Unipotent flows on products of $SL(2,K)/Γ$'s

Remarks on quantum unipotent subgroup and dual canonical basis

We prove the tensor product decomposition of the half of quantized universal enveloping algebra associated with a Weyl group element which was conjectured by Berenstein and Greenstein.

Unipotent representations as a categorical centre

Let G(F_q) be the group of rational points of a split connected reductive group G defined over the finite field F_q. In this paper we show that the category of representations of G(F_q) which are finite direct sums of unipotent representations in a fixed two-sided cell is equivalent to the centre of a certain monoidal category of sheaves on the product of two copies of the flag manifold of G. We also consider a version of this for nonsplit groups.

`Norman involutions' and tensor products of unipotent Jordan blocks

A good knowledge of the Jordan canonical form (JCF) for a tensor product of `Jordan blocks' is key to understanding the actions of $p$-groups of matrices in characteristic $p$. The JCF corresponds to a certain partition which depends on the characteristic $p$, and the study of these partitions dates back to Aitken's work in 1934. Equivalently each JCF corresponds to a certain permutation $π$ introduced by Norman in 1995. These permutations $π= π(r,s,p)$ depend on the dimensions $r$, $s$ of the Jordan blocks, and on $p$. We give necessary and sufficient conditions for $π(r,s,p)$ to be trivial, …

Correction to "Measurable quotients of unipotent translations on homogeneous spaces

The statements of Main~Theorem~1.1 and Theorem~2.1 of the author's paper [\emph{Trans.\ Amer.\ Math.\ Soc.}\ {\bf 345} (1994) 577--594] should assume that $Γ$~is discrete and $G$~is connected. (Cors.~1.3, 5.6, and~5.8 are affected similarly.) These restrictions can be removed if the conclusions of the results are weakened to allow for the possible existence of transitive, proper subgroups of~$G$. In this form, the results can be extended to the setting where $G$ is a product of real and $p$-adic Lie groups.

张量积分解的 Ennola 对偶

Ennola duality for decomposition of tensor products

本文研究有限一般线性群与有限酉群不可约特征标张量积分解的 Ennola 对偶，证明其在一般情形成立并给出箭图簇上同调的几何解释，同时构造双变量多项式以处理幂幺情形下的重数插值问题。

Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation

We show that for almost all points on any analytic curve on R^{k} which is not contained in a proper affine subspace, the Dirichlet's theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in R^{k+1}. The proof involves ergodic properties of unipotent flows on homogeneous spaces.

有限约化群的特征标重数与Drinfeld对偶

Multiplicity of characters of finite reductive groups and Drinfeld doubles

研究有限约化群\(G^F\)的几乎幂幺特征标和德利涅 - 卢茨蒂格特征标张量积重数，涉及与复不可约特征标环结构关系，还考虑Drinfeld对偶上模的弗罗贝尼乌斯 - 舒尔指标，最后研究相关重数并提出检测问题。

Limiting distributions of curves under geodesic flow on hyperbolic manifolds

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold. Moreover, if this immersed submanifold is a proper subset, then a lift of the curve to the universal co…