with Lie bracket
[((
cq
cp
)
,c)
,
((
c′q
c′p)
,c′)]
=
((
0
0
)
,cqc′p−cpc′q)
=
((
0
0
)
,Ω
((
cq
cp)
,
(
c′q
c′p)))
The linearized action of Φgat the identity ofH 3 gives the actionφgonh 3 , but
since the exponential map is the identity,φgacts onR^2 ⊕R=M⊕Rin the
same way as Φg, by
((
cq
cp
)
,c)
∈h 3 →φg(((
cq
cp)
,c))
=
(
g(
cq
cp)
,c)
Since the Lie bracket just depends on Ω, which isSL(2,R) invariant,φgpre-
serves the Lie bracket and so acts by automorphisms onh 3.
The infinitesimal version of theSL(2,R) actionφgonh 3 is an action of
sl(2,R) onh 3 by derivations. This action can be found by computing (for
L∈sl(2,R) andX∈h 3 ) using equation 16.16 to get
L·
((
cq
cp)
,c)
=
d
dt(
φetL((
cq
cp)
,c))
|t=0=
(
L
(
cq
cp)
, 0
)
(16.20)
The Poisson brackets between degree two and degree one polynomials discussed
at the beginning of this section give an alternate way of calculating this action
ofsl(2,R) onh 3 by derivations. For a generalL∈sl(2,R) (see equation 16.6)
andcqq+cpp+C∈h 3 we have
{μL,cqq+cpp+C}=c′qq+c′pp,(
c′q
c′p)
=
(
acq+bcp
ccq−acp)
=L
(
cq
cp)
(16.21)
(hereμLis given by 16.9). We see that this is the action ofsl(2,R) by derivations
onh 3 of equation 16.20, the infinitesimal version of the action ofSL(2,R) on
h 3 by automorphisms.
Note that in the larger Lie algebra of all polynomials onMof order two or
less, the action ofsl(2,R) onh 3 by derivations is part of the adjoint action of
the Lie algebra on itself, since it is given by the Poisson bracket (which is the
Lie bracket), between order two and order one polynomials.
16.3 The case of arbitrary d
For the general case of arbitraryd, the groupSp(2d,R) will act by automor-
phisms onH 2 d+1 andh 2 d+1, both of which can be identified withM ⊕R.
The group acts by linear transformations on theMfactor, preserving Ω. The
infinitesimal version of this action is computed as in thed= 1 case to be
L·
((
cq
cp)
,c)
=
d
dt(
φetL((
cq
cp)
,c))
|t=0=
(
L
(
cq
cp)
, 0
)
whereL∈sp(2d,R). This action, as in thed= 1 case, is given by taking
Poisson brackets of a quadratic function with a linear function: