Invariant functions for irreducible representations of $\mathrm{SU}(2)$.

Question

The orbits of $\mathrm{SU}(2)$ acting irreducibly on $\mathbb{C}^2$ are three-spheres centered around the origin. In other words, an orbit is uniquely specified by the Euclidean norm in $\mathbb{C}^2$. For the irreducible representation $\rho_n$ of $\mathrm{SU}(2)$ on $\mathbb{C}^n$ ($n\geq 2$), the orbits will generally be three-dimensional so I expect there are $N=2n-3$ functions $f_1,\dots,f_{N}\colon\mathbb{C}^n\to \mathbb{R}$ which are invariant under $\rho_n$ and the orbits of $\rho_n$ are equal to $\{x\in \mathbb{C}^n\mid f_i(x)=c_i\}$ for some $c_1,\dots,c_N\in \mathbb{R}$. My question is if what I have said is correct and if so, what else can be said about the functions $f_i$?

Answer 1

Quesiton: "My question is if what I have said is correct and if so, what else can be said about the functions fi?"

Response: Let $G:=SU(2)$. If $V:=k\{e_1,e_2\}$ with $k$ the complex numbers and $V^*:=k\{x_1,x_2\}$ and $R:=Sym_k^*(V^*)\cong k[x_1,x_2]$ you get a canonical action

$$G \times R \rightarrow R$$

and a general result is that since $G$ is reductive it follows the invariant ring $R^G \subseteq R$ is finitely generated. Much work has been done on determining such rings of invariants for actions of reductive groups on "rings of functions" on irreducible finite dimensional representations. Whenever you have an action

$$G \times V(\lambda) \rightarrow V(\lambda)$$

you may construct the polynomial ring $R(\lambda):=Sym_k^*(V(\lambda)^*)$ and ask about the invariant ring $R(\lambda)^G$. What are its properties?

If you want more specific information on your example you must give more information on which representation you study and what are the invarient functions - the generators of the sub ring $R^G$.

The group $SU(2)$ is a semi simple linear algebraic group and its finite dimensional irreducible representations $V(\lambda)$ have been classified. I also believe there are lists giving all generators of the invariant ring $R(\lambda)^{SU(2)}$ in many cases.

https://en.wikipedia.org/wiki/Invariant_of_a_binary_form

Note: You may define $SU(2)$ as the set of matrices

$$A:=\begin{align*} \phi= \begin{pmatrix} z & w \\ -\overline{w} & \overline{z} \end{pmatrix} \end{align*} $$

with $z,w\in \mathbb{C}$ and $\det(A):=z\overline{z}+w\overline{w}=1$. The group multiplication is algebraic, hence $SU(2)$ is a linear algebraic group. The equation $\det(A)=1$ is not a polynomial in $z,w$ hence $SU(2)$ is not a complex algebraic variety. It is a real algebraic variety.

$$SU(2) \subseteq \mathbb{C}^2 \cong \mathbb{R}^4$$

is the zero set of the polynomial

$$x_1^2+\cdots +x_4^2=1$$

hence $SU(2) \cong S^3$ is the real 3-sphere as an algebraic variety.

You are interested in complex irreducible representations of a semi simple group over the real numbers. Most introductory books (such as Fulton-Harris "Representation theory - a first course") consider complex representations of complex semi simple algebraic groups. There is a section in FH on real Lie groups and real Lie algebras and it gives some references.

The Hopf fibration: You find an elementary exposition on some aspects of the representations of $SU(2)$ in Artin's book "Algebra". If $V^*;=k\{x_0,x_1\}$ and $S^d(V^*)$ is the d'th symmetric power of $V^*$, there is a representation

$$\rho_d: SU(2) \rightarrow GL_k(S^d(V^*))$$

and I believe this representation is irreducible. There is a map (the "Hopf fibration")

$$\pi: SU(2) \rightarrow S^2 \cong \mathbb{P}^1_k$$

where $S^2$ is th real 2-sphere and $\mathbb{P}^1_k$ is the complex projective line. And the vector space $S^d(V^*) \cong H^0(\mathbb{P}^1_k, \mathcal{O}(d))$ is the vector space of global sections of the complex linebundle $\mathcal{O}(d)$ on the projective line. But it is not clear if $\rho_d$ can be constructed using the map $\pi$: If you view $\mathcal{O}(d)$ as a real rank two vector bundle $E(d)$ on $S^2$, it corresponds to a projective module $E_d$ of rank $2$ over the ring of regular functions $A(S^2)$, and $H^0(S^2,E(d)) \cong E_d$ which has infinite dimension as real vector space.