论文标题
尖峰形式在全体形态离散系列和von Neumann代数上的作用
Actions of Cusp Forms on Holomorphic Discrete Series and Von Neumann Algebras
论文作者
论文摘要
连接的半简单的真实谎言组$ g $的全态分散串联表示$（l_π，h_π）$与其最大cartact子组$ k $的不可约（π，v_π）$相关联。基础空间$h_π$可以实现为有限的对称域上的某些holomorphic $v_π$ - 价值函数$ \ mathcal {d} \ cong g/k $。通过berezin量化，我们将$ b（h_π）$转移到$ \ Mathcal {d} $上的END $（v_π）$。对于$ g $的晶格$γ$，我们在von neumann algebra $l_π（γ）'$l_π（γ）'$ $l_π（γ）上提供了忠实的正常奇特状态的公式。我们发现toeplitz运营商$ t_f $与本质上有界的end $（v_π）$ - 估价函数$ f $'s on $γ\ backslash \ mathcal {d} $生成了整个通勤$l_π（γ）'$：$：$：$：$：$：$：$：$：$：$：$：$：$：$ \ overline { l^\ infty（γ\ backSlash \ Mathcal {d}，{\ rm end}（v_π））\}}}}}}^{\ text {w.o。}} =l_π（γ）'。我们发现相关的toeplitz型运算符$ t_f $在这些方形积分表示上互动了$γ$的动作。因此，形式的$ t_g^{*} t_f $的复合操作员属于$l_π（γ）'$。我们证明了这些操作员跨度$ l^{\ infty}（γ\ backslash \ mathcal {d}）$和$$ \ overline {\ langle \ {\ text {span} _ {f，g} end}（v_π）\ rangle}^{\ text {w.o。}} =l_π（γ）'，$ f，其中$ f，g $通过holomorphic cusp表格以相同类型的$γ$进行。如果$γ$是一个无限的共轭类组，则我们从cusp表单中获得$ \ text {ii} _1 $ factor。
A holomorphic discrete series representation $(L_π,H_π)$ of a connected semi-simple real Lie group $G$ is associated with an irreducible representation $(π,V_π)$ of its maximal compact subgroup $K$. The underlying space $H_π$ can be realized as certain holomorphic $V_π$-valued functions on the bounded symmetric domain $\mathcal{D}\cong G/K$. By the Berezin quantization, we transfer $B(H_π)$ into End$(V_π)$-valued functions on $\mathcal{D}$. For a lattice $Γ$ of $G$, we give the formula of a faithful normal tracial state on the commutant $L_π(Γ)'$ of the group von Neumann algebra $L_π(Γ)''$. We find the Toeplitz operators $T_f$'s associated with essentially bounded End$(V_π)$-valued functions $f$'s on $Γ\backslash\mathcal{D}$ generate the entire commutant $L_π(Γ)'$: $$\overline{\{T_f|f\in L^\infty(Γ\backslash\mathcal{D},{\rm End}(V_π))\}}^{\text{w.o.}}=L_π(Γ)'.$$ For any cuspidal automorphic form $f$ defined on $G$ (or $\mathcal{D}$) for $Γ$, we find the associated Toeplitz-type operator $T_f$ intertwines the actions of $Γ$ on these square-integrable representations. Hence the composite operator of the form $T_g^{*}T_f$ belongs to $L_π(Γ)'$. We prove these operators span $L^{\infty}(Γ\backslash\mathcal{D})$ and $$\overline{\langle\{\text{span}_{f,g} T_g^{*}T_f\}\otimes {\rm End}(V_π)\rangle}^{\text{w.o.}}=L_π(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 $\text{II}_1$ factor from cusp forms.
