Proper r-harmonic functions from Riemannian manifolds

Abstract

We introduce a new method for constructing complex-valued r -harmonic functions on Riemannian manifolds. We then apply this for the important semisimple Lie groups $$\\mathbf{SO }(n)$$ SO ( n ) , $$\\mathbf{SU }(n)$$ SU ( n ) , $$\\mathbf{Sp }(n)$$ Sp ( n ) , $$\\mathbf{SL }_{n}({\\mathbb {R}})$$ SL n ( R ) , $$\\mathbf{Sp }(n,{\\mathbb {R}})$$ Sp ( n , R ) , $$\\mathbf{SU }(p,q)$$ SU ( p , q ) , $$\\mathbf{SO }(p,q)$$ SO ( p , q ) , $$\\mathbf{Sp }(p,q)$$ Sp ( p , q ) , $$\\mathbf{SO }^*(2n)$$ SO ∗ ( 2 n ) and $$\\mathbf{SU }^*(2n)$$ SU ∗ ( 2 n ) .