Mathematics

Koornwinder polynomials are q -orthogonal polynomials equipped with extra five parameters and the B C n -type Weyl group symmetry, which were introduced by Koornwinder (1992) as multivariate analogue of Askey-Wilson polynomials. They are now understood as the Macdonald polynomials associated to the affine root system of type ( C ∨ n , C n ) via the Macdonald-Cherednik theory of double affine Hecke algebras. In this paper we give explicit formulas of Littlewood-Richardson coefficients for Koornwinder polynomials, i.e., the structure constants of the product as invariant polynomials. Our formulas are natural ( C ∨ n , C n ) -analogue of Yip's alcove-walk formulas (2012) which were given in the case of reduced affine root systems.

Through combining the work of Jean-Loup Waldspurger (\cite{waldspurger10} and \cite{waldspurgertemperedggp}) and Raphaël Beuzart-Plessis (\cite{beuzart2015local}), we give a proof for the tempered part of the local Gan-Gross-Prasad conjecture (\cite{ggporiginal}) for special orthogonal groups over any local fields of characteristic zero, which was already proved by Waldspurger over p -adic fields.

Superpolynomials consist of commuting and anti-commuting variables. By considering the anti-commuting variables as a module of the symmetric group the theory of vector-valued nonsymmetric Jack polynomials can be specialized to superpolynomials. The theory significantly differs from the supersymmetric Jack polynomials introduced and studied in several papers by Desrosiers, Mathieu and Lapointe (Nucl. Phys. B606, 2001). The vector-valued Jack polynomials arise in standard modules of the rational Cherednik algebra and were originated by Griffeth (T.A.M.S. 362, 2010) for the family G(n,p,N) of complex reflection groups. In the present situation there is an orthogonal basis of anti-commuting polynomials which corresponds to hook tableaux arising in Young's representations of the symmetric group. The basis is then used to construct nonsymmetric Jack polynomials by specializing the machinery set up in a paper by Luque and the author (SIGMA 7,2011). There is an inner product for which these polynomials form an orthogonal basis, and the squared norms are explicitly found. Supersymmetric polynomials are obtained as linear combinations of the nonsymmetric Jack polynomials contained in a submodule; this is based on an idea of Baker and Forrester (Ann. Comb. 3, 1999). The Poincaré series for supersymmetric polynomials graded by degree is obtained and is interpreted in terms of certain minimal polynomials. There is a brief discussion of antisymmetric polynomials and an application to wavefunctions of the Calogero-Moser quantum model on the circle.

The affine Kauffmann category is a strict monoidal category and can be considered as a q -analogue of the affine Brauer category in (Rui et al. in Math. Zeit. 293, 503-550, 2019). In this paper, we prove a basis theorem for the morphism spaces in the affine Kauffmann category. The cyclotomic Kauffmann category is a quotient category of the affine Kauffmann category. We also prove that any morphism space in this category is free over an integral domain K with maximal rank if and only if the u -admissible condition holds in the sense of Definition 1.13.

We use the language of Lie pseudoalgebras to gain information about the representation theory of the simple infinite-dimensional linearly compact Lie superalgebra of exceptional type E(5,10) . This technology allows us to prove that the degree of singular vectors in minimal Verma modules is ≤14 . A few technical adjustments allow us to refine the bound, proving that the degree must always be ≤12 and it is actually, except for a finite number of cases, ≤10 .

We introduce the shifted q=0 affine algebra. It is similar to the shifted quantum affine algebra defined by Finkelberg-Tsymbaliuk arXiv:1708.01795v6. We give a definition of its categorical action. Then we prove that there is a categorical action of the shifted q=0 affine algebra on the bounded derived categories of coherent sheaves on partial flag varieties. As an application, we use it to construct a categorical action of the q=0 affine Hecke algebra on the bounded derived category of coherent sheaves on the full flag variety.

We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the monoidal category of Harish-Chandra bimodules in terms of a slight modification of the notion of a bialgebroid. Moreover, we show that the standard dynamical quantum groups F(G) and F q (G) are related to parabolic restriction functors for classical and quantum Harish-Chandra bimodules. Finally, we exhibit a natural Weyl symmetry of the parabolic restriction functor using Zhelobenko operators and show that it gives rise to the action of the dynamical Weyl group.

We prove a categorification of the stable elements formula of Cartan and Eilenberg. Our formula expresses the derived category and the stable module category of a group as a bilimit of the corresponding categories for the p -subgroups.

A pair (A,P) is called a cover of End A (P ) op if the Schur functor Hom A (P,−) is fully faithful on the full subcategory of projective A -modules, for a given projective A -module P . By definition, Morita algebras are the covers of self-injective algebras and then P is a faithful projective-injective module. Conversely, we show that A is a Morita algebra and End A (P ) op is self-injective whenever (A,P) is a cover of End A (P ) op for a faithful projective-injective module P .

In this paper, a class of super Heisenberg-Virasoro algebras is introduced on the base of conformal modules of Lie conformal superalgebras. Then we construct a class of simple super Heisenberg-Virasoro modules, which is induced from simple modules of the finite-dimensional solvable Lie superalgebras. These modules are isomorphic to simple restricted super Heisenberg-Virasoro modules, and include the highest weight modules, Whittaker modules and high order Whittaker modules.