The local structure theorem for real spherical varieties

Abstract

Abstract. Let G be an algebraic real reductive group and Z a real spherical G-variety, that is, it admits an open orbit for a minimal parabolic subgroup P . xWe prove a local structure theorem for Z. In the simplest case where Z is homogeneous, the theorem provides an isomorphism of the open P -orbit with a bundle Q× L S. Here Q is a parabolic subgroup with Levi decomposition L ⋉ U , and S is a homogeneous space for a quotient D = L/L n of L, where L n ⊆ L is normal, such that D is compact modulo center.

Geometric Counting on Wavefront Real Spherical Spaces

“…By choosing P suitably we can then arrange that its orbit through the origin z 0 = H ∈ Z is open, or equivalently that g = h + p. All symmetric spaces are known to be real spherical. 6 According to [26] there is a unique parabolic subgroup Q ⊃ P with the following two properties:…”

“…Set A H := A ∩ H and put A Z = A/A H . We recall that dim A Z is an invariant of the real spherical space, called the real rank (see [26]) and denoted by rank R Z.…”

“…It is a closed and convex subcone a − Z of a Z , defined as follows. According to [26] there exists a linear map (3.1)…”

Vanishing at infinity on homogeneous spaces of reductive type

“…We recall that dim A Z is an invariant of the real spherical space, called the real rank (see [20]). …”

“…Write Σ u for the space of a-weights of the a-module u and let u denote the corresponding sum of root spaces for −Σ u . According to [20] there exists a linear map …”