Search

Let $\pi_1,\pi_2$ be irreducible admissible generic tempered representations of $\mathrm{GL}_2(F)$ for some $p$-adic field $F$ of odd residue characteristic. We introduce a natural notion of general $(\pi_1\times\pi_2)$-integral data $(\phi,g_1,g_2)\in \mathcal{S}(F^2)\times\mathrm{GL}_2(F)^2$ at which the Rankin-Selberg integral can be evaluated. This is inspired by work of Loeffler, and previous work of the author, on unramified zeta integrals. We then establish an integral variant of a result of Jacquet-Langlands for the local Rankin-Selberg zeta integral associated to $\pi_1\times\pi_2$; i.e. we show that for any such integral datum $(\phi,g_1,g_2)$, we have $$\frac{Z(\phi,g_1W_{\pi_1}^\mathrm{new},g_2W_{\pi_2}^\mathrm{new};s)}{L(\pi_1\times\pi_2,s)}=\Phi(\phi,g_1W_{\pi_1}^\mathrm{new},g_2W_{\pi_2}^\mathrm{new};q^s)\in \mathbf{Z}[q^{-1},\Sigma^1][q^s,q^{-s}]$$for a finite set $\Sigma^1\subseteq\mathbf{C}^\times$ of roots of unity and unitary character values, depending only on $\pi_1,\pi_2$. This is compatible with the notion of integrality coming from newforms $f_1,f_2$ of even integral weights, satisfying a mild local dihedral condition at $2$. We show that if $\pi_1,\pi_2$ are local pieces of $f_1,f_2$ at any prime $p$, the coefficient algebra is $\mathcal{O}_{K}[p^{-1}]$ with $K$ a number field only depending on $f_1,f_2$. Our approach relies on a reinterpretation of the local Rankin-Selberg integral, and works of Assing and Saha on values of $p$-adic Whittaker new-vectors., Comment: 34 pages. Comments welcomed

Let $F$ be a self-dual Hecke-Maa\ss\ form for $\rm{GL}(3)$ underlying the symmetric square lift of a $\rm{GL}(2)$-newform of square-free level and trivial nebentypus. In this paper, we are interested in the first moments of the central values of $ \rm{GL}(3)\times \rm{GL}(2)$ $L$-functions and $\rm{GL}(2)$ $L$-functions. As a result, we obtain an estimate for the first moment for $L(1/2, F\otimes f)$ over a family, where $F$ is of the level $q^2$, and $f\in \mathcal{B}^\ast_k(M)$ for any primes $q,M\ge 2$ such that $(q,M)=1$. We prove the subconvex bound for $L(1/2, F\otimes f)$ involving the levels aspects simultaneously in the range $M^{5/32+\varepsilon }\le q \le M^{7/32-\varepsilon}$ and $M> q^\delta$ for any $\varepsilon, \delta>0$ for the first time. Moreover, we present a first moment estimate for $L(1/2+it, F\otimes f)$ over $K\le k\le 2K$, with $K$ being a large number. As an application, whenever $k^{10/9+\varepsilon}\le |t|\le k^{2-\varepsilon}$, we obtain a non-trivial estimate for the first moment of $L(1/2, f)L(1/2+it, F\otimes f)$ in the weight $k$ and $t$ aspects.

This paper provides a complete classification of $\mathrm{GL}_n(\mathbb{R})$-distinguished irreducible representations of $\mathrm{GL}_n(\mathbb{C})$ when the representations are either generic or unitary. Additionally, for any $\mathrm{GL}_n(\mathbb{R})$-distinguished generic representation, we demonstrate that the period does not vanish on the distinguished minimal $K$-type. Furthermore, we establish the triviality of the $\varepsilon$-factor for distinguished representations and offer some applications to the branching problem using theta correspondence., Comment: Comments are welcomed!

