Geometric Harmonic Analysis on Homogeneous Cones and Homogeneous Siegel Domains

关于齐次锥的基本相对不变量的阶数与相应的齐次管域(第一类Siegel域)的对称性之间的差距已经有了一个猜想，但我们发现了一个齐次锥来否定这个猜想。我们还发现了一个任意阶且大于或等于3的齐次锥线性同构于其对偶锥。进一步地，我们证明了基本相对不变量恰恰是对应于锥族的复化中右乘法算子行列式的不可约因子。在对称锥的情况下，我们计算了行列式到不可约因子的实际分解。

There has been a conjecture concerning the gaps between the degrees of the basic relative invariants associated to homogeneous cones and the symmetry of the corresponding homogeneous tube domains (Siegel domains of the first kind).However, we found a homogeneous cone that denies the conjecture. Moreover such a cone is found in any rank greater than or equal to 3. We also discovered a homogeneous cone of arbitrary rank greater than or equal to 3 that is linearly isomorphic to its dual cone. Further, we proved that the basic relative invariants are exactly the irreducible factors of the determinant of the right multiplication operators in the complexification of the clan corresponding to the cone. In the case of symmetric cone, we computed an actual decomposition of that determinant into irreducible factors.