C. R. Acad. Sci. Paris,

**Abstract.**We define a one-parameter
family of sigma-finite (finite on compact sets)
measures in the space of distributions. These measures
are equivalent to the laws of the classical gamma processes
and invariant under an infinite-dimensional abelian
group of certain positive multiplicators.
This family of measures
was first discovered by Gelfand-Graev-Vershik
in the context of the representation theory of current groups;
here we describe it in direct terms using some remarkable properties
of the gamma processes.
We show that the class of multiplicative measures coincides with
the class of zero-stable measures which is introduced in the paper.
We give also a new construction
of the canonical
representation of the current group SL(2,R)^X.

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