Branching rule for principal series representation and geometry of spherical varieties

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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