Spectral cones in Euclidean Jordan algebras

A spectral cone in a Euclidean Jordan algebra V of rank n is of the form K=λ−1(Q), where Q is a permutation invariant convex cone in Rn and λ:V→Rn is the eigenvalue map (which takes x to λ(x), the vector of eigenvalues of x with entries written in the decreasing order). In this paper, we describe some properties of spectral cones. We show, for example, that spectral cones are invariant under automorphisms of V, that the dual of a spectral cone is a spectral cone when V is simple or carries the canonical inner product, and characterize the pointedness/solidness of a spectral cone. We also show that for any spectral cone K in V, dim⁡(K)∈{0,1,m−1,m}, where dim⁡(K) denotes the dimension of K and m is the dimension of V.