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 ·

)−
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 ·
)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 on
such that the transition functions φαβ are fractional-linear and orientation
preserving. (This means that
=
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.
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