\section{Appendix: exactness of hyperbolic restriction}
\label{sec:append-exactn}

\newcommand\SBun{\operatorname{Bun}}
\newcommand\tcalA{\widetilde{\calA}}
\newcommand\NN{\mathbb N}
\newcommand\calU{\mathcal U}


\newcommand\calR{\mathcal R}
\newcommand\calS{\mathcal S}


\newcommand\del{\delta}

\newcommand\tilT{\widetilde T}
\newcommand\tilPhi{\widetilde\Phi}

\newcommand\CA{\mathcal A}
\newcommand\CU{\mathcal U}
\newcommand\lam{\lambda}
\newcommand\BA{\mathbb A}
\newcommand\calE{\mathcal E}
\newcommand\oS{\overline{S}}


\subsection{Zastava spaces}
Let us denote by
$\SBun_{G,B}$ the corresponding moduli space of $G$-bundles endowed with the following
structures:

a) A trivialization at the infinite line $\PP^1_{\infty} = \linf$.
\begin{NB}
    Added by H.N.
\end{NB}

b) A $B$-structure on the horizontal line $\PP^1_h = \{ y=0\}$.
\begin{NB}
    Added by H.N.
\end{NB}

\noindent
These two structures are required to be compatible at the intersection of $\PP^1_{\infty}$ and $\PP^1_h$ in the obvious way.

The connected components of $\SBun_{G,B}$ are numbered by positive elements of the coroot lattice of $G_{\aff}$ (cf.\ \cite[\S9]{BFG}); for such element
$\alp$ we denote by $\SBun_{G,B}^{\alp}$ the corresponding connected component.

We will also denote by $\calZ^{\alp}_G$\index{ZalphaG@$\calZ^\alpha_G$} the corresponding ``Zastava" space (a.k.a.\ ``flag Uhlenbeck space")
defined in \cite{BFG}. We are going to need the following properties of $\calZ^{\alp}_G$.

(Some of them are proved for the space
$\operatorname{QMap}(\proj^1_h,\mathscr G_{\g,\mathfrak p})$ of based
quasi-maps to a flag scheme $\mathscr G_{\g,\mathfrak p}$ of a
Kac-Moody Lie algebra $\g$ associated with its parabolic $\mathfrak
p$. Since $\calZ^{\alp}_G$ is the fiber product
$\operatorname{QMap}(\proj^1_h,\mathscr G_{\g,\mathfrak b})
\times_{\operatorname{QMap}(\proj^1_h,\mathscr G_{\g,\mathfrak p})}
\Uh{d}$ for a Borel subalgebra $\mathfrak b$ of an affine Lie
algebra $\g$ and a maximal parabolic $\mathfrak p$, we can deduce
assertions for $\mathcal Z^\alpha_G$ from those for $\operatorname{QMap}(\proj^1_h,\mathscr G_{\g,\mathfrak p})$.)

\begin{NB}
(all proved in \cite{BFG}):
\end{NB}

(Z1) $\calZ^{\alp}_G$ is an irreducible affine scheme of dimension
$2|\alpha|$
\begin{NB}
    $2d|\alp|$ ?
\end{NB}
endowed with an action of $T\times \CC^*\times \CC^*$ which contains $\SBun_{G,B}^{\alp}$ as an open subset
(here we set $|\alp|=\sum a_i$ if $\alp=\sum a_i\alp_i$ where $\alp_i$ are the simple coroots of $G_{\aff}$).

(Z2) There is a (factorization) map $\pi^{\alp}_{\calZ}\colon
\calZ^{\alp}_G\to S^{\alp}(\AA^1_h)$. This map is $T\times \CC^*\times
\CC^*$-equivariant if we let $T\times \CC^*\times \CC^*$ act on
$S^{\alp}(\AA^1_h)$ just through the horizontal $\CC^*$ (denoted by
$\CC^*_h$) and it admits a $T\times \CC^*\times \CC^*$-equivariant
section $\iota^{\alp}$. In particular, the fibers of
$\pi^{\alp}_{\calZ}$ are stable under $T\times \CC^*_v$ where the
$\CC^*_v = \CC^*$-action comes from the vertical action on $\AA^2$.
All of these fibers have dimension $|\alpha|$.
\begin{NB}
$d|\alp|$ ?
\end{NB}

