数物先端科学IV / Frontiers of Mathematical Sciences and Physics IV
幾何学XE / Geometry and Topolpgy XE
リー群の無限次元表現論とその手法
Lie Groups and Analytic Approach to Infinite-dimensional Representation Theory
リー群およびリー環の有限次元および無限次元の表現の理論について、代数的手 法だけではなく、解析的なアイディアおよび幾何的な手法について基本的に重要 な事柄を簡単な例を用いて解説し、時間が許せば最先端の話題にも触れる。
Inspired by the traditional concepts of generating functions for orthogonal polynomials, I have introduced a novel notion called " generating operators" for a family of differential operators between two manifolds. In the lecture, I started with classical examples like the generating functions for Catalan numbers, Jacobi polynomials, and heat kernels. Next, I presented a new explicity formula for the generating operators of Rankin–Cohen brackets using higher-dimensional residue calculus. Subsequently, we delved into some classical analysis of Hilbert spaces of holomorphic functions, including the Hardy space and weighted Bergman spaces. Following that, I explained fundamental ideas of analytic representation theory, using the infinite-dimensional representations of SL(2,R) as an example, and examined intertwining operators and symmetry breaking operators between representations. Finally, I discussed the generating operators for the Rankin-Cohen brackets from the viewpoint of representation theory.
© Toshiyuki Kobayashi