Related papers: Classification of subgroups of symplectic groups o…

Let G be a finite subgroup of SL(2,C). Let S_N#G^N be the wreath product of G by the symmetric group of degree N, acting symplectically on a complex vector space V of dimension 2N, with symplectic basis {x_i, y_i} i=1,...,N. In this paper…

We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular…

We classify up to conjugation by $\operatorname{GL}(2,\mathbb{R})$ (more precisely, block diagonal symplectic matrices) all the semidirect products inside the maximal parabolic of $\operatorname{Sp}(2,\mathbb{R})$ by means of an essentially…

This is an English translation, prepared by Wilberd van der Kallen, of the 1951 thesis by T.A. Springer. In this thesis Springer studied the classification of conjugacy classes in the symplectic group $Sp_n(K)$, where the commutative field…

We study subsets in possibly degenerate symplectic vector spaces over finite fields, which are stable under a given Coxeter/Weyl reflection group. These symplectic root systems provide crucial combinatorical data to classify…

Let $F$ be any field. We give a short and elementary proof that any finite subgroup $G$ of $PGL(2,F)$ occurs as a Galois group over the function field $F(x)$. We also develop a theory of descent to subfields of $F$. This enables us to…

The finite symplectic group Sp(2g) over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this…

Let $K$ be a 2-dimensional global field of characteristic $\neq 2$, and let $V$ be a divisorial set of places of $K$. We show that for a given $n \geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G = \mathrm{Spin}_n(q)$ of…

The genus spectrum of a finite group $G$ is the set of all $g\geq 2$ such that $G$ acts faithfully and orientation-preserving on a closed compact orientable surface of genus $g$. This article is an overview of some results relating the…

We classify the irreducible complex characters of the symplectic groups $Sp_{2n}(q)$ and the orthogonal groups $Spin_{2n}^\pm(q)$, $Spin_{2n+1}(q)$ of degrees up to the bound D, where $D=(q^n-1)q^{4n-10}/2$ for symplectic groups,…

Let $\mathbb{F}_2^\omega$ denote the countably infinite dimensional vector space over the two element field and $\operatorname{GL}(\omega, 2)$ its automorphism group. Moreover, let $\operatorname{Sym}(\mathbb{F}_2^\omega)$ denote the…

For a positive integer $g$, let $\mathrm{Sp}_{2g}(R)$ denote the group of $2g \times 2g$ symplectic matrices over a ring $R$. Assume $g \ge 2$. For a prime number $\ell$, we give a self-contained proof that any closed subgroup of…

Let G be a connected simple adjoint p-adic group not isomorphic to a projective linear group PGL(m,D) of a division algebra D, or an adjoint ramified unitary group of a split hermitian form in 3 variables. We prove that G admits an…

We classify the connected Lie subgroups of the symplectic group $Sp(2,\mathbb{R})$ whose elements are matrices in block lower triangular form. The classification is up to conjugation within $Sp(2,\mathbb{R})$. Their study is motivated by…

In this paper, we construct a quantization functor, associating a complex vector space H(V) to a finite dimensional symplectic vector space V over a finite field of odd characteristic. As a result, we obtain a canonical model for the Weil…

We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups)…

The graph of a real symplectic linear transformation is an R-Lagrangian subspace of a complex symplectic vector space. The restriction of the complex symplectic form is thus purely imaginary and may be expressed in terms of the generating…

A finite group $G$ is said to be admissible over a field $F$ if there exists a division algebra $D$ central over $F$ with a maximal subfield $L$ such that $L/F$ is Galois with group $G$. In this paper we give a complete characterization of…

We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal then it…

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex…