On the signature character of representations of p-adic general linear groups

Unipotent almost characters of simple p-adic groups

Let G be a simple adjoint group and let K=k((ε)) where k is an algebraic closure of a finite field F_q. In this paper we define some geometric objects on G(K) which are similar to the (cohomology sheaves of) the unipotent character sheaves on G(k). Using these geometric objects we define the unipotent almost characters of G(K_0) where K_0=F_q((ε)) and state some conjectures relating them to the characters of unipotent representations of G(K_0).

Supercuspidal characters of reductive p-adic groups

We compute the characters of many supercuspidal representations of reductive p-adic groups. Specifically, we deal with representations that arise via Yu's construction from data satisfying a certain compactness condition. Each character is expressed in terms of a depth-zero character of a smaller group, the (linear) characters appearing in Yu's construction, Fourier transforms of orbital integrals, and certain signs and cardinalities that are described explicitly in terms of the datum associated to the representation and of the element at which the character is evaluated.

Supercuspidal characters of $\operatorname{SL}_2$ over a $p$-adic field

The character formulas of Sally and Shalika are an early triumph in $p$-adic harmonic analysis, but, to date, the calculations underlying the formulas have not been available. In this paper, which should be viewed as a precursor of the forthcoming volume by the authors and Alan Roche, we leverage modern technology (for example, the Moy-Prasad theory) to compute explicit character tables. An interesting highlight is the computation of the 'exceptional' supercuspidal characters, i.e., those depth-zero representations not arising by inflation-induction from a Deligne-Lusztig representation of fin…

Picard groups in $p$-adic Fourier theory

Let $L$ be a proper finite extension of the field of $p$-adic numbers and let $o\subset L$ be its integers, viewed as an abelian locally $L$-analytic group. Let $\hat{o}$ be the rigid analytic group variety parametrizing the locally analytic characters of o. We study the Picard group and the Grothendieck group of this variety. This preprint is in final version, replaces the former preprint 'Vector bundles over analytic character varieties' and appears in Manuscripta Mathematica.

Branching rules for ramified principal series representations of GL(3) over a p-adic field

We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup $K=\mathrm{GL}(3,\mathscr{R})$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples.

Modulo $\ell$-representations of $p$-adic groups $\mathrm{SL_n}(F)$

Let $k$ be an algebraically closed field of characteristic $l\neq p$. We construct maximal simple cuspidal $k$-types of Levi subgroups $\mathrm{M}'$ of $\mathrm{SL}_n(F)$, when $F$ is a non-archimedean locally compact field of residual characteristic $p$. We show that the supercuspidal support of irreducible smooth $k$-representations of Levi subgroups $\mathrm{M}'$ of $\mathrm{SL}_n(F)$ is unique up to $\mathrm{M}'$-conjugation, when $F$ is either a finite field of characteristic $p$ or a non-archimedean locally compact field of residual characteristic $p$.

On the Steinberg character of a semisimple p-adic group

Let G be a semisimple almost simple algebraic group defined and split over a nonarchimedean local field K and let S be the Steinberg representation of G(K). Let t be be a very regular semisimple element of G(K). In this paper we give a simple formula (not as an alternating sum) for the value of the character of S at t; we show that this value is plus or minus an integer power of q, the cardinal of the residue field of K. We also give an explicit formula for the character at a split very regular element of any irreducible admissible representation of G(K) with nonzero vectors invariant under an…

Characters of locally analytic representations of p-adic reductive groups

We propose a definition of characters in the context of Schneider-Teitelbaum's theory of locally analytic representations of p-adic reductive groups. This character will be a function on a compact subgroup of a maximal torus of the reductive group in question. As an example we treat the locally analytic principal series of SL(2,Q_p).

Cuspidal representations of reductive p-adic groups are relatively injective and projective

Cuspidal representations of a reductive p-adic group G over a field of characteristic different from p are relatively injective and projective with respect to extensions that split by a U-equivariant linear map for any subgroup U that is compact modulo the centre. The category of smooth representations over a field whose characteristic does not divide the pro-order of G is the product of the subcategories of cuspidal representations and of subrepresentations of direct sums of parabolically induced representations.

Good product expansions for tame elements of reductive p-adic groups

We show that, under fairly general conditions, many elements of a p-adic group can be well approximated by a product whose factors have properties that are helpful in performing explicit character computations.

On unitary unipotent representations of $p$-adic groups and affine Hecke algebras with unequal parameters

We determine the unitary dual of the geometric graded Hecke algebras with {unequal} parameters which appear in Lusztig's classification of unipotent representations for {exceptional} $p$-adic groups. The largest such algebra is of type $F_4.$ Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent representations with real central character of the $p$-adic groups. In order for this correspondence to be applicable here, we show (following Lusztig's geometric classification, and Barbasch and Moy's o…

从固定点子群提升 $p$-adic 类型的半单特征

Lifting semisimple characters of $p$-adic types from fixed-point subgroups

本文研究了从固定点子群提升 $p$-adic 类型的半单特征，并通过提升 Kim-Yu 数据来实现这一过程。

Local Summability of Characters on $p$-adic Reductive Groups

In this paper we study the complex representations of reductive groups over local non-Archimedean fields. We use the building of the reductive group to give upper-bounds for the absolute value of the character of an admissible representation and for the Weyl integration formula for certain regular elements. The upper-bound for the character of a representation is based on the alternative description, depending on the building, of the character as given by R. Meyer and M. Solleveld [MS12]. Once the character and the Weyl integration formula are related to the building, the upper-bounds will fol…

On supersingular representations of $GL_2(D)$ with a division algebra $D$ over a $p$-adic field

We investigate the mod-$p$ supersingular representations of $GL_2(D)$, where $D$ is a division algebra over a $p$-adic field with characteristic 0, by computing a basis for the vector space of the pro-$p$ Iwahori subgroup invariants of a certain quotient of a compact induction. This work generalizes the results of Hendel and Schein.

