Local Langlands Conjecture for p-adic SO(4), reference request

Question

In section 10 of Gan-Gross-Prasad's paper "Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups" http://arxiv.org/pdf/0909.2999v1.pdf, the authors stated that the full local Langlands conjectures for some small ranked classical groups are known, but no reference is given. Can anybody point out a reference for the SO(4) and SO(6) case? Do we know whether the local Langlands map preserves certain local epsilon or L-factors?

Added:

I appreciated the answer provided by Professor Kimball, which is very helpful. I do want to know more details about (split) SO(4). In the last section of Gross-Prasad's paper as given by Kimball, there is an exact sequence $$0\rightarrow SO_4(k)\rightarrow GL_2(k)\times GL_2(k)/\Delta k^* \rightarrow k^* \rightarrow 0.$$

To get the local Langlands conjecture for SO(4) from that of GL(2) from this exact sequence, it seems that we need to know the relationship between irreducible admissible representations of SO(4) and irreducible admissible representations of $GL_2(k)\times GL_2(k)/\Delta k^*$. Is this something standard or even trivial?

On the other hand, in GGP, the authors stated that most local Langlands conjecture for classical groups should be established by the (forthcoming) book of Arthur and Moglin's papers. Arthur's books has already come out https://www.amazon.com/Endoscopic-Classification-Representations-Colloquium-Publications/dp/0821849905/ref=sr_1_1?s=books&ie=UTF8&qid=1465744787&sr=1-1&keywords=arthur+automorphic+representations . I never tried to read Arthur's book. What I would like to know is: whether they (Arthur and Moglin) proved the preservation of local epsilon and L-factors, or they just proved the existence of the local Langlands map.

Answer 1

I don't know an explicit reference for you, but I can tell you how these things are done and point you to some related reference. These groups are closely related to general linear groups via accidental isomorphism, so one can write down the Langlands parameters in terms of those for general linear groups and this should also give preservation of epsilon and L-factors.

• SO(4) is essentially GL(2) $\times$ GL(2). See, e.g., the last section of Gross and Prasad's original paper (Can. J, 1992).

• SO(6) is essentially GL(4). Both of the relations are essentially described in Gan and Takeda's paper Theta correspondences for GSp(4) (Rep Thy 2011). (Technically they work with describe slightly related groups, GSO, which contain the SO's.)