Classification of Invariant Differential Operators for Non-Compact Lie Algebras via Parabolic Relations

Abstract: In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduced recently the new notion of {\it parabolic relation} between two non-compact semisimple Lie algebras $\cal G$ and $\cal G'$ that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra $E_{7(7)}$ which is parabolically related to the CLA $E_{7(-25)}$. Other interesting examples are the orthogonal algebras $so(p,q)$ all of which are parabolically related to the conformal algebra $so(n,2)$ with $p+q=n+2$, the parabolic subalgebras including the Lorentz subalgebra $so(n-1,1)$ and its analogs $so(p-1,q-1)$. Further we consider the algebras $sl(2n,R)$ and for $n=2k$ the algebras $su^*(4k)$ which are parabolically related to the CLA $su(n,n)$. Further we consider the algebras $sp(r,r)$ which are parabolically related to the CLA $sp(2r,R)$. We consider also $E_{6(6)}$ and $E_{6(2)}$ which are parabolically related to the hermitian symmetric case $E_{6(-14)}$.