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Geometric structure in the representation theory of reductive \(p\)-adic groups. II. (English) Zbl 1246.22019
Doran, Robert S. (ed.) et al., Harmonic analysis on reductive, \(p\)-adic groups. Proceedings of the AMS special session, San Francisco, CA, USA, January 16, 2010. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4985-9/pbk). Contemporary Mathematics 543, 71-90 (2011).
For part I see [the authors, C. R., Math., Acad. Sci. Paris 345, No. 10, 573–578 (2007; Zbl 1128.22009)].
Let \(G\) be a connected reductive \(p\)-adic group. According to Bernstein’s theory, the admissible dual \(\hat{G}\) is partitioned into components \(\hat{G}_{\mathfrak{s}}\), indexed by the set of cuspidal irreducible representations of Levi subgroups modulo the Weyl group action. The latter set can also be described as the \(\pi_0\) of the space of primitive ideals of the Hecke algebra \(\mathcal{H}G\) equipped with the Jacobson topology. Bernstein’s theory furnishes a map \(\pi_{\mathfrak{s}}: \hat{G}_{\mathfrak{s}} \to T_{\mathfrak{s}}/W_{\mathfrak{s}}\) (“the infinitesimal character”) for some complex torus \(T_{\mathfrak{s}}\) divided by a finite extended Coxeter group, but finer structural results are certainly desirable.
The article provides a concise introduction to the ABP conjectures, named after the authors, which propose a geometric description of \(\hat{G}_{\mathfrak{s}}\). Roughly speaking, one defines an extended quotient \(T_{\mathfrak{s}}//W_{\mathfrak{s}}\) together with maps \(\rho_{\mathfrak{s}}: T_{\mathfrak{s}}//W_{\mathfrak{s}} \to T_{\mathfrak{s}}/W_{\mathfrak{s}}\) and \(T_{\mathfrak{s}}/W_{\mathfrak{s}} \hookrightarrow T_{\mathfrak{s}}//W_{\mathfrak{s}}\). Their conjectures include a bijection \(\mu_{\mathfrak{s}}\) from the extended quotient to the Bernstein component. Note that \(\pi_{\mathfrak{s}} \circ \mu_{\mathfrak{s}}\) does not coincide with \(\rho_{\mathfrak{s}}\). Nevertheless, there is a conjectured algebraic family of finite morphisms \(\theta_\tau: T_{\mathfrak{s}}//W_{\mathfrak{s}} \to T_{\mathfrak{s}}/W_{\mathfrak{s}}\), where \(\tau \in \mathbb{C}^\times\), such that \(\theta_1 = \rho_{\mathfrak{s}}\), \(\theta_{\sqrt{q}} = \pi_{\mathfrak{s}} \circ \mu_{\mathfrak{s}}\), and there is a remarkable description of the reductibility points in terms of \(\theta_{\sqrt{\tau}}\). The authors also conjectured the existence of a “correcting cocharacter” \(h_c: \mathbb{C}^\times \to T_{\mathfrak{s}}\) that describes \(\theta_\tau\) in terms of \(h_c(\tau)\). This cocharacter is expected to be closely related to the \(\mathrm{SL}(2,\mathbb{C})\)-component in the local Langlands conjecture.
Apart from Langlands’ correspondence, the admissible representations of \(G\) can also be studied via the Hecke algebra \(\mathcal{H}G\), for which tools from \(K\)-theory (for \(C^*\)-algebras) and periodic cyclic homology are available. One remarkable feature of the ABP formalism is that it interacts with both sides. For the Hecke algebra aspect, the ABP conjectures will yield finer results then the Baum-Connes conjecture [V. Lafforgue, Invent. Math. 149, No. 1, 1–95 (2002; Zbl 1084.19003)].
In the final sections of the paper, there is a summary of related recent works, especially those of Solleveld which established parts of ABP for a large number of cases. There is also a discussion on the relation between the inner structure of L-packets and the geometric structure of extended quotients.
For the entire collection see [Zbl 1216.22001].

22E50 Representations of Lie and linear algebraic groups over local fields
20G05 Representation theory for linear algebraic groups
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