GL == general linear group.
GL(1,C)=g(t,th)=e^(t+ith) 0<=th<2Pi, -infinity<t<infinity
SO(2) is a subgroup given by t=0.
th=0 is another subgroup, e^t=GL(1,R).
GL is non-compact. Integral(-infinity,infinity,dt)=infinity
Irreps of GL are 1-d.
Da(t)=e^at (a is complex)
Some of the irreps are not unitary. a=-il for l in R are the unitary irreps.
GL(n,C) is the most general lie group. All other lie groups are sub groups of it. Dim GL(n,C) = n^2 (complex dimension)
X(ab)ij=delta(a,i)delta(b,j) => characters linearly independent.
=> lie algebra with [X(ab),X(gd)]=delta(b,g)Xad-delta(a,d)Xbg
GL(n,C) is related to SL(n,C) in the following way:
Let X be a general matrix of GL.
define X~=Xab-1/n*delta(a,b)Id => Tr(X~ab)=0