P. Baum, A.-M. Aubert, R. Plymen, and M. Solleveld :
Geometric structure and the local Langlands conjecture .
Preprint ,
2012 .
ArXiv
1211.0180 techreport
Abstract
People
BibTeX
We prove that a strengthened form of the local Langlands conjecture is valid throughout the principal series of any connected split reductive \( p \) -adic group. The method of proof is to establish the presence of a very simple geometric structure, in both the smooth dual and the Langlands parameters. We prove that this geometric structure is present, in the same way, for the general linear group, including all of its inner forms. With these results as evidence, we give a detailed formulation of a general geometric structure conjecture.
@techreport {key1211.0180a,
AUTHOR = {Baum, Paul and Aubert, Anne-Marie and
Plymen, Roger and Solleveld, Maarten},
TITLE = {Geometric structure and the local {L}anglands
conjecture},
TYPE = {preprint},
YEAR = {2012},
NOTE = {ArXiv:1211.0180.},
}
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
Abstract
People
BibTeX
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.
@article {key3258616m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger and Solleveld, Maarten},
TITLE = {Geometric structure in smooth dual and
local {L}anglands conjecture},
JOURNAL = {Jpn. J. Math.},
FJOURNAL = {Japanese Journal of Mathematics},
VOLUME = {9},
NUMBER = {2},
MONTH = {September},
YEAR = {2014},
PAGES = {99--136},
DOI = {10.1007/s11537-014-1267-x},
NOTE = {Expository article based on the Takagi
lectures. MR:3258616. Zbl:1371.11097.},
ISSN = {0289-2316},
}
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
Abstract
People
BibTeX
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.
@article {key3271238m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger and Solleveld, Maarten},
TITLE = {On the local {L}anglands correspondence
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JOURNAL = {M\"unster J. Math.},
FJOURNAL = {M\"unster Journal of Mathematics},
VOLUME = {7},
NUMBER = {1},
YEAR = {2014},
PAGES = {27--50},
URL = {https://www.uni-muenster.de/FB10/mjm/vol_7/mjm_vol_7_03.pdf},
NOTE = {Dedicated to Peter Schneider on the
occasion of his 60th birthday. ArXiv:1303.0828.
MR:3271238. Zbl:06382808.},
ISSN = {1867-5778},
}
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
Abstract
People
BibTeX
Let \( \mathcal{G} \) be a split reductive \( p \) -adic group with connected centre. We show that each Bernstein block in the principal series of \( \mathcal{G} \) admits a definite geometric structure, namely that of an extended quotient. For the Iwahori-spherical block, this extended quotient has the form \( T/W \) where \( T \) is a maximal torus in the Langlands dual group of \( \mathcal{G} \) and \( W \) is the Weyl group of \( \mathcal{G} \) .
@article {key3519048m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger and Solleveld, Maarten},
TITLE = {Geometric structure for the principal
series of a split reductive \$p\$-adic
group with connected centre},
JOURNAL = {J. Noncommut. Geom.},
FJOURNAL = {Journal of Noncommutative Geometry},
VOLUME = {10},
NUMBER = {2},
YEAR = {2016},
PAGES = {663--680},
DOI = {10.4171/JNCG/244},
NOTE = {MR:3519048. Zbl:1347.22013.},
ISSN = {1661-6952},
}
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
Abstract
People
BibTeX
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.
@article {key3579297m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger and Solleveld, Maarten},
TITLE = {The local {L}anglands correspondence
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JOURNAL = {Res. Math. Sci.},
FJOURNAL = {Research in the Mathematical Sciences},
VOLUME = {3},
YEAR = {2016},
DOI = {10.1186/s40687-016-0079-4},
NOTE = {paper no. 32. MR:3579297. Zbl:06663301.},
ISSN = {2522-0144},
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A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld :
“Depth and the local Langlands correspondence 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
Abstract
People
BibTeX
@incollection {key3618046m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger and Solleveld, Maarten},
TITLE = {Depth and the local {L}anglands correspondence},
BOOKTITLE = {Arbeitstagung {B}onn 2013: {I}n memory
of {F}riedrich {H}irzebruch},
EDITOR = {Ballmann, Werner and Blohmann, Christian
and Faltings, Gerd and Teichner, Peter
and Zagier, Don},
SERIES = {Progress in Mathematics},
NUMBER = {319},
PUBLISHER = {Birkh\"auser/Springer International},
ADDRESS = {Cham, Switzerland},
YEAR = {2016},
PAGES = {17--41},
DOI = {10.1007/978-3-319-43648-7_2},
NOTE = {(Bonn, Germany, 22--28 May 2013). MR:3618046.
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ISSN = {0743-1643},
ISBN = {9783319436463},
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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
Abstract
People
BibTeX
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.
@article {key3615412m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
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TITLE = {Hecke algebras for inner forms of \$p\$-adic
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JOURNAL = {J. Inst. Math. Jussieu},
FJOURNAL = {Journal of the Institute of Mathematics
of Jussieu. JIMJ. Journal de l'Institut
de Math\'ematiques de Jussieu},
VOLUME = {16},
NUMBER = {2},
YEAR = {2017},
PAGES = {351--419},
DOI = {10.1017/S1474748015000079},
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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
Abstract
People
BibTeX
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 \( \mathfrak{s} \) ).
@article {key3653247m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger and Solleveld, Maarten},
TITLE = {The principal series of \$p\$-adic groups
with disconnected center},
JOURNAL = {Proc. Lond. Math. Soc. (3)},
FJOURNAL = {Proceedings of the London Mathematical
Society. Third Series},
VOLUME = {114},
NUMBER = {5},
YEAR = {2017},
PAGES = {798--854},
DOI = {10.1112/plms.12023},
NOTE = {MR:3653247. Zbl:06778792.},
ISSN = {0024-6115},
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A.-M. Aubert, P. Baum, R. Plymen, and M. Solleveld :
“Conjectures about \( p \) -adic groups and their noncommutative geometry 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
Abstract
People
BibTeX
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.
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AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
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TITLE = {Conjectures about \$p\$-adic groups and
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BOOKTITLE = {Around {L}anglands correspondences},
EDITOR = {Brumley, Farrell and G\'omez Aparicio,
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SERIES = {Contemporary Mathematics},
NUMBER = {691},
PUBLISHER = {American Mathematical Society},
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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
People
BibTeX
@article {key3960124m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger and Solleveld, Maarten},
TITLE = {Smooth duals of inner forms of \${\rm
GL}_n\$ and {\${\rm SL}_n\$}},
JOURNAL = {Doc. Math.},
FJOURNAL = {Documenta Mathematica},
VOLUME = {24},
YEAR = {2019},
PAGES = {373--420},
NOTE = {MR:3960124.},
ISSN = {1431-0635},
}
