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 $T_f$'s associated with essentially bounded End$(V_\pi)$-valued functions $f$'s on $\Gamma\backslash\mathcal{D}$ generate the entire commutant $L_\pi(\Gamma)'$: $$\overline{{T_f|f\in L^\infty(\Gamma\backslash\mathcal{D},{\rm End}(V_\pi))}}^{\text{w.o.}}=L_\pi(\Gamma)'.$$ 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 these operators span $L^{\infty}(\Gamma\backslash\mathcal{D})$ and $$\overline{\langle{\text{span}{f,g} T_g^{*}T_f}\otimes {\rm End}(V\pi)\rangle}^{\text{w.o.}}=L_\pi(\Gamma)',$$ where $f,g$ run through holomorphic cusp forms for $\Gamma$ of same types. If $\Gamma$ is an infinite conjugacy classes group, we obtain a $\text{II}_1$ factor from cusp forms.