Jeffrey D. Adler, Joshua M. Lansky
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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We show that for a wildly ramified torus, depth is not preserved in general under local Langlands correspondence for tori.
In this article, for nilpotent orbits of ramified quasi-split unitary groups with two Jordan blocks, we give closed formulas for their Shalika germs at certain equi-valued elements with half-integral depth previously studied by Hales. These elements are parametrized by hyperelliptic curves defined over the residue field, and the numbers we obtain can be expressed in terms of Frobenius eigenvalues on the $\ell$-adic $H^1$ of the curves, generalizing previous result of Hales on stable subregular Shalika germs. These Shalika germ formulas imply new results on stability and endoscopic transfer of …
We give an explicit construction of test vectors for $T$-equivariant linear functionals on representations $Π$ of $GL_2$ of a $p$-adic field $F$, where $T$ is a non-split torus. Of particular interest is the case when both the representations are ramified; we completely solve this problem for principal series and Steinberg representations of $GL_2$, as well as for depth zero supercuspidals over $\mathbf{Q}_p$. A key ingredient is a theorem of Casselman and Silberger, which allows us to quickly reduce almost all cases to that of the principal series, which can be analyzed directly. Our method s…
We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the underlying stratum of a given supercuspidal representation to be skew maximal simple, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a can…
On depth-zero characters of p-adic groups
本文研究了p-进群的深度零特征，证明了不同定义的等价性，并为局部朗兰兹对应的新情况提供了理论支持。
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。
In this paper we prove an explicit matching theorem for some Hecke elements in the case of (possibly ramified) cyclic base change for general linear groups over local fields of characteristic zero with odd residue characteristic under a mild assumption. A key observation, based on the works of Waldspurger and Ganapathy-Varma, is to regard the base change lifts with twisted endoscopic lifts and replace the condition for the matching orbital integrals with one for semi-simple descend in the twisted space according to Waldspurger's fundamental work "L'endoscopie tordue n'est pas si tordue".
Newforms in Cuspidal Representations
本文将 $p$-adic 群尖点表示的 newform 理论推广至 $\ell$-modular 表示，证明了导子与模 $\ell$ 同余的相容性，并在复数与 modular 设定下给出了深度零及 minimax 尖点表示的显式公式。
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 $π$.
The formal degree conjecture and the root number conjecture are verified with respect to supercuspidal representations of $Sp_{2n}(F)$ and $L$-parameters associated with tamely ramified extension $K/F$ of degree $2n$. The supercuspidal representation is constructed as a compact induction from an irreducible unitary representation of the hyper special compact group $Sp_{2n}(O_F)$, which is explicitly constructed, based upon the general theory developed by the author, by $K$ and certain character $θ$ of the multiplicative group $K^{\times}$. $L$-parameter is constructed by the data $\{K,θ\}$ by …
Based upon the general theory, developed by the author, on the parametrization of the irreducible representations of the hyper special compact groups corresponding to the regular adjoint orbit, supercuspidal representations of $SL_n(F)$ are explicitly constructed for which the formal degree conjecture and the root number conjecture are verified with respect to certain $L$-parameter defined, by means of Kaletha, that is, the local Langlands correspondence of tori and the Langlands-Schelstad procedure, by the data parametrizing the irreducible representations of the hyper special compact subgrou…
Modular functoriality for finite groups
本文推广Deligne--Lusztig理论至paraductive群，计算其Tate上同调下界，并应用模函子性计算非奇异深度0尖点表示的L-参数。
Characteristic-free approaches around Yu's construction
本文提出无特征构造扭曲Heisenberg-Weil表示，扩展Yu构造至剩余特征2，并给出几何实现及正深度Deligne-Lusztig诱导的显式描述。
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。
Ramified Quivers with Potentials, Monodromy Characters, and Cluster Bases I
本文为斜对称化簇代数构造范畴模型与簇特征，证明良好性等价于非退化可实现性，并利用三角扩张与突变得到良好实现，进而定义满足突变的$F$-多项式并构造簇基。
Langlands parameters for Moy-Prasad types
本文为温和分歧的p-adic约化群构造了深度为r的不可约光滑表示与惯性子群上编号过滤子群到对偶群的连续同态的共轭类之间的部分局部Langlands对应。
Given a finite, flat and finitely presented group scheme $G$ over some base $S$, we introduce the notion of ramified $G$-covers and study the moduli stack $G$-Cov they form. The thesis is divided in three parts. The first one concerns the case when $G$ is a diagonalizable group scheme and it essentially coincides with arxiv:1106.2347. In the second part I deal with the general case. Assuming that the base S is affine and given an $S$-scheme $T$, I interpret $G$-covers of $T$ as particolar (lax) monoidal functors from the category of finite, $G$-equivariant locally free sheaves over $S$ to the …
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。
Parametrization and reduction to depth zero of $\overline{\mathbb{Z}}[\frac{1}{p}]$-blocks of tame $p$-adic groups
研究通过参数化和降维到深度零的方法，揭示了 tame $p$-adic 群的 $\overline{\mathbb{Z}}[\frac{1}{p}]$-块结构与局部朗兰兹对应的关系。
A parametrization of irreducible representations associated with a regular adjoint orbit of a classical group over finite quotient rings of the ring of integer of a non-dyadic non-archimedean local field is presented. The parametrization is given by means of (a subset of) the character group of the centralizer of a representative of the regular adjoint orbit. Our method is based upon Weil representations over finite fields. More explicit parametrization in terms of tamely ramified extensions of the base field is given for the general linear group, the special linear group, the symplectic group…
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.
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].
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