The left-hand side of this equation isc′′q·q+c′′p·p, where
(
c′′q
c′′p)
= (LL′−L′L)
(
cq
cp)
As a result, the right-hand side is the linear map given by
{μL,μL′}=μ[L,L′]16.4 For further reading
For more on symplectic groups and the isomorphism betweensp(2d,R) and
homogeneous degree two polynomials, see chapter 14 of [37] or chapter 4 of [26].
Chapter 15 of [37] and chapter 1 of [26] discuss the action of the symplectic
group on the Heisenberg group and Lie algebra by automorphisms.