Singular conformally invariant trilinear forms, II The higher multiplicity cases

TL;DR

This paper investigates the structure and dimension of invariant trilinear forms on smooth functions over spheres under conformal group actions, providing explicit bases for various parameter cases.

Contribution

It computes the dimension and constructs explicit bases of conformally invariant trilinear forms for higher multiplicity cases on spheres.

Findings

Dimension formulas for $Tri(oldsymbol f)$ are derived.

Explicit bases of invariant trilinear forms are constructed.

Results extend understanding of conformal invariance in representation theory.

Abstract

Let $S$ be the sphere of dimension $n-1, n\geq 4$. Let $(\pi_{\lambda})_{\lambda\in \mathbb C}$ be the scalar principle series of representations of the conformal group $SO_0(1,n)$, realized on $\mathcal C^\infty(S)$. For $\boldsymbol \lambda = (\lambda_1,\lambda_2,\lambda_3) \in \mathbb C^3$, let $Tri(\boldsymbol \lambda)$ be the space of continuous trilinear forms on $\mathcal C^\infty(S) \times \mathcal C^\infty(S) \times \mathcal C^\infty(S)$ which are invariant under $\pi_{\lambda_1} \otimes \pi_{\lambda_2} \otimes \pi_{\lambda_3} $. For each value of $\boldsymbol \lambda$, the dimension of $Tri(\boldsymbol \lambda)$ is computed and a basis of $Tri(\boldsymbol \lambda)$ is described.