\section{Introduction}

Let $f: {\Bbb C}^d \to {\Bbb C}$ be a non-constant polynomial.  Fix a point
$x \in E = f^{-1} (0)$.  The {\em Milnor fiber} of $f$ at $x$ is
defined by $F_{f,x} = f^{-1} (\epsilon) \cap {\rm B}_{x,\delta}$,
where ${\rm B}_{x,\delta}$ is the $\delta$-ball around $x$, and
we assume that $0 < |\epsilon| \ll \delta \ll 1$.  Since the
publication of Milnor's seminal work [Mi], the cohomology of
$F_{f,x}$ has been a central object in the study of the
singularities of $f$.

In the language of sheaf theory, the cohomology groups
$H^* (F_{f,x})$, for all $x \in E$, fit together to form a
constructible, bounded complex $P$ of sheaves on $E$, called
the {\em sheaf of nearby cycles} of $f$ (i.e., $H^* (F_{f,x})$
is the stalk of $P$ at $x$).  The complex $P$ is a perverse
sheaf ([GM2], [KS]).

There are two kinds of questions one may ask about the
singularities of $f$.  First, one may be interested in the
cohomology of the Milnor fibers $F_{f,x}$ and of other related
spaces, as well as in the various maps between such groups.
The most important among such maps is the monodromy
transformation $\mu_x : H^* (F_{f,x}) \to H^* (F_{f,x})$,
arising from the dependence of the Milnor fiber on the choice
of the small number $\epsilon \in {\Bbb C}^*$.  Much of the work in
singularity theory since the publication of [Mi] has centered
around such concrete geometric questions.

On the other hand, one may ask: what is the structure of $P$
as an object in the abelian, artinian category of perverse
sheaves on $E$ (see [BBD])?  In particular, what are the simple
constituents of $P$?  Is $P$ semisimple?  What are the extensions
involved in building $P$ up from its simple constituents?  What
is the endomorphism ring $\rm{End} (P)$?  How does the monodromy
transformation $\mu \in \rm{Aut} (P)$ act?

These two kinds of questions are intimately related.  The
structure of $P$ as a perverse sheaf is the ``glue'' that ties
together the various local geometric invariants of $f$ at different
points.  To the author's knowledge, the only general theorem about
the structure of $P$ is the deep result of Ofer Gabber (see [BB])
describing the filtration of $P$ arising from the action of the
monodromy.  Gabber's theorem is a semisimplicity assertion about
certain subquotients of $P$.

The definition of the nearby cycles sheaf $P$ can be extended to
the setting of a dominant map $f : {\Bbb C}^d \to {\Bbb C}^r$.  In order to
assure the desired properties of $P$, one needs to impose some
technical conditions on $f$ (see Section 2.2 below).  On the
other hand, the resulting theory is richer, because instead of
a single monodromy transformation of $P$, we have an action
of the whole fundamental group of the set of regular values of
$f$ near zero.

In this paper, we consider the nearby cycles for a certain special
class of maps $f : {\Bbb C}^d \to {\Bbb C}^r$, whose components are given by
homogeneous polynomials.  This class arises naturally from the
invariant theory of reductive group actions, and our motivation
for studying it comes from the Springer theory of Weyl group
representations.

Springer theory (see [BM1], [M], [S3], and Section 2.1 below) can
be seen as the study of the singularities of the adjoint quotient
map $f: {\frak g} \to G {\backslash\!\backslash} {\frak g} = \rm{Spec} \, {\Bbb C}[{\frak g}^*]_{}^G$,
associated to a complex semisimple Lie algebra.  For this map,
both the categorical structure of $P$ and its local invariants are
well understood.  The sheaf $P$ is semisimple.  The monodromy group
acting on $P$ is the Braid group $B^{}_W$ of the Weyl group $W$ of
${\frak g}$; but the monodromy action factors through $W$.  Moreover,
this monodromy action gives an isomorphism ${\Bbb C}[W] \cong \rm{End}
(P)$.  Thus, Springer theory gives a relation between the
singularities of the nilcone ${\cal N} = f^{-1} (0) \subset {\frak g}$ and the
representations of the Weyl group.

Dadok and Kac [DK] introduced a class of rational representations
$G \, | \,V$ ($G$ reductive / ${\Bbb C}$) for which the invariant theory
works by analogy with the adjoint representations.  They call this
the class of {\em polar representations}; it includes the adjoint
representations, as well as many of the classical invariant
problems of linear algebra (see Section 3.2 for examples).  For any
polar representation $G \, | \, V$, the quotient $G {\backslash\!\backslash} V$ is
isomorphic to a vector space.  In this paper, we study the
singularities of the quotient map $f : V \to G {\backslash\!\backslash} V$, giving a
generalization of Springer theory to polar representations
satisfying a mild additional hypothesis.

Our main result (Theorem \ref{main}, part (i)) is the following.
For a polar representation $G \, | \, V$, assume that the fiber
$E = f^{-1} (0)$ consists of finitely many $G$-orbits (this
condition is called {\em visible}).  Then the nearby cycles sheaf
$P$ of the quotient map $f$ satisfies:
\begin{equation}\label{intro}
{\cal F} \, P \cong {\cal IC} ((V^*)^{rs}, {\cal L}),
\end{equation}
where ${\cal F}$ is the geometric Fourier transform functor, and the
right hand side is an intersection homology sheaf on $V^*$.
(See [KS] for a definition of the Fourier transform, and [GM1]
for a discussion of intersection homology.)  The content of this
is that the sheaf $P$ is completely encoded in the single local
system ${\cal L}$ on a certain locus in the dual space $V^*$.  In the
case of Springer theory, this result is due independently to
Ginzburg [Gi] and to Hotta-Kashiwara [HK] (see also [Br]).
The proof of (\ref{intro}) draws on a general result of the
author about the specialization of an affine variety to the
asymptotic cone (see [Gr1] and Section 2.2.2).

Further assertions of Theorem \ref{main} describe the holonomy
of the local system ${\cal L}$ and the monodromy action on $P$ of
the appropriate fundamental group.  It turns out that the
semisimplicity observed in Springer theory does not extend to
this generalization.  Associated to each polar representation
$G \, | \, V$, there is a finite complex reflection group $W$,
called the Weyl group of $G \, | \, V$ (it is not, in general,
the Weyl group of $G$).  We have $\dim \, \rm{End} (P) =
|W|$, but the algebra $\rm{End} (P)$ is not, in general,
isomorphic to ${\Bbb C} [W]$.  Instead, it is given as a kind of a
Hecke algebra associated to $W$.

We should note that our Fourier transform description of $P$,
while well suited to the study of the endomorphism ring and of
the monodromy action, falls short of giving the complete structure
of $P$ as a perverse sheaf.  This is because intersection homology
is not an exact functor from local systems to perverse sheaves.

An important special case of this theory, which includes the
adjoint representations, arises in the following way.  Let
$\theta : {\frak g} \to {\frak g}$ be an involutive automorphism of a complex
semisimple Lie algebra, and ${\frak g} = {\frak g}_+ \oplus {\frak g}_-$ the
eigenspace decomposition for $\theta$.  Then the adjoint form
$G_+$ of the Lie algebra ${\frak g}_+$ acts on the symmetric space
${\frak g}_-$.  Orbits and invariants of the representation
$G_+ \, | \, {\frak g}_-$ were studied by Kostant and Rallis in [KR].
This representation is polar and visible.  In Theorem \ref{symsp},
we explicitly compute the endomorphism ring $\rm{End} (P)$ in this
case.  It is given as a ``hybrid'' of the group algebra ${\Bbb C}[W]$
and the Hecke algebra of $W$ (which is a Coxeter group)
specialized at $q = - 1$.

I would like to thank the University of Utrecht, Holland, and the
IAS, Princeton, for their hospitality.  Throughout this work I was
supported by a Fannie and John Hertz foundation graduate fellowship.
Discussions with Tom Braden, Ian Grojnowski, Victor Kac, Peter
Magyar, David Massey, Dirk Siersma, and Tony Springer have been
of great value to me.  I am also grateful to Mark Goresky, David
Kazhdan, and Wilfried Schmid for their interest and encouragement.
Finally, I would like to thank my adviser, Robert MacPherson, for
his inspiration, guidance, and support.

\tableofcontents

\pagebreak

\section{Background}

{\bf Notations.}
We will say {\em sheaf} to mean {\em complex of sheaves}
throughout; all our sheaves will be sheaves of ${\Bbb C}$ vector
spaces.  Given a map $g : X \to Y$, the symbols $g_*$, $g_!$
will always denote the {\em derived} push-forward functors.
All perverse sheaves and intersection homology will be taken
with respect to the middle perversity (see [GM1], [BBD]);
we use the shift conventions of [BBD].  Given a sheaf $A$ on
$X$, and a pair of closed subspaces $Z \subset Y \subset X$,
we will write ${\Bbb H}^k(Y, Z; A)$ for the hypercohomology group
${\Bbb H}^k(j_! \,i^* A)$, where $i : Y \setminus Z \to X$ and
$j : Y \setminus Z \to Y$ are the inclusion maps.  We call
${\Bbb H}^k(Y, Z; A)$ the relative hypercohomology of $A$.

For an analytic function $f : M \to {\Bbb C}$, we use the notation of
[KS, Chapter 8.6] for the nearby and vanishing cycles
functors $\psi^{}_f$, $\phi^{}_f$.  Let $A$ be a sheaf on $M$
or $M \setminus f^{-1} (0)$.  We denote by $\mu : \psi^{}_f A \to
\psi^{}_f A$ the monodromy transformation of the nearby cycles.
It is the counter-clockwise monodromy for the family of sheaves
$\psi^{}_{f / \tau} A$, parametrized by the circle
$\{ \tau \in {\Bbb C} \, | \, | \tau | = 1 \}$.

When $V$ is a ${\Bbb C}$ vector space, we denote by ${\cal P}^{}_{{\Bbb C}^*} \,
(V)$ the category of ${\Bbb C}^*$-conic perverse sheaves on $V$, and
by ${\cal F} : {\cal P}^{}_{{\Bbb C}^*} \, (V) \to {\cal P}^{}_{{\Bbb C}^*} \, (V^*)$
the (shifted) Fourier transform functor.  In the notation
of [KS, Chapter 3.7], we have ${\cal F} \, P = P \, \hat{} \;
[\dim \, V]$.  To avoid cumbersome notation, we will use
the following shorthand: if $P$ is a conic perverse sheaf on a
closed, conic subvariety $X \subset V$, and $j^{}_X : X \to V$
is the inclusion, then we write ${\cal F} \, P$ instead of
${\cal F} \circ (j^{}_X)_* \, P$.  When $E$ is a $G$-space, we denote
by ${\cal P}^{}_G \, (E)$ the category of $G$-equivariant perverse
sheaves on $E$.  The symbol $\dim$, without a subscript, will
always denote the {\em complex} dimension.

\subsection{Springer Theory}

In this section we give a brief summary of the Springer theory
of Weyl group representations.  Springer theory (see [BM1], [S3])
is concerned with exhibiting the Weyl group $W$ of a complex
semisimple Lie algebra ${\frak g}$ as the symmetry group of a certain
perverse sheaf $P$ on the nilcone ${\cal N} \subset {\frak g}$.  As we
mentioned in the introduction, $P$ is the nearby cycles sheaf
for the adjoint quotient map $f : {\frak g} \to G {\backslash\!\backslash} {\frak g}$ (here
$G$ is the adjoint form of ${\frak g}$).  However, this is not the
original definition of $P$, and not the one used in the
literature to prove its properties.

We now describe the construction due to Lusztig [Lu1] and
Borho-MacPherson [BM1] of the sheaf $P$ and the $W$-action on
it.  Consider the Grothendieck simultaneous resolution diagram:

\vspace{.1in}

$$\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt}
\def\incmapright#1{\smash{
\mathop{\hookrightarrow}\limits^{#1}}}
\def\mapright#1{\smash{
\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap
{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\matrix{\tilde{{\cal N}}  & \incmapright{}   & \tilde{{\frak g}} &
\mapright{\tilde{f}} & {\frak t}              \cr
        \mapdown{p}  &                  & \mapdown{q} &
                     & \mapdown{f|_{\frak t}} \cr
        {{\cal N}}        & \incmapright{j}  & {\frak g}         &
\mapright{f}         & W \, {\backslash\!\backslash} \, {\frak t}  \cr}$$

\vspace{.2in}

\noindent
Here $\tilde{{\frak g}}$ is the variety of pairs $(x, {\frak b})$, where
${\frak b} \subset {\frak g}$ is a Borel subalgebra, and $x \in {\frak b}$.  This
variety is smooth; it is a vector bundle over the flag variety
${\cal B}$ of ${\frak g}$.  The map $f$ is given by the invariants of the
adjoint action $G \, | \, {\frak g}$; the target space is identified
with the quotient of a Cartan subalgebra ${\frak t} \subset {\frak g}$ by
the action of the Weyl group.  The map $\tilde{f}$ is the
natural morphism making the diagram commute.  Finally,
$\tilde{{\cal N}} = \{ \, (x, {\frak b}) \in \tilde{{\frak g}} \; | \;
x \in {\cal N} \, \}$.

