Related papers: Integral Bases and Invariant Vectors for Weil Repr…

Let V be a symplectic vector space over a finite or local field. We compute the character of the Weil representation of the metaplectic group Mp(V). The final formulas are overtly free of choices (e.g. they do not involve the usual choice…

We show that the space of invariants for the Weil representation for discriminant groups which contain self-dual isotropic subgroups is spanned by the characteristic functions of the self-dual isotropic subgroups. As an application, we…

Given a finite-dimensional, complex simple Lie algebra we exhibit an integral form for the universal enveloping algebra of its map algebra, and an explicit integral basis for this integral form. We also produce explicit commutation formulas…

We consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical groups instead of a product of the general linear groups and by…

We introduce the notions of symmetric and symmetrizable representations of $\text{SL}_2(\mathbb{Z})$. The linear representations of $\text{SL}_2(\mathbb{Z})$ arising from modular tensor categories are symmetric and have congruence kernel.…

By a global approach, we prove the arithmetic fundamental lemma conjecture for unitary groups in $n$ variables over $\mathbb{Q}_p$ when $p\geq n$.

Let $r$ be an odd prime and $\mathbb{F}$ a field containing a primitive $r$th root of unity. Then for all $\ell \geq 1$, there is a faithful representation $f: \operatorname{Sp}_{2\ell}(r) \rightarrow \operatorname{GL}_{r^\ell}(\mathbb{F})$…

The Weil representation of a real symplectic group $Sp(2n,R)$ admits a canonical extension to a holomorphic representation of a certain complex semigroup consisting of Lagrangian linear relations (this semigroup includes the Olshanski…

Given a minuscule representation of a simple Lie algebra, we find an algebraic model for the action of a regular element and show that these models can be glued together over the adjoint quotient, viewed as the set of all regular conjugacy…

We demonstrate that the notions of derivative representation of a Lie algebra on a vector bundle, of semi-linear representations of a Lie group on a vector bundle, and related concepts, may be understood in terms of representations of Lie…

Let $V$ be a finite abelian group of odd order, equipped with a non-degenerate, alternating form $\omega\colon V\times V \to \mathbb{Z}/m\mathbb{Z}$. We give closed formulas for the character values of the Weil representation associated…

A description is given of the image of the Weil representation of the symplectic group in the Schwartz space and in the space of tempered distributions under the Gaussian integral transform. We also discuss the problem of infinite…

We describe a bigraded generalization of the Weil algebra, of its basis and of the characteristic homomorphism which besides ordinary characteristic classes also maps on Donaldson invariants.

In a previous work the authors gave a conceptual explanation for the linearity of the Weil representation over a finite field k of odd characteristic: There exists a canonical system of intertwining operators between the Lagrangian models…

We study integrable models with $\mathfrak{gl}(2|1)$ symmetry and solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for scalar products of Bethe vectors, when the Bethe parameters obey some relations weaker…

We develop a theory of vector valued automorphic forms associated to the Weil representation $\omega_f$ and corresponding to vector valued modular forms transforming with the ``finite'' Weil representation $\rho_L$. For each prime $p$ we…

We identify $\lie{sl}_{n+1}$--isotypical components of global Weyl modules with natural subspaces in a polynomial ring, and then apply the representation theory of current algebras to classical problems in invariant theory.

We study the space of vector valued theta functions for the Weil representation of a positive definite even lattice of rank two with fundamental discriminant. We work out the relation of this space to the corresponding scalar valued theta…

Weil has generalized the Riemann-von Mangoldt explicit formula linking the prime numbers with the zeros of the zeta function to the set-up of a general algebraic number field K and Dirichlet-Hecke L-function, revealing in the process the…

Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra $X$ corresponds to a Lie algebra morphism $B\to…