In this talk, we will report on progress towards a conjecture of B. Krötz about the holomorphic extensions of non-zero K-finite vectors of irreducible admissible Banach representations of simple real Lie groups and the relation to a distinguished domain – the so-called crown domain. We will explain some of the main ideas – the Casselman–Wallach smooth globalisation, vanishing of matrix coefficients at infinity etc. Indeed we prove the conjecture with some additional growth conditions on the Banach globalisations. This is joint work with Gang Liu (Univ. Metz).

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Last updated: 06 Aug 2024