Mellin transforms of p-adic Whittaker functions

关于 $p$-adic 不完整 Mellin 变换和 $p$-adic 不完整 gamma 函数

On $p$-adic incomplete Mellin transforms and $p$-adic incomplete gamma-functions

本文提出了一种在更弱条件下构造 $p$-adic 不完整 gamma 函数的方法，利用 $p$-adic 积分变换，适用于所有除了有限多个素数外的素数。

Modular shadows and the Levy-Mellin infinity-adic transform

This paper continues the study of the structures induced on the ``invisible boundary'' of the modular tower and extends some results of math.NT/0102006. We start with a systematic formalism of pseudo-measures generalizing the well-known theory of modular symbols for SL(2). These pseudo-measures, and the related integral formula which we call the Levy-Mellin transform, can be considered as an ``infinity-adic'' version of Mazur's p-adic measures introduced in the seventies in the theory of p-adic interpolation of Mellin transforms of cusp forms. A formalism of iterated Levy-Mellin transform in t…

p-进域上$GL(n)$的Steinberg表示的Whittaker函数

Whittaker functions for Steinberg representations of $GL(n)$ over a $p$-adic field

本文通过确定Iwahori Hecke代数特征标在特定元素上的取值，完整计算了p-进域上$GL(n)$的广义Steinberg表示的Whittaker函数，并证明该表示限制到$SL_n(F)$上仍不可约，推广了Baruch-Purkait关于$GL_2(F)$的结果。

The Whittaker-Shintani functions for symplectic groups

In this note, we give a formula for the Whittaker-Shintani functions for the p-adic symplectic groups, which is a generalization of the Zonal spherical functions and Whittaker functions. We then use the formula to give an alternative proof of a conjecture given by T.Shintani on the unramified calculation of L-function for $Sp_{2n}\times GL_{1}$.

Estimates on Eisenstein distributions for reciprocals of p-adic L-functions: the case of irregular primes

We consider the p-adic distributions derived from Eisenstein series in [GMPS], whose Mellin transforms are reciprocals of the Kubota-Leopoldt p-adic L-function. These distributions were shown there to be measures when p is regular. They fail to be measures when p is irregular; in this paper we give quantitative estimates that describe their behavior more precisely.

A p-adic integral for the reciprocal of L-functions

We introduce an analog of part of the Langlands-Shahidi method to the p-adic setting, constructing reciprocals of certain p-adic L-functions using the nonconstant terms of the Fourier expansions of Eisenstein series. We carry out the method for the group SL(2), and give explicit p-adic measures whose Mellin transforms are reciprocals of Dirichlet L-functions.

Metaplectic Demazure operators and Whittaker functions

In a previous paper the first two named authors defined an action of a Weyl group on rational functions and used it to construct multiple Dirichlet series. These series are related to Whittaker functions on an n-fold metaplectic cover of a reductive group. In this paper, we define metaplectic analogues of the Demazure and Demazure-Lusztig operators. We show how these operators, together with results of McNamara, can be used to compute Whittaker functions on metaplectic groups over p-adic fields.

Whittaker functions and Demazure characters

In this paper, we consider how to express an Iwahori--Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman--Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a p-adic group; this generalizes a result of Bump--Nakasuji.

A precise description of the p-adic valuation of the number of alternating sign matrices

Following Sun and Moll, we study v_p(T(N)), the p-adic valuation of the counting function of the alternating sign matrices. We find an exact analytic expression for it that exhibits the fluctuating behaviour, by means of Fourier coefficients. The method is the Mellin-Perron technique, which is familiar in the analysis of the sum-of-digits function and related quantities.

