamma}^{\prime}\to{\mathrm{Sp}}_{\gamma},\;{\widetilde{\mathrm{Sp}}}_{\gamma}\to{\mathrm{Sp}}_{\gamma}$

the first one of which is a fibration, and the second one has fibers which are
usual Springer fibers (stratified fibration). Effectively, the two line
operators (for $(1/2,1)$ and $(1,3/2)$) in the flags get squeezed down to a
single one on the spherical level (the one for $(0,1)$).

######  Lemma 7.15.

We have $\dim{\mathrm{Sp}}_{t\gamma}=\dim{\mathrm{Sp}}_{\gamma}+\dim G/B.$

###### Proof.

By a result of Bezrukavnikov [6] the dimension of ${\mathrm{Sp}}_{\gamma}$ is
given by

(28)
$\dim{\mathrm{Sp}}_{\gamma}=\frac{1}{2}\left(\nu_{\mathrm{ad}}(\gamma)-\mathrm{rk}(\mathfrak{g})+\dim(\mathfrak{h}^{w})\right),$

where $w\in W$ is such that $Z(\gamma)$ is of type $w$, $\mathfrak{h}^{w}$
denotes the $w$-invariants in $\mathfrak{h}$ and $\nu_{\mathrm{ad}}(\gamma)$
is the valuation of

$\det\left(\mathrm{ad}\gamma:\mathfrak{g}(K)/Z(\gamma)\rightarrow\mathfrak{g}(K)/Z(\gamma)\right)$

It is easy to see that changing $\gamma$ to $t\gamma$ does not change $w$. The
matrix $\mathrm{ad}\gamma$ is multiplied by $t$ which changes
$\nu_{\mathrm{ad}}(\gamma)$ by $|\Phi|=2\dim G/B$, and the result follows. ∎

######  Lemma 7.16.

Let $\pi:{\widetilde{\mathrm{Sp}}}_{t\gamma}\to{\mathrm{Sp}}_{t\gamma}$ be the
natural projection. If $\gamma$ is elliptic then
$\pi^{-1}({\mathrm{Sp}}_{\gamma})$ is an irreducible component of
${\widetilde{\mathrm{Sp}}}_{t\gamma}$. More generally, if $\gamma$ is regular
semisimple, and $C$ is an irreducible component of ${\mathrm{Sp}}_{\gamma}$,
then $\pi^{-1}(C)$ is an irreducible component of
${\widetilde{\mathrm{Sp}}}_{t\gamma}$.

###### Proof.

By the proof of Lemma 2.2 the projection
$\pi^{-1}({\mathrm{Sp}}_{\gamma})\to{\mathrm{Sp}}_{\gamma}$ has fibers $G/B$
at every point. Since $\gamma$ is elliptic, ${\mathrm{Sp}}_{\gamma}$ is
irreducible and hence $\pi^{-1}({\mathrm{Sp}}_{\gamma})$ is irreducible as
well. If $C$ is as in the statement of the Lemma, the same proof goes through.

Furthermore, all components of $\widetilde{{\mathrm{Sp}}_{t\gamma}}$ have
dimension $\dim{\mathrm{Sp}}_{t\gamma}$. By Lemma 7.15 we have

$\dim\pi^{-1}({\mathrm{Sp}}_{\gamma})=\dim{\mathrm{Sp}}_{\gamma}+\dim
G/B=\dim{\mathrm{Sp}}_{t\gamma},$

and the result follows. ∎

Lemma 7.16 allows us to construct an important correspondence between
${\mathrm{Sp}}_{\gamma}$ and ${\mathrm{Sp}}_{t\gamma}$. By the work of Tsai
[81], there are $W$ many irreducible components up to the centralizer action
in ${\widetilde{\mathrm{Sp}}}_{t\gamma}$. Furthermore, we expect the
following:

######  Conjecture 7.17.

([81, Conjecture 8.6]) The Springer action on
$H_{*}({\widetilde{\mathrm{Sp}}}_{\gamma})$ yields a regular representation in
top-dimensional homology spanned by the classes of the irreducible components.

