Actions of cusp forms on holomorphic discrete series and von Neumann algebras

Abstract

A holomorphic discrete series representation (Lπ,Hπ) of a connected semi-simple real Lie group G is associated with an irreducible representation (π,Vπ) of its maximal compact subgroup K. The underlying space Hπ can be realized as certain holomorphic Vπ-valued functions on the bounded symmetric domain D≅G/K. By the Berezin quantization, we transfer B(Hπ) into End(Vπ)-valued functions on D. For a lattice Γ of G, we give the formula of a faithful normal tracial state on the commutant Lπ(Γ)′ of the group von Neumann algebra Lπ(Γ)″. We find the Toeplitz operators Tf that are associated with essentially bounded End(Vπ)-valued functions f on Γ﹨D generate the entire commutant Lπ(Γ)′:{Tf|f∈L∞(Γ﹨D,End(Vπ))}‾w.o.=Lπ(Γ)′. For any cuspidal automorphic form f defined on G (or D) for Γ, we find the associated Toeplitz-type operator Tf intertwines the actions of Γ on these square-integrable representations. Hence the composite operator of the form Tg⁎Tf belongs to Lπ(Γ)′. We prove these operators span L∞(Γ﹨D) and〈{spanf,gTg⁎Tf}⊗End(Vπ)〉‾w.o.=Lπ(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 II1 factor from cusp forms.