Let L be a totally real field with ring of integers OL. Let N ≥ 4 be an integer and let M(μN) be the fine moduli scheme over Z of polarized abelian varieties with real multiplication (RM) and μN-level structure, satisfying the Deligne-Pappas condition. For every scheme S, we let M(S, μN) = M(μN) ×Z S be the moduli scheme over S; see Definition 2.1. Many aspects of the geometry of the modular va...

In a seminal 1994 paper [20], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P )≥0 of an arbitrary (generalized, partial) flag variety G/P . He referred to this space as a “remarkable polyhedral subspace,” and conjectured a decomposition into cells, which was subsequently proven by the first author [27]. In this article we use discrete Morse theo...

Let K = k(B) be a function field of an integral projective variety B over the algebraically closed field k such that B is regular in codimension 1. The set of places MB is given by the prime divisors of B. We fix an ample class c on B. If we count every prime divisor Y with weight deg c (Y ), then the valuations ordY lead to a product formula on K and hence to a theory of heights (see [La] or [...

One can view a partial flag variety in $${\mathbb {C}}^n$$ as an adjoint orbit $${\mathcal {O}}_{\lambda }$$ inside the Lie algebra of $$n\times n$$ skew-Hermitian matrices. We use context to study totally nonnegative part from algebraic, geometric, and dynamical perspective. The paper has three main parts: (1) introduce , describe it explicitly several cases. define twist map on it, which gene...

Let X/K be an absolutely simple abelian variety over a number field; we study whether the reductions Xp tend to be simple, too. We show that if End(X) is a definite quaternion algebra, then the reduction Xp is geometrically isogenous to the self-product of an absolutely simple abelian variety for p in a set of positive density, while if X is of Mumford type, then Xp is simple for almost all p. ...

We consider the totally nonfree (TNF) action of a groups and the corresponding adjoint invariant (AD) measures on the lattices of the subgroups of the given group. The main result is the description of all adjoint-invariant and TNF-measures on the lattice of subgroups of the infinite symmetric group SN. The problem is closely related to the theory of characters and factor representations of gro...