My question is about p-adic Hodge-Tate theory and p-adic Galois representation.
What is the definition of $\text{weight or Hodge-Tate weight}$ in the above theory?
For example, I have the following two definitions of weights in the above context as follows:
Def 1: (https://math.uchicago.edu/~emerton/pdffiles/egh.pdf)
A $\text{weight}$ is an isomorphism class of irreducible representations $V$ of $G(k)$ over $\bar k$, where $k$ is the residue field in the finite extension of $\mathbb{Q}_p$. Since $G(k)$ is finite, there are finitely many weights.
Def 2:(http://math.bu.edu/people/bergdall/seminars/hodge-notes/week11.pdf)
A Hodge-Tate weight $i \in \mathbb{Z}$ is defined such that $Gr^{i}D_{HT}(V)=(V \otimes_{\mathbb{Q}_p} \mathbb{C}_p)^{Gal}=\frac{Fil^{i}D_{dR}(V)}{Fil^{i+1}D_{dR}(V)} \neq 0$.
This different definitions makes me into confusion though the later one is a definition of HT weight while the first one is simply weight.
Are the above two weights same or different?
Can someone explain and gives definition of weight in the above sense?