The Cherednik kernel and generalized exponents

Opdam-Cherednik核是正测度的拉普拉斯变换

The Opdam-Cherednik kernel is the Laplace transform of a positive measure

本文证明Opdam-Cherednik核可表示为支撑在其参数Weyl群轨道凸包上的正测度的拉普拉斯变换，从而三角Dunkl交织算子保持正性。

根系统 $A_1$ 的 Opdam-Cherednik $W$-不变热核的锐估计

Sharp estimates for the Opdam-Cherednik $W$-invariant heat kernel for the root system $A_1$

本文首次系统研究弯曲黎曼空间与 Opdam-Cherednik 拉普拉斯算子的非中心 Weyl 群不变热核估计，并针对任意重数 $k>0$ 的根系统 $A_1$ 证明了锐估计，同时提出任意根系统下径向热核锐估计的猜想。

W-exponentials, Schur elements, and the support of the spherical representation of the rational Cherednik algebra

Given a complex reflection group W we compute the support of the spherical irreducible module of the rational Cherednik algebra of W in terms of the simultaneous eigenfunction of the Dunkl operators and Schur elements for finite Hecke algebras.

Representations of rational Cherednik algebras of G(m,r,n) in positive characteristic

We study lowest-weight irreducible representations of rational Cherednik algebras attached to the complex reflection groups G(m,r,n) in characteristic p. Our approach is mostly from the perspective of commutative algebra. By studying the kernel of the contravariant bilinear form on Verma modules, we obtain formulas for Hilbert series of irreducible representations in a number of cases, and present conjectures in other cases. We observe that the form of the Hilbert series of the irreducible representations and the generators of the kernel tend to be determined by the value of n modulo p, and ar…

Supports of irreducible spherical representations of rational Cherednik algebras of finite Coxeter groups

We determine the support of the irreducible spherical representation (i.e., the irreducible quotient of the polynomial representation) of the rational Cherednik algebra of a finite Coxeter group for any value of the parameter c. In particular, we determine for which values of c this representation is finite dimensional. This generalizes a result of Varagnolo and Vasserot, arXiv:0705.2691, who classified finite dimensional spherical representations in the case of Weyl groups and equal parameters (i.e., when c is a constant function). As an application, we compute the zero set of the kernel of t…

Generalized exponents of small representations. II

This is the second paper in a sequence devoted to giving manifestly non-negative formulas for generalized exponents of small representations in all types. It contains a first formula for generalized exponents of small weights which extends the Shapiro-Steinberg formula for classical exponents. The formula is made possible by a computation of Fourier coefficients of the degenerate Cherednik kernel. Unlike the usual partition function coefficients, the answer reflects only the combinatorics of minimal expressions as a sum of roots.