Note that the map $q$ is a finite cover over the set ${\frak g}^{rs}$
of regular semisimple elements in ${\frak g}$.  Given a point
$x \in {\frak g}^{rs}$, the fiber $q^{-1}(x)$ can be identified with
the Weyl group $W$.  Such an identification requires a choice of
a positive Weyl chamber for the Cartan subalgebra
${\frak g}_x \subset{\frak g}$.  The Weyl group $W$, therefore, acts as the
deck transformations of the covering
$q^{rs} : \tilde{{\frak g}}^{rs} \to {\frak g}^{rs}$ (where
$\tilde{{\frak g}}^{rs} = q^{-1} ({\frak g}^{rs})$, and $q^{rs}$ is the
restriction of $q$).

The diagram above is a simultaneous resolution in the following
sense.  For any point $t \in {\frak t}$, the restriction
$q^{}_t : \tilde{f}^{-1} (t) \to f^{-1} (f(t))$ is a resolution
of singularities.  In particular, the map $p = q^{}_0$ is a
resolution of singularities of the nilcone.

We now consider the push-forward sheaves
$$P = p^{}_* \, {\Bbb C}_{\tilde{{\cal N}}} \, [\dim \, {\cal N}] \;\;
\mbox{and} \;\;
Q = q^{}_* \, {\Bbb C}_{\tilde{{\frak g}}} \, [\dim \, {\frak g}].$$
Note that $P = j^* \, Q \, [-r]$, where $r =  \dim \, {\frak t}$.

\begin{prop}\label{dimfib} {\em [Lu1]}
The map $q$ is small and the map $p$ is semi-small in the sense
of {\em [GM1]}.  Consequently, the sheaf $P$ is perverse, and
the sheaf $Q$ is the intersection cohomology extension
${\cal IC}({\frak g}^{rs}, {\cal L})$ of the local system
${\cal L} = q^{rs}_* \, {\Bbb C}_{\tilde{{\frak g}}^{rs}}$ on ${\frak g}^{rs}$.
\end{prop}

The Weyl group action on $\tilde{\frak g}^{rs}$ by deck
transformations produces an action on the local system ${\cal L}$,
and by the functoriality of intersection cohomology on the
sheaf $Q$.  In fact, we have ${\rm End} ({\cal L}) = {\rm End} (Q)
= {\Bbb C} [W]$, the group algebra of $W$.  Using the relation
$P = j^* \, Q \, [-r]$, we obtain a $W$-action on $P$ as well.
This construction of the $W$-action is due to Lusztig [Lu1].
Originally, the representations of $W$ on the stalks of $P$,
i.e., on the cohomology of the varieties $p^{-1} (x)$
($x \in {\cal N}$), were constructed by Springer ([S1], [S2]).
Other constructions of the same representations were given by
Slodowy [Sl] and Kazhdan-Lusztig [KL].  The construction of
Slodowy is essentially the nearby cycles definition, which is
the starting point for this paper.  Hotta in [Ho] showed that
all of the different constructions agree.

\begin{thm}\label{bmmain} {\em [BM1]}

(i)  $P$ is a semisimple perverse sheaf.

(ii) The action of $W$ on $P$ gives an isomorphism
${\Bbb C} [W] \cong {\rm End} (P)$.
\end{thm}

Borho and MacPherson deduce the first part of Theorem
\ref{bmmain} from the decomposition theorem of Beilinson,
Bernstein, Deligne, and Gabber [BBD], and use a counting
argument to prove the second part.  A different proof, using
the Fourier transform, was later given by Ginzburg [Gi] and
Hotta-Kashiwara [HK] (see also [Br]).  We will use the Killing
form on ${\frak g}$ to identify ${\frak g}$ with its dual, and to regard the
Fourier transform on ${\frak g}$ as a functor ${\cal F} : {\cal P}^{}_{{\Bbb C}^*}
({\frak g}) \to {\cal P}^{}_{{\Bbb C}^*} ({\frak g})$.

\begin{thm}\label{ftinspth} {\em [Gi], [HK]}

(i)  There is an isomorphism ${\cal F} \, P \cong Q$.

(ii) The Weyl group $W$ acts on each side of this isomorphism.
These two actions differ by the sign character of $W$ (i.e.,
the character that sends each simple reflection $\sigma \in W$
to $-1$.)
\end{thm}

Theorem \ref{ftinspth} implies Theorem \ref{bmmain}, because
the functor ${\cal F}$ is an equivalence of categories.

\subsection{Nearby Cycles}

\subsubsection{Families Over a Large Base}

The sheaf $P$ constructed in the previous section can be defined in
a different way, namely, as the nearby cycles of the adjoint quotient
map $f$.  One usually considers the nearby cycles functor for an
analytic function $M \to {\Bbb C}$ (see [GM2, Section 6], [KS, Chapter 8.6]).
Proposition \ref{nclb} below provides a technical basis for discussing
the nearby cycles for a sufficiently nice map $M \to {\Bbb C}^r$.

Let $M$ be a connected complex analytic manifold of dimension $d$,
${\cal U}$ a neighborhood of zero in ${\Bbb C}^r$, and $f: M \to {\cal U}$ an analytic
map onto ${\cal U}$.  Let ${\cal U}^{sing} \subset {\cal U}$ be the closure of the
set of non-regular values of $f$.  We assume that ${\cal U}^{sing}$ is a
proper analytic subvariety of ${\cal U}$.  Let ${\cal U}^{reg} =
{\cal U} \setminus {\cal U}^{sing}$, and note that the preimage
$M^\circ = f^{-1} ({\cal U}^{reg}) \subset M$ is a manifold.  We assume
that there is an analytic Whitney stratification ${\cal S}$ of the fiber
$E = f^{-1} (0)$, such that Thom's ${\rm A}_f$ condition holds for
the pair $(M^\circ, S)$, for every stratum $S \in {\cal S}$.  Recall that
the ${\rm A}_f$ condition says that for any sequence of points
$x_i \subset M^\circ$, converging to a point $y \in S$, if there
exists a limit
$$\Delta = \lim_{i \to \infty} T_{x_i} f^{-1} (f (x_i))
\subset T_y M,$$
then $\Delta \supset T_y S$.  If $r = 1$, such a stratification
${\cal S}$ can always be found.  For $r > 1$, the existence of ${\cal S}$
is an actual restriction on $f$.  It implies, in particular,
that $\dim E = d - r$.  We refer the reader to [Hi] for a detailed
discussion of the ${\rm A}_f$ condition.

Let $U$ be a neighborhood of zero in ${\Bbb C}$, and $\gamma : U \to {\cal U}$
an embedded analytic arc, such that $\gamma (0) = 0$, and
$\gamma (z) \in {\cal U}^{reg}$ for $z \neq 0$.  Form the fiber product
$M_\gamma =  M \times^{}_{{\cal U}} U$, and let $f_\gamma : M_\gamma
\to U$ be the projection map.  We may consider the nearby cycles
sheaf $P_{f_\gamma} = \psi_{f_\gamma} \, {\Bbb C}_{M_\gamma} \, [d - r]$.

\begin{prop}\label{nclb}
(i)   The sheaves $P_{f_\gamma}$ for different $\gamma$ are all
isomorphic.  We may therefore omit the subscript $\gamma$, and
call the sheaf $P_f = P_{f_\gamma}$ the nearby cycles of $f$.
It is a perverse sheaf on $E$, constructible with respect to ${\cal S}$.

(ii)  The local fundamental group $\pi_1 ({\cal U}^{reg} \cap
{\rm B}_\epsilon)$, where ${\rm B}_\epsilon \subset {\cal U}$
is a small ball around the origin, acts on $P_f$ by monodromy.  We
denote this action by $\mu : \pi_1 ({\cal U}^{reg} \cap
{\rm B}_\epsilon) \to {\rm Aut} (P_f)$.

(iii) The sheaf $P_f$ is Verdier self-dual.
\end{prop}
\begin{pf}
We refer the reader to [GM2, Section 6] for a proof that each
$P_{f_\gamma}$ is a perverse sheaf constructible with respect
to ${\cal S}$.  The self-duality of $P_{f_\gamma}$ follows from the
fact that the nearby cycles functor $\psi_{f_\gamma}$ commutes
with Verdier duality.

For the other assertions of the proposition, choose a point
$x \in E$, and let $S \subset E$ be the stratum containing $x$.  Fix
a normal slice $N \subset M$ to $S$ through the point $x$.  Also fix
a pair of numbers $0 < \epsilon \ll \delta \ll 1$ (chosen to be small
in decreasing order).  For each regular value $\lambda \in {\cal U}^{reg}$
with $|\!| \lambda |\!| < \epsilon$, consider the Milnor fiber
cohomology group
$$H^* (f^{-1} (\lambda) \cap N \cap {\rm B}_{x, \delta}),$$
where ${\rm B}_{x, \delta} \subset M$ is the $\delta$-ball around $x$
(we fix a Hermitian metric on $M$ for this).  It is enough to show
that this Milnor fiber cohomology varies as a local system in
$\lambda$.  This can be done by adapting the argument in
[L\^e1, Section 1].
\end{pf}

Returning to the notation of the previous section, it is easy to
check that the adjoint quotient map $f : {\frak g} \to W \, {\backslash\!\backslash} \, {\frak t}$
and the orbit stratification of the nilcone ${\cal N}$ satisfy the
hypotheses of Proposition \ref{nclb}.  Note that the set of regular
values of $f$ equals $f ({\frak t}^{reg})$, where ${\frak t}^{reg} =
{\frak t} \cap {\frak g}^{rs}$.  The fundamental group $\pi_1 (f ({\frak t}^{reg})
\cap {\rm B}_\epsilon)$ is the braid group $B^{}_W$ associated to $W$.
Theorem \ref{ncinspth} below is a consequence of the properties of the
simultaneous resolution (see [Sl] for an early treatment of these
ideas).

\begin{thm}\label{ncinspth} {\em [Ho], [M]}

(i)  We have $P \cong P_f$.

(ii) The monodromy action $\mu : B^{}_W \to {\rm Aut} (P_f)$ factors
through the natural homomorphism $B^{}_W \to W$, producing an action
of $W$ on $P_f$.  The isomorphism of part (i) agrees with the $W$
actions on both sides.
\end{thm}

Thus, Springer theory can be viewed as the study of the nearby cycles
of the adjoint quotient map.

\pagebreak

\subsubsection{Specialization to the Asymptotic Cone}

In this section we summarize the results of [Gr1] on which this
paper relies.

Let $V \cong {\Bbb C}^d$ be a complex vector space, and $X \subset V$
a connected, smooth, closed subvariety.  We denote by $\bar V$
the standard projective compactification of $V$, and by $\bar X$
the closure of $X$ in $\bar V$.  Set $V^\infty = \bar V
\setminus V$, and $X^\infty = \bar X \cap V^\infty$.  The
asymptotic cone ${\rm{as}} (X) \subset V$ is defined as the affine
cone over $X^\infty$.

Another way to define ${\rm{as}} (X)$ is as follows.  Let
$\tilde X^\circ = \{ (\lambda, \tilde x) \in {\Bbb C}^* \times V \, |
\, \tilde x \in \lambda X \}$, and $\tilde X$ be the closure
of $\tilde X^\circ$ in ${\Bbb C} \times V$.  Write $\tilde f :
\tilde X \to {\Bbb C}$ for the projection on the first factor.
Then ${\rm{as}} (X) = \tilde f^{-1} (0)$.  This definition shows that
the asymptotic cone is naturally equipped with a nearby cycles
sheaf $P = P^{}_X := \psi_{\tilde f} \, {\Bbb C}_{\tilde X} [n]$, where
$n = \dim X$.  The sheaf $P$ is ${\Bbb C}^*$-conic.

