Estimates of the Isovariant Borsuk-Ulam Constants of Connected Compact Lie Groups

The isovariant Borsuk-Ulam constant c(G) of a compact Lie group G is defined to be the supremum of c such that the inequality holds whenever there exists a G-isovariant map f: S(V) S(W) between G-representation spheres. In this paper, we shall discuss some properties of c(G) and provide lower estimates of c(G) of connected compact Lie groups, which leads us to some Borsuk-Ulam type results for isovariant maps. We also introduce and discuss the generalized isovariant Borsuk-Ulam constant c(G) for more general smooth G-actions on spheres. The result is considerably different from the case of linear actions.