LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS
IVAN LOSEV
11.
CalogeroMoser system and Hamiltonian reduction
11.1.
CalogeroMoser system.
The CalogeroMoser system is the system of
n
distinct
points of the same mass, say 1, on the line (we work over
C
so we consider the complex line)
with coordinates
x
1
, . . . , x
n
that interact with pairwise potentials of the form
c
(
x
i
−
x
j
)
2
, where
c
is some nonzero constant. We can rescale and assume that
c
=
−
1. The total potential is
V
=
−
∑
i<j
1
(
x
i
−
x
j
)
2
.
We want to consider the corresponding Hamiltonian system but we first need to decide
what will be the symplectic variety to “accommodate” the system. Our original configuration
space is (
C
n
)
Reg
:=
{
(
x
1
, . . . , x
n
)

x
i
̸
=
x
j
,
∀
i
̸
=
j
}
.
So we could take the variety
X
:=
T
∗
(
C
n
)
Reg
and consider the Hamiltonian
H
=
1
2
∑
n
i
=1
y
2
i
−
∑
i<j
1
(
x
i
−
x
j
)
2
.
However, there
is a better choice. Our points are indistinguishable and so we can view (
x
1
, . . . , x
n
) as an
unordered
n
tuple. So the configuration space is (
C
n
)
Reg
/
S
n
and we consider its cotangent
bundle
X
:=
T
∗
(
C
n
)
Reg
/
S
n
.
Also
S
n
acts naturally on
T
∗
(
C
n
)
Reg
, the action is induced
from (
C
n
)
Reg
and hence preserves the symplectic form.
As the following exercise shows
T
∗
((
C
n
)
Reg
/
S
n
) = (
T
∗
(
C
n
)
Reg
)
/
S
n
. So we can view
H
(that is an
S
n
invariant function)
as a function on
X
. This is a Hamiltonian of our system. Below we will write
C
Reg
instead
X
.
We will see that it is still not the best possible phase space for our system: we can
actually embed
C
Reg
to some symplectic aﬃne variety
C
(CalogeroMoser space) to “avoid
collisions”.
Exercise 11.1.
Let
X
0
be a smooth algebraic variety equipped with a free action of a finite
group
Γ
.
Show that
T
∗
(
X
0
/
Γ)
is naturally identified with
(
T
∗
X
0
)
/
Γ
(an isomorphism of
symplectic varieties).
Let us explain what kind of results regarding the CalogeroMoser (CM) system we want
to get. First of all, we want to describe the trajectories, as explicitly as possible. Second, we
want to produce 1st integrals subject to certain conditions. Namely, we want 1st integrals
H
1
, . . . , H
n
with
H
2
=
H
such that
{
H
i
, H
j
}
= 0 for all
i, j
and
d
x
H
1
, . . . , d
x
H
n
being lin
early independent for a general point
x
∈
X
. By some general results of Algebraic geometry
(generic smoothness) the last condition is equivalent to
H
1
, . . . , H
n
being algebraically inde
pendent. As the following exercise shows, here
n
is the maximal possible number for which
such first integral may exist.
Exercise 11.2.
Let
f
1
, . . . , f
m
be functions on a symplectic variety
X
such that
{
f
i
, f
j
}
= 0
for all
i, j
. Show that the dimension of the span of
d
x
f
1
, . . . , d
x
f
m
has dimension not exceeding
1
2
dim
X
.
Systems admitting such collection of functions
H
1
, . . . , H
n
(with Hamiltonian
H
being
one of them) are called
completely integrable
. The reason is the ArnoldLiouville theorem
that roughly states that (well, under some additional assumptions, and in
C
∞
setting) such
1