A Note on Pseudo-reflections

In this note, we show that if V is a finite dimensional vector space equipped with a non-degenerate bilinear form, and one has a set of pseudo-reflections on V, preserving the form and having no non-zero common fixed vector, then the group G generated by this set is ‘sufficiently large’ in the sense that for every linear transformation T : V → V, there exists an element g ∈ G such that g − G is invertible

Dimensions of group schemes of automorphisms of truncated Barsotti--Tate groups

Let $D$ be a $p$-divisible group over an algebraically closed field $k$ of characteristic $p>0$. Let $n_D$ be the smallest non-negative integer such that $D$ is determined by $D[p^{n_D}]$ within the class of $p$-divisible groups over $k$ of the same codimension $c$ and dimension $d$ as $D$. We study $n_D$, lifts of $D[p^m]$ to truncated Barsotti--Tate groups of level $m+1$ over $k$, and the numbers $\gamma_D(i):=\dim(\pmb{Aut}(D[p^i]))$. We show that $n_D\le cd$, $(\gamma_D(i+1)-\gamma_D(i))_{i\in\Bbb N}$ is a decreasing sequence in $\Bbb N$, for $cd>0$ we have $\gamma_D(1)<\gamma_D(2)<...<\gamma_D(n_D)$, and for $m\in\{1,...,n_D-1\}$ there exists an infinite set of truncated Barsotti--Tate groups of level $m+1$ which are pairwise non-isomorphic and lift $D[p^m]$. Different generalizations to $p$-divisible groups with a smooth integral group scheme in the crystalline context are also proved.Comment: 52 pages. Final version as close to the galley proofs as possible. To appear in IMR