Let $K$ be a finite extension of the field of $p$-adic numbers $\mathbb{Q}\_p$ and let $E$ be another such extension, such that all the $\mathbb{Q}\_p$ embeddings $K \to \bar{\mathbb{Q}}\_p$ are contained in $E$.


Let $V$ be an $n$-dimensional vector space over $E$ which carries a continuous action of $G\_K = Gal(\bar{K} / K)$.


Assume the corresponding representation (still denoted $V$) to be crystalline, that is the $K\otimes\_{\mathbb{Q}\_p}E$-module $D\_crys(V) = (B\_{crys} \otimes\_{\mathbb{Q}\_p} V)^{G\_K}$ is free of rank $n$. (Here $B\_{crys}$ is one of Fontaine's ring of $p$-adic periods).


There is a filtration on $B\_{crys}$ (induced by another period ring) and so, we have a filtration on $D\_{crys}(V)$ as well and so an induced graduation.


We say that an integer $i$ is a Hodge-Tate weight of $V$ if $gr^{-i} D\_{crys}(V) \neq {0}$.


If $\tau : K \to E$ is a $\mathbb{Q}\_p$-embedding, say that an integer $i$ is a $\tau$-labeled Hodge-Tate weight of $V$ if $gr^{-i} D\_{crys, \tau} (V) \neq {0}$ where $D\_{crys,\tau}(V) = (B\_{crys} \otimes\_{K, \tau}V)^{G\_K}$.


Now the question : consider $K$ as before and let $L$ be the unramified extension of $K$ of degree $d$. 


/Edit Assume $E$ contains all embeddings $L \to E$. /Edit 


Let $W$ be a crystalline representation of $G\_L$. Let $V$ be the representation $Ind\_{G\_L}^{G\_K}$(W). It is not hard to see that $V$ is crystalline, but what are the labeled Hodge-Tate weights of $V$ knowing those of $W$ ?



Hodge-Tate weights of induced representation
It's simpler (and more general) to merely assume that your representations are de Rham. If $W$ is a de Rham representation of $G\_L$, then $D = D\_{dR}(W)$ is an $L$-vector space. If $K$ is a subfield of $L$ then $D\_{dR}(Ind\_L^K W)$ is $D$, now seen as a $K$-vector space via restriction of scalars. The rest should then be an exercise in linear algebra.


Edit: in the above, you can -- and should -- replace "de Rham" by "Hodge-Tate" and $D\_{dR}$ by $D\_{HT}$, and the resulting exercise is even simpler!


