P. Baum, N. Higson, and R. Plymen :
“Equivariant homology for \( \mathrm{SL}(2) \) of a \( p \) -adic field 1–18
in
Index theory and operator algebras
(Boulder, CO, 6–10 August 1991 ).
Edited by J. Fox and P. Haskell .
Contemporary Mathematics 148 .
American Mathematical Society (Providence, RI ),
1993 .
MR
1228497 Zbl
0844.46043 incollection
Abstract
People
BibTeX
Let \( F \) be a \( p \) -adic field and let \( G = SL(2) \) be the group of unimodular \( 2{\times}2 \) matrices over \( F \) . The aim of this paper is to calculate certain equivariant homology groups attached to the action of \( G \) on its tree. They arise in connection with a theorem of M. Pimsner on the \( K \) -theory of the \( C^* \) -algebra of \( G \) [1986], and our purpose is to explore the representation theoretic content of Pimsner’s result.
@incollection {key1228497m,
AUTHOR = {Baum, Paul and Higson, Nigel and Plymen,
Roger},
TITLE = {Equivariant homology for \$\mathrm{SL}(2)\$
of a \$p\$-adic field},
BOOKTITLE = {Index theory and operator algebras},
EDITOR = {Fox, J. and Haskell, P.},
SERIES = {Contemporary Mathematics},
NUMBER = {148},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1993},
PAGES = {1--18},
DOI = {10.1090/conm/148/01246},
NOTE = {(Boulder, CO, 6--10 August 1991). MR:1228497.
Zbl:0844.46043.},
ISSN = {0271-4132},
ISBN = {9780821851524},
}
P. Baum, N. Higson, and R. Plymen :
Cosheaf homology and \( K \) theory for \( p \) -adic groups .
Preprint ,
Pennsylvania State University ,
1995 .
techreport
People
BibTeX
@techreport {key25400440,
AUTHOR = {Baum, Paul and Higson, N. and Plymen,
R.},
TITLE = {Cosheaf homology and \$K\$ theory for
\$p\$-adic groups},
TYPE = {preprint},
INSTITUTION = {Pennsylvania State University},
YEAR = {1995},
}
P. Baum, N. Higson, and R. Plymen :
“A proof of the Baum–Connes conjecture for \( p \) -adic \( \mathrm{GL}(n) \) C. R. Acad. Sci. Paris Sér. I Math.
325 : 2
(July 1997 ),
pp. 171–176 .
MR
1467072 Zbl
0918.46061 article
Abstract
People
BibTeX
@article {key1467072m,
AUTHOR = {Baum, Paul and Higson, Nigel and Plymen,
Roger},
TITLE = {A proof of the {B}aum--{C}onnes conjecture
for \$p\$-adic \$\mathrm{GL}(n)\$},
JOURNAL = {C. R. Acad. Sci. Paris S\'er. I Math.},
FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences.
S\'erie I. Math\'ematique},
VOLUME = {325},
NUMBER = {2},
MONTH = {July},
YEAR = {1997},
PAGES = {171--176},
DOI = {10.1016/S0764-4442(97)84594-6},
NOTE = {MR:1467072. Zbl:0918.46061.},
ISSN = {0764-4442},
}
P. F. Baum, N. Higson, and R. J. Plymen :
“Representation theory of \( p \) -adic groups: A view from operator algebras 111–149
in
The mathematical legacy of Harish-Chandra: A celebration of representation theory and harmonic analysis
(Baltimore, MD, 9–10 January 1998 ).
Edited by R. Doran and V. Varadarajan .
Proceedings of Symposia in Pure Mathematics 68 .
American Mathematical Society (Providence, RI ),
2000 .
MR
1767895 Zbl
0982.19006 incollection
Abstract
People
BibTeX
@incollection {key1767895m,
AUTHOR = {Baum, P. F. and Higson, N. and Plymen,
R. J.},
TITLE = {Representation theory of \$p\$-adic groups:
{A} view from operator algebras},
BOOKTITLE = {The mathematical legacy of {H}arish-{C}handra:
{A} celebration of representation theory
and harmonic analysis},
EDITOR = {Doran, R. and Varadarajan, V.},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {68},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2000},
PAGES = {111--149},
DOI = {10.1090/pspum/068/1767895},
NOTE = {(Baltimore, MD, 9--10 January 1998).
MR:1767895. Zbl:0982.19006.},
ISSN = {0082-0717},
ISBN = {9780821811979},
}
P. Baum, S. Millington, and R. Plymen :
“A proof of the Baum–Connes conjecture for reductive adelic groups C. R. Acad. Sci. Paris Sér. I Math.
