---
_id: '64275'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title>\r\n               <jats:p>We explain by
    elementary means why the existence of a discrete series representation\r\nof a
    real reductive group <jats:italic>G</jats:italic> implies the existence of a compact
    Cartan subgroup of <jats:italic>G</jats:italic>. The presented approach has the
    potential to generalize to real spherical spaces.</jats:p>"
author:
- first_name: Bernhard
  full_name: Krötz, Bernhard
  last_name: Krötz
- first_name: Job J.
  full_name: Kuit, Job J.
  last_name: Kuit
- first_name: Eric M.
  full_name: Opdam, Eric M.
  last_name: Opdam
- first_name: Henrik
  full_name: Schlichtkrull, Henrik
  last_name: Schlichtkrull
citation:
  ama: Krötz B, Kuit JJ, Opdam EM, Schlichtkrull H. Ellipticity and discrete series.
    <i>Journal für die reine und angewandte Mathematik (Crelles Journal)</i>. 2021;2022(782):109-119.
    doi:<a href="https://doi.org/10.1515/crelle-2021-0063">10.1515/crelle-2021-0063</a>
  apa: Krötz, B., Kuit, J. J., Opdam, E. M., &#38; Schlichtkrull, H. (2021). Ellipticity
    and discrete series. <i>Journal Für Die Reine Und Angewandte Mathematik (Crelles
    Journal)</i>, <i>2022</i>(782), 109–119. <a href="https://doi.org/10.1515/crelle-2021-0063">https://doi.org/10.1515/crelle-2021-0063</a>
  bibtex: '@article{Krötz_Kuit_Opdam_Schlichtkrull_2021, title={Ellipticity and discrete
    series}, volume={2022}, DOI={<a href="https://doi.org/10.1515/crelle-2021-0063">10.1515/crelle-2021-0063</a>},
    number={782}, journal={Journal für die reine und angewandte Mathematik (Crelles
    Journal)}, publisher={Walter de Gruyter GmbH}, author={Krötz, Bernhard and Kuit,
    Job J. and Opdam, Eric M. and Schlichtkrull, Henrik}, year={2021}, pages={109–119}
    }'
  chicago: 'Krötz, Bernhard, Job J. Kuit, Eric M. Opdam, and Henrik Schlichtkrull.
    “Ellipticity and Discrete Series.” <i>Journal Für Die Reine Und Angewandte Mathematik
    (Crelles Journal)</i> 2022, no. 782 (2021): 109–19. <a href="https://doi.org/10.1515/crelle-2021-0063">https://doi.org/10.1515/crelle-2021-0063</a>.'
  ieee: 'B. Krötz, J. J. Kuit, E. M. Opdam, and H. Schlichtkrull, “Ellipticity and
    discrete series,” <i>Journal für die reine und angewandte Mathematik (Crelles
    Journal)</i>, vol. 2022, no. 782, pp. 109–119, 2021, doi: <a href="https://doi.org/10.1515/crelle-2021-0063">10.1515/crelle-2021-0063</a>.'
  mla: Krötz, Bernhard, et al. “Ellipticity and Discrete Series.” <i>Journal Für Die
    Reine Und Angewandte Mathematik (Crelles Journal)</i>, vol. 2022, no. 782, Walter
    de Gruyter GmbH, 2021, pp. 109–19, doi:<a href="https://doi.org/10.1515/crelle-2021-0063">10.1515/crelle-2021-0063</a>.
  short: B. Krötz, J.J. Kuit, E.M. Opdam, H. Schlichtkrull, Journal Für Die Reine
    Und Angewandte Mathematik (Crelles Journal) 2022 (2021) 109–119.
date_created: 2026-02-19T13:27:22Z
date_updated: 2026-02-19T13:27:34Z
doi: 10.1515/crelle-2021-0063
intvolume: '      2022'
issue: '782'
language:
- iso: eng
page: 109-119
publication: Journal für die reine und angewandte Mathematik (Crelles Journal)
publication_identifier:
  issn:
  - 0075-4102
  - 1435-5345
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: Ellipticity and discrete series
type: journal_article
user_id: '52730'
volume: 2022
year: '2021'
...
