Return to Question

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.

1. $\eta$ 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.
2. The morphism $\Psi_0(G)$ is $\Gamma_{\overline{k}/k}$-equivariant. (Borel 2.5.)
3. 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.)
4. 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.)
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.

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.

1. $\eta$ 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.
2. The morphism $\Psi_0(\eta)$ is $\Gamma_{\overline{k}/k}$-equivariant. (Borel 2.5.)
3. 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.)
4. 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.)
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.

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.

1. $\eta$ 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.
2. The morphism $\Psi_0(G)$ is $\Gamma_{\overline{k}/k}$-equivariant. (Borel 2.5.)
3. 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.)
4. Once one fixes innings 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.)
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.

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.

1. $\eta$ 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.
2. The morphism $\Psi_0(G)$ is $\Gamma_{\overline{k}/k}$-equivariant. (Borel 2.5.)
3. 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.)
4. 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.)
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.