In particular, for $\gamma$ elliptic there is a distinguished component
$\pi^{-1}({\mathrm{Sp}}_{\gamma})$ and another component biregular to
${\mathrm{Sp}}_{t\gamma}$, and (assuming Conjecture 7.17) one can define a
correspondence in Borel-Moore homology sending the former to the latter (for
example, the symmetrizer $\mathbf{e}$ would suffice). More generally, fix a
component $C$ as above and note that the lattice part of the centralizer of
$\gamma$ acts transitively on the set of these components [47]. Now the class
of $\pi^{-1}(C)$ can either be sent to the class of any lattice translate of
$\pi^{-1}(C)$, or by the symmetrizer in the finite Weyl group to a one-
dimensional $W$-invariant subspace of the BM homology of
${\mathrm{Sp}}_{t\gamma}$.

This leads to the following:

######  Proposition 7.18.

Assume $\gamma$ is elliptic and $G$ is simply-connected, and assume Conjecture
7.17 holds for $\gamma$. Consider the correspondence
$\mathbf{e}[\pi^{-1}({\mathrm{Sp}}_{\gamma})]*-$ between
${\mathrm{Sp}}_{\gamma}$ and ${\mathrm{Sp}}_{t\gamma}$. The action of this
correspondence in homology corresponds to the action of some class in
${}_{i+1}\mathcal{R}_{i}$ as in Theorem 7.9, which sends the fundamental class
of ${\mathrm{Sp}}_{\gamma}$ to the fundamental class of
${\mathrm{Sp}}_{t\gamma}$.

###### Proof.

Lets construct a cycle $\Gamma^{\prime}$ in ${}_{i+1}\mathcal{R}_{i}^{\gamma}$
such that correspondence (27) with the class $\Gamma^{\prime}$ sends
$[{\mathrm{Sp}}_{\gamma}]$ to $[{\mathrm{Sp}}_{t\gamma}]$. First, we define
$\Gamma\subset{}_{i+1}\mathcal{R}_{i}^{\gamma}$ as the lift of the graph of
the embedding of ${\mathrm{Sp}}_{\gamma}$ into ${\mathrm{Sp}}_{t\gamma}$. The
lift $\Gamma$ is defined as the locus of triples
$(s_{2},g,s_{1})\in{}_{i+1}\mathcal{R}_{i}^{\gamma}$ such that
$G_{\mathcal{O}}.s_{1}=G_{\mathcal{O}}.s_{2}$.

Let $\eta=(G_{\mathcal{O}},t^{i}\mathfrak{g}(\mathcal{O}))$,
$\eta^{\prime}=(G_{\mathcal{O}},t^{i+1}\mathfrak{g}(\mathcal{O}))$ and
$\widetilde{\eta}=(\mathbf{I},t^{i}\mathfrak{g}(\mathcal{O}))$. In particular,
${}_{i+1}\mathcal{R}_{i}^{\gamma}={}_{\eta}\mathcal{R}_{\eta^{\prime}}^{\gamma}$
and on the homology of the fibers of the projection
$\widetilde{\pi}:{}_{\eta}\mathcal{R}_{\widetilde{\eta}}^{\gamma}\to{}_{i+1}\mathcal{R}_{i}^{\gamma}$
there is an action of $W$. The push-forward along the projection
$\widetilde{\pi}$ is the projection onto the $W$-invariant part of the
homology.

Let
$\widetilde{q}:{}_{\eta}\mathcal{R}_{\widetilde{\eta}}^{\gamma}\to{}_{\widetilde{\eta}}\mathbb{O}_{\gamma}$
be map from the corresponding diagram (27). The previous proposition implies
that the map $\widetilde{q}$ restricted to
$\widetilde{\Gamma}=\widetilde{\pi}^{-1}(\Gamma)$ is dominant over one of
irreducible component of ${}_{\widetilde{\eta}}\mathbb{O}_{\gamma}$. By
Conjecture 7.17 the set of irreducible components of
${}_{\widetilde{\eta}}\mathbb{O}_{\gamma}$ is a regular representation of $W$.
Thus there is $w\in W$ such $w.[\widetilde{\Gamma}]$ projects dominantly onto
${}_{\eta^{\prime}}\mathbb{O}_{\gamma}$. The class
$\Gamma^{\prime}=\widetilde{\pi}_{*}(w.[\widetilde{\Gamma}])$ satisfies
required properties.