This article will prove non-trivial estimates for the average and weighted average version of general $GL(3) \times GL(3)$ shifted convolution sums by using the circle method., Comment: 31 pages, Published in the Journal of Number Theory

Let $P,M$ be a two primes such that $(P,M)=1$. Let $\Pi$ be a normalized Hecke-Maa\ss\ form on ${\rm{GL}}(4)$ of level $P$, and $\chi$ a primitive Dirichlet character modulo $M$. In this paper, we study the hybrid subconvexity problem for $L(s, \Pi\otimes \chi)$ simultaneously in the level and conductor aspects. Among other things, we prove a hybrid subconvex bound, so long as $M^{1/5}

The second named author previously constructed a functor $\mathbb{V}^\vee\circ D^\vee_\Delta$ from the category of smooth $p$-power torsion representations of $\mathrm{GL}_n(\mathbb{Q}_p)$ to the category of inductive limits of continuous representations on finite $p$-primary abelian groups of the direct product $G_{\mathbb{Q}_p,\Delta}\times \mathbb{Q}_p^\times$ of $(n-1)$ copies of the absolute Galois group of $\mathbb{Q}_p$ and one copy of the multiplicative group $\mathbb{Q}_p^\times$. In the present work we show that this functor attaches finite dimensional representations on the Galois side to smooth $p$-power torsion representations of finite length on the automorphic side. This has some implications on the finiteness properties of Breuil's functor, too. Moreover, $\mathbb{V}^\vee\circ D^\vee_\Delta$ produces irreducible representations of $G_{\mathbb{Q}_p,\Delta}\times \mathbb{Q}_p^\times$ when applied to irreducible objects on the automorphic side and detects isomorphisms unless it vanishes. Further, we determine the kernel of $D^\vee_\Delta$ when restricted to successive extensions of subquotients of principal series. We use this to characterize representations that are parabolically induced from the product of a torus and $\mathrm{GL}_2(\mathbb{Q}_p)$. Finally, we formulate a conjecture and prove partial results on the essential image., Comment: 42 pages, comments welcome

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. If $p$ is large enough with respect to $[K:\mathbb{Q}_p]$ and under mild genericity assumptions, we prove that the admissible smooth representations of $\mathrm{GL}_2(K)$ that occur in Hecke eigenspaces of the mod $p$ cohomology are of finite length. We also prove many new structural results about these representations of $\mathrm{GL}_2(K)$ and their subquotients.

A simple abelian variety $A$ defined over a number field $k$ is called of $\mathrm{GL}_n$-type if there exists a number field of degree $2\dim(A)/n$ which is a subalgebra of $\mathrm{End}^0(A)$. We say that $A$ is genuinely of $\mathrm{GL}_n$-type if its base change $A_{\overline{k}}$ contains no isogeny factor of $\mathrm{GL}_m$-type for $m

We introduce the quantum Berezinian for the quantum affine superalgebra $\mathrm{U}_q(\widehat{\gl}_{M|N})$ and show that the coefficients of the quantum Berezinian belong to the center of $\mathrm{U}_q(\widehat{\gl}_{M|N})$. We also construct another family of central elements which can be expressed in the quantum Berezinian by a Liouville-type theorem. Moreover, we prove analogues of the Jacobi identities, the Schur complementary theorem, the Sylvester theorem and the MacMahon Master theorem for the generator matrices of $\mathrm{U}_q(\widehat{\gl}_{M|N})$., Comment: 17pp

In this paper, we use vertex operator techniques to compute character values on unipotent classes of $\GL_n(\mathbb F_q)$. By realizing the Grothendieck ring $R_G=\bigoplus_{n\geq0}^\infty R(\GL_n(\mathbb F_q))$ as Fock spaces, we formulate the Murnanghan-Nakayama rule of $\GL_n(\mathbb F_q)$ between Schur functions colored by an orbit $\phi$ of linear characters of $\overline{\mathbb F}_q$ under the Frobenius automorphism on and modified Hall-Littlewood functions colored by $f_1=t-1$, which provides detailed information on the character table of $\GL_n(\mathbb F_q)$. As applications, we use vertex algebraic methods to determine the Steinberg characters of $\GL_n(\mathbb F_q)$, which were previously determined by Curtis-Lehrer-Tits via geometry of homology groups of spherical buildings and Springer-Zelevinsky utilizing Hopf algebras., Comment: 15pp

Let $F$ be a non-Archimedean local field of characteristic zero and $G=\GL(2,F)$. Let $n\geq 2$ be a positive integer and $\widetilde{G}=\widetilde{\GL}(2,F)$ be the $n$-fold metaplectic cover of $G$. Let $\pi$ be an irreducible smooth representation of $G$ and $\pi^{\vee}$ be the contragredient of $\pi$. Let $\tau$ be an involutive anti-automorphism of $G$ satisfying $\pi^{\tau}\simeq \pi^{\vee}$. In this case, we say that $\tau$ is a dualizing involution. A well known theorem of Gelfand and Kazhdan says that the standard involution $\tau$ on $G$ is a dualizing involution. In this paper, we show that any lift of the standard involution to $\widetilde{G}$ is a dualizing involution if and only if $n=2$.

