國立清華大學 數學系

Abstract:
The field of $p$-adic numbers, $\mathbb{Q}_p$, is a completion of the field of rational numbers similar to $\mathbb{R}$. Analysis over $\mathbb{R}$ and $\mathbb{C}$ that has an algebraic flavor can often performed over $\mathbb{Q}_p$. For example, on real Lie groups we have the subject of harmonic analysis, such as Fourier analysis on $S^1$. 
When we have a Lie group acting on a geometric object, such as $\mathbb{R}^{\times}$ acting on a hyperbola, an object of interest is its asymptotic. More generally, a Lie group or $p$-adic group can act on its Lie algebra and each orbit has an asymptotic cone. These asymptotic cones are building blocks for the singularity of the characters of the group. In this talk, we give examples of the above formalism and talk about new results on asymptotic cones and characters for $p$-adic reductive groups.