论文标题
Hecke代数对双旗的AIII型的动作
Action of Hecke algebra on the double flag variety of type AIII
论文作者
论文摘要
考虑一个连接的还原代数组$ g $和对称子组$ k $。令$ \ mathfrak {x} = k/b_k \ times g/p $是双标志的有限型类型，其中$ b_k $是$ k $的borel子组，$ g $ $ g $的$ k $ a $ p $ a po $。一个一般参数表明，轨道空间$ \ mathbb {c} \，\ mathfrak {x}/k $继承了Hecke Algebra $ \ Mathscr {h} = \ Mathscr {h} = \ Mathscr {h}（k，k，b_k）$ double cosets的$。但是，找出Hecke模块的明确结构是一个完全不同的问题。 在本文中，我们在$ \ mathbb {c} \，\ mathfrak上以$ \ mathbb {c} \，\ mathfrak {x}/k $的方式确定$ \ mathscr {h} $的明确操作，使用图形为aiii类型的双标志品种。作为副产品，我们还可以在$ \ mathbb {c} \，\ mathfrak {x}/k $上获得Weyl grout的表示形式的描述，作为诱导表示的直接总和。
Consider a connected reductive algebraic group $ G $ and a symmetric subgroup $ K $. Let $ \mathfrak{X} = K/B_K \times G/P $ be a double flag variety of finite type, where $ B_K $ is a Borel subgroup of $ K $, and $ P $ a parabolic subgroup of $ G $. A general argument shows that the orbit space $ \mathbb{C}\,\mathfrak{X}/K $ inherits a natural action of the Hecke algebra $ \mathscr{H} = \mathscr{H}(K, B_K) $ of double cosets via convolutions. However, to find out the explicit structure of the Hecke module is a quite different problem. In this paper, we determine the explicit action of $ \mathscr{H} $ on $ \mathbb{C}\,\mathfrak{X}/K $ in a combinatorial way using graphs for the double flag variety of type AIII. As a by-product, we also get the description of the representation of the Weyl group on $ \mathbb{C}\,\mathfrak{X}/K $ as a direct sum of induced representations.
