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
for non-tempered representations},
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
for inner forms of \$\mathrm{SL}_n\$},
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},
}
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.
Zbl:06748683.},
ISSN = {0743-1643},
ISBN = {9783319436463},
}
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
Plymen, Roger and Solleveld, Maarten},
TITLE = {Hecke algebras for inner forms of \$p\$-adic
special linear groups},
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},
NOTE = {MR:3615412. Zbl:06704330.},
ISSN = {1474-7480},
}
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},
}
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.
@incollection {key3666049m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger and Solleveld, Maarten},
TITLE = {Conjectures about \$p\$-adic groups and
their noncommutative geometry},
BOOKTITLE = {Around {L}anglands correspondences},
EDITOR = {Brumley, Farrell and G\'omez Aparicio,
Maria Paula and Minguez, Alberto},
SERIES = {Contemporary Mathematics},
NUMBER = {691},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2017},
PAGES = {15--51},
DOI = {10.1090/conm/691/13892},
NOTE = {(Orsay, France, 17--20 June 2015). ArXiv:1508.02837.
MR:3666049.},
ISSN = {0271-4132},
ISBN = {9781470435738},
}
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},
}