Given a Hermitian inner product on $V$, we may consider for any
$x \in F$, the angle $\angle (x, T_x X) \in [0, \pi/2]$ between
the vector $x \in V$ and the subspace $T_x X \subset T_x V \cong V$.
The variety $X$ is said to be {\em transverse to infinity} if for
some (equivalently for any) inner product on $V$, there exists
a constant $k > 0$, such that for any $x \in X$, we have:
$$\angle (x, T_x X) < \frac{k}{|\!| x |\!|} \, .$$

\begin{thm}\label{asym_cone}{\em [Gr1, Theorem 1.1]}
Assume $X \subset V$ is transverse to infinity and let $P$ be
the nearby cycles sheaf on ${\rm{as}} (X)$.  Let $T_X^* V \subset T^* V
\cong V \times V^*$ be the conormal bundle to $X$, and
$p_2 : V \times V^* \to V^*$ be the projection.  Let
$Y = \overline{p_2 (T_X^* V)}$; it is an irreducible cone in $V^*$.
Then we have:
$${\cal F} \, P \cong {\cal IC} (Y^\circ, {\cal L}),$$
where ${\cal F}$ is the geometric Fourier transform functor, ${\cal L}$ is
a local system on some Zariski open subset of $Y^\circ \subset Y$,
and the right hand side is the intersection homology extension of
${\cal L}$.
\end{thm}

We will need some auxiliary facts describing the stalks of the
${\cal F} \, P$.

\begin{lemma}\label{odlim}{\em [Gr1, Proposition 3.3]}
In the situation of Theorem \ref{asym_cone}, fix $l \in V^*$,
and let $\xi = \mbox{\em Re} \, (l)$.  Also fix large positive
numbers $1 \ll \xi^{}_0 \ll \eta^{}_0$.  Then we have:
$$H^i_l \, ({\cal F} \, P) \cong H^{i+d+n} \,
( \{ \, x \in X \, | \, \xi (x) \leq \xi^{}_0 \, \}, \,
\{ \, |\!| x |\!| \geq \eta^{}_0 \, \}; \, {\Bbb C} ).$$
\end{lemma}

Assume now $l \neq 0$.  Let $\Delta \subset V$ be the kernel
of $l$, and $L \subset V$ be any line complementary to $\Delta$.
We have $V = \Delta \oplus L$.  Take the standard projective
compactification $\bar \Delta$ of $\Delta$, and let $\hat V  =
\hat V_l = \bar \Delta \times L$.  It is not hard to check that
the space $\hat V$ is canonically independent of the choice of the
line $L$.  Note that $l : V \to {\Bbb C}$ extends to a proper algebraic
function $\hat l : \hat V \to {\Bbb C}$.  Let $\hat X = \hat X_l$ be the
closure of $X$ in $\hat V$, and $j : X \to \hat X$ be the inclusion
map.  Set $\hat X^\infty = \hat X \setminus X$.

\begin{lemma}\label{auxmtw}{\em [Gr1, Proposition 3.7]}
For $l \neq 0$, the statement of Lemma \ref{odlim} can be modified
as follows:
$$H^i_l \, ({\cal F} \, P) \cong {\Bbb H}^{i+d} \, ( \{ \, x \in \hat X
\, | \, \hat \xi (x) \leq \xi^{}_0 \, \}, \, \{ \, |\!| \hat l
(x) |\!| \geq 2 \xi^{}_0 \, \}; \; j^{}_! \, {\Bbb C}^{}_X \, [n] ),$$
where $\xi^{}_0 \gg 1$, and $\hat \xi (x) = \mbox{\em Re} \,
(\hat l(x))$.
\end{lemma}

Let ${\cal S}$ be an algebraic Whitney stratification of ${\rm{as}} (X)$,
written ${\rm{as}} (X) = \bigcup_{S \in {\cal S}} S$, satisfying the following
three conditions.

(i)   ${\cal S}$ is conic, i.e., each $S \in {\cal S}$ is ${\Bbb C}^*$-invariant.

(ii)  Thom's ${\rm A}_{\tilde f}$ condition holds for the pair
$(\tilde X^\circ, S)$, for each $S \in {\cal S}$.

(iii) Let ${\cal S}^\circ = {\cal S} \setminus \{\{ 0 \}\}$.  For
$S \in {\cal S}^\circ$, let $S^\infty \subset X^\infty$ be the
projectivization of $S$.  Then the decomposition $\bar X =
X \cup \bigcup_{S \in {\cal S}^\circ} S^\infty$ is a Whitney
stratification.

The existence of such an ${\cal S}$ follows form the general results
of stratification theory.  Let $a ({\cal S}, l)$ be the dimension of
the critical locus of $l \, |_{{\rm{as}} (X)}$ with respect to ${\cal S}$.

\begin{lemma}\label{dimbound}{\em [Gr1, Section 3.4]}
There exists a stratification $\hat {\cal X}$ of $\hat X$ with the
following property.  Write $Z$ for the critical locus of
$\hat l \, |_{\hat X}$ with respect to $\hat {\cal X}$.  Then
$$\dim \, Z \cap \hat X^\infty < a ({\cal S}, l).$$
(We set $\dim \, \emptyset = -1$.)
\end{lemma}

\subsection{Polar Representations}

\subsubsection{A Summary of the Results of Dadok and Kac}

Dadok and Kac [DK] introduced and studied the class of polar
representations.  Their motivation was to find a class of
representations of reductive groups over ${\Bbb C}$, whose
invariant theory works by analogy with the adjoint
representations.  In this section we will, following [DK],
recall the definition and the main properties of polar
representations.

Let $G \, | \, V$ be a rational linear representation of a
connected reductive Lie group over ${\Bbb C}$ in a finite dimensional
vector space.  A vector $v \in V$ is called {\em semisimple} if
the orbit $G \cdot v$ is closed, and {\em nilpotent} if the
closure $\overline{G \cdot v}$ contains zero.  We say that
$v \in V$ is {\em regular semisimple} if $G \cdot v$ is closed
and of maximal dimension among all closed orbits.  We will write
$V^s$ ($V^{rs}$) for the set of all semisimple (regular semisimple)
vectors in $V$.  The representation $G \, | \, V$ is called
{\em stable} if $\dim \, G \cdot v \geq \dim \, G \cdot x$, for
any regular semisimple $v \in V$ and any $x \in V$.

For a semisimple vector $v \in V$, define a subspace
$$c_v = \{ \, x \in V \; | \; {\frak g} \cdot x \subset
{\frak g} \cdot v \, \},$$
where ${\frak g}$ is the Lie algebra of $G$.  The orbits through
$c_v$ thus have ``parallel'' tangent spaces.  The representation
$G \, | \, V$ is called {\em polar} if for some semisimple
$v \in V$, we have $\dim \, c_v = \dim \, {\Bbb C} [V^*]_{}^G$;
in this case, $c_v$ is called a {\em Cartan subspace}.
The prototype of all polar representations (responsible for the
name ``polar'') is the action of the circle $S^1$ on the plane
${\Bbb R}^2$ by rotations.  The class of polar representations
includes the adjoint representations (a Cartan subspace for an
adjoint representation is just a Cartan subalgebra) and the
representations arising from symmetric spaces, studied by Kostant
and Rallis in [KR] (see Section 6 below).  It is, in fact, much
larger.  Every representation with $\dim \, {\Bbb C} [V^*]_{}^G = 1$ is
automatically polar.  Dadok and Kac [DK] give a complete list of
all polar representations of simple $G$.  Their classification
includes many of the classical invariant problems of linear algebra.
Other examples of polar representations can be found in [K].  Note
that a polar representation need not be stable.

Before describing the invariant theory of a polar representation,
we recall the notion of a Shephard-Todd group (or a complex
reflection group).  Let $c$ be a complex vector space.  An element
$g \in GL(c)$ is called a {\em complex reflection} if in some
basis it is given by the matrix $\mbox{diag} \, (e^{2 \pi i / n},
1, \dots , 1)$, for some integer $n > 1$.  A finite subgroup
$W \subset GL(c)$ is called a {\em Shephard-Todd group} if it is
generated by complex reflections.  Shephard-Todd groups are a
natural generalization of Coxeter groups; we refer the reader to
[ST] for a discussion of their properties.  Let $W \subset GL(c)$
be a Shephard-Todd group, $g$ a complex reflection in $W$.  Fix a
basis of $c$ in which $g = \mbox{diag} \, (e^{2 \pi i / n}, 1,
\dots , 1)$.  We will say that $g$ is a {\em primitive reflection}
if there is no integer $n' > n$ such that $\mbox{diag} \,
(e^{2 \pi i / n'}, 1, \dots , 1) \in W$.

\begin{thm}\label{polar} {\em [DK]}
Let $G \, | \, V$ be a polar representation of a connected
reductive group.

(i)    All Cartan subspaces of $V$ are $G$-conjugate.

(ii)   Fix a Cartan subspace $c \subset V$.  Every vector
$v \in c$ is semisimple and every closed orbit passes
through $c$.

(iii)  Let $N^{}_G (c)$ and $Z^{}_G (c)$ be, respectively, the
normalizer and the stabilizer of $c$.  The quotient $W = N^{}_G (c)/
Z^{}_G (c)$ is a Shephard-Todd group acting on $c$; it is called the
{\em Weyl group} of the representation $G \, | \, V$ (note that, in
general, $W$ is not the Weyl group of $G$).

(iv)    Restriction to $c$ gives an isomorphism of invariant
rings ${\Bbb C} [V^*]_{}^G \cong {\Bbb C} [c^*]_{}^W$; this invariant ring
is free, generated by homogeneous polynomials.  We denote by $Q$
the categorical quotient $G \, {\backslash\!\backslash} \, V \cong W \, {\backslash\!\backslash} \, c$.
As a variety, $Q$ is isomorphic to a vector space of the same
dimension as $c$.

(v)   Let $f : V \to Q$ be the quotient map.  Denote by $c^{reg}
\subset c$ the set of regular points of $f|_c$.  It is the
complement of the union of the reflection hyperplanes in $c$.
Then the set $Q^{reg} \subset Q$ of regular values of $f$ is
equal to $f (c^{reg})$.
\end{thm}

We write $V^\circ = f^{-1} (Q^{reg})$.

\begin{rmk}\label{dkconj}
{\em  It is easy to see that $V^s \cap V^\circ \subset V^{rs}$.
Dadok and Kac conjecture that, in fact, $V^s \cap V^\circ = V^{rs}$
[DK, p. 521, Conjecture 2].}
\end{rmk}

We will call the fundamental group $\pi_1 (Q^{reg})$ the
{\em braid group} of $W$, and denote it by $B^{}_W$.

\begin{prop}\label{polardecom} {\em [DK]}
Let $G \, | \, V$ be a polar representation, as in Theorem
\ref{polar}, and $c \subset V$ a Cartan subspace.  Let
$G_c$ be the identity component of the stabilizer
$Z^{}_G (c)$.

(i)   There exists a compact form $K \subset G$, and a
$K$-invariant inner product $\langle \, , \, \rangle$ on $V$,
such that each $v \in c$ is of minimal length in its $G$-orbit.

(ii)  The group $G_c$ is reductive, and $K_c = G_c \cap K$ is a
compact form of $G_c$.

(iii) We have a $G_c$-invariant orthogonal decomposition
$$V = c \oplus U_c \oplus {\frak g} \cdot c,$$
with $\dim \, G_c {\backslash\!\backslash} U_c = 0$.

(iv)  The representation $G \, | \, V$ is stable if and only if
$U_c = 0$.
\end{prop}

We let $d = \dim \, V$, $d_0 = d - \dim \, U_c$, and
$r = \dim \, c = \dim \, Q$.  Note that $V^{rs}$ is connected and
smooth of dimension $d_0$.

The rank of a polar representation $G \, | \, V$ is defined by
$$\mathop{\rm rank}\nolimits \, G \, | \, V = \dim \, c / V^G,$$
where $V^G \subset V$ is the fixed points of $G$.  Note that
$V^G$ is contained in every Cartan subspace of $V$.
Dadok and Kac [DK, Theorem 2.12] show that any polar
representation has a decomposition into polar representations
of rank one, analogous to the root space decomposition of a
semisimple Lie algebra.  We will return to this result in
Section 5.

\subsubsection{Geometry of the General Fiber of a Polar Quotient
Map}

We now deduce from the results of [DK] some corollaries about
the conormal geometry of the general fiber of a polar quotient
map.  First, we need to discuss the dual of a polar representation.
We continue with the situation of Theorem \ref{polar} and
Proposition \ref{polardecom}.

\begin{prop}\label{polardual}
(i)  The dual representation $G \, | \, V^*$ is also polar, with
the same Weyl group $W$.  Given a Cartan subspace $c \subset V$,
as in Proposition \ref{polardecom}, a Cartan subspace for
$G \, | \, V^*$ is given by
$$c^* = ({\frak g} \cdot c \oplus U_c)^\perp \subset V^*.$$
This establishes a one-to-one correspondence between the
Cartan subspaces of $V$ and $V^*$.  Let
$\langle \, , \, \rangle^{}_{V^*}$ be the Hermitian metric on
$V^*$ induced by $\langle \, , \, \rangle$.  Then  every $l \in
c^*$ is of minimal length in its $G$-orbit.

(ii) Let $l \in c^*$ be regular semisimple.  Then the set
$\{ \, v \in V \; | \; l \, | \, _{{\frak g} \cdot v} = 0 \, \}$
equals $c \oplus U_c$.
\end{prop}
\begin{pf}
The dual representation $G \, | \, V^*$ differs from $G \, | \, V$
by an outer automorphism of $G$.  Thus, it is also polar with a
Cartan subspace of the same dimension, and the same Weyl group.

Take any $l \in c^*$.  It follows from Proposition \ref{polardecom}
that $\langle \, {\frak g} \cdot l, \, l \, \rangle^{}_{V^*} = 0$.  By a
theorem of Kempf and Ness ([KN], [DK, Theorem 1.1]), this implies
that $l$ is semisimple and of minimal length in its orbit.  The
rest of the proof is an exercise in linear algebra using
Proposition \ref{polardecom}.
\end{pf}

\begin{cor}\label{cortodual}
(i)   The set $(V^*)^{rs}$ of regular semisimple points in $V^*$ is
a connected algebraic manifold of dimension $d_0$.  We have
$(V^*)^s \subset \overline{(V^*)^{rs}}$.