Let $f$ be a newform of prime level $p$ with any central character $\chi\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$, \begin{align*} L(1/2, \, f \otimes g) \ll p^{1/2-1/524+\varepsilon}, \end{align*} where the implied constant depends on $g$, $\varepsilon$, and the archimedean parameter of $f$. This improves upon the previously best-known result by Harcos and Michel. Our method ultimately relies on non-trivial bounds for bilinear forms in Kloosterman fractions pioneered by Duke, Friedlander, and Iwaniec, with later innovations by Bettin and Chandee., Comment: 27 pages. Comments welcome!

Consider the conjugation action of $\operatorname{GL}_{2}(K)$ on the polynomial ring $K[X_{2 \times 2}]$. When $K$ is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant. We describe the ring of invariants when $K$ is a finite field, and show that it is a hypersurface., Comment: are welcome! 10 pages

Let k be an algebraically closed field of characteristic p>0, let G=GL_n be the general linear group over k, let P be a parabolic subgroup of G, and let u_P be the Lie algebra of its unipotent radical. We show that the Kumar-Lauritzen-Thomsen splitting of the cotangent bundle Gx^Pu_P of G/P has top degree (p-1)\dim(G/P). The component of that degree is therefore given by the (p-1)-th power of a function f. We give a formula for f and deduce that it vanishes on the exceptional locus of the resolution Gx^Pu_P-->\ov{\mc O} where \ov{\mc O} is the closure of the Richardson orbit of P. As a consequence we obtain that the higher cohomology groups of a line bundle on Gx^Pu_P associated to a dominant weight are zero. The splitting of Gx^Pu_P given by f^{p-1} can be seen as a generalisation of the Mehta-Van der Kallen splitting of Gx^Bu.

Let $n$ be a positive integer and $q$ be a power of an odd prime. We provide explicit formulas for calculating the orthogonal determinants $\det(\chi)$, where $\chi \in \mathrm{Irr}(\mathrm{GL}_n(q))$ is an orthogonal character of even degree. Moreover, we show that $\det(\chi)$ is "odd". This confirms a special case of a conjecture by Richard Parker., Comment: 20 pages

The conjugation representation of a finite group $G$ is the complex permutation module defined by the action of $G$ on itself by conjugation. Addressing a problem raised by Hain motivated by the study of a Hecke action on iterated Shimura integrals, Tiep proved that for $G=\operatorname{SL}_{2}(\mathbb{Z}/p^{r})$, where $r\geq1$ and $p\geq5$ is a prime, any irreducible representation of $G$ that is trivial on the centre of $G$ is contained in the conjugation representation. Moreover, Tiep asked whether this can be generalised to $p=2$ or $3$. We answer the Hain--Tiep question in the affirmative and also prove analogous statements for $\operatorname{SL}_{2}$ and $\operatorname{GL}_{2}$ over any finite local principal ideal ring with residue field of odd characteristic., Comment: 40 pages

Recent research on integrating Large Language Models (LLMs) with Graph Neural Networks (GNNs) typically follows two approaches: LLM-centered models, which convert graph data into tokens for LLM processing, and GNN-centered models, which use LLMs to encode text features into node and edge representations for GNN input. LLM-centered models often struggle to capture graph structures effectively, while GNN-centered models compress variable-length textual data into fixed-size vectors, limiting their ability to understand complex semantics. Additionally, GNN-centered approaches require converting tasks into a uniform, manually-designed format, restricting them to classification tasks and preventing language output. To address these limitations, we introduce a new architecture that deeply integrates GNN with LLM, featuring three key innovations: (1) Structure-Aware Transformers, which incorporate GNN's message-passing capabilities directly into LLM's transformer layers, allowing simultaneous processing of textual and structural information and generating outputs from both GNN and LLM; (2) Graph-Text Cross-Attention, which processes full, uncompressed text from graph nodes and edges, ensuring complete semantic integration; and (3) GNN-LLM Twin Predictor, enabling LLM's flexible autoregressive generation alongside GNN's scalable one-pass prediction. GL-Fusion achieves outstand performance on various tasks. Notably, it achieves state-of-the-art performance on OGBN-Arxiv and OGBG-Code2., Comment: under review

