This paper consists of two
parts, and in the first of these we develop the representation theory of solvable
algebraic groups over a local field of characteristic zero in analogy with the work of
Auslander and Kostant for solvable Lie groups. We show how all the representations
arise and show that the Kirillov method of orbits applies to this situation. We
find that the theory carries over completely and we discuss traces, CCR
representations and we give a version of the Kostant independence of polarization
theorem.
In the second part we take up the problems of decomposing the space of square
integrable functions on a solvable Lie group modulo a discrete cocompact
subgroup. We show how to reduce this problem to the special case when the
nilradical of the solvable group is Heisenberg. These two sections represent the
initial part of a comprehensive program in this direction to be completed
later.
