2020/10/02 by Yang, Jun
#FOS: Mathematics #Number Theory (math.NT) #Operator Algebras (math.OA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2010.00759
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's associated with essentially bounded End(Vπ)-valued functions f's on Γ\backslashD generate the entire commutant Lπ(Γ)': \Tf|f∈ L^∞(Γ\backslashD,\rm 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\backslashD) and ⟨\spanf,g Tg*Tf\⊗ \rm 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.