Related papers: Models of representations and Langlands functorial…
We use the endoscopic classification of automorphic representations of even-dimensional unitary groups to construct level-raising congruences.
We construct a family of Whittaker functions for $SL(m,\mathbb{Z})$ induced directly from Whittaker functions for $SL(n,\mathbb{Z})$, for any $2 \leq m<n$. Given Jacquet's Whittaker function $W_{\alpha,N}^{(n)}$ on the generalized upper…
We suggest a two-matrix model depending on three (infinite) sets of parameters which interpolates between all the models proposed in arXiv:2206.13038, and defined there through $W$-representations. We also discuss further generalizations of…
Distributed representations (such as those based on embeddings) and discrete representations (such as those based on logic) have complementary strengths. We explore one possible approach to combining these two kinds of representations. We…
We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2-representations…
A unified description of the relationship between the Hamiltonian structure of a large class of integrable hierarchies of equations and W-algebras is discussed. The main result is an explicit formula showing that the former can be…
In these lectures we provide a basic introduction into the topic of dispersion relation and analyticity. The properties of 2-point functions are discussed in some detail from the viewpoint of the K\"all\'en-Lehmann and general dispersion…
A number of recent papers treated the representation theory of partially ordered sets in unitary spaces with the so called orthoscalar relation. Such theory generalizes the classical theory which studies the representations of partially…
The near-completion of the program of endoscopy poses the question of what lies next. This article takes a broad view of ideas beyond the program of endoscopy, highlighting the connections among them, and emphasizing the relationship…
We use the relations between the base change representations, theta lifts and Whittaker model, to give a new proof to the period problems of $GL(2)$ over a quadratic local field extension $E/F.$ And we classify both local and global…
This is a report on the global aspects of the Langlands-Shahidi method which in conjunction with converse theorems of Cogdell and Piatetski-Shapiro has recently been instrumental in establishing a significant number of new and surprising…
For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are…
Self-adjoint Dirac systems and subclasses of canonical systems, which generalize Dirac systems are studied. Explicit and global solutions of direct and inverse problems are obtained. A local Borg-Marchenko-type theorem, integral…
A foundational investigation of the basic structural properties of two-dimensional anomalous gauge theories is performed. The Hilbert space is constructed as the representation of the intrinsic local field algebra generated by the…
By applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of ${\rm GL}_n$ over $p$-adic fields. In each case, we…
We study higher rank Jacobi partial and false theta functions (generalizations of the classical partial and false theta functions) associated to positive definite rational lattices. In particular, we focus our attention on certain Kostant's…
Whittaker functions are special functions that arise in $p$-adic number theory and representation theory. They may be defined on representations of reductive groups as well as their metaplectic covering groups: fascinatingly, many of their…
We give a fully explicit description of Lie algebra derivatives (generalizing raising and lowering operators) for representations of SL(3,R) in terms of a basis of Wigner functions. This basis is natural from the point of view of principal…
An extensive group-theoretical treatment of linear relativistic field equations on Minkowski spacetime of arbitrary dimension D>2 is presented in these lecture notes. To start with, the one-to-one correspondence between linear relativistic…
The theory of intertwining operators plays an important role in the development of the Langlands program. This, in some sense, is a very sophisticated theory, but the basic question of its singularity, in general, is quite unknown.…