332 : 3
(February 2001 ),
pp. 195–200 .
MR
1817360 Zbl
1105.19300 article
Abstract
People
BibTeX
@article {key1817360m,
AUTHOR = {Baum, Paul and Millington, Stephen and
Plymen, Roger},
TITLE = {A proof of the {B}aum--{C}onnes conjecture
for reductive adelic groups},
JOURNAL = {C. R. Acad. Sci. Paris S\'er. I Math.},
FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences.
S\'erie I. Math\'ematique},
VOLUME = {332},
NUMBER = {3},
MONTH = {February},
YEAR = {2001},
PAGES = {195--200},
DOI = {10.1016/S0764-4442(00)01810-3},
NOTE = {MR:1817360. Zbl:1105.19300.},
ISSN = {0764-4442},
}
P. Baum, S. Millington, and R. Plymen :
“Local-global principle for the Baum–Connes conjecture with coefficients \( K \) -Theory28 : 1
(2003 ),
pp. 1–18 .
MR
1988816 Zbl
1034.46073 article
Abstract
People
BibTeX
@article {key1988816m,
AUTHOR = {Baum, Paul and Millington, Stephen and
Plymen, Roger},
TITLE = {Local-global principle for the {B}aum--{C}onnes
conjecture with coefficients},
JOURNAL = {\$K\$-Theory},
FJOURNAL = {\$K\$-Theory. An Interdisciplinary Journal
for the Development, Application, and
Influence of \$K\$-Theory in the Mathematical
Sciences},
VOLUME = {28},
NUMBER = {1},
YEAR = {2003},
PAGES = {1--18},
DOI = {10.1023/A:1024197623173},
NOTE = {MR:1988816. Zbl:1034.46073.},
ISSN = {0920-3036},
}
A.-M. Aubert, P. Baum, and R. Plymen :
“The Hecke algebra of a reductive \( p \) -adic group: A geometric conjecture 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
Abstract
People
BibTeX
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.
@incollection {key2327297m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger},
TITLE = {The {H}ecke algebra of a reductive \$p\$-adic
group: {A} geometric conjecture},
BOOKTITLE = {Noncommutative geometry and number theory:
{W}here arithmetic meets geometry and
physics},
EDITOR = {Consani, C. and Marcolli, M.},
SERIES = {Aspects of Mathematics},
NUMBER = {37},
PUBLISHER = {Vieweg},
ADDRESS = {Wiesbaden, Germany},
YEAR = {2006},
PAGES = {1--34},
DOI = {10.1007/978-3-8348-0352-8_1},
NOTE = {(Bonn, Germany, August 2003 and June
2004). MR:2327297. Zbl:1120.14001.},
ISSN = {0179-2156},
ISBN = {9783834801708},
}
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)
MR
2374467 Zbl
1128.22009 article
Abstract
People
BibTeX
@article {key2374467m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger},
TITLE = {Geometric structure in the representation
theory of \$p\$-adic groups},
JOURNAL = {C. R. Math. Acad. Sci. Paris},
FJOURNAL = {Comptes Rendus Math\'ematique. Acad\'emie
des Sciences. Paris},
VOLUME = {345},
NUMBER = {10},
YEAR = {2007},
PAGES = {573--578},
DOI = {10.1016/j.crma.2007.10.011},
NOTE = {Part II was published in \textit{Harmonic
analysis on reductive,} \$p\$-\textit{adic
groups} (2011), but with ``reductive''
in the title. MR:2374467. Zbl:1128.22009.},
ISSN = {1631-073X},
}
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
Abstract
People
BibTeX
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.
@article {key2772586m,
AUTHOR = {Aubert, Anne-Marie and Baum, Paul and
Plymen, Roger},
TITLE = {Geometric structure in the principal
series of the \$p\$-adic group \$\textrm{G}_2\$},
JOURNAL = {Represent. Theory},
FJOURNAL = {Representation Theory. An Electronic
Journal of the American Mathematical
Society},
VOLUME = {15},
YEAR = {2011},
PAGES = {126--169},
DOI = {10.1090/S1088-4165-2011-00392-7},
NOTE = {MR:2772586. Zbl:1268.22015.},
ISSN = {1088-4165},
}
A.-M. Aubert, P. Baum, and R. Plymen :
“Geometric structure in the representation theory of reductive \( p \) -adic groups, II 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)
MR
2798423 Zbl
1246.22019 incollection
Abstract
People
BibTeX
@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
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
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},
}
