Dear all,
Today's speaker at the Analysis and Probability Seminar is Xiaocheng Li
from the University of Wisconsin-Madison. The title of his talk is "An
Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$".
The talk will begin at 1:30pm (Eastern Time) on Zoom, doors open at
1:25pm. The details are as follows:
Join Zoom Meeting
https://nam10.safelinks.protection.outlook.com/?url=https%3A%2F%2Fzoom.us%2Fj%2F98210782729%3Fpwd%3DQm51dnJ0N25HU0F3c2VuQ1R0dy9HQT09&data=04%7C01%7C%7C9ff86b9f5e5e4778732208d89859e97d%7C17f1a87e2a254eaab9df9d439034b080%7C0%7C1%7C637426859353951998%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=sNx%2F%2Bluv93xAV4dZaNthJWbV1J5GvZvXaI5gEkehfyw%3D&reserved=0
Meeting ID: 982 1078 2729
Passcode: AP
Abstract: We prove an estimate for spherical functions
$\varphi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing
uniform decay in the spectral parameter $\lambda$ when the group
variable $a$ is restricted to a compact subset of the abelian subgroup
$\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a
result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing
the limitation that $a$ should remain regular. As in their work, we
estimate the oscillatory integral that appears in the integral formula
for spherical functions by the method of stationary phase. However, the
major difference is that we investigate the stability of the
singularities arising from the linearized phase function by classifying
their local normal forms when $\lambda$ and $a$ vary.
Best,
Yunfeng
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