A new method is introduced to derive general recurrence relations for off-shell Bethe vectors in quantum integrable models with either type $\mathfrak{gl}_n$ or type $\mathfrak{o}_{2n+1}$ symmetries. These recurrence relations describe how to add a single parameter, $z$, to specific subsets of Bethe parameters, expressing the resulting Bethe vector as a linear combination of monodromy matrix entries that act on Bethe vectors which do not depend on $z$. We refer to these recurrence relations as \textit{rectangular} because the monodromy matrix entries involved are drawn from the upper-right rectangular part of the matrix. This construction is achieved within the framework of the zero mode method., Comment: 32 pages

In this paper power saving bounds for general Kloosterman sums for all Weyl elements for $\mathrm{GL}_n$ for $n>2$ are proven, improving the trivial bound by D\k{a}browski and Reeder. This is achieved by representing the sums in an explicit way as exponential sums and bounding these through applications of the Weil bound.

We construct four-variable $p$-adic $L$-functions for the spin Galois representation of a Siegel modular form of genus 2 twisted by the Galois representation of a cuspidal modular form as the modular forms vary in Coleman families. The main ingredient is the construction of a space of nearly overconvergent modular forms in the coherent cohomology of the Siegel threefold, extending the spaces of overconvergent modular forms appearing in higher Coleman theory. In addition to this, we construct $p$-adic distributions interpolating the Gan-Gross-Prasad automorphic periods for $(\mathrm{GSpin}(5), \mathrm{GSpin}(4))$ which, conditional on the local and global Gan-Gross-Prasad conjectures for this pair of groups, provides a construction of "square-root" $p$-adic $L$-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$ as the automorphic forms vary in Coleman families., Comment: 54 pages

Using combinations of weight-1 and weight-2 of Kronecker-Eisenstein series to construct currents in the distributional de Rham complex of a squared elliptic curve, we find a simple explicit formula for the type II $(\text{GL}_2, \text{GL}_2)$ theta lift without smoothing, analogous to the classical formula of Siegel for periods of Eisenstein series. For $K$ a CM field, the same technique applies without change to obtain an analogous formula for the $(\text{GL}_2(K),K^\times)$ theta correspondence., Comment: version for submission

Deep learning methods based on Convolutional Neural Networks (CNNs) have shown great potential to improve early and accurate diagnosis of Alzheimer's disease (AD) dementia based on imaging data. However, these methods have yet to be widely adopted in clinical practice, possibly due to the limited interpretability of deep learning models. The Explainable Boosting Machine (EBM) is a glass-box model but cannot learn features directly from input imaging data. In this study, we propose a novel interpretable model that combines CNNs and EBMs for the diagnosis and prediction of AD. We develop an innovative training strategy that alternatingly trains the CNN component as a feature extractor and the EBM component as the output block to form an end-to-end model. The model takes imaging data as input and provides both predictions and interpretable feature importance measures. We validated the proposed model on the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset and the Health-RI Parelsnoer Neurodegenerative Diseases Biobank (PND) as an external testing set. The proposed model achieved an area-under-the-curve (AUC) of 0.956 for AD and control classification, and 0.694 for the prediction of conversion of mild cognitive impairment (MCI) to AD on the ADNI cohort. The proposed model is a glass-box model that achieves a comparable performance with other state-of-the-art black-box models. Our code is publicly available at: https://anonymous.4open.science/r/GL-ICNN., Comment: 4 pages, 3 figures

The cut-elimination procedure for the provability logic is known to be problematic: a L\"ob-like rule keeps cut-formulae intact on reduction, even in the principal case, thereby complicating the proof of termination. In this paper, we present a syntactic cut-elimination proof based on nested sequents, a generalization of sequents that allows a sequent to contain other sequents as single elements. A similar calculus was developed by Poggiolesi (2009), but there are certain ambiguities in the proof. Adopting the idea of Kushida (2020) into nested sequents, our proof does not require an extra measure on cuts or error-prone, intricate rewriting on derivations, but only straightforward inductions, thus leading to less ambiguity and confusion., Comment: In Proceedings NCL'24, arXiv:2412.20053

In this article, we consider the problem of estimating the correlation of Hecke eigenvalues of GL2 automorphic forms with a class of functions of algebraic origin defined over finite fields called trace functions. The class of trace functions is vast and includes many standard exponential sums like Gauss sums, Klostermann sums, Hyperklostermann sums etc. In particular, we prove a Burgess type power saving (of 1/8) over the trivial bound for forms with level coprime to the prime p over whose residue field, the trace function is defined. For forms whose level is divisible by at most one power of p we obtain a saving of 1/12. This generalizes the results of E. Fouvry, E. Kowalski and Ph. Michel to the case of number fields, with a slightly more restrictive assumption on the Fourier-M\"obius group attached to the trace function. Moreover, the implied constant for the 1/12 power saving bound has polynomial dependence on the archimedean part of the conductor of the form. We work using the language of adeles which makes the analysis involved softer and makes the generalization to number fields more natural. The proof proceeds by studying the amplified second moment spectral average of the correlation sum using the relative trace formula.

In this paper, we establish an asymptotic formula for the twisted second moment of $L$-functions associated with Hecke--Maass cusp forms for $\rm SL(3,\mathbb{Z})$, and further deduce a weighted zero-density estimate for these $L$-functions in the spectral aspect which may have important applications in other problems., Comment: 33 pages. Comments welcome! arXiv admin note: text overlap with arXiv:1704.00314 by other authors

We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its Arthur-Clozel's base change to some real quadratic field $E$. This generalizes earlier works of Tilouine-Urban and of Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods, defined using the middle degree of the cuspidal cohomology of $\mathrm{GL}_3(E)$, instead of the top or bottom degrees. We also investigate the Rogawski's stable base change from the quasi-split unitary group $U_E$ associated with $E$ to $\mathrm{GL}_3(E)$. In this situation, we also obtain some results toward a $p$-adic divisibility of automorphic periods.

We report the discovery of a super-Earth candidate orbiting the nearby mid M dwarf Gl\,725A using the radial velocity (RV) method. The planetary signal has been independently identified using high-precision RVs from the SOPHIE and SPIRou spectrographs, in the optical and near-infrared domains, respectively. We modelled the stellar activity signal jointly with the planet using two Gaussian Processes, one for each instrument to account for the chromaticity of the stellar activity and instrumental systematics, along with a Keplerian model. The signal is significantly detected with a RV semi-amplitude of $1.67\pm0.20$ m/s. The planet Gl 725A b is found to be in an orbit compatible with circular with a period of $11.2201\pm0.0051$ days. We analysed 27 sectors of TESS photometry on which no transit event was found. We determined a minimum mass of $M_{p}\sin{i}=2.78\pm0.35\,M_{\oplus}$ which places the planet in the super-Earth regime. Using Mass-Radius relationships we predict a planetary radius to be between 1.2 and $2.0\,R_{\oplus}$. The proximity of Gl 725A, of only 3.5 pc, makes this new exoplanet one of the closest to Earth and joins the group of S-type low-mass planets in short orbits ($P<15$ d) around close M dwarfs., Comment: 26 pages, 17 figures. Accepted in A&A

We give a new proof of the existence of Whittaker functionals for principal series representation of $\text{GL}(n,\mathbb{R})$, utilizing the analytic theory of distributions. We realize Whittaker functionals as equivariant distributions on $\text{GL}(n,\mathbb{R})$, whose restriction to the open Schubert cell is unique up to a constant. Using a birational map on the Schubert cells, we show that the unique distribution on the open Schubert cell extends to a distribution on the entire space $\text{GL}(n,\mathbb{R})$. This technique gives a proof of the analytic continuation of Jacquet integrals via integration by parts., Comment: 35 pages. arXiv admin note: text overlap with arXiv:2410.13519

Let $a(n)$ be the $n$-th Dirichlet coefficient of the automorphic $L$-function or the Rankin--Selberg $L$-function. We investigate the cancellation of $a(n)$ over specific sequences linked to the Waring--Goldbach problem, by establishing a nontrivial bound for the additive twisted sums over primes on ${\mathrm{GL}}_m$ and ${\mathrm{GL}}_m\times{\mathrm{GL}}_m .$ The bound does not depend on the generalized Ramanujan conjecture or the nonexistence of Landau--Siegel zeros. Furthermore, we present an application associated with the Sato--Tate conjecture and propose a conjecture about Goldbach's conjecture on average bound., Comment: 33 pages. Comments welcome

Volume rendering in neural radiance fields is inherently time-consuming due to the large number of MLP calls on the points sampled per ray. Previous works would address this issue by introducing new neural networks or data structures. In this work, We propose GL-NeRF, a new perspective of computing volume rendering with the Gauss-Laguerre quadrature. GL-NeRF significantly reduces the number of MLP calls needed for volume rendering, introducing no additional data structures or neural networks. The simple formulation makes adopting GL-NeRF in any NeRF model possible. In the paper, we first justify the use of the Gauss-Laguerre quadrature and then demonstrate this plug-and-play attribute by implementing it in two different NeRF models. We show that with a minimal drop in performance, GL-NeRF can significantly reduce the number of MLP calls, showing the potential to speed up any NeRF model., Comment: NeurIPS 2024. Project page: https://silongyong.github.io/GL-NeRF_project_page/

