Let us fix a connected split reductive group $G$ over a local non-archimedean field $K$ with maximal torus $T$, then most notes including that of [Gross] defines the Satake isomorphism as $$\mathcal{S}:\mathcal{H}_T\otimes \mathbb{Z}[q^{-1/2},q^{1/2}]\xrightarrow{\cong}R(\hat{G})\otimes \mathbb{Z}[q^{-1/2},q^{1/2}],$$ where $q$ is the cardinality of the residue field of $K$, $\mathcal{H}_T$ is the Hecke algebra (of the torus) of functions which are bi-invariant under the action of $T(\mathcal{O}_K)$ and $R(\hat{G})$ is the Grothendieck ring of the category of complex representaions of the connected dual group $\hat{G}= {}^LG^\circ$.

A nice consequence of this isomorphism discussed in the Propositon 6.4 of [Gross], is that the Satake isomorphism gives us a bijection between unramified representations of $G(K)$ and semisimple conjugacy classes in $\hat{G}(\mathbb{C})$.

Now I have also heard people discuss the Satake isomorphism for quasi-split reductive groups as giving a bijection between unramified representations of $G(K)$, and conjugacy classes of the Langlands dual group ${}^LG$ whose projection to $\hat{G}$ is semisimple and that to the Galois group is Frob. I had a few questions:

1. Is the reformulation correct?

Assuming it is correct:

2. What does projection to $\hat{G}$ mean when the Galois group is not normal? If I just take the set theoretic projection, couldn't a conjugacy class in ${}^LG$ project to different elements in $\hat{G}$ - what if some of these are semisimple and some are not?

3. If these two formulations are the same, how is Frob hiding in the formulation given in [Gross].

My best guess is that if we take conjugates $(g,\phi)$ and $(h,\psi)$ in ${}^LG=\hat{G}\rtimes \operatorname{Gal}(\bar{K},K)$ such that $\phi$ and hence also $\psi$ is Frob (modulo inertia), then $g$ and $h$ will either both be semisimple or both not (and vice versa) - or better $g$ and $h$ are also conjugate. But, just writing these down doesn't make it obvious.

Sorry if this is too much in one question, but thanks in advance for any help.

References:

Gross, Benedict H., On the Satake isomorphism, Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 223-237 (1998). ZBL0996.11038.

Let us fix a connected split reductive group $G$ over a local non-archimedean field $K$ with maximal torus $T$, then most notes including that of [Gross] describes as a consequence of the Satake transform, an isomorphism $$\mathcal{S}:\mathcal{H}_T\otimes \mathbb{Z}[q^{-1/2},q^{1/2}]\xrightarrow{\cong}R(\hat{G})\otimes \mathbb{Z}[q^{-1/2},q^{1/2}],$$ where $q$ is the cardinality of the residue field of $K$, $\mathcal{H}_T$ is the Hecke algebra (of the torus) of functions which are bi-invariant under the action of $T(\mathcal{O}_K)$ and $R(\hat{G})$ is the Grothendieck ring of the category of complex representaions of the connected dual group $\hat{G}= {}^LG^\circ$.

A nice consequence of this isomorphism discussed in the Propositon 6.4 of [Gross], is that it gives us a bijection between unramified representations of $G(K)$ and semisimple conjugacy classes in $\hat{G}(\mathbb{C})$.

Now I have also heard people discuss a similar bijection for quasi-split reductive groups as giving a bijection between unramified representations of $G(K)$, and conjugacy classes of the Langlands dual group ${}^LG$ whose projection to $\hat{G}$ is semisimple and that to the Galois group is Frob. I had a few questions:

1. Is the reformulation correct?

Assuming it is correct:

2. What does projection to $\hat{G}$ mean when the Galois group is not normal? If I just take the set theoretic projection, couldn't a conjugacy class in ${}^LG$ project to different elements in $\hat{G}$ - what if some of these are semisimple and some are not?

3. If these two formulations are the same, how is Frob hiding in the formulation given in [Gross].

My best guess is that if we take conjugates $(g,\phi)$ and $(h,\psi)$ in ${}^LG=\hat{G}\rtimes \operatorname{Gal}(\bar{K},K)$ such that $\phi$ and hence also $\psi$ is Frob (modulo inertia), then $g$ and $h$ will either both be semisimple or both not (and vice versa) - or better $g$ and $h$ are also conjugate. But, just writing these down doesn't make it obvious.

Sorry if this is too much in one question, but thanks in advance for any help.

References:

Gross, Benedict H., On the Satake isomorphism, Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 223-237 (1998). ZBL0996.11038.

