Bernhard Krötz’s research while affiliated with Paderborn University and other places

In this article we give an overview of the Plancherel theory for Riemannian symmetric spaces Z = G/K. In particular we illustrate recently developed methods in Plancherel theory for real spherical spaces by explicating them for Riemannian symmetric spaces, and we explain how Harish-Chandra's Plancherel theorem for Z can be proven from these methods.

Our concern is with Riemannian symmetric spaces Z=G/K of the non-compact type and more precisely with the Poisson transform $\mathcal{P}_\lambda$ which maps generalized functions on the boundary $\partial Z$ to $\lambda$-eigenfunctions on Z. Special emphasis is given to a maximal unipotent group $N<G$ which naturally acts on both Z and $\partial Z$. The N-orbits on Z are parametrized by a torus $A=(\mathbb{R}_{>0})^r<G$ (Iwasawa) and letting the level $a\in A$ tend to 0 on a ray we retrieve N via $\lim_{a\to 0} Na$ as an open dense orbit in $\partial Z$ (Bruhat). For positive parameters $\lambda$ the Poisson transform $\mathcal{P}_\lambda$ is defined an injective for functions $f\in L^2(N)$ and we give a novel characterization of $\mathcal{P}_\lambda(L^2(N))$ in terms of complex analysis. For that we view eigenfunctions $\phi = \mathcal{P}_\lambda(f)$ as families $(\phi_a)_{a\in A}$ of functions on the N-orbits, i.e. $\phi_a(n)= \phi(na)$ for $n\in N$. The general theory then tells us that there is a tube domain $\mathcal{T}=N\exp(i\Lambda)\subset N_\mathbb{C}$ such that each $\phi_a$ extends to a holomorphic function on the scaled tube $\mathcal{T}_a=N\exp(i\operatorname{Ad}(a)\Lambda)$. We define a class of N-invariant weight functions ${\bf w}_\lambda$ on the tube $\mathcal{T}$, rescale them for every $a\in A$ to a weight ${\bf w}_{\lambda, a}$ on $\mathcal{T}_a$, and show that each $\phi_a$ lies in the $L^2$-weighted Bergman space $\mathcal{B}(\mathcal{T}_a, {\bf w}_{\lambda, a}):=\mathcal{O}(\mathcal{T}_a)\cap L^2(\mathcal{T}_a, {\bf w}_{\lambda, a})$. The main result of the article then describes $\mathcal{P}_\lambda(L^2(N))$ as those eigenfunctions $\phi$ for which $\phi_a\in \mathcal{B}(\mathcal{T}_a, {\bf w}_{\lambda, a})$ and $\|\phi\|:=\sup_{a\in A} a^{\operatorname{Re}\lambda -2\rho} \|\phi_a\|_{\mathcal{B}_{a,\lambda}}<\infty$ holds.

We give an example of a semisimple symmetric space G/H and an irreducible representation of G which has multiplicity 1 in L2(G/H) and multiplicity 2 in C∞(G/H).

We explain by elementary means why the existence of a discrete series representation of a real reductive group G implies the existence of a compact Cartan subgroup of G . The presented approach has the potential to generalize to real spherical spaces.

This paper lays the foundation for Plancherel theory on real spherical spaces Z = G / H Z=G/H , namely it provides the decomposition of L 2 ( Z ) L^2(Z) into different series of representations via Bernstein morphisms. These series are parametrized by subsets of spherical roots which determine the fine geometry of Z Z at infinity. In particular, we obtain a generalization of the Maass-Selberg relations. As a corollary we obtain a partial geometric characterization of the discrete spectrum: L 2 ( Z ) d i s c ≠ ∅ L^2(Z)_{\mathrm {disc}}\neq \emptyset if h ⊥ \mathfrak {h}^\perp contains elliptic elements in its interior. In case Z Z is a real reductive group or, more generally, a symmetric space our results retrieve the Plancherel formula of Harish-Chandra (for the group) as well as that of Delorme and van den Ban-Schlichtkrull (for symmetric spaces) up to the explicit determination of the discrete series for the inducing datum.

We give an example of a semisimple symmetric space G/H and an irreducible representation of G which has multiplicity 1 in $L^2(G/H)$ and multiplicity 2 in $C^\infty(G/H)$.

We define Sobolev norms of arbitrary real order for a Banach representation (π,E) of a Lie group, with regard to a single differential operator D=dπ(R2+Δ). Here, Δ is a Laplace element in the universal enveloping algebra, and R>0 depends explicitly on the growth rate of the representation. In particular, we obtain a spectral gap for D on the space of smooth vectors of E. The main tool is a novel factorization of the delta distribution on a Lie group.