∎

######  Corollary 7.19.

Under the assumption of the previous proposition we have the relation between
the fundamental classes:

$[{\mathrm{Sp}}_{t^{i+1}\gamma}]\in{}_{-i-1}\mathcal{A}_{-i}*[{\mathrm{Sp}}_{t^{i}\gamma}]$

##  8\. Finite generation and examples

###  8.1. Finite generation conjecture

As we saw in Section 7, in particular Theorem 7.9, the space

$\mathbf{F}_{\gamma}:=\bigoplus_{k=0}^{\infty}H_{*}({\mathrm{Sp}}_{t^{k}\gamma})$

is a graded module over the graded algebra
$\bigoplus_{d=0}^{\infty}{}_{0}\mathcal{A}_{d}$. Equivalently,
$\mathbf{F}_{\gamma}$ defines a quasi-coherent sheaf $\mathcal{F}_{\gamma}$ on
$\operatorname{\mathrm{Proj}}\bigoplus_{d=0}^{\infty}{}_{0}\mathcal{A}_{d}$.

######  Conjecture 8.1.

The module $\mathbf{F}_{\gamma}$ is finitely generated and the sheaf
$\mathcal{F}_{\gamma}$ is coherent.

Note that by Theorem 2.12 the homology of ${\mathrm{Sp}}_{t^{k}\gamma}$ is
finitely generated over ${}_{0}\mathcal{A}_{0}$. For $G=GL_{n}$ the graded
algebra
$\bigoplus_{d=0}^{\infty}{}_{0}\mathcal{A}_{d}=\bigoplus_{d=0}^{\infty}A^{d}$
is generated by ${}_{0}\mathcal{A}_{0}$ and ${}_{0}\mathcal{A}_{1}=A$, so
Conjecture 8.1 is equivalent to saying that for a given $\gamma$ there exists
$k_{0}$ such that $\mathbf{F}_{\gamma}$ is generated by
$\bigoplus_{k=0}^{k_{0}}H_{*}({\mathrm{Sp}}_{t^{k}\gamma})$ under the action
of ${}_{0}\mathcal{A}_{0}$ and ${}_{0}\mathcal{A}_{1}$.

Below we prove the conjecture in some special cases.

######  Theorem 8.2.

Conjecture 8.1 holds for $G=GL_{n}$ and
$\gamma=\operatorname{\mathrm{diag}}(s_{0},\ldots,s_{n})$ for $s_{i}\neq
s_{j}$.

###### Proof.

This follows from Proposition 8.11 below. ∎

######  Example 8.3.

Let $G=GL_{2}$ and

$\gamma=\begin{pmatrix}t&0\\\ 0&-t\end{pmatrix}.$

Then

${\mathrm{Sp}}_{t^{d}\gamma}\cong\bigsqcup_{{\mathbb{Z}}}C_{d+1}$

where each $C_{d+1}$ is an infinite chain of $\mathbb{P}^{d+1}$, consecutive
members of which intersect transversally along a $\mathbb{P}^{d}$. These
$\mathbb{P}^{d}$ are Spaltenstein varieties of $d$-planes in $2d$-space stable
under a nilpotent element with Jordan blocks of sizes $(d,d)$, motivically
equivalent to projective spaces $\mathbb{P}^{d}$. The inclusion maps are again
embedding the chains into one another and they are regular embeddings (because
they are effective Cartier divisors).

Note that the direct sum of homologies of these $\mathbb{P}^{d+1}$ surjects
onto the homology of ${\mathrm{Sp}}_{t^{d}\gamma}$. Let us prove that the
module $\mathbf{F}_{\gamma}$ is generated by the homology of
${\mathrm{Sp}}_{\gamma}$ under the action of ${}_{0}\mathcal{A}_{0}$ and
${}_{0}\mathcal{A}_{1}$.