We give a formula for a birational map on the Schubert cell associated to each Weyl group element of $G=\text{GL}(n)$. The map simplifies the UDL decomposition of matrices, providing structural insight into the Schubert cell decomposition of the flag variety $G/B$, where $B$ is a Borel subgroup. An application of the formula includes a new proof of the existence of Whittaker functionals for principal series representations of $\text{GL}(n,\mathbb{R})$ via integration by parts. In this paper, we establish combinatorial properties of the birational map and prove auxiliary results., Comment: 33 pages. Typos corrected, references added

Brown dwarfs provide a unique opportunity to study atmospheres and their physical and chemical processes with high precision, especially in temperature ranges relevant to exoplanets. In this study, we performed high-resolution ($R \sim 70,000$) spectroscopy using Subaru/IRD of Gl 229 B, the first discovered T-type (T7.0p) brown dwarf, which orbits an M1V host star at a separation of 33 au. We conducted atmospheric retrieval on the reduced $H$-band spectrum using the high-resolution spectrum model compatible with automatic differentiation and GPU, ExoJAX. In contrast to previous retrieval studies on medium-resolution spectra, we obtained a C/O ratio consistent with that of the host star, aligning with the expected formation process for such a massive brown dwarf. Additionally, based on the strong constraint on temperature from the high-resolution spectrum and previously measured photometric magnitude, our analysis indicates that Gl 229 B is a binary, which was also proposed by Brandt et al. (2021). Finally, we validated current molecular line lists by leveraging the obtained high-precision, high-resolution spectrum of this warm ($\sim 900$ K) atmosphere. This study highlights the importance of observing companion brown dwarfs as benchmark objects for establishing characterization techniques for low-mass objects and enhancing our understanding of their atmospheres, given the wealth of available information and the relative ease of observation., Comment: 16 pages, 9 figures, submitted to AAS

Following work of Brundan and Kleshchev (2000), which considered tensor products with the natural module (and its dual) for $\text{GL}(n)$, we take the next fundamental module and explore the relationship between multiplicities of composition factors of tensor products of simple modules for $\text{GL}(n)$ and branching multiplicities. In doing this we provide a method for determining the composition multiplicities for some composition factors of tensor products of simple modules with the wedge square of the dual natural module.

Let $F$ be a totally real field. We study the root numbers $\epsilon(1/2, \pi)$ of self-dual cuspidal automorphic representations $\pi$ of $\mathrm{GL}_{2N}/F$ with conductor $\mathfrak n$ and regular integral infinitesimal character $\lambda$. If $\pi$ is orthogonal, then $\epsilon(1/2, \pi)$ is known to be identically one. We show that for symplectic representations, the root numbers $\epsilon(1/2, \pi)$ equidistribute between~$\pm 1$ as $\lambda \to \infty$, provided that there exists a prime dividing $\mathfrak n$ with power $>N$.We also study conjugate self-dual representations with respect to a CM extension $E/F$, where we obtain a similar result under the assumption that $\mathfrak n$ is divisible by a large enough power of a ramified prime and provide evidence that equidistribution does not hold otherwise. In cases where there are known to be associated Galois representations, we deduce root number equidistribution results for the corresponding families of $N$-dimensional Galois representations. The proof generalizes a classical argument for the case of $\mathrm{GL}_2/\mathbb Q$ by using Arthur's trace formula and the endoscopic classification for quasisplit classical groups similarly to a previous work (arxiv:2212.12138). The main new technical difficulty is evaluating endoscopic transfers of the required test functions at central elements., Comment: 81 pages, this version: sign mistakes corrected in definition of J_n in section 3.4, in lemmas 3.4.2 and 4.2.3, and in analogous conjugate self-dual variants of these. Comments are welcome!