Geometrization of continuous characters of $\mathbb{Z}_p^\times$

We define the $p$-adic trace of certain rank-one local systems on the multiplicative group over $p$-adic numbers, using Sekiguchi and Suwa's unification of Kummer and Artin-Schrier-Witt theories. Our main observation is that, for every non-negative integer $n$, the $p$-adic trace defines an isomorphism of abelian groups between local systems whose order divides $(p-1)p^n$ and $\ell$-adic characters of the multiplicative group of $p$-adic integers of depth less than or equal to $n$.

Asymptotics and local constancy of characters of p-adic groups

In this paper we study quantitative aspects of trace characters $Θ_π$ of reductive $p$-adic groups when the representation $π$ varies. Our approach is based on the local constancy of characters and we survey some other related results. We formulate a conjecture on the behavior of $Θ_π$ relative to the formal degree of $π$, which we are able to prove in the case where $π$ is a tame supercuspidal. The proof builds on J.-K.~Yu's construction and the structure of Moy-Prasad subgroups.

$p$-adic Banach表示的Gelfand-Kirillov界：具有$\text{GL}_2$和四元单位的微小特征

Gelfand-Kirillov bound for $p$-adic Banach representations with infinitesimal character for $\text{GL}_2$ and quaternion units

本文研究了$p$-adic Banach表示的Gelfand-Kirillov界，针对$\text{GL}_2$和四元单位群，证明了具有微小特征的表示维度上限。

p-进群的深度零特征

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。

p-adic 约化群的 mod-$\ell$ 表示的局部特征标

Local characters of mod-$\ell$ representations of a $p$-adic reductive group

本文为 p-adic 约化群的 mod-ℓ 表示定义了提升特征标，推广了 Howe、Harish-Chandra 和 DeBacker 的局部展开结果，并证明 Moeglin-Waldspurger 和 Varma 的退化 Whittaker 模型结论仍成立。

Mod $p$ Bernstein centres of $p$-adic groups

We prove that the centre of the category of smooth mod $p$ representations with fixed central character of a split semisimple $p$-adic group is a local ring.

On p-adic Speh representations

This note contains new short proofs of known results on Speh representations of inner forms of $GL_n$ over a local non archimedean field of any characteristic.

Genericity of supercuspidal representations of p-adic Sp(4)

We describe the supercuspidal representations of Sp(4,F), for F a non-archimedean local field of residual characteristic different from 2, and determine which are generic.

关于 $p$-adic 红uctive 组局部可积性结果的研究

On local integrability results for $p$-adic reductive groups

本文基于局部特征展开法，给出了哈里-查纳关于 $p$-adic 红uctive 组复特征局部可积性的著名定理的简短证明，展示了适用于某些非复系数的代数情形，并验证了文献未覆盖的局部可积性情况。

Depth Zero Representations of Nonlinear Covers of $p$-adic Groups

We generalize the methods of Moy-Prasad, in order to define and study the genuine depth zero representations of some nonlinear covers of reductive groups over $p$-adic local fields. In particular, we construct all depth zero supercuspidal representations of the metaplectic group $Mp_{2n}$ over a $p$-adic field of odd residue characteristic.

The p-adic local Langlands correspondence for GL_2(Q_p)

The p-adic local Langlands correspondence for GL_2(Q_p) is given by an exact functor from unitary Banach representations of GL_2(Q_p) to representations of the absolute Galois group G_{Q_p} of Q_p. We prove, using characteristic 0 methods, that this correspondence induces a bijection between absolutely irreducible non-ordinary representations of GL_2(Q_p) and absolutely irreducible 2-dimensional representations of G_{Q_p}. This had already been proved, by characteristic p methods, but only for p\geq 5.

Infinite-dimensional $p$-adic groups, semigroups of double cosets, and inner functions on Bruhat--Tits builldings

We construct $p$-adic analogs of operator colligations and their characteristic functions. Consider a $p$-adic group $G=GL(α+k\infty, Q_p)$, its subgroup $L=O(k\infty,Z_p)$, and the subgroup $K=O(\infty,Z_p)$ embedded to $L$ diagonally. We show that double cosets $Γ= K\setminus G/K$ admit a structure of a semigroup, $Γ$ acts naturally in $K$-fixed vectors of unitary representations of $G$. For each double coset we assign a 'characteristic function', which sends a certain Bruhat--Tits building to another building (buildings are finite-dimensional); image of the distinguished boundary is contain…

关于GL_n的p进Deligne-Lusztig簇的模ℓ上同调

On modulo $\ell$ cohomology of $p$-adic Deligne-Lusztig varieties for $GL_n$

本文针对特征为正且ℓ≠p的系数情形，首次研究GL_n的p进Deligne-Lusztig簇的模ℓ上同调，将其与Vignéras模局部朗兰兹对应建立关联，推进p进李群表示论相关研究。

Blocks with quaternion defect group over a 2-adic ring: the case \tilde{A}_4

Except for blocks with a cyclic or Klein four defect group, it is not known in general whether the Morita equivalence class of a block algebra over a field of prime characteristic determines that of the corresponding block algebra over a p-adic ring. We prove this to be the case when the defect group is quaternion of order 8 and the block algebra over an algebraically closed field k of characteristic 2 is Morita equivalent to $k\tilde A_4$. The main ingredients are Erdmann's classification of tame blocks and work of Cabanes and Picaronny on perfect isometries between tame blocks.