  The classical Airy function has been generalised by Kontsevich to a function
of a matrix argument, which is an integral over the space of (skew) hermitian
matrices of a unitary-invariant exponential kernel. In this paper, the
Kontsevich integral is generalised to integrals over the Lie algebra of an
arbitrary connected compact Lie group, using exponential kernels invariant
under the group. The (real) polynomial defining this kernel is said to have the
Airy property if the integral defines a function of moderate growth. A general
sufficient criterion for a polynomial to have the Airy property is given. It is
shown that an invariant polynomial on the Lie algebra has the Airy property if
its restriction to a Cartan subalgebra has the Airy property. This result is
used to evaluate these invariant integrals completely and explicitly on the
hermitian matrices, obtaining formulae that contain those of Kontsevich as
special cases.