Indeed, ${\mathrm{Sp}}_{\gamma}$ is just a discrete set of points in bijection
with the affine Weyl group. Its homology is a free rank two module over the
lattice of translations.

In case of elliptic $\gamma$ the finite generation conjecture follows from the
stable of ${}_{0}\mathcal{A}_{0}$-cyclicity of the homology of
${\mathrm{Sp}}_{t^{k}\gamma}$:

######  Proposition 8.4.

Let us assume that $\gamma$ is elliptic, Conjecture 7.17 holds for
$t^{k}\gamma$ and $[{\mathrm{Sp}}_{t^{k}\gamma}]\in
H_{*}({\mathrm{Sp}}_{t^{k}\gamma})$ is the fundamental class. Then if there
exists $N$ such that
$H_{*}({\mathrm{Sp}}_{t^{k}\gamma})={\mathbb{C}}[T^{*}T^{\vee}]^{W}\cdot[{\mathrm{Sp}}_{t^{k}\gamma}]$
for $k\geq N$ then the conjecture 8.1 holds for $\mathbf{F}_{\gamma}$.

###### Proof.

The actions of ${}_{0}\mathcal{A}_{0}={\mathbb{C}}[T^{*}T^{\vee}]^{W}$ and
${}_{0}\mathcal{A}_{1}$ on $\mathbf{F}_{\gamma}$ commute. Hence by Proposition
7.18 the submodule $\oplus_{k\geq N}H_{*}({\mathrm{Sp}}_{t^{k}\gamma})$ is
generated by ${}_{0}\mathcal{A}_{0}$ and ${}_{0}\mathcal{A}_{1}$ from
$[{\mathrm{Sp}}_{t^{N}\gamma}]$. The module
$\oplus_{k<N}H_{*}({\mathrm{Sp}}_{t^{k}\gamma})$ is finite dimensional. ∎

The subalgebra ${\mathbb{C}}[\mathfrak{t}]^{W}$ is isomorphic to the
cohomology ring $H^{*}(\operatorname{Gr}_{G})$ it acts on
$H_{*}({\mathrm{Sp}}_{\gamma})$ by cap product. It is natural conjecture that
a stronger version of the condition of the previous proposition is true for
$G=PGL_{n}$.

######  Conjecture 8.5.

Let $\mathfrak{g}=\mathrm{Lie}(PGL_{n})$ and
$\gamma\in\mathfrak{g}(\mathcal{O})$ is an elliptic regular semisimple
topologically nilpotent element. Then

$H_{*}({\mathrm{Sp}}_{\gamma})=H^{*}(\operatorname{Gr}_{G})\cap[{\mathrm{Sp}}_{\gamma}].$

If $G=GL_{n}$ or $G=SL_{n}$ and $\gamma\in\mathfrak{g}(\mathcal{O})$ is an
elliptic element then ${\mathrm{Sp}}_{\gamma}$ has many connected components
and the group $\pi_{0}(G_{\gamma})$ permutes the connected components. In the
light of aforementioned Theorem 2.12 it is natural to propose

######  Conjecture 8.6.

Let $\mathfrak{g}=\mathrm{Lie}(GL_{n})$ or $\mathfrak{g}=\mathrm{Lie}(SL_{n})$
and $\gamma\in\mathfrak{g}(\mathcal{O})$ is an elliptic regular semisimple
topologically nilpotent element. Then

$H_{*}({\mathrm{Sp}}_{\gamma})={\mathbb{C}}[T^{*}T^{\vee}]^{W}[{\mathrm{Sp}}_{\gamma}].$

######  Remark 8.7.

The conjecture is false outside of type $A$ since there are examples of
elliptic affine Springer fibers with homology of not of type $(p,p)$ [47, 72].
Note however that $H_{*}({\mathrm{Sp}}_{\gamma})$ is always finitely generated
under ${\mathbb{C}}[T^{*}T^{\vee}]^{W}$ by [89] (see also Lemma 2.13).

