In 1960, Sobolev proved that for a finite reflection group $G$, a $G$-invariant cubature formula is of degree $t$ if and only if it is exact for all $G$-invariant polynomials of degree at most $t$. In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type $B$, to other finite reflection groups.

Wir entwickeln eine Strategie zum Beweis der Positivität der Koeffizienten von Kazhdan-Lusztig-Polynomen für beliebige Coxeter-Gruppen.

We develop a strategy to prove the positivity of coefficients of Kazhdan–Lusztig polynomials for arbitrary Coxeter groups.

In an earlier work, the second author proved a general formula for the equivariant Poincaré polynomial of a linear transformation g which normalises a unitary reflection group G, acting on the cohomology of the corresponding hyperplane complement. This formula involves a certain function (called a Z-function below) on the centraliser CG(g), which was proved to exist only in certain cases, for example, when g is a reflection, or is G-regular, or when the centraliser is cyclic. In this work we prove the existence of Z-functions in full generality. Applications include reduction and product formulae for the equivariant Poincaré polynomials. The method is to study the poset L(CG(g)) of subspaces which are fixed points of elements of CG(g). We show that this poset has Euler characteristic 1, which is the key property required for the definition of a Z-function. The fact about the Euler characteristic in turn follows from the ‘join-atom’ property of L(CG(g)), which asserts that if {X1,...,Xk} is any set of elements of L(CG(g)) which are maximal (set theoretically) then their setwise intersection $\bigcap_{i=1}^kX_i$ lies in L(CG(g)). 2000 Mathematical Subject Classification: primary 14R20, 55R80; secondary 20C33, 20G40.

We extend Maxwell’s representation of harmonic polynomials to $h$-harmonics associated to a reflection invariant weight function ${{h}_{k}}$. Let ${{\mathcal{D}}_{i}},\,1\,\le \,i\,\le \,d$, be Dunkl’s operators associated with a reflection group. For any homogeneous polynomial $P$ of degree $n$,we prove the polynomial ${{\left| x \right|}^{2\gamma +d-2+2n}}P\left( \mathcal{D} \right)\left\{ 1/{{\left| x \right|}^{2\gamma +d-2}} \right\}$ is a $h$-harmonic polynomial of degree $n$, where $\gamma \,=\,\sum \,ki$ and $\mathcal{D}\,=\,\left( {{\mathcal{D}}_{1}},\ldots ,{{\mathcal{D}}_{d}} \right)$. The construction yields a basis for $h$-harmonics. We also discuss self-adjoint operators acting on the space of $h$-harmonics.

We study the intertwining operator and $h$-harmonics in Dunkl's theory on $h$–harmonics associated with reflection groups. Based on a biorthogonality between the ordinary harmonics and the action of the intertwining operator $V$ on the harmonics, the main result provides a method to compute the action of the intertwining operator $V$ on polynomials and to construct an orthonormal basis for the space of $h$-harmonics.