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Some aspects of the geometric structure of the smooth dual of \(p\)-adic reductive groups. (English) Zbl 1477.22010

Dąbrowski, Ludwik (ed.) et al., Quantum dynamics. Dedicated to Professor Paul Baum. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 120, 135-150 (2020).
This expository paper reviews results and conjectures on the structure of Bernstein blocks for the smooth dual of a reductive \(p\)-adic group \(G\).
The main theme is the Aubert-Baum-Plymen-Solleveld (ABPS) conjecture. Two well-written and detailed surveys on the topic have been written by these four authors: see [the author et al., Jpn. J. Math. (3) 9, No. 2, 99–136 (2014; Zbl 1371.11097); A.-M. Aubert et al., Contemp. Math. 691, 15–51 (2017; Zbl 1468.22020)]. In the article under review, the author specializes everything to the case where \(G\) is split. In that case, many of the statements simplify. This paper is therefore a particularly accessible complement to the two substantial surveys mentioned above.
Let \(\mathrm{Irr}(G)\) denote the smooth dual of \(G\). It can be written as a disjoint union of “Bernstein blocks” \(\mathrm{Irr}(G; \mathfrak{s})\), where \(\mathfrak{s}\) runs over the Bernstein spectrum of \(G\) (the set of inertial equivalence classes of supercuspidal representations of Levi subgroups of \(G\); see for instance I. N. Bernstein [in: Représentations des groups réductifs sur un corps local. Paris: Hermann 1–32 (1984; Zbl 0599.22016)], Section 2). In the split case, the ABPS conjecture predicts (among other properties) that each Bernstein component \(\mathrm{Irr}(G; \mathfrak{s})\) of \(\mathrm{Irr}(G)\) stands in natural bijection with the “spectral extended quotient” \(T_\mathfrak{s}/\!/W_\mathfrak{s}\) attached to the action of a finite group \(W_\mathfrak{s}\) on a complex torus \(T_\mathfrak{s}\) (both coming from the Bernstein decomposition). The notion of spectral extended quotient is reviewed in Section 2, and the Bernstein decomposition and ABPS conjecture are discussed in Section 3.
Sections 4 and 5 focus on the interplay between the ABPS conjecture and the local Langlands correspondence, reviewing work of A. Moussaoui [Represent. Theory 21, 172–246 (2017; Zbl 1379.22016)] and the author et al. [Manuscr. Math. 157, No. 1–2, 121–192 (2018; Zbl 1414.11161)]. In particular, the author reviews theorems which exhibit a simple structure in the set of enhanced Langlands parameters for \(G\), given by a union of extended quotients. This gives a Galois analogue of the ABPS conjecture. A key notion is that of cuspidal (enhanced) Langlands parameters, which should (conjecturally) correspond to supercuspidal representations of \(G\) under the local Langlands correspondence. The Springer correspondence also plays an important role.
Section 6 focuses on the interplay between the ABPS conjecture and the Baum-Connes conjecture for the K-theory of the reduced \(C^\ast\)-algebra \(C^\ast_r(G)\). This algebra admits a decomposition into a direct sum of \(C^\ast\)-algebras \(C^\ast_r(G; \mathfrak{s})\); the spectrum of any one of these algebras is given by the intersection of a Bernstein block with the tempered dual of \(G\).
The information on Bernstein blocks given by the ABPS conjecture naturally leads to a prediction on the \(K\)-theory of each algebra \(C^\ast_r(G; \mathfrak{s})\): it is expected to be given by the topological equivariant \(K\)-theory for the action of \(T_\mathfrak{s}\) on the torus \(W_\mathfrak{s}\). Since the Baum-Connes conjecture gives another conjectural description of the \(K\)-theory of \(C^\ast_r(G)\), this provides a bridge between the ABPS and Baum-Connes conjectures.
The article is contained in an issue dedicated to Paul Baum, and the author includes a few reminiscences of her early interaction with Baum. This expository paper reviews results and conjectures on the structure of Bernstein blocks for the smooth dual of a reductive \(p\)-adic group \(G\).
The main theme is the Aubert-Baum-Plymen-Solleveld (ABPS) conjecture. Two well-written and detailed surveys on the topic have been written by these four authors: see [the author et al., 2014, loc. cit.; 2017, loc. cit.]. In the article under review, the author specializes everything to the case where \(G\) is split. In that case, many of the statements simplify. This paper is therefore a particularly accessible complement to the two substantial surveys mentioned above.
Let \(\mathrm{Irr}(G)\) denote the smooth dual of \(G\). It can be written as a disjoint union of “Bernstein blocks” \(\mathrm{Irr}(G; \mathfrak{s})\), where \(\mathfrak{s}\) runs over the Bernstein spectrum of \(G\) (the set of inertial equivalence classes of supercuspidal representations of Levi subgroups of \(G\); see for instance [Bernstein, loc. cit., Section 2]. In the split case, the ABPS conjecture predicts (among other properties) that each Bernstein component \(\mathrm{Irr}(G; \mathfrak{s})\) of \(\mathrm{Irr}(G)\) stands in natural bijection with the “spectral extended quotient” \(T_\mathfrak{s}/\!/W_\mathfrak{s}\) attached to the action of a finite group \(W_\mathfrak{s}\) on a complex torus \(T_\mathfrak{s}\) (both coming from the Bernstein decomposition). The notion of spectral extended quotient is reviewed in Section 2, and the Bernstein decomposition and ABPS conjecture are discussed in Section 3.
Sections 4 and 5 focus on the interplay between the ABPS conjecture and the local Langlands correspondence , reviewing work of A. Moussaoui [loc. cit.] and the author et al. [2018, loc. cit.]. In particular, the author reviews theorems which exhibit a simple structure in the set of enhanced Langlands parameters for \(G\), given by a union of extended quotients. This gives a Galois analogue of the ABPS conjecture. A key notion is that of cuspidal (enhanced) Langlands parameters, which should (conjecturally) correspond to supercuspidal representations of \(G\) under the local Langlands correspondence. The Springer correspondence also plays an important role.
Section 6 focuses on the interplay between the ABPS conjecture and the Baum-Connes conjecture for the \(K\)-theory of the reduced \(C^\ast\)-algebra \(C^\ast_r(G)\). This algebra admits a decomposition into a direct sum of \(C^\ast\)-algebras \(C^\ast_r(G; \mathfrak{s})\); the spectrum of any one of these algebras is given by the intersection of a Bernstein block with the tempered dual of \(G\).
The information on Bernstein blocks given by the ABPS conjecture naturally leads to a prediction on the \(K\)-theory of each algebra \(C^\ast_r(G; \mathfrak{s})\): it is expected to be given by the topological equivariant \(K\)-theory for the action of \(T_\mathfrak{s}\) on the torus \(W_\mathfrak{s}\). Since the Baum-Connes conjecture gives another conjectural description of the \(K\)-theory of \(C^\ast_r(G)\), this provides a bridge between the ABPS and Baum-Connes conjectures.
The article is contained in an issue dedicated to Paul Baum, and the author includes a few reminiscences of her early interaction with Baum.
For the entire collection see [Zbl 1462.46002].

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
19L47 Equivariant \(K\)-theory
20G25 Linear algebraic groups over local fields and their integers
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