Biharmonic Functions on the Classical Compact Simple Lie Groups

Abstract

Abstract. The main aim of this work is to construct several new families of proper biharmonic functions defined on open subsets of the classical compact simple Lie groups SU(n), SO(n) and Sp(n). We work in a geometric setting which connects our study with the theory of submersive harmonic morphisms. We develop a general duality principle and use this to interpret our new examples on the Euclidean sphere S 3 and on the hyperbolic space H 3 .

Biharmonic functions on the special unitary group SU(2)

“…The next result describes the first known proper biharmonic functions from the unitary group SU(n), see [7]. Proposition 4.2.…”

“…It was recently shown in [7] that there is an interesting connections between the theory of r-harmonic functions and the notion of harmonic morphisms. In the sequel, we shall often employ the two following results.…”

“…In this section we describe known proper biharmonic functions on the special unitary group SU(n) constructed in [7]. They are quotients of first order homogeneous polynomials in the matrix coefficients of the standard irredible representation π 1 of SU(2).…”

“…Theorem 9.1. [7] For the above situation we have the following duality. A complex-valued function f : W → C is proper r-harmonic if and only if its dual f * : W * → C is proper r-harmonic.…”

“…The first proper biharmonic functions, from open subsets of the classical compact simple Lie groups SU(n), SO(n) and Sp(n), were recently constructed in [7]. They are all quotients of linear combinations of the matrix coefficients for the standard irreducible representation π 1 of the corresponding group.…”