Orbits of strongly solvable spherical subgroups on the flag variety

Abstract

Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup (Formula presented.) which acts with finitely many orbits on the flag variety G / B, and we classify the H-orbits in G / B in terms of suitable root systems. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G / B, and we give a combinatorial model for this action in terms of weight polytopes

B_{n-1}-bundles on the flag variety, I

“…Our work advances on [Has04] in that our approach also describes orbits both in general linear and orthogonal cases in an essentially uniform manner, elucidates the bundle structure, and giving a complete description of the closure relation. It would also be interesting to relate our results to the work of Gandini and Pezzini [GP18], who study the general case of orbits of solvable subgroups of G with finitely many orbits on the flag variety of G. In particular, it would be of interest to understand some of the invariants they consider in their more general context in our situation.…”

P-flag spaces and incidence stratifications

“…Proposition 6.14. We have that By Theorem 6.10, the principal order ideal of Q × Q 2 < which satisfy the condition of Theorem 6.13 are the ones with maximum in the set {(1, 12), (2,12), (1,13), (2,23), (3,13), (3,23)}, which corresponds to the symmetric group S 3 . We consider S 3 with its standard Coxeter presentation given by generators {s, t}.…”

“…and the order ideal I such that max(I) = {(1, 13), (1,23), (2,13), (2, 23)}. We have that max(I) is represented by the flag (span F {e 1 + e 2 }, span F {(e 1 + e 2 ) ∧ e 3 }) .…”

“…• In Section 4 we study the action of I * (Q, F) on P -flags, where Q is any poset of cardinality n. Proposition 4.1 ensures that the double quotient I * (Q; R)\ GL(n, R)/I * (P ; R) is never finite, apart in the classical case P = Q = c n (for other general results on infiniteness of double quotients, see for instance [10], [13] and references there). In Theorem 4.8, we describe some of these orbits as homogeneous spaces: this shows that also classical Schubert cells are homogeneous spaces, where the isotropy subgroups are the incidence groups of posets whose Hasse diagrams are the graphs introduced in [5].…”

Sanya Lectures: Geometry of Spherical Varieties

“…In particular, Richardson and Springer give in [35,36] a combinatorial description of the graph Γ(X) for X a symmetric. There are very few other descriptions of these graphs, see for example [33] and [18]. See also [1] and [2] for results on the normality of B-orbit closures.…”