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

TL;DR

This paper explores the actions of cusp forms on holomorphic discrete series representations of semi-simple Lie groups, linking Toeplitz operators, von Neumann algebras, and automorphic forms to characterize their algebraic structures.

Contribution

It introduces a new framework connecting cusp forms, Toeplitz operators, and von Neumann algebra commutants in the context of holomorphic discrete series representations.

Findings

Toeplitz operators generate the entire commutant of the group von Neumann algebra.

Operators associated with cusp forms span the von Neumann algebra's commutant.

For groups with infinite conjugacy classes, a II_1 factor arises from cusp forms.

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.