For the homogeneous elements Conjecture 8.6 is known [73] and one can deduce

######  Theorem 8.8.

Conjecture 8.1 holds for $G=GL_{n}$ and equivalued $\gamma_{m,n}$ with
characteristic polynomial $x^{m}-y^{n}$, $\gcd(m,n)=1$.

###### Proof.

The affine Grassmanian $\operatorname{Gr}_{G}$ has
$\pi_{1}(GL_{n})={\mathbb{Z}}$ connected components
$\operatorname{Gr}_{G}=\operatorname{Gr}_{G}^{0}\times{\mathbb{Z}}$.
Respectively, we have
${\mathrm{Sp}}_{\gamma}={\mathrm{Sp}}_{\gamma}^{0}\times{\mathbb{Z}}$.

Observe that if $\gamma_{m,n}$ is equivalued with characteristic polynomial
$x^{m}-y^{n}$ then $t^{k}\gamma_{m,n}$ is equivalued with characteristic
polynomial $x^{kn+m}-y^{n}$. The compactified Jacobian $J_{m/n}$ of the one-
point compactification of the planar curve $\\{x^{m}-y^{n}\\}$ is irreducible
and homeomorphic to ${\mathrm{Sp}}_{\gamma_{m,n}}^{0}$ [73]. Moreover,
$\operatorname{Gr}_{G}^{0}=\operatorname{Gr}_{PGL_{n}}$ and the
${\mathrm{Sp}}_{\gamma_{m,n}}^{0}$ is the corresponding Springer fiber.

It is shown in [73] that for ${\mathrm{Sp}}_{\gamma_{m,n}}^{0}$ Conjectures
7.17 and 8.5 are true. The group $\pi_{0}(G_{\gamma_{m,n}})={\mathbb{Z}}$ acts
transitively on the connected components of ${\mathrm{Sp}}_{\gamma}$, hence
Conjecture 8.6 is true for ${\mathrm{Sp}}_{\gamma_{m,n}}$. Thus the theorem
follows from Proposition 8.4.

∎

######  Example 8.9.

For $G=GL_{2}$ and $\gamma=\gamma_{1,2}$ we recover the ${\mathbb{Z}}$-algebra
module from Example 4.17.

###  8.2. Examples in type A

In the case $G=GL_{n}$ the sheaf $\mathcal{F}_{\gamma}$ can be described in
terms of geometry of
$\operatorname{Hilb}_{n}({\mathbb{C}}^{\times}\times{\mathbb{C}})$ for some
homogeneous $\gamma$’s.

######  Proposition 8.10.

Let $\gamma$ be homogeneous of slope $(kn+1)/n$. Then $\mathcal{F}_{\gamma}$
is isomorphic to the restriction of $\mathcal{O}(k)$ to the punctual Hilbert
scheme at $(1,0)\in{\mathbb{C}}^{\times}\times{\mathbb{C}}$.

###### Proof.

The localized equivariant homology
$H_{*}^{{\mathbb{G}}_{m}}({\mathrm{Sp}}_{\gamma})$ affords the unique finite-
dimensional representation of
$\mathbf{e}\mathbb{H}_{-\frac{kn+1}{n}}\mathbf{e}$ as was checked in [72, 83].
By Lemma 2.13, $\mathcal{F}_{\gamma}$ is supported on this punctual Hilbert
scheme (i.e. the corresponding fiber of the Hilbert-Chow map). Completing our
${\mathbb{Z}}$-algebra at a neighborhood of the identity in $T^{\vee}$, we get
a completion of the rational Cherednik algebra and the corresponding
bimodules, for $GL_{n}$ with parameters given by integral shifts of
$(kn+1)/n$. The Gordon-Stafford construction then implies [36] that the
corresponding sheaf on the punctual Hilbert scheme coincides with
$\mathcal{O}(k)$.

