Bounds for asymptotic characters of simple Lie groups

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 $D{H}_{\lambda }(p)$ (continuous limit of weight multiplicities). It is shown in arXiv:2312.03101 that the best $\lambda$-independent upper bound $-c(G)$ for ${\mathrm{i}\mathrm{n}\mathrm{f}}_x\mathrm{R}\mathrm{e}X(\lambda ,x)$ 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 $D{H}_{\lambda }(0)$ when $\mathrm{\mid }\lambda \mathrm{\mid }=1$. This allows us to show that $\mathrm{\mid }X(\lambda ,x)\mathrm{\mid }\le C(G)\mathrm{\mid }\lambda {\mathrm{\mid }}^{-1}\mathrm{\mid }x{\mathrm{\mid }}^{-1}$ 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 $c(S{L}_n)\le (\frac{4}{{\pi }^2}{)}^{n-2}$. 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 $G$.