多重自由空间的交换差分算子构造

Commuting Toeplitz Operators on Bounded Symmetric Domains and Multiplicity-Free Restrictions of Holomorphic Discrete Series

For any given bounded symmetric domain, we prove the existence of commutative $C^*$-algebras generated by Toeplitz operators acting on any weighted Bergman space. The symbols of the Toeplitz operators that generate such algebras are defined by essentially bounded functions invariant under suitable subgroups of the group of biholomorphisms of the domain. These subgroups include the maximal compact groups of biholomorphisms. We prove the commutativity of the Toeplitz operators by considering the Bergman spaces as the underlying space of the holomorphic discrete series and then applying known mul…

Combinatorics and invariant differential operators on multiplicity free spaces

We study the generalization of shifted Jack polynomials to arbitrary multiplicity free spaces. In a previous paper (math.RT/0006004) we showed that these polynomials are eigenfunctions for commuting difference operators. Our central result now is the "transposition formula", a generalization of Okounkov's binomial theorem (q-alg/9608021) for shifted Jack polynomials. From this formula, we derive an interpolation formula, an evaluation formula, a scalar product, a binomial theorem, and properties of the algebra generated by the multiplication and difference operators.

Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems

We present a shuffle realization of the GKLO-type homomorphisms for shifted quantum affine, toroidal, and quiver algebras, thus generalizing its rational version of arXiv:2104.14518 and the type A construction of arXiv:1811.12137. As an application, this allows us to construct large families of commuting and q-commuting difference operators, in particular, providing a convenient approach to the Q-systems where it proves a conjecture of arXiv:1704.00154.

A pair of commuting hypergeometric operators on the complex plane and bispectrality

We consider the standard hypergeometric differential operator $D$ regarded as an operator on the complex plane $C$ and the complex conjugate operator $\overline D$. These operators formally commute and are formally adjoint one to another with respect to an appropriate weight. We find conditions when they commute in the Nelson sense and write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of the Jacobi transform (synonyms: generalized Mehler--Fock transform, Olevskii transform). We also show that the inverse transform is an operator of spectral …

B型费米子不变量环

Type $B$ fermionic coinvariant rings

该研究基于Kim–Rhoades的工作，给出B型费米子不变量环的双分次Frobenius级数显式公式，确定相关特征标重数，首次得到A、B型中对所有参数成立的非平凡特征标结果。

单位根步长差分方程的单值本征向量

Monodromy Eigenvectors for Difference Equations with Root-of-Unity Step

本文针对单位根步长的qKZ型离散平坦联络，构造其单值算子的本征截面与本征值，方法基于离散Gauss-Manin联络表示。