局部对称空间上的椭圆不变微分算子的谱
Spectra of elliptic invariant differential operators on locally symmetric spaces
AI总结：
本文针对局部对称空间Γ\G/K研究G/K不变微分算子的谱，对arXiv:2402.02530的定理给出纯分析证明，明确拟正则表示L²(Γ\G)为缓增的条件。
AI中文摘要：
针对局部对称空间Γ\G/K，本文研究G/K的不变微分算子在L²(Γ\G/K)上的谱，对arXiv:2402.02530的近期定理给出纯分析证明：若Γ的极限锥𝒸_Γ包含于Weyl腔内部，且G的实秩为1或局部同构于𝔰𝔩₃(𝕂)（𝕂=ℝ,ℂ,ℍ）或𝔢₆⁻²⁶，则拟正则表示L²(Γ\G)是缓增的。
英文摘要：
For a locally symmetric space $Γ\backslash G /K$ we study the spectrum of the invariant differential operators of $G/K$ on $L^2(Γ\backslash G/K)$. This gives a purely analytical proof of the recent theorem of arXiv:2402.02530 that the quasiregular representation $L^2(Γ\backslash G)$ is tempered if the limit cone $\mathcal{L}_Γ$ of $Γ$ is contained in the interior of the Weyl chamber except if $G$ has real rank one or is locally isomorphic to $\mathfrak{sl}_3(\mathbb K)$, $\mathbb K=\mathbb{R},\mathbb{C},\mathbb H$, or $\mathfrak{e}_6^{-26}$.
