A subgroup H of an affine
algebraic group G is observable in G if the quotient variety G∕H is quasi-affine
(equivalently, if each character on H is the character of a one-dimensional
H-submodule of an irreducible G-module). The question is how to characterize the
universally observable groups, i.e., those which are observable in every group in which
they can be embedded. We remark that the relation that G∕H should be affine for
every embedding of H is equivalent to H being reductive, by work of Cline, Parshall,
Scott. A sufficient condition for the universal observability of a solvable group is that
a certain monoid of characters for the inner operation of H on its hyperalgebra
should be a group. Here, we give a two-dimensional example (a codimension one
subgroup of a Borel subgroup of GL2) to show that this sufficient condition
is not necessary. Secondly, we give a method for testing a group for the
failure of universal observability, which we use to show the non-universal
observability of a family of codimension one subgroups of Borel subgroups
of GLn(n ≥ 3). We remark that the universal observability of an affine
algebraic group is equivalent to the universal observability of its solvable radical.
Consequently, we only need to sort the solvable groups for those that are universally
observable.
