Introduction

The idea behind the orbit method is to unite harmonic analysis with sym-

plectic geometry. This can be considered as a part of the more general idea

of the unification of mathematics and physics.

In fact, this is a post factum formulation. Historically, the orbit method

was proposed in [Ki1] for the description of the unitary dual (i.e. the set

of equivalence classes of unitary irreducible representations) of nilpotent

Lie groups. It turned out that the method not only solves this problem

but also gives simple and visual solutions to all other principal questions

in representation theory: topological structure of the unitary dual, the ex-

plicit description of the restriction and induction functors, the formulae for

generalized and infinitesimal characters, the computation of the Plancherel

measure, etc.

Moreover, the answers make sense for general Lie groups and even be-

yond, although sometimes with more or less evident corrections. I already

mentioned in [Ki1] the possible applications of the orbit method to other

types of Lie groups, but the realization of this program has taken a long

time and is still not accomplished despite the efforts of many authors.

I cannot mention here all those who contributed to the development

of the orbit method, nor give a complete bibliography: Mathematical Re-

views now contains hundreds of papers where coadjoint orbits are mentioned

and thousands of papers on geometric quantization (which is the physical

counterpart of the orbit method). But I certainly ought to mention the

outstanding role of Bertram Kostant and Michel Duflo.

As usual, the faults of the method are the continuations of its advantages.

I quote briefly the most important ones.

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