论文标题
Satake-Furstenberg紧凑和梯度图
Satake-Furstenberg compactifications and gradient map
论文作者
论文摘要
让$ g $成为一个真正的半谎言组，有有限的中心，让$ \ mathfrak g = \ mathfrak k \ oplus \ mathfrak p $是其Lie代数的cartan分解。让$ k $是$ g $的最大紧凑型亚组，其中lie代数$ \ mathfrak k $，让$τ$是复杂矢量太空$ v $的$ g $的不可约为表示。令$ h $为$ v $上的Hermitian标量产品，使$τ（g）$相对于$ \ mathrm {u}（v，h）^{\ Mathbb c} $兼容。我们用$μ_ {\ Mathfrak p}：\ Mathbb p（v）\ longrightArrow \ Mathfrak \ Mathfrak p $ $ g $ - 级别的地图和$ \ mathcal o $ unture o g $ in $ g $ in $ \ mathbb p（mathbb p（mathbb p（v）$），这是$ k $ - $ k $ of z $ of z $ of z $ n of z $ y的$ une或$ \ mathrm {u}（v，h）^{\ mathbb c} $。我们证明，直到等效于$τ$引起的$ g $的抛物线亚组的不可减至的表示，完全由极性轨道$ \ mathcal e = \ mathcal e = \ mathrm {cons}（μ_ {\ mathfrak p}（\ mathfrak p}（\ mathcal o））$。此外，任何$ g $的抛物线子组都可以很好地适应$ \ Mathcal o $和$μ_ {\ Mathfrak P} $。这些结果在复杂的还原性情况下也是新的。 $ \ Mathcal E $和$τ$之间的连接提供了没有根数据的萨克克压缩的几何描述。在这种情况下，还研究了Bourguignon-Li-Yau图的性能。给定$ \ Mathcal o $上的度量$γ$，我们从与$τ$和$ \ Mathcal e $相关的$ g/k $的$ g/k $构造了一个地图$ψ_γ$。如果$γ$是$ k $ invariant的度量，那么$ψ_γ$是Satake紧凑型和$ \ MATHCAL E $的同态。最后，我们证明，对于大量的测量，地图$ψ_γ$是冲销的。
Let $G$ be a real semisimple Lie group with finite center and let $\mathfrak g=\mathfrak k \oplus \mathfrak p$ be a Cartan decomposition of its Lie algebra. Let $K$ be a maximal compact subgroup of $G$ with Lie algebra $\mathfrak k$ and let $τ$ be an irreducible representation of $G$ on a complex vector space $V$. Let $h$ be a Hermitian scalar product on $V$ such that $τ(G)$ is compatible with respect to $\mathrm{U}(V,h)^{\mathbb C}$. We denote by $μ_{\mathfrak p}:\mathbb P(V) \longrightarrow \mathfrak p$ the $G$-gradient map and by $\mathcal O$ the unique closed orbit of $G$ in $\mathbb P(V)$, which is a $K$-orbit, contained in the unique closed orbit of the Zariski closure of $τ(G)$ in $\mathrm{U}(V,h)^{\mathbb C}$. We prove that up to equivalence the set of irreducible representations of parabolic subgroups of $G$ induced by $τ$ are completely determined by the facial structure of the polar orbitope $\mathcal E=\mathrm{conv}(μ_{\mathfrak p} (\mathcal O))$. Moreover, any parabolic subgroup of $G$ admits a unique closed orbit which is well-adapted to $\mathcal O$ and $μ_{\mathfrak p}$ respectively. These results are new also in the complex reductive case. The connection between $\mathcal E$ and $τ$ provides a geometrical description of the Satake compactifications without root data. In this context the properties of the Bourguignon-Li-Yau map are also investigated. Given a measure $γ$ on $\mathcal O$, we construct a map $Ψ_γ$ from the Satake compactification of $G/K$ associated to $τ$ and $\mathcal E$. If $γ$ is a $K$-invariant measure then $Ψ_γ$ is an homeomorphism of the Satake compactification and $\mathcal E$. Finally, we prove that for a large class of measures the map $Ψ_γ$ is surjective.