---
_id: '34790'
article_type: original
author:
- first_name: Helge
  full_name: Glöckner, Helge
  id: '178'
  last_name: Glöckner
- first_name: George A.
  full_name: Willis, George A.
  last_name: Willis
citation:
  ama: Glöckner H, Willis GA. Locally pro-p contraction groups are nilpotent. <i>Journal
    für die reine und angewandte Mathematik</i>. 2021;781:85–103. doi:<a href="https://doi.org/10.1515/crelle-2021-0050">10.1515/crelle-2021-0050</a>
  apa: Glöckner, H., &#38; Willis, G. A. (2021). Locally pro-p contraction groups
    are nilpotent. <i>Journal Für Die Reine Und Angewandte Mathematik</i>, <i>781</i>,
    85–103. <a href="https://doi.org/10.1515/crelle-2021-0050">https://doi.org/10.1515/crelle-2021-0050</a>
  bibtex: '@article{Glöckner_Willis_2021, title={Locally pro-p contraction groups
    are nilpotent}, volume={781}, DOI={<a href="https://doi.org/10.1515/crelle-2021-0050">10.1515/crelle-2021-0050</a>},
    journal={Journal für die reine und angewandte Mathematik}, author={Glöckner, Helge
    and Willis, George A.}, year={2021}, pages={85–103} }'
  chicago: 'Glöckner, Helge, and George A. Willis. “Locally Pro-p Contraction Groups
    Are Nilpotent.” <i>Journal Für Die Reine Und Angewandte Mathematik</i> 781 (2021):
    85–103. <a href="https://doi.org/10.1515/crelle-2021-0050">https://doi.org/10.1515/crelle-2021-0050</a>.'
  ieee: 'H. Glöckner and G. A. Willis, “Locally pro-p contraction groups are nilpotent,”
    <i>Journal für die reine und angewandte Mathematik</i>, vol. 781, pp. 85–103,
    2021, doi: <a href="https://doi.org/10.1515/crelle-2021-0050">10.1515/crelle-2021-0050</a>.'
  mla: Glöckner, Helge, and George A. Willis. “Locally Pro-p Contraction Groups Are
    Nilpotent.” <i>Journal Für Die Reine Und Angewandte Mathematik</i>, vol. 781,
    2021, pp. 85–103, doi:<a href="https://doi.org/10.1515/crelle-2021-0050">10.1515/crelle-2021-0050</a>.
  short: H. Glöckner, G.A. Willis, Journal Für Die Reine Und Angewandte Mathematik
    781 (2021) 85–103.
date_created: 2022-12-21T19:17:28Z
date_updated: 2026-02-27T08:34:58Z
department:
- _id: '10'
- _id: '87'
- _id: '93'
doi: 10.1515/crelle-2021-0050
intvolume: '       781'
keyword:
- 22D05
- 22A05
- '20E18'
language:
- iso: eng
page: 85–103
publication: Journal für die reine und angewandte Mathematik
publication_identifier:
  issn:
  - 0075-4102
quality_controlled: '1'
status: public
title: Locally pro-p contraction groups are nilpotent
type: journal_article
user_id: '178'
volume: 781
year: '2021'
...
---
_id: '53184'
article_type: original
author:
- first_name: Fabian
  full_name: Januszewski, Fabian
  id: '81636'
  last_name: Januszewski
citation:
  ama: Januszewski F. Modular symbols for reductive groups and p-adic Rankin–Selberg
    convolutions over number fields. <i>Journal für die reine und angewandte Mathematik
    (Crelles Journal)</i>. 2010;2011(653):1-45. doi:<a href="https://doi.org/10.1515/crelle.2011.018">10.1515/crelle.2011.018</a>
  apa: Januszewski, F. (2010). Modular symbols for reductive groups and p-adic Rankin–Selberg
    convolutions over number fields. <i>Journal Für Die Reine Und Angewandte Mathematik
    (Crelles Journal)</i>, <i>2011</i>(653), 1–45. <a href="https://doi.org/10.1515/crelle.2011.018">https://doi.org/10.1515/crelle.2011.018</a>
  bibtex: '@article{Januszewski_2010, title={Modular symbols for reductive groups
    and p-adic Rankin–Selberg convolutions over number fields}, volume={2011}, DOI={<a
    href="https://doi.org/10.1515/crelle.2011.018">10.1515/crelle.2011.018</a>}, number={653},
    journal={Journal für die reine und angewandte Mathematik (Crelles Journal)}, publisher={Walter
    de Gruyter GmbH}, author={Januszewski, Fabian}, year={2010}, pages={1–45} }'
  chicago: 'Januszewski, Fabian. “Modular Symbols for Reductive Groups and P-Adic
    Rankin–Selberg Convolutions over Number Fields.” <i>Journal Für Die Reine Und
    Angewandte Mathematik (Crelles Journal)</i> 2011, no. 653 (2010): 1–45. <a href="https://doi.org/10.1515/crelle.2011.018">https://doi.org/10.1515/crelle.2011.018</a>.'