∎

######  Proposition 8.11.

Let $\gamma$ be homogeneous of slope $k$, or more generally equivalued of
valuation $k$. Then $\mathcal{F}_{\gamma}$ is isomorphic to
$\mathcal{P}\otimes\mathcal{O}(k)$ where $\mathcal{P}$ is the Procesi sheaf on
$\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}})$.

###### Proof.

For equivalued $\gamma$ of valuation $k$ the main result of [51] identifies
the equivariant Borel-Moore homology $H^{T}_{*}({\mathrm{Sp}}_{\gamma})$ with
the space of global sections of $\mathcal{P}\otimes\mathcal{O}(k)$ on
$\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}})$ as a module
over the algebra of global functions

${}_{0}\mathcal{A}_{0}={\mathbb{C}}[T^{*}T^{\vee}]^{W}={\mathbb{C}}[x_{1}^{\pm},\ldots,x_{n}^{\pm},y_{1},\ldots,y_{n}]^{S_{n}},$

By the work of Haiman [40] we get:

(29)
$H^{0}(\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}}),\mathcal{P}\otimes\mathcal{O}(k))=\bigcap_{i\neq
j}\langle 1-x_{i}/x_{j},y_{i}-y_{j}\rangle^{k},\quad
H^{i}(\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}}),\mathcal{P}\otimes\mathcal{O}(k))=0,\quad
i>0.$

By Theorem 3.5 the graded algebra ${}_{0}\mathcal{A}_{\bullet}$ is generated
by the degree 1 component ${}_{0}\mathcal{A}_{1}=A$, where $A$ is the space of
antisymmetric polynomials in
${\mathbb{C}}[x_{1}^{\pm},\ldots,x_{n}^{\pm},y_{1},\ldots,y_{n}]$.

It is easy to see that by (29) we have a correctly defined map

$A\otimes
H^{0}(\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}}),\mathcal{P}\otimes\mathcal{O}(k))\to
H^{0}(\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}}),\mathcal{P}\otimes\mathcal{O}(k+1)),$

and it follows from [51] that this agrees with the convolution
${}_{0}\mathcal{A}_{1}\otimes H^{T}_{*}({\mathrm{Sp}}_{\gamma})\to
H^{T}_{*}({\mathrm{Sp}}_{t\gamma})$. This completes the proof. ∎

For general $\gamma$ elliptic of slope $\frac{m}{n}$, the situation is as
follows. Whilst our construction gives a sheaf $\mathcal{F}_{\gamma}$, which
is coherent by Proposition 8.4, we do not know how to identify this sheaf on
$\operatorname{Hilb}^{n}({\mathbb{C}}^{\times}\times{\mathbb{C}})$. Indeed, a
variant of this problem already appears in [36, Problem 5.5.].

###  8.3. Beyond type A

For general $G$, both the computations of the cohomology of affine Springer
fibers and the sheaves on
$\widetilde{\mathfrak{C}}_{\check{G},\check{\mathfrak{g}}}$ are very
complicated. It would be for example interesting to compute the sheaf one gets
from the Bernstein-Kazhdan example of [47, Appendix]. Nevertheless, we have
the following analogue of Proposition 8.11 for general $G$.

######  Theorem 8.12.

Let $G$ be arbitrary and $\gamma$ equivalued of valuation $k$. Then we have
the isomorphism of graded modules

$\mathbf{F}_{\gamma}=\bigoplus_{j=0}^{\infty}H^{T}_{*}({\mathrm{Sp}}_{t^{j}\gamma})\simeq\bigoplus_{j=0}^{\infty}\bigcap_{\alpha\in\Phi^{+}}\langle
1-\alpha^{\vee},y_{\alpha}\rangle^{k+j}$

over the graded algebra
$\bigoplus{}_{0}\mathcal{A}(0)_{d}\simeq\bigoplus\mathbf{e}_{d}\bigcap_{\alpha\in\Phi^{+}}\langle
1-\alpha^{\vee},y_{\alpha}\rangle^{d}$. As a consequence, the corresponding
sheaves over
$\operatorname{\mathrm{Proj}}\bigoplus{}_{0}\mathcal{A}(0)_{d}=\widetilde{\mathfrak{C}}_{\check{G}}$
are isomorphic as well.

