Actions of Cusp Forms on Holomorphic Discrete Series and Von Neumann Algebras

Abstract

A holomorphic discrete series representation $(L_{\pi},H_{\pi})$ of a connected semi-simple real Lie group $G$ is associated with an irreducible representation $(\pi,V_{\pi})$ of its maximal compact subgroup $K$. The underlying space $H_{\pi}$ can be realized as certain holomorphic $V_{\pi}$-valued functions on the bounded symmetric domain $\mathcal{D}\cong G/K$. By the Berezin quantization, we transfer $B(H_{\pi})$ into $\End(V_{\pi})$-valued functions on $\mathcal{D}$. For a lattice $\Gamma$ of $G$, we give the formula of a faithful normal tracial state on the commutant $L_{\pi}(\Gamma)'$ of the group von Neumann algebra $L_{\pi}(\Gamma)''$. We find the Toeplitz operators $T_\phi$'s with $\phi \in L^{\infty}(\Gamma\backslash\mathcal{D},\End(V_{\pi}))$ generate the entire commutant $L_{\pi}(\Gamma)'$: \begin{center} $\overline{\{T_f|f\in L^{\infty}(\Gamma\backslash\mathcal{D},\End(V_{\pi}))\}}^{\text{w.o.}}=L_{\pi}(\Gamma)'$. \end{center} For any cuspidal automorphic form $f$ defined on $G$ (or $\mathcal{D}$) for $\Gamma$, we find the associated Toeplitz-type operator $T_f$ intertwines the actions of $\Gamma$ on these square integrable representations. Hence the composite operator of the form $T_g^{*}T_f$ belongs to $L_\pi(\Gamma)'$. We prove \begin{center} $\overline{\langle\{\text{span}_{f,g} T_g^{*}T_f\}\otimes \End(V_{\pi})\rangle}^{\text{w.o.}}=L_\pi(\Gamma)'$, \end{center} where $f,g$ run through the cusp forms for $\Gamma$ of same types. If $\Gamma$ is an infinite conjugacy classes group, the cusp forms give a $\text{II}_1$ factor.