This talk presents recent developments in spectral analysis of locally symmetric spaces beyond the Riemannian framework. In particular, I will focus on settings with indefinite metrics, where both geometric and analytic challenges arise. These challenges are being addressed through a new approach based on branching theory of infinite-dimensional representations of reductive groups, which sheds light on broken symmetries in function spaces. I will outline the key ideas behind these advances in an accessible and non-technical manner.
© Toshiyuki Kobayashi