22 1. Geometry of Coadjoint Orbits

cover. This group has a unique non-trivial central extension G, the so-called

Virasoro-Bott group.

This example will be discussed in Chapter 6. Here we show only that

the classification of coadjoint orbits for this group is equivalent to each of

the following apparently non-related problems.

1. Consider the ordinary differential equation of the second order

(25) Ly ≡ cy + p(x)y = 0.

If we change the independent variable: x → φ(t), then equation (25) changes

its form: the term with y appears.

But if, at the same time, we change the unknown function: y → y ◦ φ ·

(φ

)−

1

2

, then the unwanted term with y disappears and equation (25) goes

to the equation

˜˜

L y = 0 of the same form but with a new coeﬃcient

(26) ˜ p = p ◦ φ · (φ

)2

+ cS(φ) where S(φ) =

φ

φ

−

3

2

φ

φ

2

.

Assume now that the coeﬃcient p(x) is 2π-periodic and the function φ(t)

has the property φ(t + 2π) = φ(t) + 2π. The problem is to classify the

equations (25) with respect to the transformations (26).

2. Let G be the simply connected covering of the group SL(2, R), and

let A be the group of all automorphisms of G. The problem is to classify

elements of G up to the action of A.

3. The locally projective structure on the oriented circle

S1

is de-

fined by a covering of

S1

by charts {Uα}α∈A with local parameter tα on Uα

such that the transition functions φαβ are fractional-linear and orientation

preserving. (This means that tα

=

atβ +b

ctβ +d

with ad − bc 0.)

The problem is to classify the locally projective structures on

S1

up to

the action of

Diff+(S1).

♦

Let us make a general observation about the relation between the coad-

joint orbits of a group G and of its central extension G by a 1-dimensional

subgroup A. This observation will explain the relation between the coadjoint

orbits of the Virasoro-Bott group and problem 1 in Example 9.

Let g and g be the Lie algebras of G and G. As a vector space, g can be

identified with g ⊕ R so that the commutator looks like

(27) [(X, a), (Y, b)] = ([X, Y ], c(X, Y ))

where c(X, Y ) is the cocycle defining the central extension. It is an anti-

symmetric bilinear map from g × g to R satisfying the cocycle equation:

(28) c([X, Y ], Z) = 0.