Euler characteristics and compact p-adic Lie groups

Homological multiplicities in representation theory of $p$-adic groups

We study homological multiplicities of spherical varieties of reductive group $G$ over a $p$-adic field $F$. Based on Bernstein's decomposition of the category of smooth representations of a $p$-adic group, we introduce a sheaf that measures these multiplicities. We show that these multiplicities are finite whenever the usual mutliplicities are finite, in particular this holds for symmetric varieties, conjectured for all spherical varieties and known for a large class of spherical varieties. Furthermore, we show that the Euler-Poincaré characteristic is constant in families induced from admiss…

Fourier transforms of orbital integrals on the Lie algebra of $\operatorname{SL}_2$

The Harish-Chandra--Howe local character expansion expresses the characters of reductive, $p$-adic groups in terms of Fourier transforms of nilpotent orbital integrals on their Lie algebras, and Murnaghan--Kirillov theory expresses many characters of reductive, $p$-adic groups in terms of Fourier transforms of semisimple orbital integrals (also on their Lie algebras). In many cases, the evaluation of these Fourier transforms seems intractable; but, for $\operatorname{SL}_2$, the nilpotent orbital integrals have already been computed. In this paper, we use a variant of Huntsinger's integral for…

Euler's difference table and decomposition of tensor powers of adjoint representation of $A_n$ Lie algebra

By using of Euler's difference table, we obtain simple explicit formula for the decomposition of $k$-th tensor power of adjoint representation of $A_n$ Lie algebra at $2 k \le{n+1}$.

Verma modules over p-adic Arens-Michael envelopes of reductive Lie algebras

Let K be a locally compact nonarchimedean field, g a split reductive Lie algebra over K and U(g) its universal enveloping algebra. We study the category C_g of coadmissible modules over the nonarchimedean Arens-Michael envelope of U(g). Let p be a parabolic subalgebra of g. The main result identifies a certain explicitly given highest weight category inside C_g with the classical parabolic BGG category of g relative to p. This paper is in final form, replaces and expands the former preprint 'BGG reciprocity for p-adic Arens-Michael envelopes of semisimple Lie algebras' and appears in Journal o…

$p$-adic 李代数的波前集

Wave-front sets for $p$-adic Lie algebras

本文研究 $p$-adic 约化李代数中元素的波前集，通过改编 Waldspurger 算法得到归纳计算方法，用于计算正则超尖点表示的波前集，并发现其不限于单一几何轨道的例子。

$L^r$-乘子在紧 $p$-进李群上的研究

$L^r$-Multipliers on compact $p$-adic Lie groups

本文研究了紧 $p$-进李群上 $L^r$ 乘子的理论，基于鲁扎恩基-图伦恩差异算子和萨洛夫-科斯特条件，推导了不变算子的乘子定理，并应用该理论证明了小波分解和有界函数的 $L^r$ 有界性。

Arithmetic $\D$-modules and Representations

We propose in this paper an approach to Breuil's conjecture on a Langlands correspondence between $p$-adic Galois representations and representations of $p$-adic Lie groups in $p$-adic topological vector spaces. We suggest that Berthelot's theory of arithmetic $D$-modules should give a $p$-adic analogue of Kashiwara's theory of $D$-modules for real Lie groups i.e. it should give a realization of the $p$-adic representations of a $p$-adic Lie group as spaces of overconvergent solutions of arithmetic $D$-modules which will come equipped with an action of the Galois group. We shall discuss the ca…

$\ell$-adic局部系统在$\mathcal{A}_n$上的欧拉特征

The Euler characteristic of $\ell$-adic local systems on $\mathcal{A}_n$

研究$\ell$-adic局部系统在$\mathcal{A}_n$上的欧拉特征，证明了与自守表示相关的$\mathrm{GSpin}$-valued伽罗瓦表示的存在性，并给出了紧支撑欧拉特征的显式公式。

p进李群的可饱和性

On the saturability of $p$-adic Lie groups

本文建立拉扎德理论与莫伊-普拉萨德理论的联系，在p>2且极大环面满足温和假设下，证明G的半单布吕阿-蒂茨建筑中任意点x对应的G_{x,1/(p-1)+}上存在饱和p赋值，并计算相关拉扎德李代数以重新解释莫伊-普拉萨德同构。

通过 almost mathematics 研究 $p$-进 Banach 代数的 Auslander 正则性

Auslander regularity of $p$-adic Banach algebras via almost mathematics

本文提出 Auslander 正则性的“almost”版本，并借此证明了 $p$-进局部解析表示理论中自然出现的非离散赋值域上多种 Banach 代数（如完备 Weyl 代数、完备包络代数和分布代数 $D(G, K)$ 的 Banach 完备化）的 Auslander 正则性。