(See Conjecture~2.27, which is reduced to Conjecture~15.3 and proved
for affine Lie algebras in \S15.6 in \cite{BFG}.)
\begin{NB}
    Added by H.N.
\end{NB}

(Z3) Let set $\calF^{\alp}=(\pi^{\alp}_{\calZ})^{-1}(\alp\cdot 0)$. Let $\rho\colon\mathbb C^*\to\tilT=T\times \CC^*_v$ be any one-parameter subgroup which is a regular dominant coweight of $G_{\aff}$ (i.e.\ such that $\langle \rho,\beta\rangle>0$ for any affine positive
root $\beta$).

Then the corresponding $\CC^*$-action contracts $\calZ^{\alp}_G$ to
$\iota^{\alp}(S^\alp(\AA_h^1))$, and hence $\mathcal F^\alpha$ to
$\iota^\alpha(\alpha\cdot 0)$.
\begin{NB}
    Originally, it was written
    \begin{quote}
Then the corresponding $\CC^*$-action contracts $\calF^{\alp}$ to the point $\iota^{\alp}(\alp\cdot 0)$.
    \end{quote}
    This formulation, a priori, claims only $\mathcal F^\alpha\subset
    \mathcal A_{\iota^\alpha(\alpha\cdot 0)}$, instead of $\mathcal
    F^\alpha = \mathcal A_{\iota^\alpha(\alpha\cdot 0)}$. But the
    latter is proved (and used later), as explained in Misha's message
    on May 19.
\end{NB}

(cf.\ Proposition 2.6

and Corollary 10.4
\begin{NB}
    added by H.N.
\end{NB}
in \cite{BFG}).

(Z4) Let $\alp_0$ denote the affine simple coroot and let $d$ be the coefficient of $\alp_0$ in $\alp$ (in other
words, $d=\langle \alp,\omega_0\rangle$ where $\omega_0$ denotes the corresponding fundamental weight of $G_{\aff}$). Then
there is a (``forgetting the $B$-structure") $T\times \CC^*\times\CC^*$-equivariant map $f_{\alp}\colon \calZ^{\alp}_G\to \calU^d_G$ which fits into
a commutative diagram
$$
\begin{CD}
\calZ^{\alp}_G @> f_{\alp}>> \calU^d_G\\
@V\pi^{\alp}_{\calZ}VV @VV\pi^d_G V\\
S^{\alp}\AA^1_h @>>> S^d\AA^1_h
\end{CD}
$$
where the bottom horizontal map sends a divisor $\sum \beta_i x_i$ to $\sum \langle \beta_i,\omega_0\rangle x_i$.


\subsection{Plan of the proof}
Let us discuss our strategy for proving Theorem \ref{thm:perverse}.
As we have explained in \subsecref{subsec:hypsemismall}, 
it follows from dimension estimates of attracting and repelling
sets by using arguments similar to those of \cite{MV2}.
\begin{NB}
Original : edited on Oct.5, 2014.

Using arguments similar to those of \cite{MV2}
it is not difficult to see that Theorem \ref{thm:perverse} would follow
if we could prove that for any $x\in \calU^d_L$ the dimension of $\calA_x$ and $\calR_x$ is equal to
$\frac{\dim \calU^d_G-\dim\calU^d_L}{2}$.     
\end{NB}
However, at the moment we do not know how to prove estimates
\begin{NB}
this     
\end{NB}
directly. So, our actual strategy will
be slightly different. First, recall that we have
$$
(\calU^d_G)^T=S^d(\AA^2),
$$
and that we denote by $\Uh[B]d$, $\Uh[B_-]d$ the corresponding attracting and
repelling sets. Also we denote by $p\colon \calU_B^d\to S^d(\AA^2)$ the corresponding map (sometimes we shall
denote it by $p^d$ when dependence on $d$ is important). Then we are going to proceed in the following way:

1) Prove that the dimension of the preimage of $S^d(\AA^1)\subset S^d(\AA^2)$ under the map
$p\colon \calU^d_B\to S^d(\AA^2)=\calU^d_{T,G}$ has dimension $\frac{\dim \calU^d_G}{2}$ (here $\AA^1\subset \AA^2$ is any line). The proof will involve some facts about the Zastava spaces from \cite{BFG}.

