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Linear neural network layers that are either equivariant or invariant to permutations of their inputs form core building blocks of modern deep learning architectures. Examples include the layers of DeepSets, as well as linear layers…

We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations…

The degree of the generators of invariant polynomial rings of is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite…

We consider four classes of classical groups over a non-archimedean local field F: symplectic, (special) orthogonal, general (s)pin and unitary. These groups need not be quasi-split over F. The main goal of the paper is to obtain a local…

Selfdual representations of any group fall into two classes when they are irreducible: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the…

This article has a twofold purpose. First, by recent works of Kaletha and Loke-Ma, we give an explicit description of the local theta correspondence between regular supercuspidal representations in the equal rank symplectic-orthogonal case.…

This article is a continuation of work on construction and calculation various of modifications of invariant based on the use Euclidean metric values attributed to elements of manifold triangulation. We again address the well investigated…

With a view towards applications in the theory of infinite-dimensional representations of finite-dimensional Lie supergroups, we introduce a new category of supermanifolds. In this category, supermanifolds of `maps' and `fields' (fibre…

Suppose a finite group acts on a scheme $X$ and a finite-dimensional Lie algebra $\mathfrak{g}$. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from $X$ to $\mathfrak{g}$. The irreducible…

Harris and Taylor proved that the supercuspidal part of the cohomology of the Lubin-Tate tower realizes both the local Langlands and Jacquet-Langlands correspondences, as conjectured by Carayol. Recently, Boyer computed the remaining part…

In this paper, we define compact open subgroups of quasi-split even unitary groups for each even non-negative integers, and establish the theory of local newforms for irreducible tempered generic representations with a certain condition on…

In this paper, we explicitly compute the standard epsilon factors on both sides of the local Langlands correspondence for simple supercuspidal representations of GL(n,F).

We construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the…

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in…

Let $F$ be a non-archimedean local field of odd residual characteristic. We compute the Jordan set of a simple cuspidal representation of a symplectic group over $F$, using explicit computations of generators of the Hecke algebras of covers…

We realize all irreducible unitary representations of the group $\mathrm{SO}_0(n+1,1)$ on explicit Hilbert spaces of vector-valued $L^2$-functions on $\mathbb{R}^n\setminus\{0\}$. The key ingredient in our construction is an explicit…

The superselection sectors of two classes of scalar bilocal quantum fields in D>=4 dimensions are explicitly determined by working out the constraints imposed by unitarity. The resulting classification in terms of the dual of the respective…

Theta series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group $G$ is split (or…

We show how, using different decompositions of E(11), one can calculate the representations under the duality group of the so--called "de-form" potentials. Evidence is presented that these potentials are in one-to-one correspondence to the…

We obtain universal models for several types of locally conformal symplectic manifolds via pullback or reduction. The relation with recent embedding results for locally conformal K\"ahler manifolds is discussed.