Bounds for asymptotic characters of simple Lie groups

An important function attached to a complex simple Lie group G is its asymptotic character X (, x) (where, x are real (co) weights of G) - the Fourier transform in x of its Duistermaat-Heckman function DH_ (p) (continuous limit of weight multiplicities). It is shown in arXiv: 2312. 03101 that the best -independent upper bound -c (G) for infₓ ReX (, x) for fixed is strictly negative. We quantify this result by providing a lower bound for c (G) in terms of G. We also provide upper and lower bounds for DH_ (0) when ||=1. This allows us to show that |X (, x) | C (G) ||^-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 (SLₙ) (4²) ^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.