2) Deduce Theorem \ref{thm:perverse} for $L=T$ from 1).

3) Using Proposition \ref{prop:trans} deduce Theorem \ref{thm:perverse} for arbitrary $L$ from the case $L=T$.

\subsection{Attractors and repellents on the Uhlenbeck space: maximal torus case}
Let us first look more closely at the case when $P=B$: a Borel subgroup of $G$.
In this case $L=T$: a maximal torus of $G$.


Let us also define the set $\calS^d\subset \calU^d_G$
to be the attracting set in $\calU^d_G$ with respect to the torus $T$ to
$S^d(\AA^1_v\backslash 0)$ where $\AA^1_v$ is the vertical line. In other words,
$\calS^d=p^{-1}(S^d(\AA^1_v\backslash 0))$.

\begin{Proposition}\label{prop:sd}
We have
$$
\dim \calS^d\leq dh^{\vee}=\frac{\dim \calU^d_G}{2}.
$$
\end{Proposition}
\begin{Corollary}\label{cor:dim-cor}
Let $\AA^1\hookrightarrow \AA^2$ be any linear embedding. Then
$$
\dim p^{-1}(S^d(\AA^1))\leq\frac{\dim \calU^d_G}{2}.
$$
\end{Corollary}
Corollary \ref{cor:dim-cor} clearly follows from \ref{prop:sd}. Indeed, first of all, it is clear that it is enough to prove Corollary \ref{cor:dim-cor} when $\AA^1=\AA^1_v$. In this case, $\calS^d$ is open in $p^{-1}(S^d(\AA^1_v))$,
hence we have $\dim\calS^d\leq \dim p^{-1}(S^d(\AA^1_v))$. On the other hand, (the vertical) $\AA^1$ acts naturally on
 $p^{-1}(S^d(\AA^1_v))$ by shifts and any point of $p^{-1}(S^d(\AA^1_v))$ lies in an open subset of the form
$x(\calS^d)$ for some $x\in \AA^1$, hence the opposite inequality follows.


Let us now pass to the proof of Proposition \ref{prop:sd}.

\subsection{The map \texorpdfstring{$f_d$}{fd}}
We have the natural (forgetting the flag)
birational map $f_{d\delta}\colon \calZ^{d\del}_G\to \calU^d_G$, which  we shall simply denote by $f_d$. This map gives an isomorphism
between the open subset of $\calU^d_G$ consisting of (generalized) bundles which are trivial
on the horizontal $\PP^1_h$ and the open subset of $\calZ^{d\delta}_G$ consisting of (generalized) bundles which are trivial
on the horizontal $\PP^1_h$ (and then the $B$-structure on the horizontal $\PP^1_h$ is automatically trivial).

\subsection{The central fiber}Recall that $\calF^{d\delta}$ denotes the preimage of $d\delta\cdot 0$ under the map
$\pi^{d\delta}_{\mathcal Z}\colon\calZ^{d\delta}_G\to S^{d\delta}(\AA^1_h)$. Again, to simplify the notation, we shall just write $\calF^d$
instead of $\calF^{d\delta}$.


According to (Z2),
\begin{NB}
    \cite[15.6]{BFG},
\end{NB}
$\dim \calF^d=dh^{\vee}$.


We claim that

1) $\calS^d$ lies in the open subset of $\calU^d_G$ over which $f_d$ is an isomorphism.

2) $f_d^{-1}(\calS^d)\subset \calF^d$.

The first statement is clear, since the image of $\calS^d$ in $S^d(\AA^1_v)$ under the factorization morphism $\pi^d_v$ (to the symmetric product of the vertical line)


must
lie in $S^d(\AA^1_v\backslash 0)$. To prove the second statement, let us note that
$f_d^{-1}(\calS^d)$ 
must lie in the
attracting set in $\calZ^{d\delta}_G$ with respect to the torus $T$ to
$f_d^{-1}(S^d(\AA^1_v\backslash 0))$. It is clear that
$f_d^{-1}(S^d(\AA^1_v\backslash 0))\subset \calF^d$
and thus the statement follows, since every fiber of the map
$\pi^{d\delta}_{\mathcal Z}\colon\calZ_G^{d\delta}\to S^{d\delta}(\AA^1_h)$ is stable under the
action of $T$.

