Let $E/F$ be a quadratic extension of number fields and $v$ is a place of $F$.


Let $\chi\_1,\chi\_2$ be the unramified characters of $F\_v^{\times}$.


If $B(\chi\_1,\chi\_2)$ is the unramified principal series representation of $GL\_2(F\_v)$, what is the $BC(\pi)$, the base change of $\pi$ to $GL\_2(E\_v)$?


I suppose that $BC(\pi)=B(\chi\_1 \circ \text{Norm}\_{E\_v/F\_v},\chi\_2 \circ \text{Norm}\_{E\_v/F\_v})$. Is this right?



Explicit formula of base change for GL(n)
The answer to your final question should be yes.


Let me assume that $v$ is inert in $E$, so that $E\_v / F\_v$ is a honest quadratic extension.
Assuming that $B(\chi\_1, \chi\_2)$ is irreducible, it corresponds via local Langlands (a theorem for $\mathrm{GL}\_2(K)$ and any $p$-adic field $K$) to the 2-dimensional representation of the Weil group of $F\_v$ given by the sum of the characters $\chi\_1$ and $\chi\_2$.


The operation of base change on the $p$-adic side corresponds to restriction of Galois representations, thus the base change of $B(\chi\_1, \chi\_2)$ corresponds via local Langlands to the restriction of $\chi\_1 \oplus \chi\_2$ to the Weil group of $E\_v$.


When we restrict $\chi\_1 \oplus \chi\_2$ to the Weil group of $E\_v$ and then make this homomorphism factor through the abelianization $E\_v^\*$, local class field theory tells us that this corresponds exactly to pre-composition of $\chi\_1$ and $\chi\_2$ with the Norm map. Now go back to the $p$-adic side and you have your statement.


