Let $K$ be a $p$-adic field and $\chi : Gal\_K \rightarrow \mathbb{Q}\_p^\times$ be a character. I know that $\chi$ is Hodge-Tate of weight $0$ iff $\chi(I\_K)$ is finite (by Sen's theory), and that it is Hodge-Tate of weight $k$ iff $\chi.\chi\_p^{-k}$ is HT of weight $0$.


Is there a similar description for De Rham, Semi-stable and Cristalline ? 



What is the classification of characters in $p$-adic Hodge theory?
The de Rham characters are the same as the Hodge-Tate ones. The semistable ones are the same as the crystalline ones, and in your notation they are the de Rham ones for which $(\chi \cdot \chi\_p^{-k})(I\_K)$ is trivial (and not merely finite). This can be found for example in Fontaine and Mazur's paper.