(ii)  The subspace $U_c \subset V$ of Proposition \ref{polardecom}
is canonically independent of the choice of the compact form $K$,
and the inner product $\langle \, , \, \rangle$.
\end{cor}
\begin{flushright}$\Box$\end{flushright}

Pick a regular value $\lambda \in Q^{reg}$, and let
$F = f^{-1} (\lambda) \subset V$.  Let ${\cal O} \subset F$ be the
unique closed $G$-orbit in $F$, and let $v \in c \cap {\cal O}$.

\begin{prop}\label{polarfiber}
(i)   There is a unique $G$-equivariant algebraic map
$\phi : F \to {\cal O}$, such that $\phi (v + u) = v$, for any
$u \in U_c$.

(ii)  The map $\phi : F \to {\cal O}$ is a complex algebraic vector
bundle of rank $d - d_0$.

(iii) Fix any $x \in {\cal O}$.  Then the tangent spaces $T_y F$,
for all $y \in \phi^{-1} (x)$, are parallel.
\end{prop}
\begin{pf}
Let $\psi : G \times U_c \to V$ be the map $\psi : (g, u)
\mapsto g \, (u + v)$.  By part (iii) of Proposition
\ref{polardecom}, we have $\mbox{\rm Im} \, \psi \subset F$.  It
is not hard to see that for small $u \in U_c$, the differential
$d_{(1, u)} \psi$ is surjective onto the tangent space
$T_{u+v} F$, which is parallel to $U_c \oplus {\frak g} \cdot c$.
It follows that $\mbox{\rm Im} \, \psi$ contains a neighborhood
of $v$ in $F$.  Since ${\cal O}$ is the unique closed $G$-orbit in
$F$, we have $\mbox{\rm Im} \, \psi = F$.  It also follows that
$T_{u+v} F$ is parallel to $U_c \oplus {\frak g} \cdot c$, for all
$u \in U_c$.  It is now not hard to check that setting
$\phi (\psi (g, u)) = g \, v$ gives the required definition.
\end{pf}

We can now describe the conormal geometry of $F$.  Consider the
conormal bundle $\Lambda_F = T_F^* V \subset T^* V = V \times
V^*$, and let $p_2 : \Lambda_F \to V^*$ be the projection map.

\begin{cor}\label{polarcon}
(i)   The image $p_2 (\Lambda_F) = (V^*)^s$.

(ii)  For any $l \in (V^*)^{rs}$, the critical points of the
restriction $l |_F$ form a non-degenerate (Morse-Bott) critical
manifold, which is a union of $|W|$ fibers of $\phi$.   In
particular, $l |_F$ is a (complex) Morse function if and only
if $G \, | \, V$ is stable.
\end{cor}
\begin{pf}
Part (i) follows from the $G$-equivariance of the map $p_2$,
and Propositions \ref{polardual}, \ref{polarfiber}.  For part
(ii) we may assume that $l \in c^*$.  Then the critical points
of $l \, |_F$ is the set $\phi^{-1} (W \cdot v)$.  The
$G$-equivariance of $p_2$ and Proposition \ref{polardual} imply
that the restriction of $p_2$ to $p_2^{-1} (V^*)^{rs}$ is a
smooth submersion.  The Morse-Bott property of $l \, |_F$ is
simply a restatement of that fact.
\end{pf}

\begin{prop}\label{polarangle}
The variety $F \subset V$ is transverse to infinity in the sense
of Section 2.2.2, i.e., using the inner product of Proposition
\ref{polardecom}, we can find a $k > 0$, such that:
$$\angle (x, T_x X) < \frac{k}{|\!| x |\!|},$$
for any $x \in F$.
\end{prop}
\begin{pf}
We will show that we can take $k = |\!| \, v \, |\!|$.  Let
$x \in F$ and $l \in V^*$ be such that
$(x, l) \in \Lambda_F \subset T^* V \cong V \times V^*$.  Then
it is enough to show that $| l (x) | \leq
|\!| \, v \, |\!| \cdot |\!| \, l \, |\!|$.  By Proposition
\ref{polarfiber}, we may write $x = g \, (v + u)$ and
$l = g \, l'$, where $g \in G$, $u \in U_c$, and $l' \in c^*$.
Then we have $l (x) = l' (v + u) = l' (v)$, and
$|l (x)| = |l' (v)| \leq |\!| \, v \, |\!| \cdot
|\!| \, l' \, |\!| \leq |\!| \, v \, |\!| \cdot
|\!| \, l \, |\!|$, where the last inequality follows from the
fact that $l'$ is of minimal length in its orbit (Proposition
\ref{polardual}).
\end{pf}

\section{The Main Theorem}

\subsection{Statement of the Theorem}

Let $G \, | \, V$ be a polar representation, as in Theorem
\ref{polar}.  Assume that either $\mathop{\rm rank}\nolimits \, G \, | \, V = 1$,
or $G \, | \, V$ is {\em visible}, i.e., there are finitely
many nilpotent orbits in $V$.  This assumption holds for the
representations discussed in [KR] and for all of the infinite
series listed in [DK] and [K].  As before, we write $f : V \to Q$
for the quotient map, and $E = f^{-1} (0)$ for the zero fiber.
The purpose of the additional restriction of $G \, | \, V$
is to ensure that there exists a $G$-invariant conical
stratification ${\cal S}$ of $E$, such that the ${\rm A}_f$
condition holds for the pair $(V^\circ, S)$, for every
$S \in {\cal S}$.  Indeed, if $G \, | \, V$ is visible, we can
take ${\cal S}$ to be the orbit stratification.  If, instead,
$\mathop{\rm rank}\nolimits \, G \, | \, V = 1$, we can readily reduce to the case
$\dim \, Q = 1$, then use the general result about the
existence of ${\rm A}_f$ stratifications for functions
[Hi, p. 248, Corollary 1].

We are now in the situation of Proposition \ref{nclb}, and
therefore may consider the nearby cycles sheaf $P = P_f \in
{\cal P}^{}_G \, (E)$.  In order to fix the up-to-isomorphism
ambiguity in the definition of $P$, we fix a regular value
$\lambda \in Q^{reg}$, let $F = f^{-1} (\lambda)$, and
identify $P$ with the sheaf $P^{}_F$ given by the specialization
of $F$ to ${\rm{as}} (F) = E$, as in Section 2.2.2.  This
corresponds to specializing along the path $\gamma : z \mapsto
f(z F)$.  We also have the monodromy action $\mu : B^{}_W
= \pi_1 (Q^{reg}, \lambda) \to {\rm Aut} (P)$.  Our main result
is Theorem \ref{main} below.  It gives a description of the
pair $(P, \mu)$ analogous to Theorem \ref{ftinspth} in Springer
theory.  Recall from Corollary \ref{cortodual} that the set
$(V^*)^{rs}$ of regular semisimple points in $V^*$ is a
connected algebraic manifold.

\begin{thm}\label{main}
Let $G \, | \, V$ be, as above, a polar representation which
is of rank one or visible.  Let $c \subset V$ be a Cartan
subspace, and $W$ be the Weyl group.

(i)    We have
$${\cal F} \, P \cong {\cal IC} ((V^*)^{rs}, {\cal L}),$$
where the right hand side is an intersection homology sheaf
with coefficients in a local system ${\cal L}$ on $(V^*)^{rs}$
of rank $|W|$.

(ii)   Write $A = {\rm End} (P)$, the endomorphisms of $P$
as a perverse sheaf (forgetting the $G$-equivariant structure).
We have $\dim \, A = |W|$.  The monodromy action $\mu$ gives
a surjection ${\Bbb C} [B^{}_W] \to A$.

(iii)  Let $\sigma \in  W$ be a primitive reflection of order
$n_\sigma$.  It gives rise to an element $\hat\sigma \in B^{}_W$,
represented by a loop going counterclockwise around the image
$f(c_\sigma)$ of the hyperplane $c_\sigma \subset c$ fixed by
$\sigma$.  The minimal polynomial $R_\sigma$ of
$\mu (\hat\sigma) \in A$ has integer coefficients and is of
degree $n_\sigma$ (we normalize $R_\sigma$ to have leading
coefficient $1$).

(iv)   Fix a basepoint $l \in (V^*)^{rs}$.  By part (i), we have
an action
$$A = {\rm End} (P) \cong {\rm End} ({\cal L}) \to {\rm End} ({\cal L}_l)$$
of $A$ on the fiber ${\cal L}_l$.  There is an identification
$\chi : {\cal L}_l \cong A$, such that this action is given by left
multiplication.

(v)    There is a semigroup homomorphism $\rho : \pi_1
((V^*)^{rs}, l) \to A^{\circ}$, the opposite of the algebra $A$,
such that the holonomy of ${\cal L}$ is given as $\rho$ followed by the
right multiplication action of $A^{\circ}$ on ${\cal L}_l
\stackrel{\chi} {\cong} A$.  The homomorphism $\rho$ gives a
surjection ${\Bbb C} [\pi_1 ((V^*)^{rs}, l)] \to A^{\circ}$.
\end{thm}

\begin{rmk}\label{visibleispolar}
{\em Dadok and Kac conjecture [DK, p. 521, Conjecture 4] that
every visible representation is polar.}
\end{rmk}

\begin{rmk}\label{supp}
{\em  The support $\, \mbox{supp} \, ({\cal F} \, P) =
\overline{(V^*)^{rs}} \,$ is equal to all of $V^*$ if and only if
the action $G \, | \, V$ is stable (cf. Corollary \ref{cortodual}).}
\end{rmk}

\begin{rmk}\label{stha}
{\em  It is natural to expect from claims (ii) and (iii) of
Theorem \ref{main} that the algebra $A$ is equal to the quotient
${\Bbb C} [B^{}_W] / \langle \, R_\sigma (\hat\sigma) \, \rangle$, where
$\sigma$ runs over all the primitive reflections in $W$.  This is
always true when $W$ is a Coxeter group.  In fact, it is the case
in all of the examples known to the author.}
\end{rmk}

\begin{rmk}\label{pointer}
{\em  We will give some additional information about the
polynomials $R_\sigma$ in Section 5.}
\end{rmk}

\begin{rmk}\label{genft}
{\em  One may ask whether it is always true, say, for a
homogeneous polynomial $f : {\Bbb C}^d \to {\Bbb C}$ that the Fourier
transform ${\cal F} \, P_f$ of the nearby cycles of $f$ is an
intersection homology sheaf.  The answer is ``no.''  A
counterexample is given by $f (x,y,z) = x^2 y + y^2 z$.}
\end{rmk}

\subsection{Examples}

\begin{ex}\label{qdrcs}
{\bf Quadrics.}
{\em Let $G = SO_n$ act on $V = {\Bbb C}^n$ by the standard
representation.  Any non-isotropic line $c \subset V$ can
serve as a Cartan subspace.  The Weyl group $W = {\Bbb Z} / 2 {\Bbb Z}$,
and the invariant map $f : V \to {\Bbb C}$ is just the standard
quadratic invariant of $SO_n$.  The algebra $A$ of Theorem
\ref{main} is given by $A = {\Bbb C} [z] / (z-1)^2$, if $n$ is even,
and $A = {\Bbb C} [z] / (z^2-1)$, if $n$ is odd.}
\end{ex}

\begin{ex}\label{nc}
{\bf Normal Crossings.}
{\em Consider the action of the torus $G = ({\Bbb C}^*)^{n-1}$ on
$V = {\Bbb C}^n$, given by
$$(t_1, \dots , t_{n-1}) : (x_1, \dots , x_n) \mapsto (t_1 x_1,
\, t_1^{-1} t_2 x_2, \, t_2^{-1} t_3 x_3, \, \dots , \,
t_{n-1}^{-1} x_n).$$
A Cartan subspace for this action is given by
$c = \{ \, x_1 = \dots = x_n \, \} \subset V$, the Weyl group
$W = {\Bbb Z} / n {\Bbb Z}$, and the invariant map $f : V \to {\Bbb C}$ is just the
product $f : (x_1, \dots , x_n) \mapsto x_1 \cdot \dots \cdot x_n$.
The algebra $A$ of Theorem \ref{main} is given by $A = {\Bbb C} [z] /
(z-1)^n$, functions on the $n$-th order neighborhood of a point.}
\end{ex}

\begin{ex}\label{det}
{\bf The Determinant} {\em (see [BG]).
Let $G = SL_n$ act on $V = Mat \, (n \times n; {\Bbb C})$
by left multiplication.  A Cartan subspace $c \subset V$ for this
action is given by the scalar matrices, the Weyl group
$W = {\Bbb Z} / n {\Bbb Z}$, and the invariant map $f : V \to {\Bbb C}$ is the
determinant.  As in the previous example, the algebra
$A = {\Bbb C} [z] / (z-1)^n$.}
\end{ex}

