Decomposition of the Uniform Projection of the Weil Character

Combinatorics of finite abelian groups and Weil representations

The Weil representation of the symplectic group associated to a finite abelian group of odd order is shown to have a multiplicity-free decomposition. When the abelian group is p-primary, the irreducible representations occurring in the Weil representation are parametrized by a partially ordered set which is independent of p. As p varies, the dimension of the irreducible representation corresponding to each parameter is shown to be a polynomial in p which is calculated explicitly. The commuting algebra of the Weil representation has a basis indexed by another partially ordered set which is inde…

The Weil-Steinberg character of finite classical groups

We compute the irreducible constitutents of the product of the Weil character and the Steinberg character in those finite classical groups for which a Weil character is defined, namely the symplectic, unitary and general linear groups. It turns out that this product is multiplicity free for the symplectic and general unitary groups, but not for the general linear groups. As an application we show that the restriction of the Steinberg character of such a group to the subgroup stabilizing a vector in the natural module is multiplicity free. The proof of this result for the unitary groups uses an…

Correspondance thêta locale $\ell$-modulaire I : groupe métaplectique, représentation de Weil et $Θ$-lift

Let $F$ be a field which is, either local non archimedean, or finite, of residual charcateristic $p$ but of characteristic different from $2$. Let $W$ be a symplectic space of finite dimension over $F$. Suppose $R$ is a field of characteristic $\ell \neq p$ so that there exists a non trivial smooth additive character $ψ: F \to R^\times$. Then the Stone-von Neumann theorem of the Heisenberg group $H(W)$ is still valid for representations with coefficients in $R$. It leads to a projective representation of the group $\text{Sp}(W)$ which lifts to a genuine smooth representation of a central exten…

Clifford theory of Weil representations of unitary groups

Let ${\mathcal O}$ be an involutive discrete valuation ring with residue field of characteristic not 2. Let $A$ be a quotient of ${\mathcal O}$ by a nonzero power of its maximal ideal and let $*$ be the involution that $A$ inherits from ${\mathcal O}$. We consider various unitary groups ${\mathcal U}_m(A)$ of rank $m$ over $A$, depending on the nature of $*$ and the equivalence type of the underlying hermitian or skew hermitian form. Each group ${\mathcal U}_m(A)$ gives rise to a Weil representation. In this paper, we give a Clifford theory description of all irreducible components of the Weil…

Isocategorical groups and their Weil representations

Two groups are called isocategorical over a field $k$ if their respective categories of $k$-linear representations are monoidally equivalent. We classify isocategorical groups over arbitrary fields, extending the earlier classification of Etingof-Gelaki and Davydov for algebraically closed fields. In order to construct concrete examples of isocategorical groups a new variant of the Weil representation associated to isocategorical groups is defined. We construct examples of non-isomorphic isocategorical groups over any field of characteristic different from two and rational Weil representations…

The Weil Representation in Characteristic Two

In this paper we construct a new variant of the Weil representation, associated with a symplectic vector space V defined over a finite field of characteristic two. Our variant is a representation of a bigger group than that of Weil. In the course, we develop the formalism of canonical vector spaces, which enables us to realize the bigger symmetry group and the representation in a transparent manner.

Weil representations and metaplectic groups over an integral domain

Given F a locally compact, non-discrete, non-archimedean field of characteristic different from 2 and R an integral domain such that a non-trivial smooth F-character with values in the multiplicative group of R exists, we construct the (reduced) metaplectic group attached to R. We show that it is in most cases a double cover of the symplectic group over F. Finally we define a faithful infinite dimensional R-representation of the metaplectic group analogue to the Weil representation in the complex case.

约化对偶对 $(\mathrm{U}_l,\mathrm{U}_{l'})$ 的对称破缺算子

Symmetry breaking operator for the reductive dual pair $(\mathrm{U}_l,\mathrm{U}_{l'})$

本文研究酉群对偶对在 Weil 表示中的对称破缺算子及其 Weyl 符号，通过两种正交投影计算相互匹配，重新证明重数至多为一并恢复 Howe 对应完整列表。

Realizing the local Weil representation over a number field

We show that the Weil representation of the symplectic group Sp(2n,F), where F is a non-archimedian local field, can be realized over the field obtained from the rationals by adjoining the square roots of p and -p, where p is the residue characteristic of F; p is assumed to be odd.

The character of the Weil representation of a finite abelian group of odd order

Let $V$ be a finite abelian group of odd order, equipped with a non-degenerate, alternating form $ω\colon V\times V \to \mathbb{Z}/m\mathbb{Z}$. We give closed formulas for the character values of the Weil representation associated with $(V,ω)$. These formulas generalize the ones given by S. Gurevich and R. Hadani (2007) and by T. Thomas (2008, 2013) from finite vector spaces to abelian groups. Our formulas do not involve the choice of a Lagrangian subgroup of $V$. Our proofs are based on an elementary algebraic approach introduced by H. N. Ward (1972, 2017) for finite vector spaces over field…