Let $W_{r}(\mathbb{F}_{q})$ be the ring of Witt vectors of length $r$ with residue field $\mathbb{F}_{q}$ of characteristic $p$. In this paper, we study the defining characteristic case of the representations of $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ over the principal ideal local rings $W_{r}(\mathbb{F}_{q})$ and $\mathbb{F}_{q}[t]/t^{r}$. Let ${\mathbf{G}}$ be either $\mathrm{GL}_{n}$ or $\mathrm{SL}_{n}$ and $F$ a perfect field of characteristic $p$, we prove that for most $p$ the group algebras $F[{\mathbf{G}}(W_{r}(\mathbb{F}_{q}))]$ and $F[{\mathbf{G}}(\mathbb{F}_{q}[t]/t^{r})]$ are not stably equivalent of Morita type. Thus, the group algebras $F[{\mathbf{G}}(W_{r}(\mathbb{F}_{q}))]$ and $F[{\mathbf{G}}(\mathbb{F}_{q}[t]/t^{r})]$ are not isomorphic in the defining characteristic case., Comment: 10 pages, comments are welcome!

For a fixed prime power $q$, let $\text{GL}_\bullet(q)$ denote the family of groups $\text{GL}_N(q)$ for $N \in \mathbb{Z}_{\geq 0}$. In this paper we study the $\mathbb{C}$-algebra of "stable" class functions of $\text{GL}_\bullet(q)$, and show it admits four different linear bases, each arising naturally in different settings. One such basis is that of stable irreducible characters, namely, the class functions spanned by the characters corresponding to finitely generated simple $\mathrm{VI}$-modules in the sense of [arXiv:1408.3694,arXiv:1602.00654]. A second one comes from characters of parabolic representations. The final two, one originally defined in [arXiv:1803.04155] and the other in [arXiv:2110.11099], are more combinatorial in nature. As corollaries, we clarify many properties of these four bases and prove a conjecture from [arXiv:2106.11587]., Comment: 25 pages

In this paper, we extend the results of Grantcharov and Robitaille in 2021 on mixed tensor products and Capelli determinants to the superalgebra setting. Specifically, we construct a family of superalgebra homomorphisms $\varphi_R : U(\mathfrak{gl}(m+1|n)) \rightarrow \mathcal{D}'(m|n) \otimes U(\mathfrak{gl}(m|n))$ for a certain space of differential operators $\mathcal{D}'(m|n)$ indexed by a central element $R$ of $\mathcal{D}'(m|n) \otimes U(\mathfrak{gl}(m|n))$. We then use this homomorphism to determine the image of Gelfand generators of the center of $U(\mathfrak{gl}(m+1|n))$. We achieve this by first relating $\varphi_R$ to the corresponding Harish-Chandra homomorphisms and then proving a super-analog of Newton's formula for $\mathfrak{gl}(m)$ relating Capelli generators and Gelfand generators. We also use the homomorphism $\varphi_R$ to obtain representations of $U(\mathfrak{gl}(m+1|n))$ from those of $U(\mathfrak{gl}(m|n))$, and find conditions under which these inflations are simple. Finally, we show that for a distinguished central element $R_1$ in $\mathcal{D}'(m|n)\otimes U(\mathfrak{gl}(m|n))$, the kernel of $\varphi_{R_1}$ is the ideal of $U(\mathfrak{gl}(m+1|n))$ generated by the first Gelfand invariant $G_1$.

For a $p$-adic field $F$ of residual cardinality $q$, we provide a triangulated equivalence between the bounded derived category $D^b(\mathcal{B}_{1}(G)_{fg})$ of finitely generated unipotent representations of $G=\mathrm{GL}_2(F)$ and perfect complexes over a dg enriched Schur algebra, in the non-banal case of odd characteristic $l$ dividing $q+1$. The dg Schur algebra is the dg endomorphism algebra of a projective resolution of a direct sum $V$ of the parahoric inductions of the trivial representations of the reductive quotients of $G$, and $V$ is shown to be a classical generator of $D^b(\mathcal{B}_{1}(G)_{fg})$.

For the group $\mathrm{GL}_{n}$ and the anisotropic elements, we confirm the purity hypothesis of Goresky, Kottwitz and MacPherson, which states that the affine Springer fibers are cohomologically pure in the sense of Grothendieck-Deligne.

