Abstract: Consider the action of
GL_{n}(Z) on
R^{n}. Clearly it commutes with
homotheties. One may ask whether the group of homotheties
consists of the full centralizer group of that action. More
precisely: Given a measurable self map of
R^{n} (almost everywhere defined)
which commute with every invertible integral matrix, need
that be a multiplication by a scalar?
In my talk I will explain how to affirmatively answer
the question above, and, indeed, how to prove a much more
general statement. The main ingredients of the proof are
- A certain Frobenius reciprocity relating suitable
induction and restriction functors, and
- The Borel
density theorem, essentially describing all invariant
measures on an algebraic varieties.
Both will be clearly explained during the talk. The
talk is based on a joint work with Furman, Gorodnik and
Weiss.