I am looking for a reference that explains in detail the existence of Langlands dual homomorphisms. It seems that in the literature two references are given most often. The first is Borel's article Automorphic $L$-functions in the Corvallis proceedings. The relevant sections are 1.4, 2.1, and 2.5, and these refer to 2.11 in Springer's article Reductive groups in the same proceedings. The second reference one finds is 1.8 in Kottwitz's article Stable trace formula: cuspidal tempered terms. However, both of these references are quite terse.
Let $k$ be a field and let $\eta:G\to G'$ be a homomorphism of connected reductive groups over $k$. Assume that $\eta(G)$ is normal in $G'$. (Kottwitz also assumes that $\eta:G\to\eta(G)$ is separable, but Borel and Springer do not assume this. Is it necessary?) The following are the main claims needed. It would be great if there is a reference that explains them in more detail.
- $\eta$ determines a morphism $\Psi_0(\eta):\Psi_0(G)\to\Psi_0(G')$ of the based root data. The notion of morphism is defined implicitly in 2.11 of Springer's article.
- The morphism $\Psi_0(\eta)$ is $\Gamma_{\overline{k}/k}$-equivariant. (Borel 2.5.)
- The dual morphism $\widehat{\Psi_0(\eta)}:\Psi_0(\hat{G'})\to\Psi_0(\hat{G})$ comes from a normal homomorphism $\hat{\eta}:\hat{G'}\to \hat{G}$. The homomorphism $\hat{\eta}$ is uniquely determined if it is required to be compatible with choices of pinnings for $\hat{G'}$ and $\hat{G}$. Changing the pinnings replaces $\hat{\eta}$ by a $\hat{G}^{\Gamma_{\overline{k}/k}}$-conjugate. (See Kottwitz 1.8.)
- Once one fixes pinnings for $\hat{G'}$ and $\hat{G}$, the uniquely determined $\hat{\eta}$ is equivariant with the resulting actions of $\Gamma_{\overline{k}/k}$. (See Borel 2.5.)
- One gets a unique $\hat{G}$-conjugacy class of L-homomorphisms ${}^L\eta:{}^LG'\to{}^LG$.
The parts that I would like most to get a better understanding of are 1 and the existence of $\hat{\eta}$. At the beginning of Section 5 of Steinberg's 1999 article, The Isomorphism and Isogeny Theorems for Reductive Algebraic Groups, he states that the isogeny theorem can be extended to arbitrary surjective homomorphisms. This covers the existence of $\hat{\eta}$ when $\eta$ is an embedding. However, this paper came after the articles by Borel and Kottwitz, so presumably they had something else in mind.
Best Answer
We have that $G'$ is an almost-direct product of $\eta(G)$ and the almost-direct product of the duals of the simple factors of $G'$ not contained in $\eta(G)$, which allows us to reduce (1) and (3) to the case where $\eta$ is surjective.
I think you probably want to assume that $\eta$ is a quotient by a smooth, connected, normal subgroup followed by a central isogeny; this is automatic if $\eta$ is separable. If you do not want to reduce to this case, then you have to handle something like the exceptional isogeny $\operatorname{SO}_{2n + 1, k} \to \operatorname{Sp}_{2n, k}$ when $\operatorname{char} k$ is $2$, whose dual homomorphism I guess should be a map $\operatorname{SO}_{2n + 1, \mathbb C} \to \operatorname{Sp}_{2n, \mathbb C}$; but I think that there are no non-trivial such homomorphisms.
When $\eta$ is a quotient by a smooth, connected, normal subgroup $N$ of $G$, we have a similar decomposition of $G$ as the almost-direct product of $N$ with a canonical complement. Thus, if it is OK to make the reduction above, then we need only handle the case where $\eta$ is an isogeny.
In this case, if $(B, T, \mathcal X)$ is a pinning of $G_{\overline k}$, then $(B', T', \mathcal X') \mathrel{:=} (\eta_{\overline k}(B), \eta_{\overline k}(T), \eta_{\overline k}(\mathcal X))$ is a pinning of $G'_{\overline k}$. We compute $\Psi_0(G_{\overline k})$ and $\Psi_0(G'_{\overline k})$ in terms of these pinnings, and then let $\Psi_0(\eta)$ be the obvious map. If I understand correctly how $\Gamma_{\overline k/k}$ acts, then this is $\Gamma_{\overline k/k}$-equivariant: if we choose $\sigma \in \Gamma_{\overline k/k}$ and let $g \in G(\overline k)$ be such that $\sigma(B, T, \mathcal X)$ equals $\DeclareMathOperator\Ad{Ad}(g B g^{-1}, g T g^{-1}, \Ad(g)\mathcal X)$, then $\sigma(B', T', \mathcal X') = (\eta_{\overline k}(\sigma B), \eta_{\overline k}(\sigma T), \eta_{\overline k}(\sigma\mathcal X))$ equals $(g'B'g^{\prime\,{-1}}, g'T'g^{\prime\,{-1}}, \Ad(g')\mathcal X')$, where $g'$ equals $\eta_{\overline k}(g)$. Then $\sigma$, viewed as a morphism $X_*(T) \to X_*(T)$, respectively $X_*(T') \to X_*(T')$, sends $\lambda$, respectively $\lambda'$, to $\DeclareMathOperator\Int{Int}\Int(g)^{-1} \circ \sigma \circ \lambda \circ \sigma^{-1}$, respectively $\Int(g')^{-1} \circ \sigma \circ \lambda \circ \sigma^{-1}$; and then $\Psi_0(\eta)(\sigma\lambda) = \eta \circ \Int(g)^{-1} \circ \sigma \circ \lambda \circ \sigma^{-1}$ equals $\Int(g')^{-1} \circ \sigma \circ \eta \circ \lambda \circ \sigma^{-1} = \sigma(\Psi_0(\eta)(\lambda))$.
Still in the case where $\eta$ is a central isogeny, we have that $\widehat{\Psi_0(\eta)}$ is also a $\Gamma_{\overline k/k}$-equivariant, separable isogeny of root data, so that it is dual to a (necessarily central) $\Gamma_{\overline k/k}$-equivariant isogeny $\hat\eta : \hat{G'} \to \hat G$, as you say uniquely determined by choices of $\Gamma_{\overline k/k}$-fixed pinnings. Since two different pinnings of $\hat{G'}$ are uniquely $\hat{G'}/\operatorname Z(\hat{G'})$-conjugate, the corresponding $\Gamma_{\overline k/k}$-equivariant homomorphisms are $(\hat{G'}/\operatorname Z(\hat{G'}))^{\Gamma_{\overline k/k}}$-conjugate; but it is not clear to me why the conjugacy can be lifted to $\smash{\hat{G'}}^{\Gamma_{\overline k/k}}$. If $\operatorname Z(\hat{G'})$ is connected (equivalently, if the derived group of $G'$ is simply connected), then one can probably show that $H^1(\Gamma_{\overline k/k}, \operatorname Z(\hat{G'}))$ is trivial and deduce a suitable lifting; but I don't know about the case of disconnected dual centre.