| Hörsaal D2
Vortrag: The Geometry of Characters of Reductive Groups
Let $f$ be a smooth function on a compact, connected Lie group $G$. The Fourier coefficients of $f$ are obtained by convolving the function $f$ with each irreducible character of $G$. Therefore, the irreducible characters of $G$ are the fundamental objects of harmonic analysis on $G$. More generally, for a real reductive group $G$ acting on a homogeneous space $X$, the irreducible characters of $G$ are important for Fourier analysis on $X$, though the precise relationship is more complicated.
In this talk, we discuss how to study irreducible characters geometrically. More precisely, we first recall the Harish-Chandra/Kirillov character formula which writes the irreducible characters of a compact Lie group as Fourier transforms of invariant measures on coadjoint orbits. Then we discuss generalizations to real, reductive groups including recent joint work with Yoshiki Oshima.