Signatures of invariant forms on finite-dimensional representations


主讲人:David Vogan,麻省理工学院教授




主讲人先容: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。

内容先容: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.

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