Langlands functorality in \(K\)-theory for \(C^\ast\)-algebras. I: Base change
Let \(G\) be an algebraic group over a local archimedean field \(\mathbb F\), \(W_\mathbb F\) the Weil group, \({}^LG\) the L-group, \(\Phi : W_\mathbb F \to {}^LG= W_\mathbb F \ltimes \check G\) the Langlands parametrization of irreducible admissible representations of \(G\), and \(\check G\) the Langlands dual group. Two admissible representations are in the same \(L\)-package if they have the same Langlands parameter. The local Langlands Correspondence (LLC) is a bijectjion between the set \(\Pi(G)\) of L-packages of equivalent classes of irreducible representations of \(G\) and the \(\check G\)-conjugacy classes of \(L\)-parameters \(\Phi: W_\mathbb F \to {}^LG\). The inclusion \(W_\mathbb C \to W_\mathbb R\) gives rise to a well-defined base change map \(bc: \{L-\text{parameters of }G(\mathbb R) \} \to \{L-\text{parameters of } G(\mathbb C) \}\). Therefore there is a base change map on \(L\)-packages of irreducible admissible representations of \(G(\mathbb R)\) to the same one of \(G(\mathbb C)\) that is commutative with LLC. The tempered irreducible admissible representations are extended to the corresponding irreducible admissible *-representations of the corresponding C*-algebras \(C^*_r(G)\). The main results of the paper are: {\parindent=0.7cm\begin{itemize}\item[--] Theorem 6.9: For a general linear group \(G=GL(n)\), there is a base change map on K-theory \(bc: K_i(C_r^*(G(\mathbb C))) \to K_j(C_r^*(G(\mathbb R)))\), that is compatible with the base change map on Galois equivalent topological K-theory \(K^i(\widehat{G(\mathbb C)}^{Gal(\mathbb C/\mathbb R)}_t)\to K^i(\widehat{G(\mathbb R)}_t)\), where \(K(\mathbb C) = U(n)\), \(K(\mathbb R) = O(n)\), \(i= \dim(G(\mathbb C)/K(\mathbb C))\mod 2\) and \(j= \dim(G(\mathbb R)/K(\mathbb R)\mod 2\). \item[--] Theorem 7.7: For the general linear group \(G=GL(n)\) with the maximal compact subgroup \(K(\mathbb C)= U(n)\) or \(K(\mathbb R)=O(n)\), the previous base change map and the base change map of the corresponding representation rings \(bc: R(U(n)) \to R(O(n))\) commute with the well known Connes-Kasparov isomorphisms \(R(U(n)) \to K_i(C^*_r(GL(n,\mathbb C)))\) and \(R(O(n)) \to K_j(C^*_r(GL(n,\mathbb R)))\), where \(i \equiv n^2, j \equiv \frac{n(n+1)}{2}\mod 2\). \end{itemize}}
scientific article; zbMATH DE number 6802920