Let us fix a connected split reductive group $G$ over a local non-archimedean field $K$ with maximal torus $T$, then most notes including that of [Gross] defines the Satake isomorphism as $$\mathcal{S}:\mathcal{H}_T\otimes \mathbb{Z}[q^{-1/2},q^{1/2}]\xrightarrow{\cong}R(\hat{G})\otimes \mathbb{Z}[q^{-1/2},q^{1/2}],$$ where $q$ is the cardinality of the residue field of $K$, $\mathcal{H}_T$ is the Hecke algebra (of the torus) of functions which are bi-invariant under the action of $T(\mathcal{O}_K)$ and $R(\hat{G})$ is the Grothendieck ring of the category of complex representaions of the connected dual group $\hat{G}= {}^LG^\circ$.

A nice consequence of this isomorphism discussed in the Propositon 6.4 of [Gross], is that the Satake isomorphism gives us a bijection between unramified representations of $G(K)$ and semisimple conjugacy classes in $\hat{G}(\mathbb{C})$.

Now I have also heard people discuss the Satake isomorphism for quasi-split reductive groups as giving a bijection between unramified representations of $G(K)$, and conjugacy classes of the Langlands dual group ${}^LG$ whose projection to $\hat{G}$ is semisimple and that to the Galois group is Frob. I had a few questions:

1. Is the reformulation correct?

Assuming it is correct:

2. What does projection to $\hat{G}$ mean when the Galois group is not normal? If I just take the set theoretic projection, couldn't a conjugacy class in ${}^LG$ project to different elements in $\hat{G}$ - what if some of these are semisimple and some are not?

3. If these two formulations are the same, how is Frob hiding in the formulation given in [Gross].

My best guess is that if we take conjugates $(g,\phi)$ and $(h,\psi)$ in ${}^LG=\hat{G}\rtimes \operatorname{Gal}(\bar{K},K)$ such that $\phi$ and hence also $\psi$ is Frob (modulo inertia), then $g$ and $h$ will either both be semisimple or both not (and vice versa). But, just writing these down doesn't make it obvious.

Sorry if this is too much in one question, but thanks in advance for any help.

References:

Gross, Benedict H., On the Satake isomorphism, Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 223-237 (1998). ZBL0996.11038.

Let us fix a connected split reductive group $G$ over a local non-archimedean field $K$ with maximal torus $T$, then most notes including that of [Gross] defines the Satake isomorphism as $$\mathcal{S}:\mathcal{H}_T\otimes \mathbb{Z}[q^{-1/2},q^{1/2}]\xrightarrow{\cong}R(\hat{G})\otimes \mathbb{Z}[q^{-1/2},q^{1/2}],$$ where $q$ is the cardinality of the residue field of $K$, $\mathcal{H}_T$ is the Hecke algebra (of the torus) of functions which are bi-invariant under the action of $T(\mathcal{O}_K)$ and $R(\hat{G})$ is the Grothendieck ring of the category of complex representaions of the connected dual group $\hat{G}= {}^LG^\circ$.

A nice consequence of this isomorphism discussed in the Propositon 6.4 of [Gross], is that the Satake isomorphism gives us a bijection between unramified representations of $G(K)$ and semisimple conjugacy classes in $\hat{G}(\mathbb{C})$.

Now I have also heard people discuss the Satake isomorphism for quasi-split reductive groups as giving a bijection between unramified representations of $G(K)$, and conjugacy classes of the Langlands dual group ${}^LG$ whose projection to $\hat{G}$ is semisimple and that to the Galois group is Frob. I had a few questions:

1. Is the reformulation correct?

Assuming it is correct:

2. What does projection to $\hat{G}$ mean when the Galois group is not normal? If I just take the set theoretic projection, couldn't a conjugacy class in ${}^LG$ project to different elements in $\hat{G}$ - what if some of these are semisimple and some are not?

3. If these two formulations are the same, how is Frob hiding in the formulation given in [Gross].

My best guess is that if we take conjugates $(g,\phi)$ and $(h,\psi)$ in ${}^LG=\hat{G}\rtimes \operatorname{Gal}(\bar{K},K)$ such that $\phi$ and hence also $\psi$ is Frob (modulo inertia), then $g$ and $h$ will either both be semisimple or both not (and vice versa) - or better $g$ and $h$ are also conjugate. But, just writing these down doesn't make it obvious.

Sorry if this is too much in one question, but thanks in advance for any help.

References:

Gross, Benedict H., On the Satake isomorphism, Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 223-237 (1998). ZBL0996.11038.

