Related papers: Character Varieties of Abelian Groups

Let $G$ be an abelian group. We prove that a group $G$ admits a Hausdorff group topology $\tau$ such that the von Neumann radical $\mathbf{n}(G, \tau)$ of $(G, \tau)$ is non-trivial and finite iff $G$ has a non-trivial finite subgroup. If…

Let $G$ be a $(2,m,n)$-group and let $x$ be the number of distinct primes dividing $\chi$, the Euler characteristic of $G$. We prove, first, that, apart from a finite number of known exceptions, a non-abelian simple composition factor $T$…

Let $\mathcal{A}(q)$ be a finite-dimensional nilpotent algebra over a finite field $\mathbb{F}_{q}$ with $q$ elements, and let $G(q) = 1+\mathcal{A}(q)$. On the other hand, let $\Bbbk$ denote the algebraic closure of $\mathbb{F}_{q}$, and…

Assume $G$ is a connected reductive algebraic group defined over $\bar{\mathbb{F}_p}$ such that $p$ is good prime for $G$. Furthermore we assume that $Z(G)$ is connected and $G/Z(G)$ is simple of classical type. Let $F$ be a Frobenius…

We develop a simple algebraic approach to the study of the Weil representation associated to a finite abelian group. As a result, we obtain a simple proof of a generalisation of a well-known formula for the absolute value of its character.…

Let G be a reductive group over a non-Archimedean local field. Then the canonical functor from the derived category of smooth tempered representations of G to the derived category of all smooth representations of G is fully faithful. Here…

This is an exposition of our joint work with Kakde, Silliman, and Wang, in which we prove a version of Ribet's Lemma for $\mathrm{GL}_2$ in the residually indistinguishable case. We suppose we are given a Galois representation taking values…

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak c$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak c \oplus {\mathfrak c}^*/W \to V\oplus V^*/\!\!/\!\!/ G$. This morphism…

Let U be a smooth quasi-projective variety over a field k that is finite, the algebraic closure of a finite field or algebraically closed of characteristic 0. Let X be a suitable projective compactification of U, and D an effective divisor…

Let k be a non-archimedean local field with residual characteristic p. Let G be a connected reductive group over k that splits over a tamely ramified field extension of k. Suppose p does not divide the order of the Weyl group of G. Then we…

Let $X$ be a hyperk\"ahler variety, and let $G$ be a group of finite order non-symplectic automorphisms of $X$. Beauville's conjectural splitting property predicts that each Chow group of $X$ should split in a finite number of pieces. The…

Let $A$ be an Artinian local ring with algebraically closed residue field $k$, and let $\mathbf{G}$ be an affine smooth group scheme over $A$. The Greenberg functor $\mathcal{F}$ associates to $\mathbf{G}$ a linear algebraic group…

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman…

Let $G$ be a complex reductive group, $T$ be a maximal torus of $G$, $B$ be a Borel subgroup of $G$ containing $T$, $W$ be the Weyl group of $G$ with respect to $T$. To each element $w$ of $W$ one can associate the Schubert subvariety $X_w$…

Let $G$ be a finite group, and $\pi$ be a set of primes. The $\pi$-core $\mathbf{O}_\pi(G)$ is the unique maximal normal $\pi$-subgroup of $G$, and $b(G)$ is the largest irreducible character degree of $G$. In 2017, Qian and Yang proved…

Let $ G $ be a finite non-solvable group with a primitive irreducible character $ \chi $ that vanishes on one conjugacy class. We show that $ G $ has a homomorphic image that is either almost simple or a Frobenius group. We also classify…

We consider a finite abelian group $M$ of odd exponent $n$ with a symplectic form $\omega: M\times M\to \mu_n$ and the Heisenberg extension $1\to \mu_n\to H\to M\to 1$ with the commutator $\omega$. According to the Stone - von Neumann…

Real forms of a complex reductive group are classified by Galois cohomology H^1(Gamma,G_ad) where G_ad is the adjoint group. Cartan's classification of real forms in terms of maximal compact subgroups can be stated in terms of H^(Z/2Z,G_ad)…

A theorem of Cantat and Urech says that an analog of the classical Tits alternative holds for the group of birational automorphisms of a compact complex Kaehler surface. We established in our previous paper the following Tits-type…

For a finite group $G$ and complex character $\chi\in\mathrm{Irr}(G)$ that restricts irreducibly to a normal subgroup $N\vartriangleleft G,$ we prove a theorem about Clifford correspondences between the characters of subgroups of $G$ that…