## Characterizing the Local Langlands Correspondence

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In the p-adic case, is there any hope for a set of conditions on the local Langlands correspondence which would make it unique? In the case of GL(n) this is provided by L and epsilon factors. For classical groups, can one make a precise statement (even conjecturally), using lifting to GL(n)?

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## 1 Answer

This answer will necessarily be full of conjecture, but I'll try to make things more concrete for some classical groups.

For a reductive group $G$, and smooth irreducible representation $(\pi, V)$ of $G$, and L-group homomorphism $\sigma: {}^L G \rightarrow {}^L GL_n$, one may conjecturally associate a local L-function and $\epsilon$-factor $L(\pi, \sigma, s)$ and $\epsilon(\pi, \sigma, s)$.

As $\sigma$ varies over all L-group representations of ${}^L G$, this family of L-functions and $\epsilon$-factors will uniquely determine the (conjectural) Langlands parameter $$\phi_\pi: W' \rightarrow {}^L G,$$ where $W'$ is the Weil-Deligne group of the $p$-adic field. Namely, there will be a unique $\phi_\pi$ such that $$L(\sigma \circ \phi_\pi, s) = L(\pi, \sigma, s), \epsilon(\sigma \circ \phi_\pi, s) = \epsilon(\pi,\sigma,s)$$ where the L and $\epsilon$ on the left sides are Artin L-functions (adapted to the Weil-Deligne group instead of Galois group in a simple way).

In this way, the L- and $\epsilon$-factors determine the Langlands parameter of a smooth irrep of $G$. This partitions the smooth irreps of $G$ into the "L-packets".

Now, for classical groups, one can get away with less information (or at least more computable information) in many cases. I refer to the recent paper of Wee Teck Gan and Takeda, "The Local Langlands Correspondence for Sp(4)" (avaialble on Wee Teck's home page, for example), where they prove that the Langlands parameterization is uniquely determined by L- and $\epsilon$-factors (and $\gamma$-factors) of pairs (i.e. twists by $GL(n)$ with $n \leq 3$) for generic representations and a fact about Plancherel measure for certain nongeneric representations (this fact is analogous to a fact about Shahidi-type L-functions, which are only understood in the generic case).

For any classical group, I imagine one (probably not me) could come up with an analogous set of twists, etc.. to find a reasonable set of L-functions and $\epsilon$-factors, and Plancherel measures to determine the Langlands parameterization. Perhaps this is done by others who prove functoriality for classical groups (P.S., Cogdell, Rallis, Jiang, Kim, Shahidi, and many others).

Refining the local Langlands conjectures (I appreciate Vogan's survey most here), one should be able to parameterize the contents of an L-packet (with parameter $\phi: W' \rightarrow {}^L G$) by the representations of a finite group $S_\phi$. This finite group has a general description, which in some nice cases (split, adjoint perhaps, I don't recall), is that $S_\phi$ is the component group of the centralizer of the image of $\phi$ in the dual group $\hat G$.)

So to pin down the Langlands correspondence, one must characterize this parameterization of the elements of an L-packet as well. For this, it seems that characters are key -- there is certainly no telling the difference with L-functions or $\epsilon$-factors. I believe this parameterization of the elements of an L-packet should be uniquely determined by the stability of an associated sum of character distributions of elements of the L-packet. I tend to avoid characters, but maybe the work of DeBacker and Reeder -- who prove such a stability result for depth zero representations -- is a good place to start.

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Except in special cases the definition of $L(\pi,\sigma,s)$ and $\varepsilon(\pi,\sigma,s)$ assume the LLC. For GL(n) these can be defined independently using Whittaker models. This gives a set of conditions one can impose on the (conjectural) LLC for GL(n) which (if it exists) make it unique. This is now a theorem.   What I'm asking about is something similar for other groups. Do not assume LLC exists, and give a set of conditions on it which, if it exists, make it unique. – Jeffrey Adams May 13 2010 at 12:26
A great example in this spirit is the Sp4 paper of Gan-Takeda that I referenced.  They used L-functions in the generic case -- when suitable L-functions can be defined without LLC -- and Plancherel measure in the non-generic case -- when suitable L-functions are harder to define (though there are options using Bessel instead of Whittaker models).  My personal hope is that someday, someone will be able to define L-functions for generic irreps of general (quasisplit?) groups, without LLC, by understanding the structure of some Hecke algebras.  Such a discovery might answer your wishes as well. – Marty May 13 2010 at 15:37
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