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 $D\cong G/K$. By the Berezin quantization, we transfer $B\left(H_{\pi }\right)$ into $\mathrm{End}\left(V_{\pi }\right)$-valued functions on $D$. For a lattice Γ of G, we give the formula of a faithful normal tracial state on the commutant $L_{\pi }{\left(\Gamma \right)}^{\prime }$ of the group von Neumann algebra $L_{\pi }{\left(\Gamma \right)}^{″}$. We find the Toeplitz operators $T_f$ that are associated with essentially bounded $\mathrm{End}\left(V_{\pi }\right)$-valued functions f on $\Gamma ﹨D$ generate the entire commutant $L_{\pi }{\left(\Gamma \right)}^{\prime }$:${\overline{\left\{T_f\right|f\in L^{\infty }(\Gamma ﹨D,\mathrm{End}(V_{\pi }\left)\right)\}}}^{\text{w.o.}}=L_{\pi }{\left(\Gamma \right)}^{\prime }.$ For any cuspidal automorphic form f defined on G (or $D$) 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 $L_{\pi }{\left(\Gamma \right)}^{\prime }$. We prove these operators span $L^{\infty }(\Gamma ﹨D)$ and${\stackrel{‾}{〈\left\{{\text{span}}_{f,g}{T}_g^{⁎}{T}_f\right\}\otimes \mathrm{End}\left(V_{\pi }\right)〉}}^{\text{w.o.}}=L_{\pi }{\left(\Gamma \right)}^{\prime },$ 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.