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• # Signatures of invariant forms on finite-dimensional representations

发布者：文明办发布时间：2019-06-10浏览次数：218

主讲人：David Vogan，麻省理工学院教授

时间：2019年7月30日15：00

地点：徐汇校区西部3号楼301

举办单位：数理学院

主讲人介绍：David Vogan教授目前在麻省理工学院担任Norbert Wiener Professor of  Mathematics。他是美国两院院士，于2013-2014年期间任美国数学会主席。他在李群表示领域四十余年的研究工作重塑了这个学科的面貌。

内容介绍：Suppose $G$ is a real reductive algebraic group, and $\pi$ is an irreducible  complex representation of $G$. It often happens that $\pi$ admits a non-zero  $G$-invariant Hermitian form $\langle\cdot,\cdot\rangle_\pi$. Schur's lemma  guarantees that the form is nondegenerate and unique up to a real scalar; so  Sylvester's theorem says that the only possible signatures are $(p,q)$ and  $(q,p)$. Write $\text{Sig}(\pi) = |p-q|$; the smallness of $\text{Sig}(\pi)$  measures how thoroughly indefinite the form is. The Weyl dimension formula says  that $\dim(\pi)$ is a polynomial of degree equal to $(\dim G - \text{rank}(G))/2$ in the highest weight. I'll prove that $\text{Sig}(\pi)$ is a  quasipolynomial of degree $(\dim K - \text{rank}(K))/2$ in the highest weight,  with $K$ a maximal compact subgroup of $G$. This says (for noncompact $G$) that  the signature is much smaller'' than the dimension, meaning that the form is  very indefinite. This is joint work with MIT undergraduate Christopher Xu and  his grad student mentor Daniil Kalinov.