Representations of the Möbius group and pairs of homogeneous operators in the Cowen-Douglas class

Abstract

Let Möb be the biholomorphic automorphism group of the unit disc of the complex plane, $\mathcal{H}$ be a complex separable Hilbert space and $\mathcal{U}(\mathcal{H})$ be the group of all unitary operators. Suppose $\mathcal{H}$ is a reproducing kernel Hilbert space consisting of holomorphic functions over the poly-disc ${\mathbb{D}}^n$ and contains all the polynomials. If $π: \mbox{Möb} \to \mathcal{U}(\mathcal{H})$ is a multiplier representation, then we prove that there exist ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n>0$ such that $\pi$ is unitarily equivalent to $(\otimes_{i=1}^{n} D_{λ_i}^+)|_{\mbox{Möb}}$, where each $D_{{\lambda }_i}^{+}$ is a holomorphic discrete series representation of Möb. As an application, we prove that if $(T_1,T_2)$ is a Möb - homogeneous pair in the Cowen - Douglas class of rank $1$ over the bi-disc, then each $T_i$ posses an upper triangular form with respect to a decomposition of the Hilbert space. In this upper triangular form of each $T_i$, the diagonal operators are identified. We also prove that if $\mathcal{H}$ consists of symmetric (resp. anti-symmetric) holomorphic functions over ${\mathbb{D}}^2$ and contains all the symmetric (resp. anti-symmetric) polynomials, then there exists $\lambda >0$ such that $\pi \cong {\oplus }_{m=0}^{\infty }{D}_{\lambda +4m}^{+}$ (resp. $\pi \cong {\oplus }_{m=0}^{\infty }{D}_{\lambda +4m+2}^{+}$).