## 2023冬学期

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