On the transfer congruence between $p$-adic Hecke $L$-functions

We prove the transfer congruence between $p$-adic Hecke $L$-functions for CM fields over cyclotomic extensions, which is a non-abelian generalization of the Kummer's congruence. The ingredients of the proof include the comparison between Hilbert modular varieties, the $q$-expansion principle, and some modification of Hsieh's Whittaker model for Katz' Eisenstein series. As a first application, we prove explicit congruence between special values of Hasse-Weil $L$-function of a CM elliptic curve twisted by Artin representations. As a second application, we prove the existence of a non-commutative…

$p$-进ABV-包的Whittaker归一化与 tempered 表示的Vogan猜想

Whittaker normalization of $p$-adic ABV-packets and Vogan's conjecture for tempered representations

本研究证明$p$-进群ABV-包不依赖Whittaker基准选取，得出其到基本群映射的依赖关系，提出并证明拟分裂典型群ABV-包通有性猜想，验证开参数ABV-包为L-包，最终证明tempered表示的Vogan猜想。

A note on regularized Bernoulli distributions and p-adic Dirichlet expansions

We consider Bernoulli distributions and their regularizations, which are measures on the $p$-adic integers $\mathbb{Z}_p$. It is well known that their Mellin transform can be used to define $p$-adic $L$-functions. We show that for $p>2$ one of the regularized Bernoulli distributions is particularly simple and equal to a measure on $\mathbb{Z}_p$ that takes the values $\pm \frac{1}{2}$ on clopen balls. We apply this to $p$-adic $L$-functions for Dirichlet characters of $p$-power conductor and obtain Dirichlet series expansions similar to the complex case. Such expansions were studied by D. Delb…

Motivic Mellin变换

Motivic Mellin transforms

本文将Mellin变换引入motivic积分领域，构建了在Mellin和Fourier变换下稳定的新函数类，提供Fubini定理与变量替换公式，并推广了前人理论与转移原理。

Symmetric genuine Spherical Whittaker functions on the metaplectic double cover of GSp(2n,F)

Let F be a p-adic field of odd residual characteristic. Let G(n) and G`(n) be the metaplectic double covers of the general symplectic group and the symplectic group attached to the 2n dimensional symplectic space over F. Let T be a genuine, possibly reducible, unramified principal series representation of G(n). In these notes we give an explicit formulas for a spanning set for the space of Spherical Whittaker functions attached to T. For odd n, and generically for even n, this spanning set is a basis. The signicant property of this set is that each of its elements is unchanged under the action…

酉群上 Fourier-Jacobi 模型的 Whittaker-Shintani 函数

Whittaker-Shintani functions for Fourier-Jacobi models on unitary groups

本文陈述并证明了 p 进酉群和一般线性群上 Fourier-Jacobi 模型的 Whittaker-Shintani 函数公式，并以此证明了非分歧的 Ichino-Ikeda 猜想。

Archimedean zeta integrals on $GL_n \times GL_m$ and $SO_{2n+1} \times GL_m$

In this paper, we evaluate archimedean zeta integrals for automorphic $L$-functions on $GL_n \times GL_{n-1+\ell}$ and on $ SO_{2n+1} \times GL_{n+\ell}$, for $\ell=-1$, $0$, and $1$. In each of these cases, the zeta integrals in question may be expressed as Mellin transforms of products of class one Whittaker functions. Here, we obtain explicit expressions for these Mellin transforms in terms of Gamma functions and Barnes integrals.

Metaplectic Covers of $p$-adic Groups and Quantum Groups at Roots of Unity

We describe the structure of the Whittaker or Gelfand-Graev module on a $n$-fold metaplectic cover of a $p$-adic group $G$ at both the Iwahori and spherical level. We express our answer in terms of the representation theory of a quantum group at a root of unity attached to the Langlands dual group of $G$. To do so, we introduce an algebro-combinatorial model for these modules and develop for them a Kazhdan-Lusztig theory involving new generic parameters. These parameters can either be specialized to Gauss sums to recover the $p$-adic theory or to the natural grading parameter in the representa…

