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

 dc.contributor.advisor Bisch, Dietmar dc.contributor.advisor Jones, Vaughan F.R. dc.creator Yang, Jun dc.date.accessioned 2021-07-09T03:52:12Z dc.date.created 2021-08 dc.date.issued 2021-06-03 dc.date.submitted August 2021 dc.identifier.uri http://hdl.handle.net/1803/16741 dc.description.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 $T_\phi$'s with $\phi \in L^{\infty}(\Gamma\backslash\mathcal{D},\End(V_{\pi}))$ generate the entire commutant $L_{\pi}(\Gamma)'$: \begin{center} $\overline{\{T_f|f\in L^{\infty}(\Gamma\backslash\mathcal{D},\End(V_{\pi}))\}}^{\text{w.o.}}=L_{\pi}(\Gamma)'$. \end{center} 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 \begin{center} $\overline{\langle\{\text{span}_{f,g} T_g^{*}T_f\}\otimes \End(V_{\pi})\rangle}^{\text{w.o.}}=L_\pi(\Gamma)'$, \end{center} where $f,g$ run through the cusp forms for $\Gamma$ of same types. If $\Gamma$ is an infinite conjugacy classes group, the cusp forms give a $\text{II}_1$ factor. dc.format.mimetype application/pdf dc.language.iso en dc.subject Representation Theory, Number Theory, Operator Algebras dc.title Actions of Cusp Forms on Holomorphic Discrete Series and Von Neumann Algebras dc.type Thesis dc.date.updated 2021-07-09T03:52:12Z dc.type.material text thesis.degree.name PhD thesis.degree.level Doctoral thesis.degree.discipline Mathematics thesis.degree.grantor Vanderbilt University Graduate School local.embargo.terms 2022-02-01 local.embargo.lift 2022-02-01 dc.creator.orcid 0000-0002-6736-0655 dc.contributor.committeeChair Bisch, Dietmar
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