Hence we get $\dim \calS^d\leq dh^{\vee}=\dim \calF^d$.


\subsection{Good coweights}\label{good}


Let $X$ be an affine variety endowed with an action of $T\times \CC^*$ (here $T$ can be any torus).
Let $x$ be any $T\x\CC^*$-fixed point (in practice this point will always be unique, but this is not needed formally
for what follows) and let $Y\subset X^T$ be the $\CC^*$-attractor to $x$ inside $X^T$.
Let now $\lam:\CC^*\to T$ be any coweight. Let us denote by $\calA_{\lam}$ the attractor to $Y$ with respect to the
$\CC^*$-action given by $\lambda$. Let us also denote by $\tcalA_{\lam}$ the attractor to $x$ with respect to the
$\CC^*$-action given by the cocharacter $(\lam,1)$ of $T\times \CC^*$.

We say that $\lambda$ is {\em good} if $\calA_{\lam}=\tcalA_{\lam}$.

\begin{Lemma}\label{lem:good}
For any $\lam$ as above, the coweight $n\lam$ is good for $n\in \NN$ large enough.
\end{Lemma}
\begin{proof}
Obviously,  there exists a closed $T$-equivariant embedding of $X$ into a vector space $V$ such that
the action of $T\x\CC^*$ on $V$ is linear and such that $x$ corresponds to $0\in V$.
Then it is clear that if $\lam$ is good for $V$, then it is also good for $X$.
Hence we may assume that $X=V$.

In this case, we see that $n\lambda$
\begin{NB}
    $\lam$
\end{NB}
is good if and only if for every weight of $T\times \CC^*$ on $V$ of the
form $(\theta,k)$ the following condition is satisfied:


\begin{quote}
$n\langle\lam,\theta\rangle +k>0$
\begin{NB}
$\langle\lam,\theta\rangle +k>0$
\end{NB}
if and only if either $\langle\lam,\theta\rangle>0$, or
$\langle\lam,\theta\rangle =0$ and $k>0$.
\end{quote}


Now, every $n\in \NN$ such that $n|\langle\lam,\theta\rangle|>|k|$
\begin{NB}
    $|\langle\lam,\theta\rangle|>|k|$
\end{NB}
for any $(\theta,k)$ as above such that
$\langle\lam,\theta\rangle \neq 0$ will satisfy the conditions of the Lemma.
\end{proof}

Let $\lam$ be as before and assume in addition that

(i) $x$ is the only fixed point of $\CC^*$ acting
by means of the coweight $(\lam,1)$;

(ii) $X^{\lam(\CC^*)}=X^T$

\noindent
(in this case we automatically have $(X^T)^{\CC^*}=\{x\}$).
Let us
 denote by $\tilPhi$ the hyperbolic restriction for $(\lam,1)$ (acting from sheaves on $X$ to sheaves on $\{x\}$), by $\Phi$ the hyperbolic restriction for $\lam\colon\CC^*\to T$ (acting from sheaves on $X$ to sheaves on $X^T$) and by $\Phi_0$ the hyperbolic restriction
 for the action of $\CC^*$ on $X^T$ (from sheaves on $X^T$ to sheaves on $\{x\}$). Then the definition of ``goodness" implies
\begin{Lemma}\label{lem:comp-of-hyp}
Assume that $\lam$ is good and satisfies the conditions (i) and (ii).
Then we have $\tilPhi=\Phi_0\circ\Phi$.
\end{Lemma}

\subsection{Exactness of twisted hyperbolic restriction}
Let $\tilT=T\x\CC^*$ and let us make it act on $\calU^d_G$ so that the action of $\CC^*$ comes from the
hyperbolic action of $\CC^*$ on $\AA^2$ of the form $z(x,y)=(z^{-1}x,zy)$.
Note that
$(\calU^d_G)^{\tilT}$ consists of one point.