Rankin-Selberg local factors modulo $\ell$

After extending the theory of Rankin-Selberg local factors to pairs of $\ell$-modular representations of Whittaker type, of general linear groups over a non-archimedean local field, we study the reduction modulo $\ell$ of $\ell$-adic local factors and their relation to these $\ell$-modular local factors. While the $\ell$-modular local $γ$-factor we associate to such a pair turns out to always coincide with the reduction modulo $\ell$ of the $\ell$-adic $γ$-factor of any Whittaker lifts of this pair, the local $L$-factor exhibits a more interesting behaviour; always dividing the reduction modul…

局部新形式、Paškūnas-Stevens Whittaker函数以及$\mathrm{GL}(n)\times \mathrm{GL}(n)$的显式测试向量

Local newforms, Paškūnas-Stevens Whittaker functions, and explicit test vectors for $\mathrm{GL}(n)\times \mathrm{GL}(n)$

本文为p-adic GL(n)的尖点表示建立Paškūnas-Stevens Whittaker函数与Whittaker新形式间的双向积分转换公式，并据此给出GL(n)×GL(n)对在Rankin-Selberg L-因子非恒为1时的显式测试向量。

Irrationality of special values of formal Laurent series represented by the formal Mellin transform of $G$-functions

Let $p$ be a prime number and $\mathbb{C}_p$ the completion of algebraic closure of $\mathbb{Q}_p$. Let $K$ be an algebraic number field. We fix an embedding $ι_p:\overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ and denote $K_p$ the completion of $K$ with respect to the embedding $ι_p$. Let $g(z)\in K[[z]]$ and denote by $\mathcal{M}(g)(z)\in \tfrac{1}{z}K\left[\left[\tfrac{1}{z}\right]\right]$ the formal Mellin transform of $g(z)$. In this article, we prove that if $\mathcal{M}(g)(z)$ has a good Padé approximation, the special values $\mathcal{M}(g)(α)$ are convergent in $K_p$ and irrationa…

Small automorphic representations and degenerate Whittaker vectors

We investigate Fourier coefficients of automorphic forms on split simply-laced Lie groups G. We show that for automorphic representations of small Gelfand-Kirillov dimension the Fourier coefficients are completely determined by certain degenerate Whittaker vectors on G. Although we expect our results to hold for arbitrary simply-laced groups, we give complete proofs only for G=SL(3) and G=SL(4). This is based on a method of Ginzburg that associates Fourier coefficients of automorphic forms with nilpotent orbits of G. Our results complement and extend recent results of Miller and Sahi. We also …

Whittaker 格的 Tate 上同调与 ${\rm GL}_n$ 一般表示的基变换

Tate cohomology of Whittaker lattices and base change of generic representations of ${\rm GL}_n$

本文研究 $p$-进域 $l$ 次 Galois 扩张中 ${\rm GL}_n$ 一般模 $l$ 表示的基变换，利用整 Whittaker 格的 Tate 上同调证明底域表示的 Frobenius 扭作为子商出现。

Measuring the Space of Metaplectic Whittaker Functions

Whittaker functions are special functions that arise in $p$-adic number theory and representation theory. They may be defined on representations of reductive groups as well as their metaplectic covering groups: fascinatingly, many of their number theoretic applications survive the transition between the reductive and metaplectic cases. However, one notable difference is that the space of Whittaker functions on a reductive group over a nonarchimedean local field $F$ is one-dimensional, whereas this is no longer true in the metaplectic case. In a previous paper, the second author showed that the…

Test vectors for Rankin-Selberg $L$-functions

We study the local zeta integrals attached to a pair of generic representations $(π,τ)$ of $GL_n\times GL_m$, $n>m$, over a $p$-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of $π$ and $τ$. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin-Selberg (local) $L$-function.

