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 $L_{\pi }(\Gamma {)}^{\prime }$ of the group von Neumann algebra $L_{\pi }(\Gamma {)}^{\prime \prime }$. 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 $L_{\pi }(\Gamma {)}^{\prime }$: $${\overline{\{T_f|f\in L^{\infty }(\Gamma \backslash \mathcal{D},\mathrm{End}(V_{\pi }))\}}}^{\text{w.o.}}=L_{\pi }(\Gamma {)}^{\prime }.$$ 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 $T_g^{\ast }{T}_f$ belongs to $L_{\pi }(\Gamma {)}^{\prime }$. We prove these operators span $L^{\infty }(\Gamma \backslash \mathcal{D})$ and $${\overline{⟨\{{\text{span}}_{f,g}{T}_g^{\ast }{T}_f\}\otimes \mathrm{End}(V_{\pi })⟩}}^{\text{w.o.}}=L_{\pi }(\Gamma {)}^{\prime },$$ 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.

arXiv comment43 pages