Bounds for asymptotic characters of simple Lie groups

Abstract

An important function attached to a complex simple Lie group $G$ is its asymptotic character $X(λ,x)$ (where $λ,x$ are real (co)weights of $G$) - the Fourier transform in $x$ of its Duistermaat-Heckman function $DH_λ(p)$ (continuous limit of weight multiplicities). It is shown in arXiv:2312.03101 that the best $λ$-independent upper bound $-c(G)$ for ${\rm inf}_x{\rm Re}X(λ,x)$ for fixed $λ$ 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 $DH_λ(0)$ when $|λ|=1$. This allows us to show that $|X(λ,x)|\le C(G)|λ|^{-1}|x|^{-1}$ 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 $c(SL_n)\le (\frac{4}{π^2})^{n-2}$. 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 $G$.