Iwahori-metaplectic duality

We construct a family of solvable lattice models whose partition functions include $p$-adic Whittaker functions for general linear groups from two very different sources: from Iwahori-fixed vectors and from metaplectic covers. Interpolating between them by Drinfeld twisting, we uncover unexpected relationships between Iwahori and metaplectic Whittaker functions. This leads to new Demazure operator recurrence relations for spherical metaplectic Whittaker functions. In prior work of the authors it was shown that the row transfer matrices of certain lattice models for spherical metaplectic Whitta…

The local period integrals and essential vectors

By applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of ${\rm GL}_n$ over $p$-adic fields. In each case, we show that the integrals achieve local formal $L$-functions defined by Langlands parameters, when the test vector is associated to the newform. We give the relation between local periods involving essential Whittaker functions and special values of formal $L$-factors at $s=1$ for certain distinguished or unitary representations. The period integrals are also…

Identities involving q-Genocchi numbers and polynomials

In this paper, we focus on the q-Genocchi numbers and polynomials. We shall introduce new identities of the q-Genocchi numbers and polynomials by using the fermionic p-adic integral on Zp which are very important in the study of Frobenius-Genocchi numbers and polynomials. Also, we give Cauchy-integral formula for the q-Genocchi polynomials and moreover by using measure theory on p-adic integral we derive the distribution formula q-Genocchi polynomials. Finally, we present a new definition of q-Zeta-type function by using Mellin transformation which is the interpolation function of the q-Genocc…

Twisted p-adic (h,q)-L-functions

By using q-Volkenborn integral on Z_{p}, we (simsek, simsekCanada) constructed new generating functions of the (h,q)-Bernoulli polynomials and numbers. By applying the Mellin transformation to the generating functions, we constructed integral representation of the twisted (h,q)-Hurwitz function and twisted (h,q)-two-variable L-function. By using these functions, we construct twisted new (h,q)-partial zeta function which interpolates the twisted (h,q)-Bernoulli polynomials and generalized twisted (h,q)-Bernoulli numbers at negative integers. We give relation between twisted (h,q)-partial zeta f…

论 $p$-进谱 zeta 函数

On $p$-adic spectral zeta functions

本文针对 $p$-进数域，利用局部解析函数的 $p$-进 Mellin 变换构造谱 zeta 函数，并研究其函数行列式。

p-adic interpolating function associated with modified Dirichlet's type of twisted q-euler numbers and polynomials with weight alpha

The q-calculus theory is a novel theory that is based on finite difference re-scaling. The rapid development of q-calculus has led to the discovery of new generalizations of q-Euler polynomials involving q-integers. The present paper deals with the modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of q-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between mod…

Test vectors for local cuspidal Rankin-Selberg integrals of GL(n), and reduction modulo $\ell$

Let $π_1,π_2$ be a pair of cuspidal complex, or $\ell$-adic, representations of the general linear group of rank $n$ over a non-archimedean local field $F$ of residual characteristic $p$, different to $\ell$. Whenever the local Rankin-Selberg $L$-factor $L(X,π_1,π_2)$ is nontrivial, we exhibit explicit test vectors in the Whittaker models of $π_1$ and $π_2$ such that the local Rankin-Selberg integral associated to these vectors and to the characteristic function of $\mathfrak{o}_F^n$ is equal to $L(X,π_1,π_2)$. We give an initial application of the test vectors to reduction modulo $\ell$ of $\…

关于秩2与秩3的单变量Witten zeta函数

On single-variable Witten zeta functions of rank two and three

该研究通过引入Mellin变换的新型积分核，探究秩2与秩3的单变量Witten zeta函数性质，得到极点、留数等相关结果，提出原点导数猜想，并讨论其与Eisenstein级数的关联及p进性质。

Local Langlands correspondence for $GL_n$ and the exterior and symmetric square $\varepsilon$--factors

Let $F$ be a $p$--adic field, i.e., a finite extension of $\mathbb Q_p$ for some prime $p$. The local Langlands correspondence attaches to each continuous $n$--dimensional $Φ$-semisimple representation $ρ$ of $W'_F$, the Weil--Deligne group for $\bar F/F$, an irreducible admissible representation $π(ρ)$ of $GL_n(F)$ such that, among other things, the local $L$- and $\varepsilon$-factors of pairs are preserved. This correspondence should be robust and preserve various parallel operations on the arithmetic and analytic sides, such as taking the exterior square or symmetric square. In this paper,…