Let us fix $d$ and
let us choose a dominant regular coweight $\lam:\CC^*\to T$ which is good in the sense of Subsection \ref{good} (such $\lam$
exists because of Lemma \ref{lem:good}).
Then the fact that $\lam$ is regular implies that it satisfies the conditions (i) and (ii). Consider the corresponding functors $\tilPhi,\Phi$ and $\Phi_0$.
Obviously we have $\Phi=\Phi^d_{T,G}$, so we shall write $\tilPhi^d_{T,G}$ instead of $\tilPhi$.
Also, to emphasize the dependence on $d$ we set $\Phi^d_0$ instead of $\Phi_0$.
According to Lemma \ref{lem:comp-of-hyp} we have $\tilPhi^d_{T,G}=\Phi^d_0\circ\Phi^d_{T,G}$.


\begin{Theorem}\label{thm:twistedmain}
The complex of vector spaces $\tilPhi^d_{T,G}(\IC(\calU^d_G))$ is concentrated in degree $0$.
\end{Theorem}

\begin{proof}
We will use the same notations as before for $L=T$ replaced with $\tilT$,
such as $i_{\tilT,G}$, $j_{\tilT,G}$, $p_{\tilT,G}$,
$i^-_{\tilT,G}$, $j^-_{\tilT,G}$, $p^-_{\tilT,G}$.

The attracting set is denoted by $\CA_{\lam,\tilT,G}^d$.

According to~\cite[Theorem~1]{Braden}, the natural morphism
$(p^-_{\tilT,G})_*(j^-_{\tilT,G})^!\IC(\calU^d_G)\to
(p_{\tilT,G})_!(j_{\tilT,G})^*\IC(\calU^d_G)=\tilPhi^d_{T,G}(\IC(\calU^d_G))$
is an isomorphism. We will prove that $(p_{\tilT,G})_!(j_{\tilT,G})^*\IC(\calU^d_G)$
is concentrated in nonpositive degrees. A similar (dual) argument proves that
$(p^-_{\tilT,G})_*(j^-_{\tilT,G})^!\IC(\calU^d_G)$ is concentrated in nonnegative
degrees. In other words, we must prove that
$H^\bullet_c(\CA_{\lam,\tilT,G}^d,\IC(\calU^d_G))$ lives in nonpositive cohomological
degrees.

Now $\IC(\CU^d_G)$ is smooth along the stratification
$$
\CU^d_G=\bigsqcup\limits_{m+|\lam|=d}\SBun_G^m\times S_{\lam}(\BA^2),
$$
the dimension of
a stratum being equal to $2l(\lam)+2mh^\vee$.
Here for a partition $\lambda=(\lambda_1,\ldots,\lambda_l)$ we set
$l(\lam)=l$.
The perverse sheaf $\IC(\CU_G^d)$ lives in cohomological degrees
$\leq-2l(\lambda)-2mh^\vee$ on the stratum
$\SBun_G^m\times  S_{\lam}(\BA^2)$.
We have $\CA_{\lam,\tilT,G}^d\cap\left(\SBun_G^m\times S_{\lam}(\BA^2)\right)=
(\CA_{\lam,\tilT,G}^m\cap \SBun_G^m)\times S_{\lam}(\BA^1_v)$.


Now it follows from Corollary \ref{cor:dim-cor} and

the goodness assumption on $\lambda$ that
$\dim(\CA_{\lam,\tilT,G}^m)\leq mh^\vee$.