###### Proof.

The proof is similar to Proposition 8.11. By the main result of [51] the
isomorphism holds for each $j$ separately on the level of modules over
${}_{0}\mathcal{A}(0)_{0}\cong{\mathbb{C}}[T^{*}T^{\vee}]^{W}$. The comparison
of the action of ${}_{0}\mathcal{A}(0)_{d}$ follows from Theorem 3.8 and the
constructions in [51, 26]. More precisely, the result in [51] identifies
$\Delta^{j}H^{T}_{*}({\mathrm{Sp}}_{t^{j}\gamma})$ with
$\bigcap_{\alpha\in\Phi^{+}}\langle 1-\alpha^{\vee},y_{\alpha}\rangle^{k+j}$
inside ${\mathbb{C}}[T^{*}T^{\vee}]\cong H_{*}^{T}(\operatorname{Gr}_{T})$
using GKM localization. The latter has a multiplication structure which
coincides with convolution on the Coulomb branch for $T$ with zero matter. The
fact that the convolution action for ${}_{0}\mathcal{A}(0)_{0}$ respects the
localization is [26, Proposition 4.15.]. ∎

Let $G$ be quasisimple of adjoint type. Respectively, let $cox\in W$ be the
Coxeter element of the Weyl group of $G$ and $n$ be the order of $cox$. For
any $m$ co-prime with $n$ there is a regular semisimple element
$\gamma_{m,n}\in\mathfrak{g}(\mathcal{O})$ which is homogeneous:
$\gamma_{m,n}(\lambda\cdot
t)=\lambda^{m/n}\operatorname{Ad}_{g(\lambda)}\gamma_{m,n}(t)$, $g(\lambda)\in
G$. The element $\gamma_{m.n}$ is unique up to rescaling and conjugation, an
explicit construction of $\gamma_{m,n}$ can found for example in [72]. The
element $\gamma_{m,n}$ is equivalued of valuation $m/n$.

The stabilizer in ${\widetilde{G}}_{\mathcal{K}}\rtimes{\mathbb{G}}_{m}$ is
given by $L_{\gamma_{m,n}}=\mathbb{G}_{m}$, and it acts naturally on
${\mathrm{Sp}}_{\gamma_{m,n}}$. It is shown in [72] that
$\dim{\mathrm{Sp}}_{\gamma_{m,n}}^{\mathbb{G}_{m}}=0$, the fixed points are
isolated and that the localized homology
$H_{*}^{\mathbb{G}_{m}}({\mathrm{Sp}}_{\gamma_{m,n}})\otimes{\mathbb{C}}(\hbar)$
is generated by tautological classes
$H^{*}_{\mathbb{G}_{m}}(\operatorname{Gr}_{G})$ from the fundamental class
$[{\mathrm{Sp}}_{\gamma_{m,n}}]$. We expect the generation statement in the
non-equivariant setting:

######  Conjecture 8.13.

Let $G$,
$\mathrm{Lie}(\mathfrak{g})$,$\gamma_{m,n}\in\mathfrak{g}(\mathcal{O})$ are as
above, then

$H_{*}({\mathrm{Sp}}_{\gamma_{m,n}})=H^{*}(\operatorname{Gr}_{G})\cap[{\mathrm{Sp}}_{\gamma_{m,n}}].$

Note also that this ”Coxeter case” gives the so called spherical simple
modules of the dDAHA, as first observed in [83]. More generally, the slopes
with so called regular elliptic denominators yield (spherical and other)
finite-dimensional modules of the dDAHA [83, 72]. Since $\gamma$ elliptic
implies $t\gamma$ elliptic, one sees that the tensor products by the shift
bimodules ${}_{i-1}\mathcal{B}_{i}$ send finite-dimensional modules to finite-
dimensional modules, which one could also deduce from the theory of shift
functors for dDAHA like in [4]. As far as the authors are aware, this theory
is still undeveloped, but would potentially give insight on the $m=1,n=h$ case
of Proposition 8.10 for other groups.

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