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?



On local Rankin-Selberg L-functions for Steinberg representations
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)](http://www.numdam.org/item?id=ASENS_1978_4_11_4_471_0), joint with S. Gelbart. With the expression obtained in that paper we do have the expected equality $L(St\_G \times St\_G,s)=L(\sigma \otimes \sigma,s)$.


