Bounds for asymptotic characters of simple Lie groups
Pavel Etingof, Eric Rains
TL;DR
This work provides quantitative bounds on the asymptotic character $X(\lambda,x)$ of complex simple Lie groups by relating it to the Duistermaat-Heckman measure $DH_\lambda$ via the Kirillov formula. It establishes a dimension-dependent universal lower bound for the minimax $c(G)$, derives upper bounds for $DH_\lambda$ and a uniform decay bound $|X(\lambda,x)|\le C(G)|\lambda|^{-1}|x|^{-1}$, and proves a sharp bound $c(SL_n)\le (4/\pi^2)^{n-2}$. The decay rate of $X(\lambda,x)$ is analyzed, with precise values computed in several families (e.g., $\gamma(SL_n)=1$, $\gamma(Sp_4)=2$), and a corollary confirming a conjecture of Goh–Garibaldi–Röhrle (GGR) for broad parameter ranges. An appendix strengthens the theory with Mittag-Leffler-type sums for root systems, proving the positivity of certain $F_{k,\xi}$ and their decomposition into irreducible characters, including cases related to the Coquereaux–Zuber conjecture.
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 $G$.