\begin{ex}\label{symmat}
{\bf Symmetric Matrices} {\em (see [BG]).
Let $G = SL_n$ act on the space $V$ of symmetric $n\times n$
matrices by $g : x \mapsto g x g^t$.  A Cartan subspace
$c \subset V$ is again given by the scalar matrices, the Weyl group
$W = {\Bbb Z} / n {\Bbb Z}$, and the invariant map $f : V \to {\Bbb C}$ is again
the determinant.  However, the algebra $A$ is now given by
$A = {\Bbb C} [z] / (z-1)^{\lceil n/2 \rceil} (z+1)^{\lfloor n/2
\rfloor}$.}
\end{ex}

\begin{ex}\label{ran}
{\bf A ``Real Analog'' of Springer Theory for $SL_n$}
{\em (see [Gr2] and Section 6 below).
Let $G = SO_n$ act on the space $V$ of symmetric $n\times n$
matrices by conjugation.  A Cartan subspace $c \subset V$ is
given by the diagonal matrices.  The Weyl group $W = \Sigma_n$,
the symmetric group on $n$ letters.  The invariant map
$f : V \to {\Bbb C}^{n-1}$ is given by sending a matrix $x \in V$ to
the characteristic polynomial $\mbox{char} \, (x)$.  The algebra
$A = {\cal H}^{}_{-1} (\Sigma_n)$, the Hecke algebra of $\Sigma_n$
specialized at $q = -1$.}
\end{ex}

\begin{ex}\label{fullstgp}
{\bf Maps from an Orthogonal to a Symplectic Vector Space.}
{\em Let $(U_1, \nu)$ be an orthogonal vector space of dimension
$2n+1$ ($\nu$ is a non-degenerate quadratic form), and
$(U_2, \omega)$ a symplectic vector space of dimension $2n$.  Set
$V = \mbox{Hom}^{}_{\Bbb C} \, (U_1, U_2)$.  The group $G = Sp_{2n}
\times SO_{2n+1}$ acts on $V$ by left-right multiplication.  This
action is polar, with a Cartan subspace of dimension $n$.  The Weyl
group $W \subset GL_n$ is the semi-direct product of $({\Bbb Z} / 4 {\Bbb Z})^n
\subset GL_n$, acting by diagonal matrices of fourth roots of unity,
and the symmetric group $\Sigma_n \subset GL_n$, acting by
permutation matrices.  Let $\sigma \in \Sigma_n \subset W$ be any
simple reflection, and let $\tau = \mbox{diag} \, (i, 1, \dots, 1)
\in ({\Bbb Z} / 4 {\Bbb Z})^n \subset W$.  The algebra $A$ is given by
$A = {\Bbb C} [B^{}_W] / \langle (\sigma^2 - 1), \,
(\tau^2 - 1)^2 \rangle$.}
\end{ex}

In all of the above examples, the representation $G \, | \, V$ is
stable.  Here is an example where it is not.

\begin{ex}\label{nonstab}
{\bf An Evaluation Map.}
{\em  Let $U \cong {\Bbb C}^{2n}$.  Set $V = \Lambda^2 U^* \oplus U
\oplus U$.  The linear group $G = GL(U)$ acts on $V$.  The only
invariant is the evaluation map $f : (\omega, u_1, u_2) \mapsto
\omega (u_1, u_2)$.  A non-zero vector $(\omega, u_1, u_2) \in V$
is semisimple, if and only if $\omega$ is of rank $2$ and
$\omega (u_1, u_2) \neq 0$.  The Weyl group $W = {\Bbb Z} / 3 {\Bbb Z}$, and
the algebra $A = {\Bbb C} [z] / (z - 1)^3$.}
\end{ex}

\subsection{A Remark on the Degree of Generality}

The reason we assume in Theorem \ref{main} that $G \, | \, V$ is
of rank one or visible, is to insure that the ${\rm A}_f$ condition
holds and the sheaf $P$ is well defined.  If we dropped this
assumption, we could still pick a $\lambda \in Q^{reg}$, and
consider a sheaf $P_\lambda$ coming from the specialization of
$F = f^{-1} (\lambda)$ to the asymptotic cone.  Using the techniques
of this paper and some algebro-geometric generalities, we could then
show that the sheaves $P_\lambda$ for {\em generic} $\lambda$ are
isomorphic, and that they form a local system over some open
set $Q' \subset Q^{reg}$.  Furthermore, all of the assertions
of Theorem \ref{main} could be given meaning and proved in this
context.  This is somewhat unsatisfactory, since we would
have to use the full power of our methods just to show that there
is a well posed question.  Conjecture \ref{polaraf} below implies
that the subset $Q'$ is a phantom, and that nearby cycles are well
defined for any polar representation.

\begin{conj}\label{polaraf}
Let $G \, | \, V$ be a polar representation, $f : V \to Q$ the
quotient map, and $E = f^{-1} (0)$.  Then there exists an
${\rm A}_f$ stratification of $E$ (see Section 2.2.1).
\end{conj}

If this conjecture is true, all our results extend automatically
to an arbitrary polar representations.

\section{Proof of the Main Theorem}

\subsection{A Preliminary Lemma}

We begin with a preliminary lemma.  Recall the $G$-invariant
conical stratification ${\cal S}$ of $E$ discussed at the beginning
of Section 3.  By passing if necessary to a refinement, we
may assume that ${\cal S}$ satisfies conditions (i) - (iii) of
Section 2.2.2.

\begin{lemma}\label{prelim}
Let $l \in (V^*)^{rs}$, and let $a ({\cal S}, l)$ be the dimension
of the critical locus of $l \, |_E$ with respect to
${\cal S}$, as in Lemma \ref{dimbound}.  Then $a ({\cal S}, l) \leq
d - d_0$, where $d = \dim V$, and $d_0 = \dim \, V^{rs}$.
\end{lemma}
\begin{pf}
Let $\Lambda_E \subset T^* V = V \times V^*$ be the conormal
variety to the stratification ${\cal S}$, and $p_2 : V \times V^*
\to V^*$ be the projection.  Using Proposition \ref{polardual}
and the $G$-invariance of ${\cal S}$, it is not hard to show that
$\dim \, p_2^{-1} (l) \cap \Lambda_E$ is independent of $l$
for $l \in (V^*)^{rs}$.  Therefore,
$$a ({\cal S}, l) + \dim \, (V^*)^{rs} \leq \dim \, \Lambda_E = d.$$
Together with Corollary \ref{cortodual}, this proves the lemma.
\end{pf}

\subsection{The Stable Case}

In this section we prove Theorem \ref{main} in the case when
the representation $G \, | \, V$ is stable.  Then, in Section
4.3, we will indicate how to modify the argument in the
nonstable case.

Assuming $G \, | \, V$ is stable, choose a basepoint
$l \in (V^*)^{rs}$.  Note that $(V^*)^{rs}$ is open in $V^*$.
We may assume that $l \in c^*$, the Cartan subspace of
Proposition \ref{polardual}.  By Proposition \ref{polarangle},
all the constructions of Section 2.2.2 apply to $X = F$.  As
in that section, we consider the compactification $\hat F$ of
$F$ relative to $l$, and denote by $Z$ the set of critical
points of the restriction $\hat l \, |_{\hat F}$, as in Lemma
\ref{dimbound}.  By Lemmas \ref{dimbound}, \ref{prelim}, and
Corollary \ref{polarcon}, we have $Z \subset c \subset V$.
Furthermore, $Z$ is a single $W$-orbit in $c$, and each
critical point in $Z$ is Morse. Fix a point $e^{}_0 \in Z$,
and write $Z = \{ e^{}_w \}^{}_{w \in W}$, where $e^{}_w =
w \, e^{}_0$.

Recall the description of the stalk $H^{*}_l \, ({\cal F} \, P)$ in
Lemma \ref{auxmtw}.  Since $\dim \, Z = 0$, we have
$H^{i}_l \, ({\cal F} \, P) = 0$, unless $i = -d$.  Furthermore,
by duality:
$$H^{-d}_l \, ({\cal F} \, P) \cong H_{d-r} \, ( F, \, \{ \, \xi (y)
\geq \xi^{}_0 \, \}; \; {\Bbb C}),$$
where $\xi^{}_0 > | \, l (e) \, |$, for all $e \in Z$.

We will need a standard Picard-Lefschetz construction
of classes in the relative homology group above (see,
for example, [BG, Section 7.2]).  Let $e \in Z$ be a critical
point.  Fix a smooth path $\gamma : [0, 1] \to {\Bbb C}$ such that:

(i)   $\gamma (0) = l (e)$, and $\gamma (1) = \xi^{}_0$;

(ii)  $\gamma (t) \notin l (Z)$, for $t > 0$;

(iii) $\gamma (t_1) \neq \gamma (t_2)$, for $t_1 \neq t_2$;

(iv)  $\gamma' (t) \neq 0$, for $t \in [0, 1]$.

\noindent
Let ${\cal H}_e : T_e F \rightarrow {\Bbb C}$ be the Hessian of
$l \, |_F$ at $e$, and let $T_e \, [\gamma] \subset T_e F$ be the
positive eigenspace of the (non-degenerate) real quadratic form
$$\mbox{Re} \, (\, {\cal H}_e \, / \, \gamma'(0) \, ) :
T_e F \to {\Bbb R}.$$
Note that $\dim_{\Bbb R} \, T_e \, [\gamma] = d-r$.  Fix an orientation
${\cal O}$ of $T_e \, [\gamma]$.  The triple $(e, \gamma, {\cal O})$
defines a homology class
$$u = u (e, \gamma, {\cal O}) \in H_{d-r} \, ( F, \,
\{ \, \xi (y) \geq \xi^{}_0 \, \}; \; {\Bbb C}).$$
Namely, the class $u$ is represented by an embedded $(d-r)$-disc
$$\kappa : ({\rm D}^{d-r}, \, \partial \, {\rm D}^{d-r}) \to
(F, \, \{ \, \xi (y) \geq \xi^{}_0 \, \}),$$
such that the image of $\kappa$ projects onto the image of
$\gamma$, is tangent to $T_e \, [\gamma]$ at $e$, and does not
pass through any point of $Z$ except $e$.  The sign of $u$ is given
by the orientation ${\cal O}$.  It is a standard fact that $u \neq 0$.

We now fix a path $\gamma^{}_0 : [0, 1] \to {\Bbb C}$, satisfying
conditions (i) - (iv) above for $e = e^{}_0$, and an orientation
${\cal O}^{}_0$ of the space $T_{e^{}_0} \, [\gamma^{}_0]$.  Let
$u^{}_0 = u (e^{}_0, \gamma^{}_0, {\cal O}^{}_0)$; we will use the
same symbol for the corresponding element of $H^{-d}_l \,
({\cal F} \, P)$.

\begin{lemma}\label{monsurj}
The image of $u^{}_0 \in H^{-d}_l \, ({\cal F} \, P)$ under the
monodromy action of $B^{}_W$ generates the stalk
$H^{-d}_l \, ({\cal F} \, P)$ as a vector space.  We have
$\dim \, H^{-d}_l \, ({\cal F} \, P) = |W|$.
\end{lemma}
\begin{pf}
For each $w \in W$, pick a lift $b^{}_w \in B^{}_W$.  By a
standard Picard-Lefschetz theory argument, we have
$$\mu (b^{}_w) \, u^{}_0 = u \, (e^{}_w, \gamma^{}_w, {\cal O}^{}_w),$$
where $\gamma^{}_w$ is some path satisfying conditions (i) - (iv)
for $e = e^{}_w$, and ${\cal O}^{}_w$ is an orientation of
$T_{e^{}_w} \, [\gamma^{}_w]$.  Since $\{ e^{}_w \}^{}_{w \in W}$
are the only critical points of the function $\hat l \, |_{\hat F}$,
the elements $\{ \mu (b^{}_w) \, u^{}_0 \}^{}_{w \in W}$ form a
basis of $H^{-d}_l \, ({\cal F} \, P)$.
\end{pf}

Consider now the restriction of ${\cal F} \, P$ to the manifold
$(V^*)^{rs}$.  By Lemma \ref{monsurj} and the preceding arguments,
it is a perverse sheaf whose stalks only live in degree $-d$ and
all have rank $|W|$.  Therefore, it is a rank $|W|$ local system
with a degree shift.  Together with Theorem \ref{asym_cone}, this
proves part (i) of Theorem \ref{main}.  The local system ${\cal L}$ of
that theorem is given by ${\cal L} = {\cal F} \, P \, |_{\, (V^*)^{rs}} \,
[-d]$.  Let $h :  \pi_1 ((V^*)^{rs}, l) \to {\rm End} ({\cal L}_l)$ be
the holonomy of ${\cal L}$.

\begin{lemma}\label{holsurj}
The image of $u^{}_0 \in {\cal L}_l$ under the action $h$ generates
the stalk ${\cal L}_l$ as a vector space.
\end{lemma}
\begin{pf}
The proof of this is analogous to the proof of Lemma
\ref{monsurj}.  Let $Q' = G {\backslash\!\backslash} V^*$, and $f' : V^* \to Q'$
be the quotient map.  Let $(Q')^{reg} \subset Q'$ be the set
of regular values of $f'$.   Set $(V^*)^\circ = {f'}^{\, -1}\,
((Q')^{reg})$; it is a Zariski open subset of $(V^*)^{rs}$ (see
Remark \ref{dkconj}).  We may assume without loss of generality
that $l \in (V^*)^\circ$.  Let ${\cal L}^\circ_{}$ be the restriction
of the local system ${\cal L}$ to $(V^*)^\circ$, and let $h^\circ :
\pi_1 ((V^*)^\circ, l) \to {\rm End} ({\cal L}_l)$ be the holonomy of
${\cal L}^\circ$.  It is enough to show that the image of $u^{}_0$
under $h^\circ$ generates the stalk ${\cal L}_l$.

