Explicit Jacquet-Langlands correspondence for real reductive groups

Question

Let $G$ be a connected reductive group over $\mathbb R$. Let $G'$ over $\mathbb R$ be an inner form of $G$ with ${}^LG={}^LG'$. By local Langlands correspondence over $\mathbb R$, if a $L$-packet of $G$ and another $L$-packet of $G'$ share the same Langlands parameter $\phi: W_{\mathbb R} \to {}^LG$, we may say that these two L-packets are Jacquet-Langlands transfers.

I am interested in cases where the Jacquet-Langlands correspondence could be made more explicit. In my case, $G=SO(2n+1,0)$ (resp. $Spin(2n+1)$) is compact, and $G'=SO(p,q)$ (resp. $Spin(p,q)$). So they are pure inner forms and have discrete series. And irreducible representations of $G$ are all finite-dimensional. Note $G'(\mathbb R)$ has $2$ connected components in general, and may not be quasi-split (unless $G'=SO(n+1,n)$).

1. Starting from an irreducible representation $\pi$ of $G(\mathbb R)$ (say a discrete series), could we specify an irreducible rep of $G'(\mathbb R)$ that is "the Jacquet-Langlands transfer" of $\pi$? In my case, it seems when $G'$ is quasi-split we could use the unique generic member in L-packets.
2. Do we know a relation between Harish-Chandra characters of Jacquet-Langlands transfers as in the case of $GL_2$ (with suitable signs)?
3. Do we know a relation between $K$-types for $G$ and $K'$-types for $G'$? How about the case of discrete series?

Thank you very much.

Edit: As the comment says, in Gross-Reeder's survey paper (Section 4-5), there is a construction of discrete L-packets. Let $G$ be a connected compact Lie group of rank $n$ with finite center and $S$ be a maximal torus with Weyl group $W=N_G(S)/S$. Starting with a positive character $\chi \in \rho_{G_\mathbb C} + X^*(S_\mathbb C)$ (e.g. $\chi=\rho_{G_\mathbb C}$) and $s \in S[2] \cong (\mathbb Z/2)^n$, there is a construction of $\pi(\chi,s)$ in the L-packet for $\chi$ (consisting of discrete series) on the real form $G_s$ of $G$, where $K_s=C_{G}(s) \leq G$ is a maximal compact subgroup of $G_s$. Here $\pi(\chi,s)$ is the non-vanishing coherent cohomology of line bundle $O(-\chi-\rho)$ on the open complex domain $D_s=G_s/S \hookrightarrow Fl=G_\mathbb C/B_\mathbb C$. There is a fibration $K_s/S \to D_s \to G_s/K_s$ of real manifolds ($G_s/K_s$ is not algebraic in general). Schmid proves, $\pi(\chi,s)$ is the unique discrete series of $G_s$ with (multiplicity one) lowest $K_s$-type of highest weight $\chi-\rho_{s}$. Moreover, the construction gives exactly $|W.s|$ distinct discrete series of $G_s$, hence the total $L$-packet for $\chi$ (on all $G_s$) has size $2^n$. The reference is W. Schmid, Some properties of square integrable representations of semisimple Lie groups, Ann. Math., 102 (1975), pp. 535–564. It seems that construction shall also work in the above algebraic setup, and $\chi$ is the infinitesimal character.

But there is no description of characters or how to transfer between representations. Moreover, is there a way of dealing with the limits of discrete series?

Answer 1

The short answer to all of your questions is: Jacquet-Langlands for $GL(n,F)$ is special because $L$-packets for $GL(n,F)$ are singletons. Consequently the kinds of things you are asking about for other groups do not exist at the level of individual representations, only packets.

In particular suppose $\phi$ is a discrete series parameter for an inner class of real forms of $G$. Then, for each real form, $\Pi(G(\mathbb R),\phi)$ is a set of discrete series representations of $G(\mathbb R)$. Define $S(G(\mathbb R),\phi)$ to be the sum of these representations. The character of this sum is the basic example of a "stable" distribution. What is true is that the characters of $S(G(\mathbb R),\phi)$ are related over the different real forms.

The basic case is the quasisplit group. In some sense the analogue of Jacquet-Langlands is that the character of $S(G(\mathbb R)_{qs},\phi)$ determines all of the others $S(G(\mathbb R),\phi)$. However there is no character identity of this type relating individual discrete series on different real forms. (Also, there is no reasonable way to specify a single discrete series in an L-packet, aside from a generic discrete series of the quasisplit form.)

A good starting point is Lemma 5.2 and Theorem 6.3 in Character and inner forms of a quasisplit group over $\mathbb R$ by Shelstad (Compositio 1979).

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