1
vote
0answers
26 views

Real-valued Irreducible Representations of Lie Groups

I'm interested in the real-valued irreducible representations of a number of Lie groups. For concreteness I'll use the group $M(2)$ of Euclidean motions, which can be parameterized as follows: $$ g(t, ...
0
votes
1answer
28 views

References on “relative Lie groups”

I'm trying to read Deligne's Formes modulaires et représentations $\ell$-adiques. In section 2, he briefly goes over a number of facts about (complex-analytic) elliptic curves in a relative setting. I ...
2
votes
1answer
174 views

Automorphism group of Annulus

Could any one tell me precisely how to compute the automorphism group of an annulus say $r<|z|<R$? Thank you!
4
votes
2answers
136 views

Automorphism group any bounded domain of $\mathbb{C}$

So far the automorphism group I have calculated for known domain is a Lie Group,so Automorphism group any bounded domain of $\mathbb{C}$ is a lie group?
1
vote
0answers
101 views

Continuous but not uniformly continuous in $\text{GL}(2,\Bbb{C})$

Essentially, I'm trying to find an example of a function that is continuous but not uniformly continuous on $\text{GL}(2,\Bbb{C})$. I'm aware that this group is isomorphic (up to constant multiples) ...
6
votes
1answer
306 views

Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$

Consider the fractional integro-derivative $\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi ...
0
votes
2answers
291 views

How to show that the unitary group $U(n)$ is not isomorphic to the semidirect product $U^n(1)\rtimes S_n$?

I came to this problem when doing the exercise that the polydisc $\Delta(0,1)^n=\prod\limits_{n}\Delta(0,1)$ in $\mathbb{C}^n$ is not biholomorphic to the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$.We ...
10
votes
2answers
546 views

The action of SU(2) on the Riemann sphere

One way to get the famous double cover $\text{SU}(2) \to \text{SO}(3)$ is to note that $\text{SU}(2)$ is isomorphic to the group of unit quaternions and to let unit quaternions $q$ act on the subspace ...
0
votes
2answers
463 views

Fundamental theorem of algebra using Lie Theory

It is said that there is a proof of fundamental theorem of algebra using Lie Theory. I have seen this claim at various places. But I could never find such a proof. Can anybody help me out?