Typical representations for level zero Bernstein components of ${\rm GL}_n(F)$

The Bernstein center of the category of smooth $W(k)[GL_n(F)]$-modules

We consider the category of smooth $W(k)[GL_n(F)]$-modules, where F is a p-adic field and k is an algebraically closed field of characteristic l different from p. We describe a factorization of this category into blocks, and show that the center of each block is a reduced, finite type, l-torsion free W(k)-algebra. Moreover, the k-points of the center of each block are in bijection with the possible "supercuspidal supports" of the smooth $k[GL_n(F)]$-modules that lie in the block. Finally, we describe a large, explicit subalgebra of the center of each block and give a description of the action …

$p$-adic 有限简约群的 Hecke 代数与伯恩斯坦块的局部朗兰兹对应关系

Hecke algebras for $p$-adic reductive groups and Local Langlands Correspondence for Bernstein blocks

本文研究了 $p$-adic 有限简约群的 Hecke 代数与伯恩斯坦块的局部朗兰兹对应关系，通过构建增强的 $L$-参数集并应用到各种群上，证明了某些猜想并简化了深度零情况。

On classification of typical representations for ${\rm GL}_3(F)$

Let $F$ be any non-Archimedean local field with residue field of cardinality $q_F$. In this article, we obtain a classification of typical representations for the Bernstein components associated to the inertial classes of the form $[{\rm GL}_n(F)\times F^\times, σ\otimesχ]$ with $q_F>2$, and for the principal series components with $q_F>3$. With this we complete the classification of typical representations for ${\rm GL}_3(F)$, for $q_F>2$.

On semisimple l-modular Bernstein-blocks of a p-adic general linear group

Let $G_n=\operatorname{GL}_n(F)$, where $F$ is a non-archimedean local field with residue characteristic $p$. Our starting point is the Bernstein-decomposition of the representation category of $G_n$ over an algebraically closed field of characteristic $\ell \neq p$ into blocks. In level zero, we associate to each block a replacement for the Iwahori-Hecke algebra which provides a Morita-equivalence just as in the complex case. Additionally, we will explain how this gives rise to a description of an arbitrary $G_n$-block in terms of simple $G_m$-blocks (for $m\leq n$), paralleling the approach …

On irreducible representations of compact $p$-adic analytic groups

We prove that the canonical dimension of a coadmissible representation of a semisimple $p$-adic Lie group in a $p$-adic Banach space is either zero or at least half the dimension of a non-zero coadjoint orbit. To do this we establish analogues for $p$-adically completed enveloping algebras of Bernstein's inequality for modules over Weyl algebras, the Beilinson-Bernstein localisation theorem and Quillen's Lemma about the endomorphism ring of a simple module over an enveloping algebra.

A categorical approach to the stable center conjecture

The stable center conjecture asserts that the space of stable distributions in the Bernstein center of a reductive p-adic is closed under convolution. It is closely related to the notion of an L-packet and endoscopy theory. We describe a categorical approach to the depth zero part of the conjecture. As an illustration of our method, we show that the Bernstein projector to the depth zero spectrum is stable.

通过Hecke代数同构实现驯顺p进群的深度零约化

Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms

该研究针对驯分歧p进连通约化群，证明Kim-Yu型对应的Hecke代数同构于某扭Levi子群的深度零型Hecke代数，在p不整除绝对Weyl群阶的条件下实现所有Bernstein块到深度零块的约化，且同构显式、系数域可推广。

A comparison of endomorphism algebras

Let $F$ be a non-archimedean local field and $G$ be a connected reductive group over $F$. For a Bernstein block in the category of smooth complex representations of $G(F)$, we have two kinds of progenerators: the compactly induced representation $\text{ind}_{K}^{G(F)} (ρ)$ of a type $(K, ρ)$, and the parabolically induced representation $I_{P}^{G}(Π^{M})$ of a progenerator $Π^{M}$ of a Bernstein block for a Levi subgroup $M$ of $G$. In this paper, we construct an explicit isomorphism of these two progenerators. Moreover, we compare the description of the endomorphism algebra $\text{End}_{G(F)}…

整Bernstein中心上的族与GL(n, F)局部基提升的Tate上同调

Families over the integral Bernstein Center and Tate cohomology of local Base change lifts for GL(n, F)

该研究针对p-进域上GL(n,F)的泛光滑整表示的局部基提升，利用Rankin-Selberg卷积与局部γ-因子恒等式，证明其零次Tate上同调的泛子商同构于模l约化的Frobenius扭，去除了先前工作的素数整除性假设。

一个扭曲的海丘代数，过去与现在，以及一个温特表示的克莱因瓶

A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations

本文研究了扭曲海丘代数及其与温特表示的克莱因瓶模型的联系，并展示了伯恩斯坦谱中点的同构性。

关于 $p$-进域 $F$ 上 $\mathrm{GL}_n(F)$ ($n>2$) 的 Hecke 代数的一个问题

A problem on Hecke algebras for $\mathrm{GL}_n(F)$ for $n>2$ over $p$-adic field $F$

研究 $p$-进域上 $\mathrm{GL}_n$ ($n>2$) 的 Hecke 代数，通过 Bernstein 分解和交织代数证明不同域上的 Hecke 代数 Morita 等价。

关于 $\mathrm{Sp}_4(F)$ 的典型表示

Typical representations for $\mathrm{Sp}_4(F)$

针对奇剩余特征非阿基米德局部域上的辛群Sp₄(F)，本文刻画其超特殊极大紧子群的典型不可约表示，证明它们恰为某类s-型诱导表示的不可约分量，该s-型是Levi子群上Bushnell-Kutzko极大简单型的G-覆盖。

GL_N(F)的所有零深度表示的Zelevinsky段与超特殊分支律

Zelevinsky Segments and Hyperspecial Branching Laws for All Depth-Zero Representations of GL_N(F)

该研究针对非阿基米德局部域上GL_N(F)的不可约零深度表示，结合Bushnell-Kutzko型等工具，确定了超特殊极大紧子群幂幺根基固定向量空间的分解，建立了p进与有限侧对偶的交织关系，得到了完整有效的超特殊分支律。

Categorification via blocks of modular representations II

Bernstein, Frenkel and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak{sl}_2$ using singular blocks of category $\mathcal{O}$ for $\mathfrak{sl}_n$. In earlier work, we construct a positive characteristic analogue using blocks of representations of $\mathfrak{sl}_n$ over a field $\textbf{k}$ of characteristic $p > n$, with zero Frobenius character, and singular Harish-Chandra character. In the present paper, we extend these results and construct a categorical $\mathfrak{sl}_k$-action, following Sussan's approach, by considering more …