We introduce an infinite-dimensional [Formula: see text]-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However, it is possible to define its action on some classes of functions.

中文翻译：

---

无限维p进仿射群的表示

我们引入了一个无限维[公式：见正文]-进数仿射群，并构造了它的不可约酉表示。我们的方法遵循 Vershik、Gelfand 和 Graev 用于微分同胚群的方法，但由于群不作用于相空间这一事实而进行了必要的修改。但是，可以在某些类函数上定义它的作用。