Since we assume that $G$ is connected, the push-forward
homomorphism $f'_* : \pi_1 ((V^*)^\circ, l) \to
\pi_1 ((Q')^{reg}, f' (l))$ is surjective.  By Proposition
\ref{polardual}, we have an isomorphism
$\eta : \pi_1 ((Q')^{reg}, f' (l)) \cong B^{}_W$ (it is not
canonical).  The composition $\eta \circ f'_*$, followed by the
natural map $B^{}_W \to W$, gives a surjection
$\theta : \pi_1 ((V^*)^\circ, l) \to W$.  For each $w \in W$,
choose a lift $\alpha^{}_w \in \theta^{-1} (w)$.  Then a
Picard-Lefschetz theory argument similar to the proof of Lemma
\ref{monsurj} shows that the elements
$\{ \, h^\circ (\alpha^{}_w) \, u^{}_0 \, \}^{}_{w \in W}$
form a basis of ${\cal L}_l$.
\end{pf}

Claims (ii), (iv), and (v) of Theorem 3.1 follow easily from
Lemmas \ref{monsurj} and \ref{holsurj}.  We define the vector
space map $\chi : A \to {\cal L}_l$ of claim (iv) by $\chi : a \mapsto
a \, u^{}_0$.  By Lemma \ref{monsurj}, $\chi$ is surjective.  On
the other hand, the action of any $a \in A$ on ${\cal L}_l$ must
commute with the holonomy representation $h$.  Therefore, Lemma
\ref{holsurj} implies that $\chi$ is injective.  This proves that
$\chi$ is a vector space isomorphism.  Claim (ii) now follows from
Lemma \ref{monsurj}; claim (iv) from the definition of $\chi$; and
claim (v) from Lemma \ref{holsurj}, and the fact that every
endomorphisms of $A$ commuting with the left action of $A$ on
itself is of the form $a \mapsto a \, a'$, for some $a' \in A$
(this is true for any associative algebra with unit).

It remains to prove claim (iii) of the theorem.  The idea is to
use a regular value $\lambda \in Q^{reg}$ which is very near the
image $f (c_\sigma)$, then apply a standard carousel argument
(see [L\^e2] and [BG, Section 7.3]).

Pick a point $v_1 \in c_\sigma \subset c$ which is not fixed
by any element of $W$, other than the powers of $\sigma$.  The
orbit $W \cdot v_1$ consists of $|W| / n_\sigma$ points.  Choose
a point $v \in c^{reg}$ which is very near $v_1$.  The orbit
$W \cdot v$ consists of $|W|$ points which are grouped into
$|W| / n_\sigma$ clusters.  Each cluster consists of $n_\sigma$
points surrounding a point in $W \cdot v_1$, and the action of
$\sigma \in W$ cyclically permutes the points in each cluster.
We may assume that the fiber $F = f^{-1} (\lambda)$ passes
through $v$, so that $Z = W \cdot v$.  We may also assume that
the choice of $l \in c^* \cap (V^*)^{rs}$ is sufficiently generic,
so that the image $l (Z)$ appears in the ${\Bbb C}$-plane as
$|W| / n_\sigma$ disjoint clusters, each consisting of $n_\sigma$
points arranged in a circle (see Figure 1).

Choose a path $\gamma_0 : [0, 1] \to {\Bbb C}$, satisfying conditions
(i) - (iv) above for $e = v$, which does not ``come near'' any
of the clusters in $l (Z)$, other than the one containing
$l (v)$.  Fix an orientation ${\cal O}_0$ of the space
$T_{v} \, [\gamma]$.  Let $u_0 = u (v, \gamma_0, {\cal O}_0)$.
The element $\hat \sigma \in B^{}_W = \pi_1 (Q^{reg}, \lambda)$ is
represented by a loop going once counterclockwise around the image
$f (c_\sigma)$, which stays in a small neighborhood of the critical
value $f (v_1)$.  Then, by a standard carousel argument, the vectors
$\{ \, \mu(\hat \sigma)^k \, u_0 \, \}_{k = 0}^{n_\sigma - 1}$
are linearly independent, and we have
$$\mu(\hat \sigma)^{n_\sigma} \, u_0 = \pm u_0 +
\sum_{k = 1}^{n_\sigma - 1} g_k \cdot \mu(\hat \sigma)^k \, u_0,$$
where the $\{ g_k \}$ are some integers.  Together with Lemma
\ref{holsurj}, this implies claim (iii) of Theorem \ref{main}.

\begin{center}
\leavevmode{}
\epsfbox{figure.eps}
Figure 1
\end{center}
\vspace{.1in}

\subsection{The Nonstable Case}

The proof of Theorem \ref{main} in the case when $G \, | \, V$ is
not stable is analogous to the argument of Section 4.2, with Morse
critical points of $l$ replaced by Morse-Bott critical manifolds
(see part (ii) of Corollary \ref{polarcon}).  We briefly indicate
the changes that have to be made in the nonstable case.

As before, we pick a basepoint $l \in (V^*)^{rs}$, lying in the
Cartan subspace $c^*$ of Proposition \ref{polardual}, and write $Z$
for the critical points of the restriction $\hat l \, |_{\hat F}$.
Fix a point $e \in (F \cap c) \subset Z$, and a path
$\gamma: [0,1] \to {\Bbb C}$, satisfying conditions (i) - (iv) of Section
4.2.  The Hessian ${\cal H}_e : T_e F \rightarrow {\Bbb C}$ of $l \, |_F$ at
$e$ will now be degenerate.  However, we may still consider the
positive eigenspace $T_e \, [\gamma] \subset T_e F$ of the
real quadratic form
$\mbox{Re} \, (\, {\cal H}_e \, / \, \gamma'(0) \, )$.
By Corollary \ref{polarcon}, we have $\dim_{\Bbb R} \, T_e \, [\gamma] =
d_0 - r$.  Fix an orientation ${\cal O}$ of $T_e \, [\gamma]$.  The
triple $(e, \gamma, {\cal O})$ defines a Picard-Lefschetz class
$$u = u (e, \gamma, {\cal O}) \in H_{d_0 - r} \, ( F, \,
\{ \, \xi (y) \geq \xi^{}_0 \, \}; \; {\Bbb C}),$$
where $\xi_0$ is large, exactly as before.  We may regard $u$ as an
element of $H^{-d_0}_l \, ({\cal F} \, P)$.

The main distinction from the stable case is that there is no
a priori geometric reason for $u$ to be non-zero.  This is because
the Morse-Bott manifold containing $e$ is non-compact.  However,
we may use Lemmas \ref{auxmtw}, \ref{dimbound}, \ref{prelim}, and
an argument as in Lemma \ref{monsurj}, to show that $u$ generates
the stalk $H^{-d_0}_l \, ({\cal F} \, P)$ under the monodromy action of
$B^{}_W$.  Thus, if we had $u = 0$, it would follow that
$H^{-d_0}_l \, ({\cal F} \, P) = 0$, and by Theorem \ref{asym_cone},
that $P = 0$.  This contradiction shows that $u \neq 0$.  The rest of
the proof is exactly as in the stable case.

\section{Rank One Representations and the Polynomials
$R_\sigma$}

As we mentioned in Section 2, there is an analog of root space
decomposition for polar representation.  Namely, any polar
representation has a decomposition into polar representations
of rank one.  In this section, we use these rank one
representations to give an interpretation (see Theorem
\ref{rankone}) of the polynomials $R_\sigma$ appearing in part
(iii) of Theorem \ref{main}.  This interpretation will be used
in Section 6.2 to analyze the polar representations arising from
symmetric spaces.

Let $G \, | \, V$ be a polar representation, $\sigma \in  W$ be
a primitive reflection of order $n_\sigma$, and $c_\sigma
\subset c$ be the hyperplane fixed by $\sigma$ (cf. Theorem
\ref{main}).  Denote by $c_\sigma^\circ$ the set of all
$v \in c_\sigma$ such that the stabilizer $Z^{}_W (v)$ is
generated by $\sigma$; it is an open subset of the hyperplane
$c_\sigma$.  Let ${\frak g}_\sigma \subset {\frak g}$ be the stabilizer of
$c_\sigma$, and $G_\sigma \subset G$ the adjoint form of
${\frak g}_\sigma$.  In terms of Proposition \ref{polardecom}, let
$$V_\sigma = c \oplus U_c \oplus {\frak g}_\sigma \cdot c \subset V.$$
If ${\frak g}_\sigma \cdot c = 0$, we say that $\sigma$ is {\em of global
type}.  Otherwise, we say that $\sigma$ is {\em of local type}.

\begin{prop}\label{polarslice} {\em [DK]}
(i)   The representation $G \, | \, V$ restricts to a
representation $G_\sigma \, | \, V_\sigma$, which is polar
with Cartan subspace $c$.  The rank of $G_\sigma \, | \, V_\sigma$
is equal to zero if $\sigma$ is of global type, and to one if
$\sigma$ is of local type.

(ii)  Choose a point $v_1 \in c_\sigma^\circ$.  Let ${\frak g}_{v^{}_1}
\subset {\frak g}$ be the stabilizer of $v_1$.  Then we have
${\frak g}_{v^{}_1} = {\frak g}_\sigma$, ${\frak g} \cdot v_1 = {\frak g} \cdot c_\sigma$,
and  $V = V_\sigma \oplus {\frak g} \cdot v_1$.
\end{prop}

Proposition \ref{polarslice} implies that $\sigma$ is of
local type if and only if $c_\sigma \cap V^{rs} = \emptyset$.
Dadok and Kac conjecture (cf. Remark \ref{dkconj}) that $\sigma$ is
always of local type.  We do not know of any counterexamples, but
allow for the possibility that they exist.  The ``root space
decomposition'' theorem of [DK] involves the representations
$G_\sigma \, | \, V_\sigma$, for all primitive reflections $\sigma$
of local type.

The Weyl group $W_\sigma$ of $G_\sigma \, | \, V_\sigma$ is
naturally a subgroup of $W$.  Moreover, it must be a cyclic group,
generated by $\sigma^{m_\sigma}$ for some $m_\sigma \in {\Bbb Z}_+$, a
divisor of $n_\sigma$.  Note that when $\sigma$ is of global type,
we have $m_\sigma = n_\sigma$ and $W_\sigma = \{ 1 \}$.  Let
$f_\sigma : V_\sigma \to Q_\sigma = G_\sigma {\backslash\!\backslash} V_\sigma$ be the
quotient map, $E_\sigma = f_\sigma^{-1} (0)$ be the zero fiber,
and $P_\sigma \in {\cal P}_{G_\sigma} (E_\sigma)$ be the nearby cycles
of $f_\sigma$.  Let $B_\sigma$ be the braid group of $W_\sigma$.
If $\sigma$ is of local type, then $B_\sigma \cong {\Bbb Z}$, and we have
a monodromy transformation  $\mu_\sigma : P_\sigma \to P_\sigma$,
given by the action of the counterclockwise generator of $B_\sigma$.
If $\sigma$ is of global type, then $B_\sigma = \{ 1 \}$, and we set
$\mu_\sigma = \mbox{Id} : P_\sigma \to P_\sigma$.  Let $\tilde R$ be
the minimal polynomial of $\mu_\sigma \in {\rm End} (P_\sigma)$.  It
is a monic polynomial with integer coefficients of degree
$n_\sigma / m_\sigma$.  Note that if $\sigma$ is of global type,
then $\tilde R_\sigma (z) = z - 1$.  By Theorem \ref{main}, we have
${\rm End} (P_\sigma) \cong {\Bbb C} [z] / \tilde R_\sigma (z)$.

\begin{thm}\label{rankone}
Let $G \, | \, V$ be as in Theorem \ref{main}, and $G_\sigma
\, | \, V_\sigma$ be as in Proposition \ref{polarslice}.  Then the
polynomial $R_\sigma$ of part (iii) of Theorem \ref{main} is given
by $R_\sigma (z) = \tilde R_\sigma (z^{m_\sigma})$.
\end{thm}
\begin{pf}
Choose a point $v_1 \in c_\sigma$, as in part (ii) of Proposition
\ref{polarslice}.  Let $\lambda_1 = f (v_1) \in Q$.  Choose a
regular value $\lambda \in Q^{reg}$ very near $\lambda_1$.  The
idea of this proof is to break up the specialization of $\lambda$
to $0$ into two steps: first specialize $\lambda$ to $\lambda_1$,
then specialize $\lambda_1$ to $0$.  In order to compute the
polynomial $R_\sigma$, we will only need to understand the first
step.

