We study $K$-orbits in $G/P$ where $G$ is a complex connectedreductive group, $P subseteq G$ is a parabolic subgroup, and $Ksubseteq G$ is the fixed point subgroup of an involutiveautomorphism $ heta$. Generalizing work of Springer, weparametrize the (finite) orbit set $K setminus G slash P$ and wedetermine the isotropy groups. As a consequence, we describe theclosed (resp. affine) orbits in terms of $ heta$-stable(resp. $ heta$-split) parabolic subgroups. We also describe thedecomposition of any $(K,P)$-double coset in $G$ into$(K,B)$-double cosets, where $B subseteq P$ is a Borel subgroup.Finally, for certain $K$-orbit closures $X subseteq G/B$, and forany homogeneous line bundle $mathcal{L}$ on $G/B$ having nonzeroglobal sections, we show that the restriction map $ es_X colonH^0 (G/B, mathcal{L}) o H^0 (X, mathcal{L})$ is surjective andthat $H^i (X, mathcal{L}) = 0$ for $i geq 1$. Moreover, wedescribe the $K$-module $H^0 (X, mathcal{L})$. This givesinformation on the restriction to $K$ of the simple $G$-module $H^0(G/B, mathcal{L})$. Our construction is a geometric analogue ofVogan and Sepanski's approach to extremal $K$-types.
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机译：我们研究了$ G / P $中的$ K $轨道，其中$ G $是一个复杂的连通归约组，$ Psubseteq G $是一个抛物线子组，$ Ksubseteq G $是一个渐进自同构$ heta $的不动点子组。推广Springer的工作，对（有限的）轨道集合$ K set减去G斜线P $进行参数化，并确定各向同性群。结果，我们用稳定的抛物线子群描述了闭合的（仿射）轨道。我们还描述了将$ G $中的任何$（K，P）$-双陪集分解为$（K，B）$-双陪集，其中$ Bsubseteq P $是Borel子组。最后，对于某些$ K $轨道闭合$ Xsubseteq G / B $，以及具有非零全局截面的$ G / B $上的任何同质线束$ mathcal {L} $，我们显示了限制图$ es_X ColonH ^ 0（G / B，mathcal { L}）o H ^ 0（X，mathcal {L}）$是射影，对于$ i geq 1 $，$ H ^ i（X，mathcal {L}）= 0 $。此外，我们描述了$ K $模块$ H ^ 0（X，mathcal {L}）$。这给出了简单$ G $-模块$ H ^ 0（G / B，mathcal {L}）$对$ K $的限制的信息。我们的构造是Vogan和Sepanski运用极值$ K $类型的方法的几何模拟。
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