François Rodier
We correct two errors in the paper "Sur les représentations non ramifiées des groupes réductifs p-adiques; l'exemple de GSp(4)": the first in the study of an involution on the irreducible unramified representations of a semi-simple group, the second in the description of representations of the group GSp(4).
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Consider the Grothendieck group of finite type projective modular representations of the symmetric groups on n letters, or more generally, of its wreath product with a finite group. They form a graded group, with a product defined using induction. We show that the resulting graded ring is a polynomial ring.
Let G be a special orthogonal group SO(2n+1) defined over a p-adic field F. Let $π$ be an admissible irreducible representation of G(F) which is tempered and of unipotent reduction. We prove that $π$ has a wave front set. In some particular cases, for instance if $π$ is of the discrete series, we give a method to compute this wave front set.
Sur la structure des repr{é}sentations g{é}n{é}riques des groupes lin{é}aires infinis
本文研究从小加性范畴到模范畴的函子范畴结构，证明在温和假设下有限生成函子可由多项式函子构建或通过模余有限理想约化分解，并推得此类函子是noetherian的且具有有限生成射影分解。
We find the irreducible decomposition of the Weil representation of the unitary group $\mathrm{U}_{2n}(A)$, where $A$ is a ramified quadratic extension of a finite, commutative, local, principal ideal ring $R$ and the nilpotency degree of the maximal ideal of $A$ is odd. We show in particular that this Weil representation is multiplicity free. Restriction to the special unitary group $\mathrm{SU}_{2n}(A)$ preserves irreducibility and multiplicity freeness provided $n>1$.
We find all irreducible constituents of the Weil representation of a unitary group $U_m(A)$ of rank $m$ associated to a ramified quadratic extension $A$ of a finite, commutative, local and principal ring $R$ of odd characteristic. We show that this Weil representation is multiplicity free with monomial irreducible constituents. We also find the number of these constituents and describe them in terms of Clifford theory with respect to a congruence subgroup. We find all character degrees in the special case when $R$ is a field.
On unipotent representations of ramified $p$-adic groups
本文分析了分歧 $p$-进群的单幂表示，为其建立了满足 Borel 期望及形式次数猜想的局部 Langlands 对应，推广了前人工作至非分歧分裂约化群，并用刚性内扭曲解释结果。
We will give an explicit construction of irreducible suparcuspidal representations of the special linear group over a non-archimedean local field and will speculate its Langlands parameter by means of verifying the Hiraga-Ichino-Ikeda formula of the formal degree of the supercuspidal representations.
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…
Let F be a totally real number field, n a prime integer, and G a unitary group of rank n defined over F that is compact at every infinite place. We prove an asymptotic formula for the number of cuspidal automorphic representations of G whose factors at finitely many places are prescribed up to inertia. The results and the methods used are a generalization to this setting of those used by Weinstein for GL_2.
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.
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 …
Spherical representations of unitary groups at ramified places and the arithmetic inner product formula
本文研究了局部域上偶单位群的可允许表示，其中二次扩张是分歧的，并考虑了在unimodular晶格稳定子作用下不变的向量以及相应单位Shimura品种积分模型的一些性质。作为直接应用，本文改进了算术内积公式，允许局部根数为(-1)的处为分歧。
On a tamely ramified local relative Langlands conjecture via categorical representations
该研究在非分歧相对局部朗兰兹猜想等假设成立的前提下，用相对朗兰兹对偶给出环路空间上伊瓦霍里等变D-模满子范畴的谱描述，证实了驯分歧局部相对朗兰兹猜想的一个轻微变体。
Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with Weil group $\Cal W_F$. Let $σ$ be an irreducible smooth complex representation of $\Cal W_F$, realized as the Langlands parameter of an irreducible cuspidal representation $π$ of a general linear group over $F$. In an earlier paper, we showed that the ramification structure of $σ$ is determined by the fine structure of the endo-class $\varTheta$ of the simple character contained in $π$, in the sense of Bushnell-Kutzko. The connection is made via the {\it Herbrand function} $Ψ_\varTheta$ of $\varTheta$. In th…
Let $F$ be a non-Archimedean locally compact field. We show that the local Langlands correspondence over $F$ has a strong property generalizing the higher ramification theorem of local class field theory. If $π$ is an irreducible cuspidal representation of a general linear group $GL_n(F)$ and $σ$ the corresponding irreducible representation of the Weil group $W_F$ of $F$, the restriction of $σ$ to a ramification subgroup of $W_F$ is determined by a truncation of the simple character $θ_π$ contained in $π$, and conversely. Numerical aspects of the relation are governed by a Herbrand-like functi…
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。
We describe the image, under the local Langlands correspondence for tori, of the characters of a torus which are trivial on its Iwahori subgroup. Let $k$ be a non-archimedian local field. Let $\boldsymbol{G}$ be a connected reductive group defined over $k$, which is quasi-split and split over a tamely ramified extension. Let $K$ be a special maximal parahoric subgroup of $\boldsymbol{G}(k)$. To the class of representations of $\boldsymbol{G}(k)$, having a non-zero vector fixed under $K$, we establish a bijection, in a natural way, with the twisted semi-simple conjugacy classes of the inertia f…
In this note, we introduce the notion of almost unramified representations of quasi-split unitary groups of even ranks with respect to an unramified quadratic extension of local fields, and study their behavior under the local theta correspondence. These representations are in some sense the least ramified representations that have root number $-1$.
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