Let ${\rm D} = \{ \, z \in {\Bbb C} \; | \; |z| < 2 \, \}$ and $\gamma^{}_1 :
{\rm D} \to Q$ be an embedded holomorphic arc such that:

(i)   $\gamma^{}_1 (0) = \lambda_1$;

(ii)  $\gamma^{}_1 (1) = \lambda$;

(iii) $\gamma^{}_1 ({\rm D} \setminus 0) \subset Q^{reg}$;

(iv)  $\gamma^{}_1$ is transverse to the image $f (c_\sigma)$.

\noindent
Form the fiber product $V_1 = V \times^{}_Q {\rm D}$, and let $f_1 : V_1
\to {\rm D}$ be the projection map.  Let $E_1 = f^{-1} (\lambda_1)$.  We
have $\dim \, E_1 = \dim \, E = d - r$.  Let $P_1 = \psi_{f_1} \,
{\Bbb C}^{}_{V_1} \, [d - r]$; we have $P_1 \in {\cal P}^{}_G (E_1)$.  Write
$\mu_1 : P_1 \to P_1$ for the monodromy transformation of $P_1$.

{\bf Claim:}
There is a functor $\Psi : {\cal P}_G (E_1) \to {\cal P}_G (E)$ such
that $\Psi \, P_1 = P$ and $\Psi (\mu_1) = \mu (\hat\sigma)$.

To prove the claim, define an arc $\gamma_2 : {\rm D} \to Q$ by
$\gamma_2 : z \mapsto f (z \, v_1)$.  Form $V_2 = V \times^{}_Q {\rm D}$,
and let $f_2 : V_2 \to {\rm D}$ be the projection.  We have a map
$\pi : V_2 \setminus E \to E_1$, defined by $\pi (z \, e, \, z) = e$,
for $z \in {\rm D} \setminus 0$, $e \in E_1$.  Set $\Psi = \psi_{f_2}
\circ \pi^*$.  The claim follows form the definition of nearby
cycles.

Because of the claim, it will suffice to show that
$\tilde R_\sigma (\mu_1^{m_\sigma}) = 0$.  For this, consider
the inclusion $j_\sigma : V_\sigma \to V$ given by
$j_\sigma : v \mapsto v + v_1$.  By Proposition \ref{polarslice},
$j_\sigma$ exhibits $V_\sigma$ as a $G_\sigma$-invariant normal
slice to the orbit $G \cdot v_1$.  Consider the perverse
restriction to the normal slice functor ${{{}^{\scriptscriptstyle p}} \hspace{-.02in} j}_\sigma^*  =
j_\sigma^* \, [d_\sigma - d] : {\cal P}_G (E_1) \to {\cal P}_{G_\sigma}
(E_\sigma)$, where $d_\sigma = \dim \, V_\sigma$.  It is not
hard to see that ${{{}^{\scriptscriptstyle p}} \hspace{-.02in} j}_\sigma^*$ is injective on morphisms.  Set
$P_2 = {{{}^{\scriptscriptstyle p}} \hspace{-.02in} j}_\sigma^* \, P_1$, and $\mu_2 = {{{}^{\scriptscriptstyle p}} \hspace{-.02in} j}_\sigma^* (\mu_1) :
P_2 \to P_2$.  Then it will suffice to show that
\begin{equation}\label{EE}
\tilde R_\sigma (\mu_2^{m_\sigma}) = 0.
\end{equation}

Let $g_\sigma : Q_\sigma \to Q$ be the natural map.   Note
that $j_\sigma \circ f = f_\sigma \circ g_\sigma$.  Note also
that near the point $f_\sigma (v_1)$, the map $g_\sigma$ is an
$m_\sigma$-fold cover, ramified along the smooth hypersurface
$f_\sigma (c_\sigma)$.  It follows that
$P_2 \cong \oplus_{k = 1}^{m_\sigma} P_\sigma$,
and that $\mu_2$ is given by the matrix below:
$$
\pmatrix{0      & 0 & \dots  & 0      & \mu_\sigma \cr
         1      & 0 &        & 0      & 0          \cr
         0      & 1 &        & 0      & 0          \cr
         \vdots &   & \ddots &        & \vdots     \cr
         0      & 0 & \dots  & 1      & 0          \cr}
$$
This implies (\ref{EE}).
\end{pf}

\begin{rmk}\label{partres}
{\em Our technique of first specializing $\lambda$ to $\lambda_1$,
then specializing $\lambda_1$ to $0$ has a counterpart in the
resolution approach to Springer theory.  Namely, it corresponds
to Borho and MacPherson's idea of first ``forgetting the
complete flag partially,'' then ``forgetting the partial flag
completely'' (see [BM2, p. 28]).}
\end{rmk}

\begin{cor}
All the roots of $R_\sigma$ are roots of unity.
\end{cor}
\begin{pf}
This follows form Theorem \ref{rankone} and the quasiunipotence
of the monodromy transformation $\mu_\sigma$ (see, for example,
[L\^e2]).
\end{pf}

\section{Symmetric Spaces}

Both Springer theory and Example \ref{ran} above are special
cases of representations arising from symmetric spaces studied
by Kostant and Rallis in [KR].  Let ${\frak g}$ be a semisimple Lie
algebra over ${\Bbb C}$, and $\theta : {\frak g} \to {\frak g}$ an involutive
automorphism.  Let ${\frak g} = {\frak g}^+ \oplus {\frak g}^-$ be the eigenspace
decomposition for $\theta$, so that $\theta |_{{\frak g}^\pm} = \pm 1$.
Then ${\frak g}^+$ is a Lie algebra, and the adjoint form $G^+$ of
${\frak g}^+$ acts on the symmetric space ${\frak g}^-$ by conjugation.  By
the results of [KR], the representation $G^+ \, | \, {\frak g}^-$ is
polar and visible.  Theorem \ref{symsp} below gives a recipe for
computing the algebra $A$ of Theorem \ref{main} for this
representation.

Let ${\frak g}^{}_{\Bbb R}$ be a real form of ${\frak g}$ with a Cartan
decomposition ${\frak g}^{}_{\Bbb R} = {\frak g}^+_{\Bbb R} \oplus {\frak g}^-_{\Bbb R}$,
such that ${\frak g}^\pm$ is the complexification of
${\frak g}^\pm_{\Bbb R}$ (see [OV] for a discussion of real forms
of complex semisimple Lie algebras).  Let
$c^{}_{\Bbb R} \subset {\frak g}^-_{\Bbb R}$ be a maximal abelian
subalgebra.  Then the complexification
$c \subset {\frak g}^-$ of $c^{}_{\Bbb R}$ is a Cartan subspace
for $G^+ \, | \, {\frak g}^-$.  The Weyl group $W$ of this
representation is just the small Weyl group associated to
${\frak g}^{}_{\Bbb R}$.  It is a Coxeter group acting on $c^{}_{\Bbb R}$
with its Euclidean structure induced by the Killing
form on ${\frak g}^{}_{\Bbb R}$.

Fix a Weyl chamber $C \subset c^{}_{\Bbb R}$.  Let
$\{ \, \sigma_i \, \}_{i=1}^r$ ($r = \dim^{}_{\Bbb R} \, c^{}_{\Bbb R}$)
be the reflections in the walls of $C$; they give a set
of generators for $W$.  Fix a basepoint $b \in C$.  Consider
the braid group $B^{}_W = \pi_1 (Q^{reg}, f(b))$ (we use
the notation of Theorem \ref{polar}).  Let $\hat\sigma_i \in
B^{}_W$ be the element represented by the $f$-image of a
path from $b$ to $\sigma_i (b)$ in $c^{reg}$, which is almost
straight but passes counterclockwise half-way around the
hyperplane $c_{\sigma_i} \subset c$ fixed by $\sigma_i$.  The
braid group $B^{}_W$ is generated by the elements
$\{ \, \hat\sigma_i \, \}_{i=1}^r$.

To each reflection $\sigma_i$ we associate a number $s(i)$
as follows.  Let ${\frak g}^+_c \subset {\frak g}^+$ be the stabilizer
of $c$, and let ${\frak g}^+_i \subset {\frak g}^+$ be the stabilizer
of $c_{\sigma_i}$.  Then we set
$$s(i) = \dim \, {\frak g}^+_i - \dim \, {\frak g}^+_c.$$
(This is just half the sum of the dimensions of the root
spaces in ${\frak g}^{}_{\Bbb R}$ corresponding to $\sigma_i$.)

The Killing form on ${\frak g}$ restricts to a non-degenerate
$G^+$-invariant quadratic form on ${\frak g}^-$.  We will use it
to identify ${\frak g}^-$ with $({\frak g}^-)^*$, and to regard the
Fourier transform on ${\frak g}^-$ as a functor
${\cal F} : {\cal P}^{}_{{\Bbb C}^*} ({\frak g}^-) \to {\cal P}^{}_{{\Bbb C}^*} ({\frak g}^-)$.
The set of regular semisimple vectors in ${\frak g}^-$ is given by
$({\frak g}^-)^{rs} = f^{-1} (Q^{reg})$.

\begin{thm}\label{symsp}
Let $G^+ \, | \, {\frak g}^-$ be the representation arising from an
involution $\theta$ as above.  Then, in terms of Theorem
\ref{main}, we have:

(i)   The endomorphism algebra $A = {\rm End} (P)$ is given as
\begin{equation}\label{FF}
A = {\Bbb C} [B^{}_W] / \langle \, (\hat\sigma_i - 1)
(\hat\sigma_i + (-1)^{s(i)}) \, \rangle_{i=1}^r.
\end{equation}

(ii)  Take $l = b$ as a basepoint for $({\frak g}^-)^{rs}$.
The homomorphism $\rho : \pi_1 (({\frak g}^-)^{rs}, \, b) \to
A^\circ$ of part (v) of Theorem \ref{main} is given by
$\rho = \alpha \circ f_*$, where
$f_* : \pi_1 (({\frak g}^-)^{rs}, \, b) \to \pi_1 (Q^{reg}, f(b)) =
B^{}_W$ is the push-forward by $f$, and $\alpha : B^{}_W \to
A^\circ$ is defined by $\alpha : \hat\sigma_i \mapsto
(-1)^{s(i) - 1} \cdot \hat\sigma_i$.
\end{thm}
\begin{pf}
Because $W$ is a Coxeter group, the braid group quotient in the
right hand side of (\ref{FF}) has the correct dimension, namely
the order of $W$.  Thus, for part (i) it is enough to show that
for $\sigma = \sigma_i$, we have $R_\sigma (z) = (z - 1)
(z + (-1)^{s(i)})$.  This is an easy application of Theorem
\ref{rankone}.  It follows from the root space decomposition for
${\frak g}^{}_{\Bbb R}$ that, in terms of Proposition \ref{polarslice}, we
have $G_\sigma = SO (s(i) + 1)$, $V_\sigma = {\Bbb C}^{s(i) + r}$, and
$G_\sigma$ acts by the standard representation on the first
$s(i) + 1$ coordinates.  The computation of $R_\sigma$ is now
given by Theorem \ref{rankone} and Example \ref{qdrcs}.

Part (ii) of the theorem is proved by a Picard-Lefschetz argument
as in Section 5 (cf. Lemma \ref{holsurj}).
\end{pf}

\begin{rmk}\label{hecvsgrp}
{\em  Note that $A$ is isomorphic to the group algebra
${\Bbb C} [W]$, when $s(i)$ is even for all $i$, and to the
Hecke algebra ${\cal H}^{}_{-1} (W)$ specialized at $q = -1$,
when $s(i)$ is odd for all $i$.  In general, it is a ``hybrid''
of the two.  Group algebra deformations of this kind were
considered by Lusztig in [Lu2].}
\end{rmk}


\section{Introduction}

A Compton polarimeter analyzes either the scattered beam electrons or the
back-scattered gamma rays from the collision of a longitudinally polarized
electron beam with a circularly polarized laser beam.  The cross-section for
this process\cite{Comptxsect} is given by:
\begin{eqnarray}
\sigma_C   & = & \sigma_C^0 + {\cal P}_e {\cal P}_\gamma \sigma_C^1
\label{eq:compton}
\end{eqnarray}
where $\sigma_C^0$ is the unpolarized cross-section and $\sigma_C^1$ is the
polarized cross-section; ${\cal P}_e$ is the electron linear polarization and
${\cal P}_\gamma$ is the laser circular polarization.
The electron beam polarization can then be deduced from measurements of the
relative Compton-scattering rates for the J=3/2 (electron and photon spins
parallel) and J=1/2 (electron and photon spins anti-parallel) initial states,
given accurate determinations of the laser polarization and the detector
analyzing power.  This is reflected in Equation~(\ref{eq:asym}):
\begin{eqnarray}
A^{meas}= {R(\rightarrow\rar)-R(\rightarrow\leftarrow) \over R(\rightarrow\rar)+R(\rightarrow\leftarrow)} =
         {\cal P}_e{\cal P}_\gamma A_C^{det}
\label{eq:asym}
\end{eqnarray}
where
$A_C^{det}$ is the detector analyzing power
determined from Equation~(\ref{eq:compton}) and the detector acceptance.

