Dennis The, Australian National University
"The gap phenomenon in parabolic geometries"
Many
geometric
structures
(such
as
Riemannian,
conformal,
CR,
projective,
systems
of
ODE,
and
various
types
of
generic
distributions)
admit
an
equivalent
description
as
Cartan
geometries.
For
Cartan
geometries
of
a
given
type,
the
maximal
amount
of
symmetry
is
realized
by
the
flat
model.
However,
if
the
geometry
is
not
(locally)
flat,
how
much
symmetry
can
it
have?
Understanding
this
"gap"
between
maximal
and
submaximal
symmetry
in
the
context
of
parabolic
geometries
is
the
subject
of
this
talk.
We
show
how
representation-theoretic
considerations
involving
Kostant's
version
of
the
Bott-Borel-Weil
theorem
and
Tanaka
prolongation
lead
to
restrictions
on
the
submaximal
dimension.
In
particular,
I'll
discuss
conformal
geometry
as
well
as
the
(G2)
geometry
of
generic
rank
two
distributions
in
dimension
five.
(Joint
work
with
Boris
Kruglikov.)