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This is a basic question concerning the local Langlands correspondence for $GL_2(\mathbb{Q}_p)$, in particular the compatibility with tensor products . I apologize if this is too basic but I hope someone with more experience can answer it quickly.

Let $G=GL_2(\mathbb{Q}_p)$, $B \subset G$ the subgroup of upper triangular matrices and $St_G$ be the Steinberg representation of $G$; this is the unique irreducible quotient of the representation $Ind_B^G(|\cdot|^{1/2},|\cdot|^{-1/2})$ (normalized induction).

According to the local Langlands correspondence for $G$, we can attach to $St_G$ a Weil-Deligne representation $(V=\mathbb{C}^2,\sigma:W_F \rightarrow GL(V),N \in End(V))$: in this case the inertia acts trivially, $\sigma(Fr) = \begin{pmatrix} p^{-1/2} & 0 \\\ 0 & p^{+1/2} \end{pmatrix}$ as a representation of $W_F$ and $N=\begin{pmatrix} 0 & 1 \\\ 0 & 0 \end{pmatrix}$.

Now consider the equality of $L$-factors $L(St_G \times St_G,s)=L(\sigma \otimes \sigma,s)$. According to Jacquet (Automorphic Forms on GL(2), Part II, p. 23), we have $L(St_G \times St_G,s)=(1-q^{-s-1})^{-1}$, an $L$-function of degree $1$. On the RHS, the tensor product of Weil-Deligne representations is defined by $(V,\sigma,N) \otimes (V',\sigma',N')=(V\otimes V',\sigma \otimes \sigma',N \otimes 1 + 1 \otimes N')$. But, if I am not mistaken, $\dim_\mathbb{C} Ker(N \otimes 1 + 1 \otimes N')=2$ so that the RHS seems to be of degree $2$! Where did I go wrong?

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For what it's worth: if instead you use the "representation of SL_2" language rather than the "N" language then the tensor product of two copies of the standard representation has length 2, so again the RHS sounds like it should have degree 2. –  user30035 Mar 5 '13 at 7:28
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up vote 5 down vote accepted

I apologize for answering my own question, but I think that the answer might be of some interest to others.

Since I was getting no answers I asked Professor Jacquet directly today. He explained that the expression for $L(St_G \times St_G,s)$ in the book cited above is not correct. The correct $L$-function appears in Proposition 1.4 of his paper A relation between automorphic forms of GL(2) and GL(3), joint with S. Gelbart. With the expression obtained in that paper we do have the expected equality $L(St_G \times St_G,s)=L(\sigma \otimes \sigma,s)$.

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