Celebratio Mathematica

A.-M. Aubert, P. Baum, and R. Plymen: “The Hecke algebra of a reductive \( p \)-adic group: A geometric conjecture,” pp. 1–34 in Noncommutative geometry and number theory: Where arithmetic meets geometry and physics (Bonn, Germany, August 2003 and June 2004). Edited by C. Consani and M. Marcolli. Aspects of Mathematics 37. Vieweg (Wiesbaden, Germany), 2006. MR 2327297 Zbl 1120.14001 incollection

Let \( \mathcal{H}(G) \) be the Hecke algebra of a reductive \( p \)-adic group \( G \). We formulate a conjecture for the ideals in the Bernstein decomposition of \( \mathcal{H}(G) \). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig’s conjecture on the asymptotic Hecke algebra. We prove our conjecture for \( \mathrm{SL}(2) \) and \( \mathrm{GL}(n) \). We also prove part (1) of the conjecture for the Iwahori ideals of the groups \( \mathrm{PGL}(n) \) and \( \mathrm{SO}(5) \). The conjecture, if true, leads to a parametrization of the smooth dual of \( G \) by the points in a complex affine locally algebraic variety.

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Caterina Consani	Related
Matilde Marcolli	Related

A.-M. Aubert, P. Baum, and R. Plymen: “Geometric structure in the representation theory of \( p \)-adic groups,” C. R. Math. Acad. Sci. Paris 345 : 10 (2007), pp. 573–578. Part II was published in Harmonic analysis on reductive, \( p \)-adic groups (2011), but with “reductive” in the title. MR 2374467 Zbl 1128.22009 article

Anne-Marie Aubert	Related
Roger J. Plymen	Related

A.-M. Aubert, P. Baum, and R. Plymen: “Geometric structure in the principal series of the \( p \)-adic group \( \textrm{G}_2 \),” Represent. Theory 15 (2011), pp. 126–169. MR 2772586 Zbl 1268.22015 article

In the representation theory of reductive \( p \)-adic groups \( G \), the issue of reducibility of induced representations is an issue of great intricacy. It is our contention, expressed as a conjecture in [2007], that there exists a simple geometric structure underlying this intricate theory.

We will illustrate here the conjecture with some detailed computations in the principal series of \( \mathrm{G}_2 \).

A feature of this article is the role played by cocharacters \( h_{\mathbf{c}} \) attached to two-sided cells \( \mathbf{c} \) in certain extended affine Weyl groups.

The quotient varieties which occur in the Bernstein programme are replaced by extended quotients. We form the disjoint union \( \mathfrak{A}(G) \) of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space \( \mathfrak{A}(G) \) is a model of the smooth dual \( \textrm{Irr}(G) \). In this respect, our programme is a conjectural refinement of the Bernstein programme.

The algebraic deformation is controlled by the cocharacters \( h_{\mathbf{c}} \). The cocharacters themselves appear to be closely related to Langlands parameters.

Anne-Marie Aubert	Related
Roger J. Plymen	Related

A.-M. Aubert, P. Baum, and R. Plymen: “Geometric structure in the representation theory of reductive \( p \)-adic groups, II,” pp. 71–90 in Harmonic analysis on reductive, \( p \)-adic groups (San Francisco, 16 January 2010). Edited by R. S. Doran, P. J. Sally, Jr., and L. Spice. Contemporary Mathematics 543. American Mathematical Society (Providence, RI), 2011. Part I was published in C. R. Math. Acad. Sci. Paris 345:10 (2007), doesn’t include “reductive” in title. MR 2798423 Zbl 1246.22019 incollection

Paul J. Sally	Related
Robert Stuart Doran	Related
Anne-Marie Aubert	Related
Roger J. Plymen	Related

@incollection {key2798423m,
	AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
		 Plymen, Roger},
	TITLE = {Geometric structure in the representation
		 theory of reductive \$p\$-adic groups,
		 {II}},
	BOOKTITLE = {Harmonic analysis on reductive, \$p\$-adic
		 groups},
	EDITOR = {Doran, Robert S. and Sally, Jr., Paul
		 J. and Spice, Loren},
	SERIES = {Contemporary Mathematics},
	NUMBER = {543},
	PUBLISHER = {American Mathematical Society},
	ADDRESS = {Providence, RI},
	YEAR = {2011},
	PAGES = {71--90},
	DOI = {10.1090/conm/543/10730},
	NOTE = {(San Francisco, 16 January 2010). Part
		 I was published in \textit{C. R. Math.
		 Acad. Sci. Paris} \textbf{345}:10 (2007),
		 doesn't include ``reductive'' in title.
		 MR:2798423. Zbl:1246.22019.},
	ISSN = {0271-4132},
	ISBN = {9780821849859},
}

P. Baum, A.-M. Aubert, R. Plymen, and M. Solleveld: Geometric structure and the local Langlands conjecture. Preprint, 2012. ArXiv 1211.0180 techreport

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Geometric structure in smooth dual and local Langlands conjecture,” Jpn. J. Math. 9 : 2 (September 2014), pp. 99–136. Expository article based on the Takagi lectures. MR 3258616 Zbl 1371.11097 article

This expository paper first reviews some basic facts about \( p \)-adic fields, reductive \( p \)-adic groups, and the local Langlands conjecture. If \( G \) is a reductive \( p \)-adic group, then the smooth dual of \( G \) is the set of equivalence classes of smooth irreducible representations of \( G \). The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual is the disjoint union of subsets known as the Bernstein components. According to a conjecture due to ABPS (Aubert–Baum–Plymen–Solleveld), each Bernstein component has a geometric structure given by an appropriate extended quotient. The paper states this ABPS conjecture and then indicates evidence for the conjecture, and its connection to the local Langlands conjecture.

