Generically Stable Type in an NIP Theory

Let $$T$$ be a complete NIP theory, with monster model $$\mathbb{U}$$. A generically stable type is a global type $$p(x)$$ on $$\mathbb{U}$$ which is $$C$$-invariant for some small set $$C$$, satisfying one of the following equivalent conditions:


 * $$p(x)$$ is definable, and finitely satisfiable in some small set
 * $$p(x) \otimes p(y) = p(y) \otimes p(x)$$.  Here, $$p \otimes q$$ denotes the product of two global invariant types.
 * $$p(x) \otimes q(y) = q(y) \otimes p(x)$$ for every global invariant type $$q(y)$$.
 * The Morley sequence for $$p(x)$$ is totally indiscernible.

These conditions generalize the properties of types in stable theories.

In a stable theory, all global types are generically stable. In an o-minimal theory, there are essentially no totally indiscernible sequences, so the only generically stable types are constant. In a general NIP theory, stable types are generically stable, as are stably dominated types. In ACVF, the generically stable types turn out to be exactly the same as the stably dominated types.