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