Evidently, $\dim S_{\lam}(\BA^1_v)=l(\lam)$.
So the restriction of $\IC(\CU_G^d)$ to
$\CA_{\lam,\tilT,G}^d\cap\left(\SBun_G^m\times S_{\lam}(\BA^2)\right)$ lives in
degrees $\leq-2\dim\left(\CA_{\lam,\tilT,G}^d\cap
\left(\SBun_G^m\times S_{\lam}(\BA^2)\right)\right)$.
Now an application of
the Cousin spectral sequence for the stratification of $\CA_{\lam,\tilT,G}^d$ finishes
the proof.
\end{proof}

The following corollary is not needed for the rest, but we include it for the sake of completeness.
\begin{Corollary}
$\dim\calS^d=\dim p^{-1}(S^d(\AA^1))=dh^{\vee}$.
\end{Corollary}

\begin{proof}
We need to show that $\dim \CA_{\lam,\tilT,G}^d$ is at least $dh^{\vee}$. By induction on $d$ we may assume that this is true for
all $d'<d$. Assume that $\dim \CA_{\lam,\tilT,G}^d<dh^{\vee}$.
Then repeating the argument from the above proof we see that $\tilPhi^d_{T,G}(\IC(\calU^d_G))$ is concentrated in strictly negative cohomological degrees, which contradicts Theorem \ref{thm:twistedmain}.
\end{proof}

\begin{Remark}\label{rem:dimofirr}
    The above argument only shows that the dimension of the whole of $\calS^d$ is equal to $dh^{\vee}$, but doesn't show that
this is true for each of its irreducible components (however, we believe that this is true).
\end{Remark}


\subsection{Exactness of \texorpdfstring{$\Phi_{T,G}$}{PhiT,G}}\label{exborel}
We can now show that $\Phi^d_{T,G}(\IC(\calU^d_G))$ is perverse. Indeed, using the factorization argument and induction on $d$,
we may assume that $\Phi^d_{T,G}(\IC(\calU^d_G))$ is perverse away from the main diagonal $\AA^2\subset S^d(\AA^2)$.
Since according to \cite{Braden} the complex  $\Phi^d_{T,G}(\IC(\calU^d_G))$ is semi-simple and since
it is also equivariant with respect to
the action of $\AA^2$ on $S^d(\AA^2)$ by shifts, it follows that we just need to prove that $\Phi^d_{T,G}(\IC(\calU^d_G))$
doesn't have any direct summands which are isomorphic to constant sheaves on $\AA^2$ sitting in cohomological degrees
$\neq-2$. But if such a direct summand existed, it would imply that $\Phi^d_0(\Phi^d_{T,G}(\IC(\calU^d_G)))=\tilPhi^d_{T,G}(\IC(\calU^d_G))$ has
non-zero cohomology in degree $\neq 0$, which contradicts Theorem \ref{thm:twistedmain}.

\subsection{Exactness of \texorpdfstring{$\Phi_{L,G}$}{PhiL,G}}
Let us now show that $\Phi_{L,G}^d(\IC(\calU^d_G))$ is perverse. Indeed, first of all, according to Braden's theorem \cite{Braden},
$\Phi_{L,G}^d(\IC(\calU^d_G))$ is a semi-simple complex, which is constructible with respect to the stratification
(\ref{eq:strat}). In other words, it is a direct sum of (possibly shifted) simple perverse
sheaves, where each such sheaf is isomorphic to the Goresky-MacPherson extension of a local system $\calE$ on
$\SBun_L^{d_1}\x S_{\lam}(\AA^2)$ for some $d_1$ and $\lam$ as in \ref{eq:strat}.

