What is this formula saying?
If f(x) is an
Schwartz function on the group Sl(2,R),
we can express its value at the identity element as an
"sum" (i.e. Plancherel integral) of distributions
or 
.
These distributions can be regarded as Fourier transforms at
various points of the unitary dual. The formula identifies the explicit
portion of the unitary dual where the Plancherel measure is
supported and
computes it explicitly.
You can learn more about this by clicking
here and reading the books referenced therein.