Tensor products of unipotent characters of general linear groups over finite fields

Given unipotent characters U ₁, . . . , U _k of GL _n $ \left( {{{\mathbb{F}}_q}} \right) $ , we prove that $ \left\langle {{U_1} \otimes \cdots \cdots \otimes {U_k},1} \right\rangle $ is a polynomial in q with non-negative integer coefficients. We study the degree of this polynomial and give a necessary and sufficient condition in terms of the representation theory of symmetric groups and root systems for this polynomial to be non-zero.     

Invariant measures for actions of unipotent groups over local fields on homogeneous spaces

On leave from the Institute of Information Transmission of Russian Academy of Science     

Special unipotent groups are split

We show that over any field k, a smooth unipotent algebraic k-group is special if and only if it is k-split.     

On exact actions of unipotent groups

Any affine variety with a d-exact action of a unipotent group can be embedded in an affine space preserving d-exactness. Furthermore, we can find such an ambient space which has some other good properties. The key idea of the proof is describing the property “d-exact” by means of inequalities.     

Witt groups and unipotent elements in algebraic groups

Let G be a semisimple algebraic group defined over an algebraicallyclosed field K of good characteristic p>0. Let u be a unipotentelement of G of order p^t, for some t N. In this paper it isshown that u lies in a closed subgroup of G isomorphic to theit Witt group W^t(K), which is a t-dimensional connected abelianunipotent algebraic group. 2000 Mathematics Subject Classification:20G15.     

Constructing arithmetic subgroups of unipotent groups

Let G be a unipotent algebraic subgroup of some defined over . We describe an algorithm for finding a finite set of generators of the subgroup . This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists.     

Modular categories associated with unipotent groups

Let $G$ be a unipotent algebraic group over an algebraically closed field $\mathtt{k }$ of characteristic $p>0$ and let $l\ne p$ be another prime. Let $e$ be a minimal idempotent in $\mathcal{D }_G(G)$ , the $\overline{\mathbb{Q }}_l$ -linear triangulated braided monoidal category of $G$ -equivariant (for the conjugation action) $\overline{\mathbb{Q }}_l$ -complexes on $G$ under convolution (with compact support) of complexes. Then, by a construction due to Boyarchenko and Drinfeld, we can associate to $G$ and $e$ a modular category $\mathcal{M }_{G,e}$ . In this paper, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class $\mathfrak{C }_p^{\pm }$ .     

Unit groups of some multiquadratic number fields and 2-class groups

Periodica Mathematica Hungarica - Let $$pequiv -q equiv 5pmod 8$$ be two prime integers. In this paper, we investigate the unit groups of the fields $$ L_1 =mathbb {Q}(sqrt{2}, sqrt{p},...     

Weakly closed unipotent subgroups in Chevalley groups

The goal of this note is to classify the weakly closed unipotent subgroups in the split Chevalley groups. In an application we show under some mild assumptions on the characteristic that the Lie algebra of a connected simple algebraic group fails to be a so-called 2F-module.     

Normal,unipotent subgroup schemes of reductive groups

Let G be a reductive group over a field k of characteristic p. Let $k^{\mathrm{sep}}$ be a separable closure of k. If $p\ne 2$, there exists a linear representation of G that is faithful and semisimple; moreover, any unipotent, normal subgroup scheme of G is trivial. For $p=2$, these two properties hold if and only if $G_{k^{\mathrm{sep}}}$ has no direct factor that is isomorphic to ${\mathbf{SO}}_{2n+1}$ for some $n?1$. To cite this article: A. Vasiu, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Some small unipotent representations of indefinite orthogonal groups

We study a family of small unitary representations of indefinite orthogonal groups. These representations arise as analytic continuations of the discrete series and were studied extensively by Knapp in [K3]. We complete Knapp's analysis by proving that they are irreducible. In order to do so we prove that the representations are unipotent and have irreducible associated cycles in which all multiplicities are exactly one. Moreover, we prove that the K-type structure of each representation matches (up to a shift) the K-type structure of the ring of functions on the closure a nilpotent orbit on .