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “On the local Langlands correspondence for non-tempered representations,” Münster J. Math. 7 : 1 (2014), pp. 27–50. Dedicated to Peter Schneider on the occasion of his 60th birthday. MR 3271238 Zbl 06382808 ArXiv 1303.0828 article

Let \( G \) be a reductive \( p \)-adic group. We study how a local Langlands correspondence for irreducible tempered \( G \)-representations can be extended to a local Langlands correspondence for all irreducible smooth representations of \( G \). We prove that, under a natural condition involving compatibility with unramified twists, this is possible in a canonical way.

To this end we introduce analytic \( \mathrm{R} \)-groups associated to non-tempered essentially square-integrable representations of Levi subgroups of \( G \). We establish the basic properties of these new \( \mathrm{R} \)-groups, which generalize Knapp–Stein \( \mathrm{R} \)-groups.

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Peter Schneider	Related
Maarten Solleveld	Related

A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Geometric structure for the principal series of a split reductive \( p \)-adic group with connected centre,” J. Noncommut. Geom. 10 : 2 (2016), pp. 663–680. MR 3519048 Zbl 1347.22013 article

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “The local Langlands correspondence for inner forms of \( \mathrm{SL}_n \),” Res. Math. Sci. 3 (2016). paper no. 32. MR 3579297 Zbl 06663301 article

Let \( F \) be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group \( \mathrm{SL}_n(F) \). It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for \( \mathrm{SL}_n(F) \) enhanced with an irreducible representation of an \( \mathrm{S} \)-group and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of \( \mathrm{SL}_n(F) \) up to equivalence. An analogous result is shown in the archimedean case. For \( p \)-adic fields, this is based on the work of Hiraga and Saito. To settle the case where \( F \) has positive characteristic, we employ the method of close fields. We prove that this method is compatible with the local Langlands correspondence for inner forms of \( \mathrm{GL}_n(F) \), when the fields are close enough compared to the depth of the representations.

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Depth and the local Langlands correspondence,” pp. 17–41 in Arbeitstagung Bonn 2013: In memory of Friedrich Hirzebruch (Bonn, Germany, 22–28 May 2013). Edited by W. Ballmann, C. Blohmann, G. Faltings, P. Teichner, and D. Zagier. Progress in Mathematics 319. Birkhäuser/Springer International (Cham, Switzerland), 2016. MR 3618046 Zbl 06748683 incollection

F. Hirzebruch	Related	Volume
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Christian Blohmann	Related
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A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Hecke algebras for inner forms of \( p \)-adic special linear groups,” J. Inst. Math. Jussieu 16 : 2 (2017), pp. 351–419. MR 3615412 Zbl 06704330 article

Let \( F \) be a non-Archimedean local field, and let \( G^{\sharp} \) be the group of \( F \)-rational points of an inner form of \( \mathrm{SL}_n \). We study Hecke algebras for all Bernstein components of \( G^{\sharp} \), via restriction from an inner form \( G \) of \( \mathrm{GL}_n(F) \).

For any packet of \( \mathrm{L} \)-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth \( G^{\sharp} \)-representations. This algebra comes from an idempotent in the full Hecke algebra of \( G^{\sharp} \), and the idempotent is derived from a type for \( G \). We show that the Hecke algebras for Bernstein components of \( G^{\sharp} \) are similar to affine Hecke algebras of type \( A \), yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “The principal series of \( p \)-adic groups with disconnected center,” Proc. Lond. Math. Soc. (3) 114 : 5 (2017), pp. 798–854. MR 3653247 Zbl 06778792 article

Let \( \mathscr{G} \) be a split connected reductive group over a local non-Archimedean field. We classify all irreducible complex \( \mathscr{G} \)-representations in the principal series, irrespective of the (dis)connectedness of the center of \( \mathscr{G} \). This leads to a local Langlands correspondence for principal series representations of \( \mathscr{G} \). It satisfies all expected properties, in particular it is functorial with respect to homomorphisms of reductive groups. At the same time, we show that every Bernstein component \( \mathfrak{s} \) in the principal series has the structure of an extended quotient of Bernstein’s torus by Bernstein’s finite group (both attached to \( \mathfrak{s} \)).

Anne-Marie Aubert	Related
Roger J. Plymen	Related
Maarten Solleveld	Related

A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Conjectures about \( p \)-adic groups and their noncommutative geometry,” pp. 15–51 in Around Langlands correspondences (Orsay, France, 17–20 June 2015). Edited by F. Brumley, M. P. Gómez Aparicio, and A. Minguez. Contemporary Mathematics 691. American Mathematical Society (Providence, RI), 2017. MR 3666049 ArXiv 1508.02837 incollection

Let \( G \) be any reductive \( p \)-adic group. We discuss several conjectures, some of them new, that involve the representation theory and the geometry of \( G \).

At the heart of these conjectures are statements about the geometric structure of Bernstein components for \( G \), both at the level of the space of irreducible representations and at the level of the associated Hecke algebras. We relate this to two well-known conjectures: the local Langlands correspondence and the Baum–Connes conjecture for \( G \). In particular, we present a strategy to reduce the local Langlands correspondence for irreducible \( G \)-representations to the local Langlands correspondence for supercuspidal representations of Levi subgroups.

Farrell Brumley	Related
Alberto Mínguez	Related
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A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld: “Smooth duals of inner forms of \( \mathrm{ GL}_n \) and \( \mathrm{ SL}_n \),” Doc. Math. 24 (2019), pp. 373–420. MR 3960124 article

Anne-Marie Aubert	Related
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