Non-archimedean Local Field Research Articles

An explicit formula for the Siegel series of a quadratic form over a non-archimedean local field

Abstract Let F be a non-archimedean local field of characteristic 0, and 𝔬 {{\mathfrak{o}}} the ring of integers in F. We give an explicit formula for the Siegel series of a half-integral matrix over 𝔬 {{\mathfrak{o}}} . This formula expresses the Siegel series of a half-integral matrix B explicitly in terms of the Gross–Keating invariant of B and its related invariants.

On the construction of tame supercuspidal representations

Let $F$ be a non-archimedean local field of residual characteristic $p \neq 2$. Let $G$ be a (connected) reductive group over $F$ that splits over a tamely ramified field extension of $F$. We revisit Yu's construction of smooth complex representations of $G(F)$ from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in Yu [Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), 579–622], whose proofs relied on a typo in a reference.

Nonuniform low-pass filters on non Archimedean local fields

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of this types of signals by a stable mathematical tool. Gabardo and Nashed (J. Funct. Anal. 158:209-241, 1998) filled this gap by the concept of nonuniform multiresolution analysis. In this setting, the associated translation set \(\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z\) is no longer a discrete subgroup of \(\mathbb R\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. The main aim of this article is to provide the characterization of nonuniform low-pass filters on non-Archimedean local fields.

A remark on conductor, depth and principal congruence subgroups

In this paper we study a relation between conductor, depth, and the level of principal congruence subgroups for irreducible admissible representations of GLn(F) for a non-archimedean local field F of characteristic zero. As a global application, we estimate conductors of unitary irreducible automorphic cuspidal representations of GLn(AQ) in terms of principal congruence subgroups. We also give an explicit formula for the dimension of fixed vectors with respect to principal congruence subgroups for irreducible admissible representations of GL2(F) as a local application.

DISTANCES ON A MASURE

A masure (also known as an affine ordered hovel) is a generalization of the Bruhat-Tits building that is associated with a split Kac-Moody group G over a nonarchimedean local field. This is a union of affine spaces called apartments. When G is a reductive group, is a building and there is a G-invariant distance inducing a norm on each apartment. In this paper, we study distances on inducing the affine topology on each apartment. We construct distances such that each element of G is a continuous automorphism of and we study their properties (completeness, local compactness, …).

$$L^p$$ properties of non-Archimedean fractional differentiation operators

Let \(D^{\alpha }\) be the Vladimirov–Taibleson fractional differentiation operator acting on complex-valued functions on a non-Archimedean local field. The identity \(D^{\alpha }D^{-\alpha }f=f\) was known only for the case where f has a compact support. Following a result by Samko about the fractional Laplacian of real analysis, we extend the above identity in terms of \(L^p\)-convergence of truncated integrals. Differences between real and non-Archimedean cases are discussed.

A new proof of the Jacquet–Rallis fundamental lemma

We give a new proof of the so-called Lie algebra version of the Jacquet–Rallis fundamental lemma for local non-Archimedean fields of characteristic 0. This proof is local and based on a previous result of Zhang on the compatibility of smooth transfer with a (partial) Fourier transform.

On the Moy–Prasad filtration

Let K be a maximal unramified extension of a nonarchimedean local field with arbitrary residual characteristic p. Let G be a reductive group over K which splits over a tamely ramified extension of K. We show that the associated Moy-Prasad filtration representations are in a certain sense independent of p. We also establish descriptions of these representations in terms of explicit Weyl modules and as representations occurring in a generalized Vinberg-Levy theory. As an application, we use these results to provide necessary and sufficient conditions for the existence of stable vectors in Moy-Prasad filtration representations, which extend earlier results by Reeder and Yu (which required p to be large) and by Romano and the author (which required G to be absolutely simple and split). This yields new supercuspidal representations. We also treat reductive groups G that are not necessarily split over a tamely ramified field extension.

Gross–Prasad periods for reducible representations

Abstract We study GL 2 ⁡ ( F ) {\operatorname{GL}_{2}(F)} -invariant periods on representations of GL 2 ⁡ ( A ) {\operatorname{GL}_{2}(A)} , where F is a non-archimedean local field and A / F {A/F} a product of field extensions of total degree 3. For irreducible representations, a theorem of Prasad shows that the space of such periods has dimension ⩽ 1 {\leqslant 1} , and is non-zero when a certain ε-factor condition holds. We give an extension of this result to a certain class of reducible representations (of Whittaker type), extending results of Harris–Scholl when A is the split algebra F × F × F {F\times F\times F} .

Restricting supercuspidal representations via a restriction of data

Let $F$ be a non-archimedean local field of residual characteristic $p$. Let $\mathbb{G}$ be a reductive group defined over $F$ which splits over a tamely ramified extension and set $G=\mathbb{G}(F)$. We assume that $p$ does not divide the order of the Weyl group of $\mathbb{G}$. Given a closed connected $F$-subgroup $\mathbb{H}$ that contains the derived subgroup of $\mathbb{G}$, we study the restriction to $H$ of an irreducible supercuspidal representation $\pi=\pi_G(\Psi)$ of $G$, where $\Psi$ is a $G$-datum as per the J.K. Yu Construction. We provide a full description of $\pi|_H$ into irreducible components, with multiplicity, via a restriction of data which constructs $H$-data from $\Psi$. Analogously, we define a restriction of Kim-Yu types to study the restriction of irreducible representations of $G$ which are not supercuspidal.

