Variants of the classical Satake classfication

Question

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.

Answer 1

(1) Borel's article in the Corvallis proceedings does this slightly differently: he chooses a specific Frobenius element $\sigma$, and then looks at the subset $\widehat{G} \times \{\sigma\}$ of ${}^L G$, up to conjugacy by the subgroup $\widehat{G} \times \{1\}$. (I haven't checked carefully whether this is equivalent to what you wrote, but it's not obviously so.)

(2) Clearly $(g, \sigma)$ is semi-simple iff its image in the "finite Galois form" of the $L$-group is so, where the "finite Galois form" is defined as $\widehat{G} \rtimes \Gamma$ where $\Gamma$ the finite quotient of $Gal(\overline{K} / K)$ through which the action on $\widehat{G}$ factors. This group is a reductive group, of finite type over $\mathbb{C}$, and hence we can apply the standard result that semi-simplicity in reductive groups is invariant under conjugacy.

(3) It is hiding in plain sight, as the identity. If $G$ is split, then the action of $Gal(\overline{K} / K)$ on $\widehat{G}$ is trivial, so ${}^L G \cong \widehat{G} \times Gal(\overline{K} / K)$ and the construction simplifies to conjugacy classes in $\widehat{G}$.

Incidentally, I find it much easier to keep track of this construction by thinking about it in terms of the finite Galois form of the $L$-group (the "full" form is useful for considering functoriality, generalisations to ramified representations, etc but it is much less concrete). The example of $\operatorname{Res}_{L / K} GL_2$, where $L / K$ is an unratified extension, is a particularly nice one: then the reduced $L$-group is $(GL_2 \mathbb{C})^d \rtimes C_d$, where $C_d$ is a cyclic group permuting the factors in the obvious way. It's a fun exercise to classify the semisimple conj classes in this group.