Jeffrey D. Adler, Joshua M. Lansky
We give an explicit description of L-packets and quadratic base change for depth-zero representations of unramified unitary groups in two and three variables. We show that this base change is compatible with unrefined minimal K-types.
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Let $G$ be an unramified group over a $p$-adic field. This article introduces a base change homomorphism for Bernstein centers of depth-zero principal series blocks for $G$ and proves the corresponding base change fundamental lemma. This result is used in the approach to Shimura varieties with $Γ_1(p)$-level structure initiated by M. Rapoport and the author in [HR2].
Green functions for positive-depth Deligne--Lusztig induction
本文在剩余域充分大条件下显式描述了非分歧椭圆对的正深度 Deligne--Lusztig 诱导，给出 Kaletha 正则 $L$-包的几何实现，并通过比较代数与几何两类 Green 函数证明正深度 Springer 假设。
By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation $π$ of a classical group (which may be not-quasi-split) over a nonarchimedean local field of odd residual characteristic. From this, we can explicitly describe all the irreducible cuspidal representations in the union of one, two, or four L-packets, containing $π$. These results generalize the work of DeBacker-Reeder (in the case of classical groups) from regular to …
In this paper we provide a geometric framework for the study of characters of depth-zero representations of unramified groups over local fields with finite residue fields which is built directly on Lusztig's theory of character sheaves for groups over finite fields and uses ideas due to Schneider-Stuhler. Specifically, we introduce a class of coefficient systems on Bruhat-Tits buildings of perverse sheaves sheaves on affine algebraic groups over an algebraic closure of a finite field and to each supercuspidal depth-zero representation of an unramified $p$-adic group we associate a formal sum o…
We prove the conjectural endoscopic transfer of L-packets for the local Langlands correspondence for pure inner forms of unramified p-adic groups and depth-zero parameters established by DeBacker and Reeder. More precisely, we show that under mild conditions on the residual characteristic, endoscopic induction identifies an unstable character of such an L-packet with the stable character of the corresponding endoscopic L-packet.
Let $G$ be a quasi-split tamely ramified connected reductive group defined over a $p$-adic field $F$. We show that if $-1$ is in the $F$-points of the absolute Weyl group of $G$, then self-dual supercuspidal representations of $G(F)$ exist. Now assume further that $G$ is unramified and that the center of $G$ is connected. Let $π$ be a generic self-dual depth-zero regular supercuspidal representation of $G(F)$. We show that the Frobenius-Schur indicator of $π$ is given by the sign by which a certain distinguished element of the center of $G(F)$ of order two acts on $π$.
Siegel Vectors for Nongeneric Depth Zero Supercuspidals of $GSp(4)$
本文计算了非阿基米德局部域上 $GSp(4)$ 的深度零非一般超尖表示中 Siegel 不变向量空间的维数，覆盖所有级别 $\mathfrak{p}^n$。
Base Change Fundamental Lemma for Central Elements in Depth-zero Hecke Algebras over Local Function Fields
研究局部函数域上非分歧群\(G\)，在正特征下引入抽象范数映射，通过（扭曲）迹公式稳定化过程，证明正则半单元素的基变换基本引理。
A Pinned Local Langlands Correspondence for Depth-Zero Supercuspidal Representations
本文构建了深度零超 cuspidal 表示的固定 pinning 局部 Langlands 对应，通过固定 pinning 分裂的 quasi-split 内形式，得到深度零超 cuspidal 表示与相关 cuspidal 增强深度零 Langlands 参数的双射。
Depth-zero local Langlands via shtukas in mixed characteristic
本文假设混合特征 shtuka 空间的光滑性等条件，确定不可约零深度表示的 Fargues-Scholze L 参数的惯性限制，通过 shtuka 空间上同调的惯性作用证明，并应用于证明两类 L 参数在 \u00af\u211a_\u2113 上一致。
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。
Let $E/F$ be a quadratic extension of non archimedean local fields of odd residual characteristic. We prove a conjecture of Prasad and Takloo-Bighash, in the case of cuspidal representations of depth zero of $\mathrm{GL}(2m,F)$. This conjecture characterizes distinction for the pair $(\mathrm{GL}(2m,F),\mathrm{GL}(m,E))$ with respect to a character $μ\circ \mathrm{det}$ of $\mathrm{GL}(m,E)$, in terms of certain conditions on Langlands paremeters, including an epsilon value. We also compute the multiplicity of the involved equivariant linear forms when $E/F$ is unramified, and also when $μ$ is…
Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$
本文研究p进拟分裂约化群在$\bar Z[1/p]$上的深度0表示范畴，证明其块与驯分歧Langlands参数空间连通分支一一对应，并应用于半单局部Langlands对应的深度0表示性质。
On Tate cohomology and base change of representations of $D^\times$
该研究针对非阿基米德局部域上中心可除代数的乘法群表示，计算其零深度等性质的Tate上同调群，验证了Treumann–Venkatesh关于模ℓ函子性的猜想。
We give an explicit formula for the twisted gamma factor for a pair of irreducible supercuspidal representations of level zero. We also obtain an explicit formula for the unramified base change of level zero supercuspidal representations.
Clozel and Labesse proved the base change fundamental lemma for spherical Hecke algebras attached to an unramified group over a p-adic field. This paper proves an analogous fundamental lemma for centers of parahoric Hecke algebras attached to the same class of groups. This provides an ingredient needed for the author's program to study Shimura varieties with parahoric level structure at p.
Affine Springer fibers and depth zero L-packets
本文通过仿射Springer纤维的同调给出深度零L-包中表示特征标及其内窥组合的几何描述，证明其稳定性与内扭转相容性，建立Lusztig猜想的类比。
On parameters of Hecke algebras for $p$-adic groups
本文证明了深度零类型的仿射 Hecke 代数与幂幺类型的同构，从而可显式计算其参数，并在特定假设下证明了 Lusztig 猜想关于任意 Bernstein 块与幂幺块参数一致的版本。
In this article, we study Prasad's conjecture for regular supercuspidal representations based on the machinery developed by Hakim and Murnaghan to study distinguished representations, and the fundamental work of Kaletha on parameterization of regular supercuspidal representations. For regular supercuspidal representations, we give some new interpretations of the numerical quantities appearing in Prasad's formula, and reduce the proof to the case of tori. The proof of Prasad's conjecture then reduces to a comparison of various quadratic characters appearing naturally in the above process. We al…
Base change fundamental lemma for Bernstein centers of principal series blocks
本文研究了主系列块伯恩斯坦中心的基础变换同态，证明了相关函数的关联性，并为稳定伯恩斯坦中心扭曲摄动转移猜想提供了证据。
We give an explicit description of L-packets and quadratic base change for depth-zero representations of ramified unitary groups in two and three variables. We show that this base change lifting is compatible with a certain lifting of families of representations of finite groups. We conjecture that such a compatibility is valid in much greater generality.
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