My today's and tomorrow's lectures are in some sense a continuation of the lectures delivered here in the winter of 1997/98. Those two lectures were entitled “The Orbit Method and Finite Groups,” and these two lectures are entitled “The Orbit Method beyond Lie Groups.” I shall not dwell on the orbit method. I only mention that it is applied to Lie groups. Thus, the main object under consideration is Lie groups. But the orbit method applies also to other groups, which are not Lie groups. I have prepared three series of such examples:

1. infinite-dimensional groups;

2. finite groups;

3. quantum groups.

My last-year lectures were concerned with the second series, finite groups; so I shall not talk about them, although an interesting progress has been made in this direction. Today I shall talk about infinite-dimensional groups, and tomorrow, about quantum groups. Quantum groups is a very fashionable direction in modern mathematics. Their success is largely due to the sonorous name. The fine point is that the quantum groups are not groups; this is an object of a different nature. But they still have some group features, and we could try to apply the orbit method to them. I shall talk about these attempts tomorrow. Today I shall talk about infinite-dimensional groups, which are not Lie groups either.

The usual Lie groups are (finite-dimensional) manifolds endowed with a group structure which is compatible in a certain sense with the structure of a manifold.