Bounds for asymptotic characters of simple Lie groups

TL;DR

This paper establishes bounds on the asymptotic characters of complex simple Lie groups, providing new quantitative estimates and confirming a conjecture related to their decay properties.

Contribution

It offers explicit bounds for the asymptotic character function and its related Duistermaat-Heckman function, advancing understanding of their asymptotic behavior and confirming a key conjecture.

Findings

Lower bound for $c(G)$ in terms of $ ext{dim } G$

Upper and lower bounds for $DH_ ext{lambda}(0)$ when $| ext{lambda}|=1$

Proof of a conjecture on Mittag-Leffler sums for $G$

Abstract

An important function attached to a complex simple Lie group $G$ is its asymptotic character $X(\lambda,x)$ (where $\lambda,x$ are real (co)weights of $G$) - the Fourier transform in $x$ of its Duistermaat-Heckman function $DH_\lambda(p)$ (continuous limit of weight multiplicities). It is shown in arXiv:2312.03101 that the best $\lambda$-independent upper bound $-c(G)$ for ${\rm inf}_x{\rm Re}X(\lambda,x)$ for fixed $\lambda$ is strictly negative. We quantify this result by providing a lower bound for $c(G)$ in terms of $\dim G$. We also provide upper and lower bounds for $DH_\lambda(0)$ when $|\lambda|=1$. This allows us to show that $|X(\lambda,x)|\le C(G)|\lambda|^{-1}|x|^{-1}$ for some constant $C(G)$ depending only on $G$, which implies the conjecture in Remark 17.16 of arXiv:2312.03101. We also show that $c(SL_n)\le (\frac{4}{\pi^2})^{n-2}$. Finally, in the appendix, which subsumes our previous paper arXiv:1811.05293, we prove Conjecture 1 in arXiv:1706.02793 about Mittag-Leffler type sums for $G$.