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Description of irreducible representations of $SL(2,mathbb R)$.

Mathematics Asked on December 5, 2021

Let $rho: SL(2,mathbb R) to GL(V)$ be an irreducible representation of $SL(2,mathbb R)$. A paper I am reading says $rho$ can be described as a symmetric tensor product in $Sym^n(mathbb R^2)$, which has $A^n, A^{n-1}B,dots, B^n$ as a basis (over $mathbb R$), where $A=(1,0)^T$ and $B=(0,1)^T$.

But I am very confused about how this is actually connected to a representation $rho: SL(2,mathbb R) to GL(V)$. Say if I have a homogeneous polynomial

$$p(A,B):=c_0 A^n + c_1 A^{n-1}B + cdots c_n B^n,$$

then how can we view $p(A,B)$ as a representation
$rho: SL(2,mathbb R) to GL(V)$ literally?

It would be great if someone could also show me some reference to the proof of this result

One Answer

So $V$ is the set of homogeneous polynomials in $A$ and $B$ of degree $n$, so is a vector space of dimension $n+1$.

So how does $text{SL}_2(Bbb R)$ act on this? Let $f(A,B)in V$ and $M=pmatrix{a&b\c&d}in text{SL}_2(Bbb R)$. Then $M$ takes $f$ to $f(aA+bB,cA+dB)$.

Answered by Angina Seng on December 5, 2021

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