Related papers: Computational proof of the Mackey formula for q > …

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring $k$. One considers a split reductive group scheme $G$ acting on a $k$-algebra $A$ and leaving invariant a subalgebra $R$. If $R^U=A^U$ then the…

Let $G$ be a connected reductive algebraic group defined over an algebraic closure of a finite field and let $F : G \to G$ be an endomorphism such that $F^d$ is a Frobenius endomorphism for some $d \geq 1$. Let $P$ be a parabolic subgroup…

When $G$ is a real semisimple group, there is a surprising interplay between its representation theory and that of its motion group $G_0$, known as the Mackey analogy. The present paper extends this analogy to the framework of…

Every nonabelian finite simple group of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, has a presentation with a bounded number of generators and relations and total length $O(\log n…

A conjecture due to Zassenhaus asserts that if $\ G$ is a finite group then any torsion unit in $\mathbb{Z}G$ is conjugate in $\mathbb{Q}G$ to an element of $\ G$. We present a weaker form of this conjecture for some infinite groups.

Let (G,S) be a finitely generated Coxeter group, such that the Coxeter system is indecomposable and the canonical bilinear form is indefinite but non-degenerate. We show that the reduced C-*-algebra of G is simple with unique normalised…

A key result in a 2004 paper by S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg (ABG) gives an equivalence of the bounded derived category of finite dimensional modules for the principal block of a Lusztig quantum algebra at an $\ell^{th}$…

In this paper we consider the inductive Alperin-McKay condition for quasi-isolated 2-blocks of exceptional groups of Lie type. Thereby, we complete the proof of the Alperin-McKay conjecture for the prime 2.

Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells, which have important applications in representation theory. We study the case where $W$ is an affine…

We prove Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$. Specifically, let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$. If $E/\mathbb{Q}$ has analytic rank at most $1$, then we prove that…

Let K be an algebraically closed field. For a finitely generated graded K algebra R, let cmdef R := dim R - depth R denote the Cohen-Macaulay-defect of R. Let G be a linear algebraic group over K that is reductive but not linearly…

Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors…

Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by 3 elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper we consider subgroups at the next…

In this paper we study higher Deligne--Lusztig representations of reductive groups over finite quotients of discrete valuation rings. At even levels, we show that these geometrically constructed representations coincide with certain induced…

For a fixed finite group $G$, we study the fields of definition of geometrically irreducible components of Hurwitz moduli schemes of marked branched $G$-covers of the projective line. The main focus is on determining whether components…

Our aim is to find the irreducibility criteria for the Koopman representation, when the group acts on some space with a measure (Conjecture 1.5). Some general necessary conditions of the irreducibility of this representation are…

A consequence of Rapinchuk et al. is that for $S$ a closed surface of genus $g\geq 2$, the set of Hitchin representations of $\pi_1(S)$ with image in $\mathrm{SL}(n,\mathbb{Q})$ is dense in the Hitchin component. We give a dynamical proof…

Mockenhaupt and Tao (Duke 2004) proved a finite field analogue of the Stein--Tomas restriction theorem, establishing a range of $q$ for which $L^q\to L^2$ restriction estimates hold for a given measure $\mu$ on a vector space over a finite…

For any large prime $q$, $x \leq 1$ and any real $k\geq 2$, we prove a lower bound for the following $2k$-th moment \begin{equation*} \sum_{\substack{\chi \in X_q^*}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k}, \end{equation*} where…

Fix a matroid N. A matroid M is N-fragile if, for each element e of M, at least one of M\e and M/e has no N-minor. The Bounded Canopy Conjecture is that all GF(q)-representable matroids M that have an N-minor and are N-fragile have branch…