One-parameter subgroup

Let X be a smooth projective variety over k with an action of a torus H which has only finitely many fixed points. A one-parameter subgroup Gm —> H of H which does not lie in a finite set of given hyperplanes in the lattice of one-parameter groups of H will have the same fixed points as H. In future we call such a one-parameter group general . Thus the induced action of a general one-parameter group Gm — G has only finitely many fixed points on P. ... 

Theorem 3.6 (Birkes [8], Kempf [44]). If x e V and Y is a nonempty closed invariant subset contained in the closure of x, then there exists a one-parameter subgroup A C G such that limt o A(t)x = y for some y eY. 

One can define a universal mapping exp G 0, under which exp(tX) R — 0 set a one-parameter subgroup with a velocity vector X at a unit element. 

One can see that the basic matrix units can be considered as the infinitesimal generators of the unitary group. A more precise derivation is done by considering a set of one-parameter subgroups of U(n) (see, e.g. Pauncz 1979). 

---