  ieee: 'F. Januszewski, “Modular symbols for reductive groups and p-adic Rankin–Selberg
    convolutions over number fields,” <i>Journal für die reine und angewandte Mathematik
    (Crelles Journal)</i>, vol. 2011, no. 653, pp. 1–45, 2010, doi: <a href="https://doi.org/10.1515/crelle.2011.018">10.1515/crelle.2011.018</a>.'
  mla: Januszewski, Fabian. “Modular Symbols for Reductive Groups and P-Adic Rankin–Selberg
    Convolutions over Number Fields.” <i>Journal Für Die Reine Und Angewandte Mathematik
    (Crelles Journal)</i>, vol. 2011, no. 653, Walter de Gruyter GmbH, 2010, pp. 1–45,
    doi:<a href="https://doi.org/10.1515/crelle.2011.018">10.1515/crelle.2011.018</a>.
  short: F. Januszewski, Journal Für Die Reine Und Angewandte Mathematik (Crelles
    Journal) 2011 (2010) 1–45.
date_created: 2024-04-03T16:47:27Z
date_updated: 2024-04-03T17:13:10Z
doi: 10.1515/crelle.2011.018
extern: '1'
intvolume: '      2011'
issue: '653'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
page: 1-45
publication: Journal für die reine und angewandte Mathematik (Crelles Journal)
publication_identifier:
  issn:
  - 0075-4102
  - 1435-5345
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: Modular symbols for reductive groups and p-adic Rankin–Selberg convolutions
  over number fields
type: journal_article
user_id: '81636'
volume: 2011
year: '2010'
...
---
_id: '64680'
article_type: original
author:
- first_name: Helge
  full_name: Glöckner, Helge
  id: '178'
  last_name: Glöckner
- first_name: George A.
  full_name: Willis, George A.
  last_name: Willis
citation:
  ama: Glöckner H, Willis GA. Classification of the simple factors appearing in composition
    series of totally disconnected contraction groups. <i>Journal für die reine und
    angewandte Mathematik</i>. 2010;643:141–169. doi:<a href="https://doi.org/10.1515/CRELLE.2010.047">10.1515/CRELLE.2010.047</a>
  apa: Glöckner, H., &#38; Willis, G. A. (2010). Classification of the simple factors
    appearing in composition series of totally disconnected contraction groups. <i>Journal
    Für Die Reine Und Angewandte Mathematik</i>, <i>643</i>, 141–169. <a href="https://doi.org/10.1515/CRELLE.2010.047">https://doi.org/10.1515/CRELLE.2010.047</a>
  bibtex: '@article{Glöckner_Willis_2010, title={Classification of the simple factors
    appearing in composition series of totally disconnected contraction groups}, volume={643},
    DOI={<a href="https://doi.org/10.1515/CRELLE.2010.047">10.1515/CRELLE.2010.047</a>},
    journal={Journal für die reine und angewandte Mathematik}, author={Glöckner, Helge
    and Willis, George A.}, year={2010}, pages={141–169} }'
  chicago: 'Glöckner, Helge, and George A. Willis. “Classification of the Simple Factors
    Appearing in Composition Series of Totally Disconnected Contraction Groups.” <i>Journal
    Für Die Reine Und Angewandte Mathematik</i> 643 (2010): 141–169. <a href="https://doi.org/10.1515/CRELLE.2010.047">https://doi.org/10.1515/CRELLE.2010.047</a>.'
