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Let $F$ be a totally real field with real primes $\tau_1,\ldots,\tau_n$ and $\pi$ a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbf{A}_F)$. My rough understanding is that in order for $\pi$ to be attached to a cuspidal Hilbert newform of weight $(k_1,\ldots,k_n)$ with $k_i$ all of the same parity and $k_i\geq 2$ (perhaps these assumptions aren't necessary but I'm not sure), the infinite components $\pi_{\tau_1},\ldots,\pi_{\tau_n}$ of $\pi$ must be discrete series whose lowest non-negative $K$-type is related to the $k_i$. But precisely how the $K$-type is related to the weight seems to me to differ from source to source.

My understanding of what is meant by $\pi_{\tau_i}$ being discrete series is that it arises as the space of $K$-finite vectors in the normalized parabolic induction $\mathrm{Ind}(\mu,\nu)$ for characters $\mu,\nu:\mathbf{R}^\times\rightarrow\mathbf{C}^\times$ satsifying $\mu\nu^{-1}(t)=\mathrm{sgn}(t)^\epsilon\vert t\vert^{k-1}$ where $k$ is an integer bigger than $1$ and $\epsilon\equiv k\pmod{2}$ (this is the definition from Bump's book on automorphic representations). In two references I've looked at (Carayol's paper on the Galois representations attached to Hilbert modular forms) as well as the paper of Ohta referenced in that paper give seemingly different conditions on the $\pi_{\tau_i}$ to ensure that $\pi$ comes from a Hilbert modular form. Carayol asks that $\pi_{\tau_i}$ come from the characters $t\mapsto\mathrm{sgn}(t)^{k_i}\vert t\vert^{(1/2)(k_i-1-w)}$ and $t\mapsto\vert t\vert^{(1/2)(-k_i+1-w)}$ for some integer $w\equiv k_i$ mod $2$. On the other hand, Ohta, changes the exponents on the two characters to $(1/2)(k_i+1-w)$ and $(1/2)(-k_i-1-w)$.

Is there a conceptual difference between these two sets of exponents? Are these just different normalizations? Is one of them a typo? Carayol's makes sense to me because the difference of the exponents is $k_i-1$, which, at least using Bump's conventions, would mean that $\pi_{\tau_i}$ has lowest non-negative $K$-type $k_i$, and I've read in other sources that this is how things are supposed to work.

I really hope this question isn't completely non-sensical, and I apologize if it is.

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1 Answer 1

This is mostly an issue of normalization, although, in any case the holomorphic discrete series is not the whole principal series, but a proper subrepresentation, missing the $K$-types between the "weight" and its negative. There are at least two choices about normalization: whether to include the half-sum of positive roots automatically, or to use the "un-normalized" parametrization of the character; there is also the question of whether a "2" is built into the $k_j$'s or not.

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