Langlands classification of reductive groups (Q1201617)

Let \(G\) be a reductive algebraic group, defined over \(\mathbb{R}\), i.e. there is a homomorphism \(\alpha: \text{Gal}(\mathbb{C}/\mathbb{R})\to\Aut(G)\) such that \(\alpha(\sigma)\) is anti-holomorphic, where \(\sigma\) is the non-trivial element of the Galois group \(\Gamma\) of the field extension \(\mathbb{C}/\mathbb{R}\), \(G(\mathbb{R})\) its subgroup of real points. The dual group \(^ L G^ 0\) is the algebraic group whose simple root system \(\pi^ \lor\) is the dual of the simple root system \(\pi\) of \(G\) and whose characteristic lattice \(X_ *(T)\) is the dual to the characteristic lattice \(X^*(T)\), where \(T\) is a given maximal \(\theta\)-invariant torus of \(G\), with respect to the Cartan involution \(\theta\). The \(L\)-group of \(G\) is a pair consisting of an order 2 extension \({^ L G}\) of \({^ L G^ 0}\) and a pair of \({^ L G^ 0}\)-conjugacy class of an element \(\sigma\) of \({^ L G}\backslash{^ L G^ 0}\) and a Borel subgroup \({^ L B^ 0}\). Let \(W_{\mathbb{R}}\) be the Weil group of \(\mathbb{R}\), i.e. an extension of the Galois group \[ 1\longrightarrow\mathbb{C}^ \times\longrightarrow W_{\mathbb{R}}\longrightarrow \Gamma=\text{Gal}(\mathbb{C}/\mathbb{R})\longrightarrow 1. \] A homomorphism \(\Phi: W_{\mathbb{R}}\to {^ LG}\) is quasi-admissible iff it is continuous and lifts via the extensions \[ 1\longrightarrow \mathbb{C}^ \times\longrightarrow W_{\mathbb{R}}\longrightarrow \Gamma=\text{Gal}(\mathbb{C}/\mathbb{R})\longrightarrow 1\text{ and }1\longrightarrow {^ LG^ 0}\longrightarrow {^ LG}\longrightarrow\text{Gal}(\mathbb{C}/\mathbb{R})\longrightarrow 1 \] to the identity homomorphism of \(\Gamma\), and maps \(\mathbb{C}^ \times\) to semisimple elements in \({^ L G^ 0}\). Following the well-known Adams- Vogan's version of the Langlands classification of admissible representations there is a one-to-one correspondence between the equivalent classes of \(L\)-packets \(\pi(\Phi)\) and the conjugacy classes of quasi-admissible homomorphisms from the Weil group to \(G(\mathbb{R})\). Moreover, \(L\)-packets are non-empty iff \(\Phi\) is admissible. In the paper under review, the author gives a clear description of the correspondence between the parametrization of the Langlands classification by using discrete series representations and this modified Adams-Vogan's version.