$A^{meas}$ is maximal at the kinematic edge corresponding to $180^{\circ}$
backscatter in the center of mass frame,\footnote{The energy of the
Compton-scattered electron has a minimum in the laboratory frame at the
kinematic edge.} \hspace{1mm}
and is zero for $90^{\circ}$
scattering.  An electron polarimeter typically determines the beam
polarization from asymmetry measurements at the kinematic edge, while a gamma
polarimeter measures the energy flow asymmetry integrated over the full
gamma spectrum.

\section{The Compton Polarimeter at the SLAC Linear Collider (SLC)}
We begin by considering the performance of the Compton polarimeter for the SLC,
and the experience from its operations during the SLD experiment.
This polarimeter,\cite{Woods1} shown in Figure~\ref{fig:sldcompton}, detects
both Compton-scattered electrons and Compton-backscattered gammas
from the collision of the longitudinally polarized 45.6 GeV electron
beam\cite{Woods2} with a circularly polarized photon beam. The photon beam is
produced
from a pulsed Nd:YAG laser with a wavelength of 532 nm.  The laser is pulsed
once for every 7 electron beam pulses.\footnote{Once every 7 seconds,
the laser is pulsed on the 6th electron pulse rather than the 7th electron
pulse to avoid any synching of the laser pulse with instabilities in the
electron beam.  The electron beam pulse rate is 120 Hz.}  \hspace{1mm}
Laser off pulses
are used to measure backgrounds in the polarimeter detectors.
After the Compton
Interaction Point (CIP), the electrons and backscattered gammas pass through
a dipole spectrometer.  A nine-channel threshold Cherenkov detector (CKV)
measures electrons in the range 17 to 30 GeV.\footnote{The kinematic edge with
maximal asymmetry is at an electron energy of 17.4 GeV, and the zero-asymmetry
point is at 25.2 GeV.} \hspace{1mm}
Two detectors, a single-channel
Polarized Gamma Counter (PGC)\cite{PGC}
 and a multi-channel Quartz Fiber Calorimeter (QFC),\cite{QFC}
are located in the neutral beamline to measure the counting rates of
the Compton-scattered gammas.

\begin{wrapfigure}{r}{7cm}
\epsfig{figure=SLD_Compton.eps,width=7cm}
\fcaption{Compton Polarimeter for the SLD Experiment}
\label{fig:sldcompton}
\end{wrapfigure}

The CKV, PGC and QFC are all
threshold Cherenkov detectors, which are readout by photomultiplier tubes and
charge-sensitive ADCs.  The CKV detector uses propane as the Cherenkov
radiator, while the PGC uses ethylene and the QFC uses quartz fibers.
Table~\ref{tab:detector}
summarizes the index of refraction and Cherenkov threshold energy for each
of these detectors.  A high threshold energy is desirable to discriminate
against background sources of gammas from synchrotron radiation produced in
the dipole spectrometer and beamstrahlung produced in the collision process.
In fact, the PGC and QFC located in the neutral beamline only make
polarization measurements when the beams are not in collision due to the large
beamstrahlung backgrounds.  Typical signal and background sources of gammas
are summarized in Table~\ref{tab:rates}.\footnote{These rates correspond to
running
conditions during the 1997 SLD run with beam intensities of $4.0 \cdot 10^{10}$
per pulse and luminosities of $1.5 \cdot 10^{30} cm^{-2}s^{-1}$.}
\hspace{1mm} The dipole
spectrometer is composed of a
soft bend (1 mrad bend angle) and a hard bend (17 mrad bend angle), and the
synchrotron radiation from each is listed separately.  The soft bend magnet
does not produce a significant background source, but the hard bend magnet
does and especially so for the QFC detector.  There is a separation at the
QFC, however, between the swath of hard bend synchrotron radiation
and the Compton gammas due to the presence of the soft bend.  The geometry
of the detector takes advantage of this, and with careful work on shielding
against scattered hard bend gammas from flanges and beampipes, acceptable
backgrounds can be achieved in the QFC.  Shielding against gammas scattered
from beampipes, flanges and apertures is also important for the PGC and
CKV detectors.  Additionally, the CKV detector makes
polarization measurements during beam collisions; it must be shielded against
beamstrahlung radiation and
tails of the disrupted electron beam which can hit apertures near the
detector.  It is necessary to heavily shield these detectors and their
phototubes with lead.
Typical Compton signal:background
levels in the polarimeter detectors are 2:1 in the CKV detector during
collisions, 10:1 in the PGC detector during electron-only running, and 1:1
in the QFC detector during electron-only running.

\begin{table} [tbp]
\tcaption{Threshold Cherenkov Counters}
\vspace{2mm}
\begin{center}
\begin{tabular}{|c|clc|}
\hline
Detector	& Material	& Index of	& Threshold Energy	\\
		&		& Refraction 	&			\\
\hline
CKV		& Propane	& 1.0011	& 11 MeV		\\
PGC		& Ethylene	& 1.0007	& 14 MeV		\\
QFC		& Quartz	& 1.5		& 0.2 MeV		\\
\hline
\end{tabular}
\end{center}
\label{tab:detector}
\end{table}

\begin{table} [tbp]
\tcaption{Neutral Beam Gammas}
\vspace{2mm}
\begin{center}
\begin{tabular}{|c|crr|}
\hline
Gamma Source	& Gammas/Pulse	& $<E_{\gamma}>$	& $E_{TOTAL}$ 	\\
\hline
Compton		& 1000		& 15 GeV		& 15 TeV	\\
Soft Bend	& $4 \cdot 10^{10}$ 	& 15 keV	& 600 TeV	\\
Hard Bend	& $6.7 \cdot 10^{11}$	& 0.5 MeV	& $3.3 \cdot
10^5$ TeV \\
Beamstrahlung	& $3.6 \cdot 10^{10}$	& 30 MeV	& $1.1 \cdot 10^6$ TeV
\\
\hline
\end{tabular}
\end{center}
\label{tab:rates}
\end{table}

Because the CKV detector is the only one which can make polarization
measurements during beam collisions, it is the primary detector and the
most carefully analyzed.  A summary of the systematic errors associated
with its polarization measurement is given in Table~\ref{tab:systematic}.
The error noted for the SLC IP vs Compton IP reflects that the
luminosity-weighted polarization at the SLC IP can differ from the average
polarization measured at the Compton IP.  This
includes the effects of depolarization from the
beam-beam interaction, chromatic effects, and steering effects.\cite{Woods2}
Analysis of data from the PGC and QFC detectors is not yet complete, but
preliminary results are consistent with the CKV result to within 1$\%$.
Typical beam polarizations for the SLD experiment have been in the range
$74-78\%$.

\begin{table} [tbp]
\tcaption{Systematic Errors for CKV Polarization Measurement}
\vspace{2mm}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Item		&	Error	\\
\hline
Analyzing Power		&	$0.3\%$	\\
Detector Linearity	&	$0.5\%$	\\
Electronics Linearity	&	$0.2\%$	\\
Laser Polarization	&	$0.2\%$	\\
SLC IP vs Compton IP	&	$0.2\%$	\\
\hline
Total			&	$0.7\%$	\\
\hline
\end{tabular}
\end{center}
\label{tab:systematic}
\end{table}

\section{Polarimetry at a 1 TeV Linear Collider (TLC)}

The primary polarimeter at a TLC should be a Compton
polarimeter in the extraction line from the IP.  Additionally, there should
be a Mott polarimeter at the polarized electron source and a polarimeter at
or following the Damping Ring.

\begin{figure}
\begin{center}
\epsfig{figure=endpoint.eps,width=6.2cm}\hfill
\epsfig{figure=endpointasym.eps,width=6.2cm}
\end{center}
\fcaption{Compton endpoint energy and endpoint asymmetry vs beam energy}
\label{fig:endpoint}
\end{figure}

Following the experience of the SLD Compton polarimeter, we expect to measure
the beam polarization primarily from measurements of the Compton rate asymmetry
of the Compton-scattered electrons at the kinematic endpoint.
The endpoint and endpoint asymmetry are given by the following equations:
\begin{eqnarray}
y & = & (1+\frac{4E_eE_\gamma}{m_e^2})^{-1} \nonumber \\
E_C(endpoint) & = & E_e \cdot y \nonumber \\
A_C(endpoint) & = & \frac{y^2-1}{y^2+1}. \nonumber
\end{eqnarray}
They are plotted versus the beam energy in Figure~\ref{fig:endpoint}
for a laser photon energy
of 1.165eV (corresponding to a 1064nm Nd:YAG laser).  At high beam energies,
the Compton endpoint is well separated from the beam energy; this is
important and helps allow a layout of the extraction line that achieves a
good suppression of background to Compton signal.  The Compton
asymmetry is also very large, facilitating quick and accurate polarization
measurements.  For the example of a 1.165eV laser photon
scattering from a 500 GeV beam electron, the Compton cross-sections
for the J=3/2 and J=1/2 polarization states are
plotted in Figure~\ref{fig:Compxsect}.

\begin{wrapfigure}{r}{6.2cm}
\epsfig{figure=Compxsect.eps,width=6.2cm}
\fcaption{Compton scattering cross-sections for the J=3/2 and J=1/2
polarization states}
\label{fig:Compxsect}
\end{wrapfigure}

The primary backgrounds to contend with are collision-related.  The outgoing
beams from the IP are severely disrupted and there is a large flux of
beamstrahlung.  To illustrate this, we reproduce two plots from SLAC's
Zeroth-order Design Report (ZDR)\cite{ZDR} for the NLC in
Figure~\ref{fig:disrupted}.  The
average energy loss per incident beam particle is about $10\%$, significantly
greater than the $0.1\%$ loss at the SLC.

At a TLC, the beam power is roughly 10 MW with a corresponding beamstrahlung
power of 1 MW to be compared with 30 kW beam power and 30 W
beamstrahlung power at the SLC.  This imposes a challenging environment for
beam
diagnostics at the TLC.  Very careful design and simulation of the
extraction lines from the IP to the beam dumps is needed to transport both the
disrupted incident beam and the beamstrahlung with minimal losses, while
allowing for sufficient beam diagnostics.  For SLAC's ZDR a proposed layout
for the extraction line was developed, which is shown in
Figure~\ref{fig:extline}.

\begin{figure}
\begin{center}
\epsfig{figure=zdr1156.eps,width=6.2cm} \hfill
\epsfig{figure=zdr1157.eps,width=6.2cm}
\end{center}
\fcaption{Disrupted electron energy and beamstrahlung energy distributions
at a TLC}
\label{fig:disrupted}
\end{figure}

The Compton IP is chosen to be at a location of high dispersion with
$\eta=20$mm.  This assists separating the Compton signal from backgrounds at
the polarimeter detector.  It also allows studying the dependence of the beam
polarization on the disrupted electron energy by varying the targeting of
the laser beam on the dispersed electron beam.  This requires a good
measurement of the disrupted energy distribution, which should be achievable
with a conventional wire scanner.

The polarization of the incident electron beam prior to colliding is easily
determined from measurements where the opposing colliding beam is absent.
At the TLC there can be significant depolarization at the level of a few
percent in the collision process.\cite{Chen}  Though this can be calculated
given a good knowledge of the beam parameters, it is important to measure
the depolarization directly.  By comparing polarization measurements with and
without collisions and under differing luminosity configurations, it should be
possible to understand the depolarization loss  to better than $1\%$.

\begin{figure}
\centering
\epsfig{figure=NLCd2.eps,width=12cm}
\fcaption{Extraction Line from IP to Beam Dump}
\label{fig:extline}
\end{figure}

A new feature of the TLC compared to the SLC is the use of bunch trains,
with typically 90 bunches per train and an interbunch spacing of 2.8ns.
It is of interest to measure how the polarization varies within the train.
This can be done by colliding a short ($<2$ns) laser pulse with an individual
electron bunch.  A fast photomultiplier tube and readout gate can then be
used to minimize background from other bunches in the train.  The laser and
gate timing can be adjusted to map out the polarization within the train.

\section{Conclusions}
A Compton polarimeter located in the extraction line from the IP should achieve
$1\%$ accuracy at a TLC.  It is important to take polarization
measurements parasitic to beam collisions and detector data logging.  The
outgoing beams from the IP are highly disrupted from the collision process.
This and the high power beamstrahlung produced provide significant
challenges to designing the extraction line.  This is a more difficult
problem for the electron-electron collider than for the electron-positron
collider.  Detailed system design and simulations are needed to ensure
adequate signal to background in the polarimeter detector.  Detailed studies
are also needed to evaluate how well the luminosity-weighted
polarization for a given collision process can be determined from the Compton
polarization measurements.

\nonumsection{References}