Journal of Lie Theory

Let G be an inductive limit group of general finite-dimensional linear groups which is acting from the right on the space X of three infinite rows equipped with a Gaussian measure μ. The involved action "respects" the measure; that is the right action is admissible. The unitary representation of the group G on the space L²(X, μ) appears naturally. We give an irreducibility criterion in terms of the action of the group GL(3, R) from the left on X. Namely, we prove that this representation is irreducible if and only if all non trivial left actions are not admissible. This is also a manifestation of a phenomenon predicted by the Ismagilov conjecture, see below. To prove the irreducibility we show that the von Neumann algebra generated by the representation contains certain abelian subalgebras. This is a consequence of the orthogonality and can be seen as a kind of ergodic theorem (comparable to the Law of Large Numbers, but more subtle). More precisely, the elements of the corresponding commutative subalgebras can be approximated (in the strong resolvent sense) by combinations of generators of one-parameter groups. This approximation being optimal at every finite step, represents the best possible outcome under the given conditions. Its construction relies mainly on the properties of the generalized characteristic polynomial, an explicit expression for the minimum of the quadratic form on a hyperplane, and a theorem regarding the height of an infinite parallelotope.