Generalizations

•One can make a slicker
proof using the Kirillov model,
in which t(x) = ån¹0 andn(x).

•In this model t(x) has group
translates

•When ta,c(x) is
integrated against a test function f(x), one gets __exactly__ the LHS of the Voronoi formula.

•The righthand side is
(almost tautologically) equivalent
to the automorphy of t(x) under SL(2,Z) under
the G-action in the Kirillov model.

•However, the analytic
justification of this argument – and
especially its generalizations – gets somewhat technical.