xviii Introduction

MERITS VERSUS DEMERITS

1. Universality: the method works 1. The recipes are not accurately

for Lie groups of any type and precisely developed.

over any field.

2. The rules are visual, 2. Sometimes they are wrong

and are easy to memorize and need corrections

and illustrate by a picture. or modifications.

3. The method explains 3. It could be diﬃcult

some facts which otherwise to transform this explanation

look mysterious. into a rigorous proof.

4. It provides a great amount of 4. Most of the completely integrable

symplectic manifolds and dynamical systems were

Poisson commuting families discovered earlier

of functions. by other methods.

5. The method introduces new 5. The description of coadjoint

fundamental notions: coadjoint orbits and their structures

orbit and moment map. is sometimes not an easy problem.

For the reader’s convenience we formulate the ideology of the orbit

method here in the form of a “User’s Guide” where practical instructions

are given as to how to get answers to ten basic questions in representation

theory.

These simple rules are applicable literally for all connected and simply

connected nilpotent groups. For groups of general type we formulate the

“ten amendments” to these rules in the main text of the book.

Throughout the User’s Guide we use the following notation:

G – a connected simply connected Lie group;

H ⊂ G – a closed connected subgroup;

g, h – Lie algebras of G, H, respectively;

g∗, h∗

– the dual spaces to g, h, respectively;

p:

g∗

→

h∗

– the canonical projection;

σ – the canonical 2-form (symplectic structure) on a coadjoint orbit;

πΩ – the unirrep of G corresponding to the orbit Ω ⊂ g∗;