  ieee: 'H. Glöckner and G. A. Willis, “Classification of the simple factors appearing
    in composition series of totally disconnected contraction groups,” <i>Journal
    für die reine und angewandte Mathematik</i>, vol. 643, pp. 141–169, 2010, doi:
    <a href="https://doi.org/10.1515/CRELLE.2010.047">10.1515/CRELLE.2010.047</a>.'
  mla: Glöckner, Helge, and George A. Willis. “Classification of the Simple Factors
    Appearing in Composition Series of Totally Disconnected Contraction Groups.” <i>Journal
    Für Die Reine Und Angewandte Mathematik</i>, vol. 643, 2010, pp. 141–169, doi:<a
    href="https://doi.org/10.1515/CRELLE.2010.047">10.1515/CRELLE.2010.047</a>.
  short: H. Glöckner, G.A. Willis, Journal Für Die Reine Und Angewandte Mathematik
    643 (2010) 141–169.
date_created: 2026-02-26T11:15:48Z
date_updated: 2026-02-27T08:22:51Z
department:
- _id: '10'
- _id: '87'
- _id: '93'
doi: 10.1515/CRELLE.2010.047
intvolume: '       643'
keyword:
- 22D05
- 22A05
- 22D45
language:
- iso: eng
page: 141–169
publication: Journal für die reine und angewandte Mathematik
publication_identifier:
  issn:
  - 0075-4102
quality_controlled: '1'
status: public
title: Classification of the simple factors appearing in composition series of totally
  disconnected contraction groups
type: journal_article
user_id: '178'
volume: 643
year: '2010'
...
---
_id: '34895'
abstract:
- lang: eng
  text: 'We obtain strong information on the asymptotic behaviour of the counting
    function for nilpotent Galois extensions with bounded discriminant of arbitrary
    number fields. This extends previous investigations for the case of abelian groups.
    In particular, the result confirms a conjecture by the second author on this function
    for arbitrary groups in the nilpotent case. We further prove compatibility of
    the conjecture with taking wreath products with the cyclic group of order 2 and
    give examples in degree up to 8. '
author:
- first_name: Jürgen
  full_name: Klüners, Jürgen
  id: '21202'
  last_name: Klüners
- first_name: G.
  full_name: Malle, G.
  last_name: Malle
citation:
  ama: Klüners J, Malle G. Counting nilpotent Galois extensions. <i>Journal für die
    reine und angewandte Mathematik (Crelles Journal)</i>. 2006;2004(572):1-26. doi:<a
    href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>
  apa: Klüners, J., &#38; Malle, G. (2006). Counting nilpotent Galois extensions.
    <i>Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)</i>, <i>2004</i>(572),
    1–26. <a href="https://doi.org/10.1515/crll.2004.050">https://doi.org/10.1515/crll.2004.050</a>
  bibtex: '@article{Klüners_Malle_2006, title={Counting nilpotent Galois extensions},
    volume={2004}, DOI={<a href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>},
    number={572}, journal={Journal für die reine und angewandte Mathematik (Crelles
    Journal)}, publisher={Walter de Gruyter GmbH}, author={Klüners, Jürgen and Malle,
    G.}, year={2006}, pages={1–26} }'
  chicago: 'Klüners, Jürgen, and G. Malle. “Counting Nilpotent Galois Extensions.”
    <i>Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)</i> 2004,
    no. 572 (2006): 1–26. <a href="https://doi.org/10.1515/crll.2004.050">https://doi.org/10.1515/crll.2004.050</a>.'
  ieee: 'J. Klüners and G. Malle, “Counting nilpotent Galois extensions,” <i>Journal
    für die reine und angewandte Mathematik (Crelles Journal)</i>, vol. 2004, no.
    572, pp. 1–26, 2006, doi: <a href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>.'
  mla: Klüners, Jürgen, and G. Malle. “Counting Nilpotent Galois Extensions.” <i>Journal
    Für Die Reine Und Angewandte Mathematik (Crelles Journal)</i>, vol. 2004, no.