\begin{Lemma}
Any such $\calE$ is necessarily of the form $\CC_{\SBun_L^{d_1}}\boxtimes \calE'$ where $\calE'$ is some local
system on $S_{\lam}(\AA^2)$.
\end{Lemma}
\begin{proof}
To prove this it is enough to show that the restriction of $\Phi_{L,G}(\IC(\calU^d_G))$
 to $\SBun_L^{d_1}\x S^{d_2}(\AA^2)$ (here $d=d_1+d_2$) is isomorphic to the exterior tensor product of the constant sheaf
 of $\SBun_L^{d_1}$ and some complex on $S^{d_2}(\AA^2)$. Moreover, it is enough to
 construct such an isomorphism on some Zariski open subset $U$ of
$\SBun_L^{d_1}\x S^{d_2}(\AA^2)$ (this follows from the fact that a local system which is constant on a Zariski dense subset is constant
everywhere). Let us choose a projection $a:\AA^2\to \AA^1$ and let $\pi_{a,L}^{d_1}:\SBun_L^{d_1}\to S^{d_1}(\AA^1)$ be the corresponding map. Let $U$ be the open subset of $\SBun_L^{d_1}\x S^{d_2}(\AA^2)$ consisting
of pairs $(\calF,x)$ such that $\pi_{a,L}^{d_1}$ is disjoint from the projection of $x$ to $S^{d_2}(\AA^1)$. Then locally in \'etale topology near
every point of $U$ the scheme $\calU^d_G$ looks like the product $\SBun_G^{d_1}\x \calU_G^{d_2}$ and the statement follows.
\end{proof}
Now, we can finish the proof. Indeed, recall that the closure of $\SBun_L^{d_1}\x S_{\lam}(\AA^2)$ admits a finite
birational map from $\calU^{d_1}_L\x\oS^{\lam}(\AA^2)$, where $\oS^{\lam}(\AA^2)$ stands for the closure of
$S_{\lam}$ in $S^{d_2}(\AA^2)$. Thus for any $\calE$ as above we see that $\IC(\calE)$ is the direct image
of $\IC(\calU^{d_1}_L)\boxtimes \IC(\calE')$ under this map. Moreover, the complex
$\Phi_{T,L}(\IC(\calE))$ is equal to the direct image of $\Phi_{T,L}(\IC(\calU^{d_1}_L))\boxtimes \IC(\calE')$. Hence, we see
that it is perverse and non-zero. Thus, if for some $i\neq 0$ the complex $\IC(\calE)[i]$ is a direct summand of $\Phi_{L,G}(\IC(\calU^d_G))$, then $\Phi_{T,L}(\Phi_{L,G}(\calU^d_G))$ is not perverse. Since $\Phi_{T,L}\circ\Phi_{L,G}\simeq \Phi_{T,G}$, this contradicts Subsection \ref{exborel}.
\qed

Recall $\Uh[P,0]d \defeq p^{-1}(d\cdot 0)$, see~(\ref{247}).
\begin{Corollary}\label{cor:dimest}
$\dim\Uh[P,0]d\leq dh^\vee-1$.
\end{Corollary}
\begin{proof}
We will argue by induction in $d$. We assume the claim for all $d'<d$.
We know that the dual space
$(U^d)^*\simeq H_c^\bullet(p^{-1}(d\cdot0),\tilde{j}{}^*\IC(\cUh{d}))$ lives
in degree 0. We consider the Cousin spectral sequence for the stratification
$\Uh[P,0]d=\bigsqcup_{d'\leq d}(\Uh[P,0]{d'}\cap\Bun{d'})$. By the induction
assumption, all the strata for $d'<d$ contribute to nonpositive degrees
of $H_c^\bullet(p^{-1}(d\cdot0),\tilde{j}{}^*\IC(\cUh{d}))$ only.
If we had $\dim\Uh[P,0]d>dh^\vee-1$, the fundamental classes of the top
dimensional components of $\Uh[P,0]d$ would contribute to the strictly
positive degrees in $H_c^\bullet(p^{-1}(d\cdot0),\tilde{j}{}^*\IC(\cUh{d}))$,
and nothing would cancel their contribution. This would contradict to
$H_c^{>0}(p^{-1}(d\cdot0),\tilde{j}{}^*\IC(\cUh{d}))=0$.
\end{proof}

\begin{NB}
    Added on Oct.~5, 2014.
\end{NB}

This is the estimate of the attracting set for the most singular point
$d\cdot 0$. The exactness also implies estimates for attracting sets
of other points, more precisely their intersection with the open locus
$\Bun{d}$.
\begin{NB}
    Corrected, Oct. 10, 2014
\end{NB}
Since any stratum of $\Uh{d}$ is of the form $\Bun{d_1}\times
S_\lambda(\AA^2)$, we have the corresponding dimension estimate for
other strata from the perversity of $\Phi_{L,G}(\IC(\Uh{d_1}))$ for
any $d_1$.
\begin{NB}
    Added, Oct. 10, 2014.
\end{NB}
Therefore we see that $\Phi_{L,G}$ is hyperbolic semi-small in
the sense of Definition~\ref{def:hypsemismall}.
{"config": "arxiv", "file": "1406.2381/perverse.tex"}