Related papers: Parabolic induction and restriction functors for r…
The Geometrical Lemma is a classical result in the theory of (complex) smooth representations of $p$-adic reductive groups, which helps to analyze the parabolic restriction of a parabolically induced representation by providing a filtration…
In this paper we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R. We discuss the *-representation theory of such *-algebras on pre-Hilbert spaces over C and develop the…
We prove a conjecture of Rouquier relating the decomposition numbers in category $\mathcal{O}$ for a cyclotomic rational Cherednik algebra to Uglov's canonical basis of a higher level Fock space. Independent proofs of this conjecture have…
We show that quasi-projective relation algebras and directed cylindric algebras are equivalent categorialy. We work out a Godels second incompleteness theorem for finite varibale fragments of first order logic. We show that distinct set…
(Affine) $\mathcal{W}$-algebras are a family of vertex algebras defined by the generalized Drinfeld-Sokolov reductions associated with a finite-dimensional reductive Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, a nilpotent element $f$ in…
The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring…
In arXiv:2011.03313, the author has constructed a category of abstractly automorphic representations for $\mathrm{GL}(2)$ over a function field $F$. This is a symmetric monoidal Abelian category, constructed with the goal of having the…
Let $\mathbf{G}$ be a connected reductive group defined over a locally compact non-archimedean field $F$, let $\mathbf{P}$ be a parabolic subgroup with Levi $\mathbf{M}$ and compatible with a pro-$p$ Iwahori subgroup of $G :=…
We introduce a Cherednik kernel and a hypergeometric function for integral root systems and prove their relation to spherical functions associated with Riemannian symmetric spaces of reductive Lie groups. Furthermore, we characterize the…
The representation theory of deformed oscillator algebras, defined in terms of an arbitrary function of the number operator~$N$, is developed in terms of the eigenvalues of a Casimir operator~$C$. It is shown that according to the nature of…
We introduce the notion of a totally aspherical parameter for a Rational Cherednik algebra. We get an explicit construction of the projective object defining the KZ functor for such parameters. We establish the existence of sufficiently…
We give a new interpretation and proof of the "quasi-particle" type character formulas for integrable representations of the simply-laced affine Kac-Moody algebras through a new "semi-infinite" construction of such representations. We…
Recently quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov [20]--[28] to solve the so-called "inverse Born's rule problem": to construct a representation of probabilistic data by a complex or more general (in…
The \begin{it} Invariance Theorem \end{it} of M. Gerstenhaber and S. D. Schack states that if $\mathbb{A}$ is a diagram of algebras then the subdivision functor induces a natural isomorphism between the Yoneda cohomologies of the category…
We construct and study an explicit simultaneous $\mathscr{Y}$ eigenbasis of Ion and Wu's standard representation of the $^+$stable-limit double affine Hecke algebra for the limit Cherednik operators $\mathscr{Y}_i$. This basis arises as a…
We introduce the concept of a restriction semigroupoid S, which unifies the notion of restriction semigroups and restriction categories within a single structure. We prove a representation theorem, showing that every restriction…
We prove a combinatorial rule for a complete decomposition, in terms of Langlands parameters, for representations of p-adic $GL_n$ that appear as parabolic induction from a large family (ladder representations). Our rule obviates the need…
In this article we continue the study of holonomic modules over sheaves of Cherednik algebras, initiated by the third author in [Tho18]. Working with arbitrary parameters, we first develop a theory of $b$-functions to prove that…
In this article, we give a proof of multiplicativity for $\gamma$-factors, an equality of parabolically induced and inducing factors, in the context of the Braverman-Kazhdan/Ngo program, under the assumption of commutativity of the…
In this paper we describe the characters of irreducible objects in category O for the rational Cherednik algebra associated to GL_2(F_p) over an algebraically closed field of positive characteristic p, for any value of the parameter t and…