    572, Walter de Gruyter GmbH, 2006, pp. 1–26, doi:<a href="https://doi.org/10.1515/crll.2004.050">10.1515/crll.2004.050</a>.
  short: J. Klüners, G. Malle, Journal Für Die Reine Und Angewandte Mathematik (Crelles
    Journal) 2004 (2006) 1–26.
date_created: 2022-12-23T09:50:49Z
date_updated: 2023-03-06T09:11:16Z
department:
- _id: '102'
doi: 10.1515/crll.2004.050
external_id:
  arxiv:
  - math/0112318
intvolume: '      2004'
issue: '572'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
page: 1-26
publication: Journal für die reine und angewandte Mathematik (Crelles Journal)
publication_identifier:
  issn:
  - 0075-4102
  - 1435-5345
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: Counting nilpotent Galois extensions
type: journal_article
user_id: '93826'
volume: 2004
year: '2006'
...
---
_id: '64712'
article_type: original
author:
- first_name: Helge
  full_name: Glöckner, Helge
  id: '178'
  last_name: Glöckner
- first_name: Karl-Hermann
  full_name: Neeb, Karl-Hermann
  last_name: Neeb
citation:
  ama: Glöckner H, Neeb K-H. Banach-Lie quotients, enlargibility, and universal complexifications.
    <i>Journal für die reine und angewandte Mathematik</i>. 2003;560:1–28. doi:<a
    href="https://doi.org/10.1515/crll.2003.056">10.1515/crll.2003.056</a>
  apa: Glöckner, H., &#38; Neeb, K.-H. (2003). Banach-Lie quotients, enlargibility,
    and universal complexifications. <i>Journal Für Die Reine Und Angewandte Mathematik</i>,
    <i>560</i>, 1–28. <a href="https://doi.org/10.1515/crll.2003.056">https://doi.org/10.1515/crll.2003.056</a>
  bibtex: '@article{Glöckner_Neeb_2003, title={Banach-Lie quotients, enlargibility,
    and universal complexifications}, volume={560}, DOI={<a href="https://doi.org/10.1515/crll.2003.056">10.1515/crll.2003.056</a>},
    journal={Journal für die reine und angewandte Mathematik}, author={Glöckner, Helge
    and Neeb, Karl-Hermann}, year={2003}, pages={1–28} }'
  chicago: 'Glöckner, Helge, and Karl-Hermann Neeb. “Banach-Lie Quotients, Enlargibility,
    and Universal Complexifications.” <i>Journal Für Die Reine Und Angewandte Mathematik</i>
    560 (2003): 1–28. <a href="https://doi.org/10.1515/crll.2003.056">https://doi.org/10.1515/crll.2003.056</a>.'
  ieee: 'H. Glöckner and K.-H. Neeb, “Banach-Lie quotients, enlargibility, and universal
    complexifications,” <i>Journal für die reine und angewandte Mathematik</i>, vol.
    560, pp. 1–28, 2003, doi: <a href="https://doi.org/10.1515/crll.2003.056">10.1515/crll.2003.056</a>.'
  mla: Glöckner, Helge, and Karl-Hermann Neeb. “Banach-Lie Quotients, Enlargibility,
    and Universal Complexifications.” <i>Journal Für Die Reine Und Angewandte Mathematik</i>,
    vol. 560, 2003, pp. 1–28, doi:<a href="https://doi.org/10.1515/crll.2003.056">10.1515/crll.2003.056</a>.
  short: H. Glöckner, K.-H. Neeb, Journal Für Die Reine Und Angewandte Mathematik
    560 (2003) 1–28.
date_created: 2026-02-26T12:16:39Z
date_updated: 2026-02-27T07:46:29Z
department:
- _id: '10'
- _id: '87'
- _id: '93'
doi: 10.1515/crll.2003.056
extern: '1'
intvolume: '       560'
keyword:
- '22E65'
- '22E15'
- '22E10'
language:
- iso: eng
page: 1–28
publication: Journal für die reine und angewandte Mathematik
publication_identifier:
  issn:
  - 0075-4102
quality_controlled: '1'
status: public
title: Banach-Lie quotients, enlargibility, and universal complexifications
type: journal_article
user_id: '178'
volume: 560
year: '2003'
...