Consider the KPZ equation on a spatial interval $[0,1]$ with mixed Neumann boundary conditions at 0 and 1. For each given pair of boundary parameters $(\mathrm{u}, \mathrm{v})$, there should exist a unique stationary measure for the height profile differences (i.e., for the derivative of the KPZ equation). In this talk I will describe recent work in which we show that for each pair $(u, v)$ satisfying $u+v>0$, certain exponentially reweighted Brownian paths measures are stationary measures for the corresponding open KPZ equation. Along the way, we will also encounter the open ASEP, as well as Askey-Wilson processes and $q$ function asymptotics. This is mainly based on my recent work with Alisa Knizel, though also relies on earlier work with Hao Shen as well as earlier work of Wlodzimierz Bryc, Jacek Wesolowski and Yizao Wang. I will also touch on some recent related work of Wlodzimierz Bryc, Alexey Kuznetsov, Jacek Wesolowski and Yizao Wang; as well as work of Guillaume Barraquand and Pierre Le Doussal.[-]

Superpolynomials are formed with $N$ commuting and anti-commuting (skew) variables. By considering the space of skew variables of fixed degree as a module of the symmetric group $\mathcal{S}_{N}$ the theory of generalized Jack polynomials constructed by S Griffeth can be used to define nonsymmetric Jack superpolynomials. We present the theory, give details about the structure and derive norm formulas. Denote the parameter by $\kappa$ then the norm is positive-definite for $-\frac{1}{N}<\kappa<\frac{1}{N}$. Analogously there is a structure as Hecke algebra $\mathcal{H}_{N}(t)$-module on the skew polynomials and this allows the use of the theory of vectorvalued $(q, t)$-Macdonald polynomials studied by J-G Luque and the author. We outline the theory and present norm formulas and evaluations at special points. The norm is positive-definite for $q>0$ and min $(q^{1 / N}, q^{-1 / N}) < t < max (q^{1 / N}, q^{-1 / N} )$. As in the scalar case the evaluations use $(q, t)$-hook products.[-]

The goal of the talk is to present selected results in real harmonic analysis in the rational Dunkl setting. We shall start by deriving estimates for the generalized translations$$\tau_{\mathbf{x}} f(-\mathbf{y})=c_{k}^{-1} \int_{\mathbb{R}^{N}} E(\mathbf{x}, i \xi) E(\mathbf{y},-i \xi) \mathcal{F} f(\xi) d w(\xi)$$of certain radial and non-radial functions $f$ on $\mathbb{R}^{N}$, including estimates for the integral kernel of the heat Dunkl semigroup. Here $d w(\mathbf{x})=$ $\prod_{\alpha \in R}|\langle\alpha, \mathbf{x}\rangle|^{k(\alpha)} d \mathbf{x}$ denotes the associated measure, $E(\mathbf{x}, \mathbf{y})$ is the Dunkl kernel, and $\mathcal{F} f(\xi)=c_{k}^{-1} \int_{\mathbb{R}^{N}} f(\mathbf{x}) E(-i \xi, \mathbf{x}) f(\mathbf{x}) d w(\mathbf{x})$ is the Dunkl transform. The obtained estimates will be given by means of the distance $d(\mathbf{x}, \mathbf{y})$ of the orbit of $\mathbf{x}$ to the orbit of $\mathbf{y}$ under the action of the reflection group $G$, that is,$$d(\mathbf{x}, \mathbf{y})=\min _{\sigma \in G}\|\sigma(\mathbf{x})-\mathbf{y}\|$$the Euclidean distance $\|\mathbf{x}-\mathbf{y}\|$, and $d w$-volumes of the Euclidean balls and they will be in the spirit of estimates needed in real harmonic analysis on spaces of homogeneous type.Then, if time permits, we shall discuss selected results, parallel to classical ones, which are proved by utilizing the obtained estimates for the generalized translation. In particular, we will be interested in:- boundedness of maximal functions on various function spaces,- characterizations of the real Hardy space $H^{1}$ in the Dunkl setting- boundedness of the Dunkl transform multiplier operators,- boundedness of singular integral operators,- upper and lower bounds for Littlewood-Paley square functions. The results are joint works with Jean-Philippe Anker and Agnieszka Hejna.[-]

Matrix spherical functions associated to the symmetric pair $(G, K)=$ $\left(\mathrm{SU}(m+2), \mathrm{S}(\mathrm{U}(2) \times \mathrm{U}(m))\right.$, having reduced root system of type $\mathrm{BC}_{2}$ are studied. We consider a $K$-representation $\left(\pi, V_{\pi}\right)$ arising from the $\mathrm{U}(2)$-part of $K$, then the induced representation $\operatorname{Ind}_{K}^{G} \pi$ is multiplicity free. The corresponding spherical functions, i.e. $\Phi: G \rightarrow \operatorname{End}\left(V_{\pi}\right)$ satisfying $\Phi\left(k_{1} g k_{2}\right)=\pi\left(k_{1}\right) \Phi(g) \pi\left(k_{2}\right)$ for all $g \in G, k_{1}, k_{2} \in K$, are studied by studying certain leading coefficients. This is done explicitly using the action of the radial part of the Casimir operator on these functions and their leading coefficients. To suitably grouped matrix spherical functions we associate two-variable matrix orthogonal polynomials giving a matrix analogue of Koornwinder's 1970 s two-variable orthogonal polynomials, which are Heckman-Opdam polynomials for $\mathrm{BC}_{2}$. In particular, we find explicit orthogonality relations and the polynomials being eigenfunctions to a second order matrix partial differential operator. This is joint work with Jie Liu (Radboud $\mathrm{U}$ ).[-]