## Tempered Homogeneous Spaces. Special lecture at the functional analysis session, MSJ Spring meeting 2021. Keio University (online), 16 March 2021. ɑȓ. 2021Nx{wN316iFc`mwjCʍui͊wȉj

Let $G$ be a reductive Lie groups, $H$ an algebraic subgroup, and $X =G/ H$. Joint with Y.~Benoist, we have established a geometric criterion which detects whether the regular representation of $G$ in $L^2(X)$ is tempered. The proof employs analytic and dynamical approaches. Moreover, we have given a complete description for which $L^2(X)$ is tempered by algebraic and combinatorial method. If time permits, I would like to discuss also its relations with deformation of Lie algebras, and with geometric quantization from the orbit philosophy. {\textbf{Reference:}} Y.~Benoist and T.~Kobayashi, Tempered Homogeneous Spaces I (J. Euro. Math, 17 (2015), 3015--3036); II (Margulis Festschrift, Univ Chicago Press, to appear, available also at arXiv 1706.10131); III (preprint 2020, arXiv 2009.10389); IV (preprint 2020, arXiv 2009.10391).