Geometric construction of Heisenberg-Weil representations for finite unitary groups and Howe correspondences

We give a geometric construction of the Heisenberg-Weil representation of a finite unitary group by the middle \'{e}tale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for $(\mathit{Sp}_{2n},O_2^-)$ over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.Comment: 27 page

The orthosymplectic Lie supergroup in harmonic analysis

The study of harmonic analysis in superspace led to the Howe dual pair (O(m) x Sp(2n); sl2). This Howe dual pair is incomplete, which has several implications. These problems can be solved by introducing the orthosymplectic Lie supergroup OSp(m|2n). We introduce Lie supergroups in the supermanifold setting and show that everything is captured in the Harish-Chandra pair. We show the uniqueness of the supersphere integration as an orthosymplectically invariant linear functional. Finally we study the osp(m|2n)-representations of supersymmetric tensors for the natural module for osp(m|2n)

Al-Andalus Rediscovered: Iberia's New Muslims [Book Review]

This article reviews the book 'Al-Andalus Rediscovered: Iberia’s New Muslims', by Marvine Howe

The Howe-Moore property for real and p-adic groups

We consider in this paper a relative version of the Howe-Moore Property, about vanishing at infinity of coefficients of unitary representations. We characterize this property in terms of ergodic measure-preserving actions. We also characterize, for linear Lie groups or p-adic Lie groups, the pairs with the relative Howe-Moore Property with respect to a closed, normal subgroup. This involves, in one direction, structural results on locally compact groups all of whose proper closed characteristic subgroups are compact, and, in the other direction, some results about the vanishing at infinity of oscillatory integrals.Comment: 25 pages, no figur