A projection formula for the ind-Grassmannian

Let $X = \bigcup_k X_k$ be the ind-Grassmannian of codimension $n$ subspaces of an infinite-dimensional torus representation. If $\cE$ is a bundle on $X$, we expect that $\sum_j (-1)^j Λ^j(\cE)$ represents the $K$-theoretic fundamental class $[\cO_Y]$ of a subvariety $Y \subset X$ dual to $\cE^*$. It is desirable to lift a $K$-theoretic "projection formula" from the finite-dimensional subvarieties $X_k$, but such a statement requires switching the order of the limits in $j$ and $k$. We find conditions in which this may be done, and consider examples in which $Y$ is the Hilbert scheme of points…

Extended Weil representations: the finite field cases

It is well known(cf. Weil, Gérardin's works) that there are two different Weil representations of a symplectic group over an odd finite field. By a twisted action, we show that one can reorganize them as a representation of a related projective symplectic similitude group. We also discuss the even field case by following Genestier-Lysenko and Gurevich-Hadani's works on geometric Weil representations in characteristic two. As a result, we approach some of their results from the lattice model, which is inspired by MVW, Prasad and Takeda's works.

张量积马尔可夫链与Weil表示

Tensor product Markov chains and Weil representations

本文研究了有限经典群不可约表示上由Weil表示张量积诱导的马尔可夫链，并给出了其收敛速度的尖锐界限。

Weil 表示限制到开紧子群上的不可约分解

An irreducible decomposition of the Weil representation restricted to open compact subgroups

本文利用 Schrödinger 模型，对任意剩余特征的 p-进域，给出了 Weil 表示限制到极大开紧子群和 Iwahori 子群的不可约分解，并证明其无重数性。

酉群Weil表示的环面周期

On tori periods of Weil representations of unitary groups

本文确定了酉群Weil表示到极大环面的限制，证明了局部相容特征与全局环面周期非零的充要条件，并利用跷跷板论证和theta提升结果为构造G2的Arthur包提供了基础。

扩展 Weil 表示：非二进局部域情形

Extended Weil representations: the non-dyadic local field cases

本文通过扭作用将 Meteplectic 覆盖群 Mp(W) 的不同 Weil 表示重组为与射影相似辛群相关的覆盖群 $PGMp^{\pm}(W)$ 的一个表示。

关于Weil表示的格模型

On the lattice model of the Weil representation

本文通过引入有理Metaplectic群和Metaplectic群，将基于非自对偶格的Heisenberg-Weil表示扩展至Jacobi子群与Jacobi群，并推广至非阿基米德局部域等情形。

New Sequences Design from Weil Representation with Low Two-Dimensional Correlation in Both Time and Phase Shifts

For a given prime $p$, a new construction of families of the complex valued sequences of period $p$ with efficient implementation is given by applying both multiplicative characters and additive characters of finite field $\mathbb{F}_p$. Such a signal set consists of $p^2(p-2)$ time-shift distinct sequences, the magnitude of the two-dimensional autocorrelation function (i.e., the ambiguity function) in both time and phase of each sequence is upper bounded by $2\sqrt{p}$ at any shift not equal to $(0, 0)$, and the magnitude of the ambiguity function of any pair of phase-shift distinct sequences…

Exponential sums and total Weil representations of finite symplectic and unitary groups

We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for all $n \ge 2$, and others whose geometric monodromy groups are the special unitary groups $SU_n(q)$ for all odd $n \ge 3$, and $q$ any power of $p$, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one-parameter families of exponential sums of a very simple, i.e., easy to remember, form. We also exhibit hypergeometric sheaves on $G_m$, whose geometric monodro…

有限酉群的 Heisenberg-Weil 表示的几何构造与 Howe 对应

Geometric construction of Heisenberg-Weil representations for finite unitary groups and Howe correspondences

本文通过代数簇的中阶 étale 上同调几何构造了有限酉群的 Heisenberg-Weil 表示，并利用 Frobenius 作用实现了任意有限域上 $(\mathit{Sp}_{2n},O_2^-)$ 的 Howe 对应，证明了幂单性在该对应下保持不变。

On the regularized Siegel-Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups

We derive a (weak) second term identity for the regularized Siegel-Weil formula for the even orthogonal group, which is used to obtain a Rallis inner product formula in the "second term range". As an application, we show the following non-vanishing result of global theta lifts from orthogonal groups. Let $π$ be a cuspidal automorphic representation of an orthogonal group $O(V)$ with $\dim V=m$ even and $r+1\leq m\leq 2r$. Assume further that there is a place $v$ such that $π_v\congπ_v\otimes\det$. Then the global theta lift of $π$ to $Sp_{2r}$ does not vanish up to twisting by automorphic dete…

Borel-Weil Theory for Root Graded Banach-Lie groups

In this paper we introduce (weakly) root graded Banach--Lie algebras and corresponding Lie groups as natural generalizations of group like $\GL_n(A)$ for a Banach algebra $A$ or groups like $C(X,K)$ of continuous maps of a compact space $X$ into a complex semisimple Lie group $K$. We study holomorphic induction from holomorphic Banach representations of so-called parabolic subgroups $P$ to representations of $G$ on holomorphic sections of homogeneous vector bundles over $G/P$. One of our main results is an algebraic characterization of the space of sections which is used to show that this spac…

Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras

In this paper, we use geometrical methods adapted from the Borel-Weil-Bott theory to compute the character of every finite dimensional simple module over a basic classical Lie superalgebra.