Related papers: Repr\'esentations des quaternions de norme 1

A relative one-relator presentation has the form P = < X,H ; R > where X is a set, H is a group, and R is a group word on X and H. We show that if the group word on X obtained from R by deleting all the terms from H has what we call the…

Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad…

Let $G$ be a simply connected, connected completely solvable Lie group with Lie algebra $\mathfrak{g}=\mathfrak{p}+\mathfrak{m}.$ Next, let $\pi$ be an infinite-dimensional unitary irreducible representation of $G$ obtained by inducing a…

We give a complete classification of simple representations of the braid group B_3 with dimension $\leq 5$ over any algebraically closed f ield. In particular, we prove that a simple d-dimensional representation $\rho: B_3 \to GL(V)$ is…

We study the representations of a class of non-commutative polynomial algebras truncated at degree 3, with one additional relation. We determine the irreducible components of their varieties of representations. We do this by showing that…

Let n be a positive integer, F be a non-Archimedean locally compact field of odd residue characteristic p and G be an inner form of GL(2n,F). This is a group of the form GL(r,D) for a positive integer r and division F-algebra D of reduced…

Let $D$ denote a quaternion division algebra over a non-archimedean local field $F$ with characteristic zero. Let $Sp_n(D)$ be the unique non-split inner form of the symplectic group $Sp_{2n}(F)$. An irreducible admissible representation…

Let \lambda be a partition of a positive integer n. Let C be a symmetric rigid tensor category over a field k of characteristic 0 or char(k)>n, and let V be an object of C. In our main result (Theorem 4.3) we introduce a finite set of…

We study the existence of symplectic resolutions of quotient singularities V/G where V is a symplectic vector space and G acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the…

This Ph.D. thesis belongs to the realm of mod $p$ representation theory of $p$-adic groups. The main object of study is the inner form of the general linear group $\mathrm{GL}(m,D)$ where $D$ is a division algebra over a non-Archimedean…

For $G$ an algebraic group of type $A_l$ over an algebraically closed field of characteristic $p$, we determine all irreducible rational representations of $G$ in defining characteristic with dimensions $\le (l+1)^s$ for $s = 3, 4$,…

We prove that the restriction of any nontrivial representation of the Ree groups $^2F_{4}(q), q=2^{2n+1}\geq8$ in odd characteristic to any proper subgroup is reducible. We also determine all triples $(K, V, H)$ such that $K \in \{^2F_4(2),…

In this paper we construct finite dimensional representations of the wreath product symplectic reflection algebra H(k,c,N,G) of rank N attached to a finite subgroup G of SL(2,C) (here k is a number and c a class function on the set of…

We prove that a complete local or graded one-dimensional domain of prime characteristic has finite F-representation type if its residue field is algebraically closed or finite, and present examples of a complete local or graded…

We establish vanishing results for limits of characters in various discrete groups, most notably irreducible lattices in higher rank semisimple Lie groups. As an application, we show that any sequence of finite-dimensional representations…

Let $F$ be a quadratic extension of $\mathbb{Q}_p$. We prove that smooth irreducible supersingular representations with central character of $\mathrm{GL}_2(F)$ are not of finite presentation.

For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$, or more generally, which representations remain homogeneous in characteristic $p$. In this paper we…

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…

Let (G,V) be a regular prehomogeneous vector space (abbreviated to PV), where G is a connected reductive algebraic group over C. If $V= \oplus_{i=0}^{n}V_{i}$ is a decomposition of V into irreducible representations, then, in general, the…

Singular vectors of a representation of a finite-dimensional simple Lie algebra are weight vectors in the underlying module that are nullified by positive root vectors. In this article, we use partial differential equations to find all the…