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We rewrite the standard 4-dimensional Dirac equation in terms of quaternionic 2-component spinors, leading to a formalism which treats both massive and massless particles on an equal footing. The resulting unified description has the…
In this paper we study the Gan-Gross-Prasad problem for unitary groups over finite fields. Our results provide complete answers for unipotent representations, and we obtain the explicit branching of these representations.
We study representations $G\to H$ where $G$ is either a simple Lie group with real rank at least 2 or an infinite dimensional orthogonal group of some quadratic form of finite index at least 2 and $H$ is such an orthogonal group as well.…
We prove that when $q$ is a power of $2$, every complex irreducible representation of $\mathrm{Sp}(2n, \mathbb{F}_q)$ may be defined over the real numbers, that is, all Frobenius-Schur indicators are 1. We also obtain a generating function…
Lorentz's group represented by the hypercomplex system of numbers, which is based on dirac matrices, is investigated. This representation is similar to the space rotation representation by quaternions. This representation has several…
We examine various consequences of the existence of exceptional representations of an irreducible Weyl group. (These are notes from a talk in the MIT Lie groups seminar.)
We classify the simple modules of the exceptional algebraic supergroups over an algebraically closed field of prime characteristic.
It is generally believed (and for the most part is probably true) that Lie theory, in contrast to the characteristic zero case, is insufficient to tackle the representation theory of algebraic groups over prime characteristic fields.…
Let $G$ be a complex reductive algebraic group. In arXiv:2108.03453, we have defined a finite set of irreducible admissible representations of $G$ called `unipotent representations', generalizing the special unipotent representations of…
Starting from the defining transformations of complex matrices for the SO(4) group, we construct the fundamental representation and the tensor and spinor representations of the group SO(4). Given the commutation relations for the…
For a quadratic field K, we investigate continuous mod p representations of the absolute Galois groups of K that are unramified away from p and infinity. We prove that for certain pairs (K,p), there are no such irreducible representations.…
We characterize groups with Guoliang Yu's property A (i.e., exact groups) by the existence of a family of uniformly bounded representations which approximate the trivial representation.
Positive energy ray representations of the Poincar\'e group are naturally subdivided into three classes according to their mass and spin content: m>0, m=0 finite helicity and m=0 infinite helicity. For a long time the localization…
We provide a unified treatment of several results concerning full groups of ample groupoids and paradoxical decompositions attached to them. This includes a criterion for the full group of an ample groupoid being amenable as well as…
Quasirational presentations ($QR$-presentations) of (pro-$p$)groups are studied. Such presentations include, in particular, aspherical presentations of discrete groups and their subpresentations and pro-$p$-groups with a single defining…
We classify the automorphic representations (over number fields) and the irreducible admissible representations (over local fields) of unitary groups which are not quasi-split, under the assumption that the same is known for quasi-split…
We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of…
As discussed in a previous article, any (real) Lorentz algebra element possess a unique orthogonal decomposition as a sum of two mutually annihilating decomposable Lorentz algebra elements. In this article, this concept is extended to the…
For infinite reductive groups with Frobenius maps, we show that certain subquotients of abstract representations of the groups induced from 1-dimensional representations of Borel subgroups are irreducible.
We prove that certain Fuchsian triangle groups are profinitely rigid in the absolute sense, i.e. each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. We also develop a method…