Given a Lie group G with Lie algebra g, and a manifold M^n of dimension n >0, what invariants determine whether there is an effective action A of G, or g, on M^n? If A exists what can be said about fixed points? Can A be analytic? If not, what can be said about its kernel?

Typical results:

1. The identity component of the group of upper triangular n-by-n real matrices has effective smooth actions on every M^n. But such an action cannot be analytic, because the fixed point set of some 1-dimensional central subgroup has nonempty interior.

2. Assume for some X in g that ad X has m>0 eigenvalues whose imaginary parts are linearly independent over the rationals. Let A be
an effective analytic action of g on M^n.

(a) If n < 2m then A(X) has no fixed points.

(b) Assume n=2m and M^n is compact. Then the number of fixed points of A(X) equals the Euler characteristic Char(M), which is therefore nonnegative.

EXAMPLES: If M^n is compact and Char(M)