Robinson–Schensted–Knuth correspondence in the representation theory of the general linear group over a non-archimedean local field

We construct new “standard modules” for the representations of general linear groups over a local non-archimedean field. The construction uses a modified Robinson–Schensted–Knuth correspondence for Zelevinsky’s multisegments. Typically, the new class categorifies the basis of Doubilet, Rota, and Stein (DRS) for matrix polynomial rings, indexed by bitableaux. Hence, our main result provides a link between the dual canonical basis (coming from quantum groups) and the DRS basis.

Comparison of the depths on both sides of the local Langlands correspondence for Weil-restricted groups (with an appendix by Jessica Fintzen)

Let E/F be a finite and Galois extension of non-archimedean local fields. Let G be a connected reductive group defined over E and let M:=RE/FG be the reductive group over F obtained by Weil restriction of scalars. We investigate depth, and the enhanced local Langlands correspondence, in the transition from G(E) to M(F). We obtain a depth-comparison formula for Weil-restricted groups.

Explicit Plancherel Formula for the Space of Symplectic Forms

Abstract We provide a Plancherel decomposition for the space of symplectic bilinear forms of rank $2n$ over a local non-Archimedean field $F$ in terms of that of $\operatorname{GL}_n(F)$.

Equivalence of categories between coefficient systems and systems of idempotents

The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of Rep R ⁡ ( G ) \operatorname {Rep}_R(G) , the category of smooth representations of a p p -adic group G G with coefficients in R R . In particular, they were used to construct level 0 decompositions when R = Z ¯ ℓ R=\overline {\mathbb {Z}}_{\ell } , ℓ ≠ p \ell \neq p , by Dat for G L n GL_{n} and the author for a more general group. Wang proved in the case of G L n GL_{n} that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of G L n GL_{n} and a unipotent block of another group. In this paper, we generalize Wang’s equivalence of category to a connected reductive group on a non-archimedean local field.

On the motivic oscillation index and bound of exponential sums modulo pm via analytic isomorphisms

Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12]. In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of ℓ-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A−D−E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

Smooth representations of unit groups of split basic algebras over non-Archimedean local fields

Abstract We consider smooth representations of the unit group G = A × G=\mathcal{A}^{\times} of a finite-dimensional split basic algebra 𝒜 over a non-Archimedean local field. In particular, we prove a version of Gutkin’s conjecture, namely, we prove that every irreducible smooth representation of 𝐺 is compactly induced by a one-dimensional representation of the unit group of some subalgebra of 𝒜. We also discuss admissibility and unitarisability of smooth representations of 𝐺.

Principal series component of Gelfand-Graev representation

Let $G$ be a connected reductive group defined over a non-archimedean local field $F$. Let $B$ be a minimal $F$-parabolic subgroup with Levi factor $T$ and unipotent radical $U$. Let $\psi$ be a non-degenerate character of $U(F)$ and $\lambda$ a character of $T(F)$. Let $(K,\rho)$ be a Bushnell-Kutzko type associated to the Bernstein block of $G(F)$ determined by the pair $(T,\lambda)$. We study the $\rho$-isotypical component $(c\text{-ind}_{U(F)}^{G(F)}\psi)^{\rho}$ of the induced space $c\text{-ind}_{U(F)}^{G(F)}\psi$ of functions compactly supported mod $U(F)$. We show that $(c\text{-ind}_{U(F)}^{G(F)}\psi)^{\rho}$ is cyclic module for the Hecke algebra $\mathcal{H}(G,\rho)$ associated to the pair $(K,\rho)$. When $T$ is split, we describe it more explicitly in terms of $\mathcal{H}(G,\rho)$. We make assumptions on the residue characteristic of $F$ and later also on the characteristic of $F$ and the center of $G$ depending on the pair $(T,\lambda)$. Our results generalize the main result of Chan and Savin in \cite{CS18} who treated the case of $\lambda=1$ for $T$ split.

Finiteness Theorems for Kac–Moody Groups over Non-archimedean Local Fields

Abstract We prove the finiteness of formal analogs of the spherical function (Spherical Finiteness), the ${\textbf c}$-function (Gindikin–Karpelevich Finiteness), and obtain a formal analog of Harish-Chandra’s limit (Approximation Theorem) relating spherical and ${\textbf c}$-function in the setting of $p$-adic Kac–Moody groups. The finiteness theorems imply that the formal analog of the Gindikin–Karpelevich integral is well defined in local Kac–Moody settings. These results extend Braverman–Garland–Kazhdan–Patnaik’s affine Gindikin–Karpelevich finiteness theorems from [ 4] and provide an algebraic analog of the combinatorial results of Gaussent–Rousseau [10] and Hébert [14].

FORMAL DEGREES AND LOCAL THETA CORRESPONDENCE: QUATERNIONIC CASE

In this paper, we determine a constant occurring in a local analogue of the Siegel-Weil formula, and describe the behavior of the formal degrees under the local theta correspondence for quaternionic dual pairs of almost equal rank over a non-Archimedean local field of characteristic $0$. As an application, we prove the formal degree conjecture of Hiraga-Ichino-Ikeda for the non-split inner forms of ${\rm Sp}_4$ and ${\rm GSp}_4$.

Generic cuspidal representations of 𝑈(2, 1)

Abstract Let 𝐹 be a non-Archimedean local field, and let 𝜎 be a non-trivial Galois involution with fixed field F 0 F_{0} . When the residue characteristic of F 0 F_{0} is odd, using the construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of U ⁢ ( 2 , 1 ) ⁢ ( F / F 0 ) U(2,1)(F/F_{0}) .