Representations of the M\"obius group and pairs of homogeneous operators in the Cowen-Douglas class

Abstract

Let M\"ob 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 $\pi : \mbox{M\"ob} \to \mathcal{U}(\mathcal{H})$ is a multiplier representation, then we prove that there exist $\lambda_1, \lambda_2, \ldots, \lambda_n > 0$ such that $\pi$ is unitarily equivalent to $(\otimes_{i=1}^{n} D_{\lambda_i}^+)|_{\mbox{M\"ob}}$, where each $D_{\lambda_i}^+$ is a holomorphic discrete series representation of M\"ob. As an application, we prove that if $(T_1, T_2)$ is a M\"ob - 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. $\pi \cong \oplus_{m=0}^\infty D^+_{\lambda + 4m + 2}$).