Branching rule for principal series representation and geometry of spherical varieties

本文的目的是描述限制在G_{n-1}上的G_n的非酉主表示的分支规则，这里G_n是真实的数域或复数域上的一般线性群或正交群，G_{n-1}自然嵌入G_n中。由于主级数表示是由其Borel子群B_n的一维表示导出的，因此需要描述旗簇G_n/B_n中B_{n-1}-轨道的封闭关系.一般地，对于约化李群G及其Borel子群B，以及G正则作用于其上的正规代数簇X，如果B在X中有稠密开轨道，则G-空间X称为球面簇。旗形变种和对称变种就是这样的例子。已知G_{n-1}-空间G_n/B_n是球面的，G_n/B_n上B_{n-1}-轨道的封闭关系称为Bruhat-Chevalley序（BC-序）。为了以组合的方式描述BC序， ...更多信息 需要获得（右）弱序，这可以通过观察轨道和最小抛物子群的乘积如何分解为轨道来完成。在这个研究项目中，我发现存在关于弱序的不闭合的极小轨道，因此，不可能确定总的BC-序。这种现象在旗型品种和对称型品种中不存在。然而，通过计算一些例子，我得到了一个猜想，即左弱序和右弱序都产生全BC-序，并证明了这个猜想是正确的G/B上的B-轨道称为Schubert簇，由G的Weyl群参数化，记为X_w，其中w是Weyl群的元素，如果G是A型，则X_w在G中的原像在n × n矩阵集合的坐标环中的闭包的行列式理想由次行列式生成.因此，为了描述主级数表示的分支规则，这是我们在这个研究项目中的目标，有必要获得切出每个B_{n-1}-轨道的Schubert簇的坐标环中的理想的显式表示。少

The purpose of this research project is to describe the branching rule for the non-unitary principal representation of G_n restricted to G_{n-1}, where G_n is the general linear group or the orthogonal group over the field of real numbers or complex numbers, and G_{n-1} is naturally embedded in G_n. Since the principal series representations are induced from the one-dimensional representations of its Borel subgroup B_n, one needs to describe the closure relation of the B_{n-1}-orbits in the flag variety G_n/B_n. In general, for a reductive Lie group G and its Borel subgroup B, and a normal algebraic variety X, on which G acts regularly, if B has a dense open orbit in X, then the G-space X is called a spherical variety. The flag varieties and symmetric varieties are such examples. It was known that the G_{n-1}-space G_n/B_n is spherical.The closure relation of B_{n-1}-orbits on G_n/B_n is called Bruhat-Chevalley order (BC-order). In order to describe the BC-order in a combinatorial way, … More one needs to obtain the (right) weak order, which can be done by looking at how the product of the orbit and the minimal parabolic subgroup decomposes into the orbits. In this research project, I found that there exist minimal orbits with respect to the weak order that are NOT closed, and therefore, it is impossible to determine the total BC-order. This phenomenon does not occur in the cases of the flag varieties or symmetric varieties. However, computing some examples, I obtained a conjecture that the left weak order as well as the right weak order yields the total BC-order, and have proved it is true (not yet published, but in preparation).The B-orbits on G/B are called Schubert varieties, and are parameterized by the Weyl group of G.Denoting it by X_w, with w an element of the Weyl group, if G is of type A, then the determinantal ideal of the closure of the preimage of X_w in G in the coordinate ring of the set of n-by-n matrices are generated by minor determinants. Therefore, to describe the branching rule for principal series representations, which is our goal in this research project, it is necessary to obtain explicit presentation of the ideal in the coordinate ring of the Schubert variety that cuts out each B_{n-1}-orbit. Less