I'm trying to figure out the arithmetic analogue of a key conjecture in the geometric local Langlands correspondence. Briefly, one expects for $K=\mathbb{C}((t))$ an equivalence of dg categories $$\operatorname{BiWhit}(\mathcal{D}\operatorname{-mod}(G(K)))\cong\operatorname{QCoh}(\operatorname{LocSys}_{\hat{G}}(D^\times)).$$ Here, $G$ is some reductive group with Langlands dual $\hat{G}$ and $\operatorname{LocSys}_{\hat{G}}(D^\times)$ is the stack of local systems on the punctured disk. The left hand side is the dg category of Bi-Whittaker $\mathcal{D}$-modules on the loop group. This conjecture is what allows you to formulate the compatibility condition the (tempered) geometric local Langlands correspondence should satisfy.

Now let's move to the arithmetic setting (where I really understand very little - hopefully not all of this will be nonsense.) Take $G$ some group over $\mathbb{Q}_p$, let me assume split for now.

The naive decategorified translation of this is an isomorphism of two rings. Fixing some Whittaker character $\chi$ of $N(\mathbb{Q}_p)$, I can take the set $R$ of locally constant functions on $G(\mathbb{Q}_p)$ which transform under the left and right actions of $N(\mathbb{Q}_p)$ by $\chi$ and $-\chi$, respectively (I am not 100% sure of these signs.) The ring structure on $R$ will be given by convolution, not multiplication of functions (and I probably need Rodier's compact approximation mentioned below to define it.)

The second ring $S$ will be a subalgebra of the algebra $S_0$ of complex-valued functions on the set of Galois representations into $\hat{G}$. Namely, for every element $g\in\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ (EDIT: The Weil group is almost certainly the better object to take here) and every conjugation invariant function $f$ on $\hat{G}(\mathbb{C})$, I get an element of $S_0$ given by evaluation of $f$ on the conjugacy class of $g.$ Take $S$ to be the subalgebra generated by these elements, varied over all $f$ and $g$.

Question 1: Are $R$ and $S$ (conjecturally) isomorphic?

Assuming a positive answer, one might try to formulate a compatibility condition for (tempered) local Langlands as follows. Given a tempered irreducible $G(\mathbb{Q}_p)$-representation $V$, the LLC assigns to it some Galois representation $\sigma_V$. The analogy with the geometric case suggests that the space of Whittaker functionals should be one-dimensional, and that there should be an action of $R$ on this space which corresponds to the one-dimensional representation of $S$ coming from $\sigma_V.$ My impressions are that the action of $R$ is definable using Rodier's compact approximation to the Whittaker model (to get around the non-compact-supportedness of bi-Whittaker functions), and that it is a theorem that for any $V$, this space of Whittaker functionals is at most one-dimensional.

Question 2: Is there anything like this phrasing of the properties that LLC is should satisfy in the literature?

The closest thing I can find is Scholze's characterization of LLC in "The Local Langlands Conjecture for $GL_n$ over $p$-adic fields." There he describes (up to something about cutoff functions that I don't yet understand) a map from $n$-dimensional Galois representations to compactly supported functions on $GL_n(\mathbb{Q}_p)$ which seems like it should describe a compact approximation of a map $S\rightarrow R$.

  • $\begingroup$ Is it obvious that question 1 has answer yes even in the easiest case of $\mathrm G = \mathbb G_m$? Then $R$ consists of l.c. functions $f: \mathbb Q_p^* \to \mathbb C$. On the other hand, $\hat{\mathrm G}(\mathbb C)= \mathbb C^*$. After fixing a normalization of the Artin map, your algebra $S$ is generated by varying $g \in \widehat{\mathbb Q_p^*}$ and $f: \mathbb C^* \to \mathbb C$ and defining $F_{g,f}$ that maps a Galois character $\chi$ to $f(\chi(g))$. Now, $R$ is 'spanned' by countably many translates of the constant support subalgebra. Is there an equivalent structure on $S$? $\endgroup$ – user94041 Nov 28 '17 at 0:43
  • $\begingroup$ Side question: do you know a reference for this bi-Whittaker expectation in geometric Langlands? I thought it was in Gaitsgory's Quantum Langlands Correspondence (arXiv:1601.05279) but found there only a more complicated statement about the isomonodromy groupoid on opers..though depending on how one understands his statement it could be the same as yours by Frenkel-Zhu arXiv:0811.3186. $\endgroup$ – David Ben-Zvi Nov 28 '17 at 18:18
  • $\begingroup$ @user94041 That's a good question - that reminds me that 1. I forgot to mention the multiplication on R is by convolution 2. I should probably be considering the Weil group in the definition of $S$. I think with those two fixes the $\mathbb{G}_m$ case should just be a Fourier transform? $\endgroup$ – dhy Nov 28 '17 at 20:25
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    $\begingroup$ @DavidBen-Zvi To my knowledge, the only place where this is written is in Raskin's "W-algebras and Whittaker categories", arxiv.org/abs/1611.04937v1, where it is briefly mentioned in Example 1.23.1. Probably why it wasn't stated earlier is because the results of that paper are needed to make sure the statement is actually reasonable? $\endgroup$ – dhy Nov 28 '17 at 20:28
  • $\begingroup$ @dhy maybe you also want some continuity conditions on the Galois representations? Also, again in the case of $\mathbb G_m$, how do you know that the integral $\int \chi(g) f(g) \mathrm{d}g$ converges? $f$ is not assumed to be compactly supported, and $\chi$ does not need to be unitary. $\endgroup$ – user94041 Nov 29 '17 at 1:26

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