Bernhard Krötz

The image of the heat kernel transform on Riemannian symmetric spacesof the noncompact type

This overview is essentially the short article [13] written jointly with Olafsson and Stanton. The only dierence lies in the addition of a new section in the beginning where some classical examples (real line and circle) are discussed.

Invariant Stein domains in Stein symmetric spaces anda nonlinear complex convexity theorem

Let X = G/K be a semisimple noncompact Riemannian symmetric space. We may assume that G is algebraic. By our assumption, G sits in its universal complexification GC and so X ⊆ XC := GC/KC. Note that XC is a Stein symmetric space. Observe that the group G does not act properly (i.e., with compact isotropy subgroups) on GC/KC since KC is not compact. In our considerations, a central role is played by certain G-invariant neighborhood Ξ of X in XC which was first studied in [1]. We recall its definition. Write G = NAK for an Iwasawa decomposition of G. We consider G-orbits in XC. Those have a quite complicated parametrization. We restrict ourselves to G-orbits which intersect exp(ia)KC/KC where a: =Lie(A). Among those G-orbits define Ξ as the maximal connected union of Gorbits with compact isotropy subgroups. In [1], it was shown that

Multiplicity bounds and the subrepresentation theorem for real spherical spaces

Let G be a real semi-simple Lie group and H a closed subgroup which admits an open orbit on the flag manifold of a minimal parabolic subgroup. Let V be a Harish-Chandra module. A sharp finite bound is given for the dimension of the space of H-fixed distribution vectors for V and a related subrepresentation theorem is derived. Extended final version. To appear in Trans. Amer. Math. Soc.

The local structure theorem for real spherical varieties

Let

Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces

G

Holomorphic H-spherical distribution vectors in principal series representations

be an algebraic real reductive group and

Finite orbit decomposition of real flag manifolds

Z

On Function Spaces on Symmetric Spaces

a real spherical

Domains of holomorphy for irreducible unitary representations of simple Lie groups

G

Vanishing at infinity on homogeneous spaces of reductive type

-variety, that is, it admits an open orbit for a minimal parabolic subgroup