“Harmonic
analysis on complex semisimple groups and symmetric spaces from point of view of complex
analysis”
Abstract
In
the classical conception of harmonic analysis on semisimple groups of E. Cartan
and H. Weyl there was very essential interaction between algebraic and
transcendental (analytic) methods. It looks that one more possibility - to start
from the complex analysis on groups - was never systematically considered. Since
all complex semisimple groups are Stein manifolds and all spherical functions
are holomorphic, this way deserves an attention.
If
to choose this way then we will start not from characters or irreducible
representations but from Cauchy integral formula on these groups. It turns out
that there is a beautiful explicit Cauchy formula on all semisimple Lie groups
and, more general, on all symmetric Stein manifolds and this formula will be the
basic subject of this talk. This formula, similarly to the situation in
one-dimensional complex analysis, is a base of complex analysis on these
symmetric spaces, which incorporates the classical harmonic analysis and adds
some new facts.
ore
17.00
Aula Bianchi
PISA