Let $\mathfrak{g}=\mathfrak{gl}_{M|N}(\mathbb{k})$ be the general linear Lie superalgebra over an algebraically closed field $\mathbb{k}$ of characteristic zero. Fix an arbitrary even nilpotent element $e$ in $\mathfrak{g}$ and let $U(\mathfrak{g},e)$ be the finite $W$-superalgebra associated to the pair $(\mathfrak{g},e)$. In this paper we will give a complete classification of one-dimensional representations for $U(\mathfrak{g},e)$. To achieve this, we use the tool of shifted super Yangians to determine the commutative quotients of the finite $W$-superalgebras., Comment: 22 pages. Comments are welcome

Let $k$ be a global field and $\mathbb{A}_k$ be its ring of adeles. Let $\ell$ be a prime number and fix a field isomorphism from $\mathbb{C}$ to $\overline{\mathbb{Q}}_{\ell}$. Let $\Pi_1$ and $\Pi_2$ be cuspidal automorphic representations of ${\rm GL}_n(\mathbb{A}_k)$ for some integer $n\geq1$. In this paper, we study the following question: assuming that there is a finite set $S$ of places of $k$ containing all Archimedean places and all finite places above $\ell$ such that, for all $v\notin S$, the local components $\Pi_{1,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $\Pi_{2,v} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ are unramified and their Satake parameters are congruent mod $\ell$, are the local components $\Pi_{1,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ and $\Pi_{2,w} \otimes_{\mathbb{C}} \overline{\mathbb{Q}}_{\ell}$ integral, and do their reductions mod $\ell$ share an irreducible factor for all non-Archimedean places $w$ not dividing $\ell$? We show that, under certain conditions on $\Pi_1$ and $\Pi_2$, the answer is yes. We also give a simple proof when $k$ is a function field.

In this paper, we relate $L(1,\pi,\mathrm{Ad}^\circ)$ to the congruence ideals for cohomological cuspidal automorphic representations $\pi$ of $\mathrm{GL}_n$ over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint $L$-functions. For CM fields, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of $L(1,\pi,\mathrm{Ad}^\circ)$., Comment: 33 pages, minor errors corrected

For $G=\mathrm{SL}_2$ or $\mathrm{GL}_2$, we present explicit formulas for the nonabelian Fourier kernels on $G$, as conjectured by A. Braverman and D. Kazhdan. Additionally, we furnish explicit formulas for the orbital Hankel transform on $G$, a topic investigated by the second author, and provide an explicit formula for the stable orbital integral of the basic function. These results are applicable to local fields with residual characteristics other than two., Comment: 106pp. Any comments are very welcome

Let m be a cube-free positive integer and let p be a prime such that p does not divide m. In this paper we find the number of conjugacy classes of completely reducible solvable cube-free subgroups in GL(2, q) of order m, where q is a power of p.

Let $m$ be a positive integer such that $p$ does not divide $m$ where $p$ is prime. In this paper we find the number of conjugacy classes of completely reducible cyclic subgroups in GL$(2, q)$ of order $m$, where $q$ is a power of $p$.

In this paper we realize the supersymmetric classical $W$-algebras $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ and $\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ as differential algebras generated by the coefficients of a monic superdifferential operator $L$. In the case of $\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ (resp. $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$) this operator is even (resp. odd). We show that the supersymmetric Poisson vertex algebra bracket on these supersymmetric W-algebras is the supersymmetric analogue of the quadratic Gelfand-Dickey bracket associated to the operator $L$. Finally, we construct integrable hierarchies of evolutionary Hamiltonian PDEs on both W-algebras. A key observation is that to construct these hierarchies on the algebra $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ one needs to introduce a new concept of even supersymmetric Poisson vertex algebras.

Every double coset in $\text{GL}_m(k[[z]])\backslash \text{GL}_m(k((z)))/\text{GL}_m(k((z^2)))$ is uniquely represented by a block diagonal matrix with diagonal blocks in $\{1,z, \begin{pmatrix} 1& z\\ 0 &z^i \end{pmatrix} (i>1)\}$ if $char(k) \neq 2$ and $k$ is a finite field. These cosets form a (spherical) Hecke module $\mathcal{H}(G,H,K)$ over the (spherical) Hecke algebra $\mathcal{H}(G,K)$ of double cosets in $K\backslash G/H$, where $K=\text{GL}_m(k[[z]])$ and $H=\text{GL}_m(k((z^2)))$ and $G=\text{GL}_m(k((z)))$. Similarly to Hall polynomial $h_{\lambda,\nu}^{\mu}$ from the Hecke algebra $\mathcal{H}(G,K)$, coefficients $h_{\lambda,\nu}^{\mu}$ arise from the Hecke module. We will provide a closed formula for $h_{\lambda,\nu}^\mu$, under some restrictions over ${\lambda,\nu,\mu}$.