Let us fix a connected split reductive group $G$ over a local non-archimedean field $K$ with maximal torus $T$, then most notes including that of [Gross] defines the Satake isomorphism as $$\mathcal{S}:\mathcal{H}_T\otimes \mathbb{Z}[q^{-1/2},q^{1/2}]\xrightarrow{\cong}R(\hat{G})\otimes \mathbb{Z}[q^{-1/2},q^{1/2}],$$ where $q$ is the cardinality of the residue field of $K$, $\mathcal{H}_T$ is the Hecke algebra (of the torus) of functions which are bi-invariant under the action of $T(\mathcal{O}_K)$ and $R(\hat{G})$ is the Grothendieck ring of the category of complex representaions of the connected dual group $\hat{G}= {}^LG^\circ$.

A nice consequence of this isomorphism discussed in the Propositon 6.4 of [Gross], is that the Satake isomorphism gives us a bijection between unramified representations of $G(K)$ and semisimple conjugacy classes in $\hat{G}(\mathbb{C})$.

Now I have also heard people discuss the Satake isomorphism for quasi-split reductive groups as giving a bijection between unramified representations of $G(K)$, and conjugacy classes of the Langlands dual group ${}^LG$ whose projection to $\hat{G}$ is semisimple and that to the Galois group is Frob. I had a few questions:

1. Is the reformulation correct?

Assuming it is correct:

2. What does projection to $\hat{G}$ mean when the Galois group is not normal? If I just take the set theoretic projection, couldn't a conjugacy class in ${}^LG$ project to different elements in $\hat{G}$ - what if some of these are semisimple and some are not?

3. If these two formulations are the same, how is Frob hiding in the formulation given in [Gross].

My best guess is that if we take conjugates $(g,\phi)$ and $(h,\psi)$ in ${}^LG=\hat{G}\rtimes \operatorname{Gal}(\bar{K},K)$ such that $\phi$ and hence also $\psi$ is Frob (modulo inertia), then both $g$ and $h$ will also be conjugates in $\hat{G}$ (and vice versa). But, just writing these down doesn't make it obvious.

Sorry if this is too much in one question, but thanks in advance for any help.

References:

Gross, Benedict H., On the Satake isomorphism, Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 223-237 (1998). ZBL0996.11038.

Let us fix a connected split reductive group $G$ over a local non-archimedean field $K$ with maximal torus $T$, then most notes including that of [Gross] defines the Satake isomorphism as $$\mathcal{S}:\mathcal{H}_T\otimes \mathbb{Z}[q^{-1/2},q^{1/2}]\xrightarrow{\cong}R(\hat{G})\otimes \mathbb{Z}[q^{-1/2},q^{1/2}],$$ where $q$ is the cardinality of the residue field of $K$, $\mathcal{H}_T$ is the Hecke algebra (of the torus) of functions which are bi-invariant under the action of $T(\mathcal{O}_K)$ and $R(\hat{G})$ is the Grothendieck ring of the category of complex representaions of the connected dual group $\hat{G}= {}^LG^\circ$.

A nice consequence of this isomorphism discussed in the Propositon 6.4 of [Gross], is that the Satake isomorphism gives us a bijection between unramified representations of $G(K)$ and semisimple conjugacy classes in $\hat{G}(\mathbb{C})$.

Now I have also heard people discuss the Satake isomorphism for quasi-split reductive groups as giving a bijection between unramified representations of $G(K)$, and conjugacy classes of the Langlands dual group ${}^LG$ whose projection to $\hat{G}$ is semisimple and that to the Galois group is Frob. I had a few questions:

1. Is the reformulation correct?

Assuming it is correct:

2. What does projection to $\hat{G}$ mean when the Galois group is not normal? If I just take the set theoretic projection, couldn't a conjugacy class in ${}^LG$ project to different elements in $\hat{G}$ - what if some of these are semisimple and some are not?

3. If these two formulations are the same, how is Frob hiding in the formulation given in [Gross].

My best guess is that if we take conjugates $(g,\phi)$ and $(h,\psi)$ in ${}^LG=\hat{G}\rtimes \operatorname{Gal}(\bar{K},K)$ such that $\phi$ and hence also $\psi$ is Frob (modulo inertia), then $g$ and $h$ will either both be semisimple or both not (and vice versa). But, just writing these down doesn't make it obvious.

Sorry if this is too much in one question, but thanks in advance for any help.

References:

Gross, Benedict H., On the Satake isomorphism, Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 223-237 (1998). ZBL0996.11038.