March 6, 1997
Ivan Penkov, UC Riverside
"Shadows of representations"
Some most important classical categories of representations are associated
with certain subalgebras in the Lie algebra: highest weight modules are
defined in terms of a triangular decomposition and Harish-Chandra modules
are defined in terms of a subalgebra. But until recently it had not been
realized that there exists also a nice "inverse procedure". Namely every
irreducible weight representation M of a reductive Lie algebra g defines
a canonical decomposition of g into four subalgebras with very nice
properties. This is "the shadow of M on g". The shadow carries valuable
information about M, for instance it displays the directions in which M
is finite-dimensional. The case of g=sl(3) is already non-trivial and it
will be explained in detail.