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Compatible intertwiners for representations of finite nilpotent groups
Author(s):
Masoud
Kamgarpour;
Teruji
Thomas
Journal:
Represent. Theory
15
(2011),
407-432.
MSC (2010):
Primary 20C15
Posted:
May 16, 2011
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Abstract:
We sharpen the orbit method for finite groups of small nilpotence class by associating representations to functionals on the corresponding Lie rings. This amounts to describing compatible intertwiners between representations parameterized by an additional choice of polarization. Our construction is motivated by the theory of the linearized Weil representation of the symplectic group. In particular, we provide generalizations of the Maslov index and the determinant functor to the context of finite abelian groups.
References:
-
- [BD06]
- D. Boyarchenko and V. Drinfeld.
A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic.
arXiv:math/0609769v1, 2006. - [BLLV74]
- J. Barge, J. Lannes, F. Latour, and P. Vogel.
 -sphères.
Ann. Sci. École Norm. Sup. (4), 7:463-505 (1975), 1974. MR 0377939 (51:14108) - [Del87]
- P. Deligne.
Le déterminant de la cohomologie.
In Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), volume 67 of Contemp. Math., pages 93-177. Amer. Math. Soc., Providence, RI, 1987. MR 902592 (89b:32038) - [Gai07]
- D. Gaitsgory.
On de Jong's conjecture.
Israel J. Math., 157:155-191, 2007. MR 2342444 (2008j:14021) - [Gér77]
- P. Gérardin.
Weil representation associated to finite fields.
J. Algebra, 46(1):54-101, 1977. MR 0460477 (57:470) - [GH07]
- S. Gurevich and R. Hadani.
The geometric Weil representation.
Selecta Math. (N.S.), 13(3):465-481, 2007. MR 2383602 (2009e:11078) - [GH08]
- S. Gurevich and R. Hadani.
Quantization of symplectic vector spaces over finite fields. J. Symplectic Geom. 7(4):475-502, 2009. MR 2552002 - [How73]
- R. Howe.
On the character of Weil's representation.
Trans. Amer. Math. Soc., 177:287-298, 1973. MR 0316633 (47:5180) - [How77]
- R. Howe.
On representations of discrete, finitely generated, torsion-free, nilpotent groups.
Pacific J. Math., 73(2):281-305, 1977. MR 0499004 (58:16984) - [Kam05]
- M. Kamgarpour.
Weil representations over finite fields.
www.masoudkamgarpour.com/ Media Files/masterthesis.pdf, 2005. - [Kaz77]
- D. Kazhdan.
Proof of Springer's hypothesis.
Israel J. Math., 28(4):272-286, 1977. MR 0486181 (58:5959) - [Kir62]
- A. A. Kirillov.
Unitary representations of nilpotent lie groups.
Uspehi Mat. Nauk, 17:57-110, 1962. MR 0142001 (25:5396) - [Kir04]
- A. A. Kirillov.
Lectures on the orbit method.
American Math. Soc., 2004. MR 2069175 (2005c:22001) - [Lam05]
- T.Y. Lam.
Introduction to quadratic forms over fields, volume 67 of Graduate Studies in Mathematics.
Amer. Math. Soc., 2005. MR 2104929 (2005h:11075) - [Lan05]
- S. Lang.
Algebra, volume 211 of Graduate Text in Mathematics.
Springer-Verlag, 2005. MR 1878556 (2003e:00003) - [Laz54]
- M. Lazard.
Sur les groupes nilpotents et anneaux de Lie.
Ann. Sci. Eco. Norm. Sup., (31) 71:101-190, 1954. MR 0088496 (19:529b) - [LP81]
- G. Lion and P. Perrin.
Extension des représentations de groupes unipotents  -adiques. Calculs d'obstructions.
In Noncommutative harmonic analysis and Lie groups (Marseille, 1980), volume 880 of Lecture Notes in Math., pages 337-356. Springer, Berlin, 1981. MR 644839 (83h:22032) - [LV80]
- G. Lion and M. Vergne.
The Weil representation, Maslov index and theta series, volume 6 of Progress in Mathematics.
Birkhäuser Boston, Mass., 1980. MR 573448 (81j:58075) - [Per81]
- P. Perrin.
Représentations de Schrödinger, indice de Maslov et groupe metaplectique.
In Noncommutative harmonic analysis and Lie groups (Marseille, 1980), volume 880 of Lecture Notes in Math., pages 370-407. Springer, Berlin, 1981. MR 644841 (83m:22027) - [PPS00]
- R. Parimala, R. Preeti, and R. Sridharan.
Maslov index and a central extension of the symplectic group.
 -Theory, 19(1):29-45, 2000. MR 1740881 (2001c:11053a) - [Qui73]
- D. Quillen.
Higher algebraic  -theory. I.
In Algebraic  -theory, I: Higher  -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 85-147. Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973. MR 0338129 (49:2895) - [RY02]
- Z. Reichstein and B. Youssin.
A birational invariant for algebraic group actions.
Pacific J. Math., 204(1):223-246, 2002. MR 1905199 (2003c:14052) - [Tho06]
- T. Thomas.
The Maslov index as a quadratic space.
Math. Res. Lett., 5-6:985-999, 2006. MR 2280792 (2007j:53093) - [Tho08]
- T. Thomas.
The character of the Weil representation.
J. LMS, 77(2):221-239, 2008. MR 2389926 (2008k:11049) - [Wei64]
- A. Weil.
Sur certains groupes d'opérateurs unitaires.
Acta Math., 111:143-211, 1964. MR 0165033 (29:2324)
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Additional Information:
Masoud
Kamgarpour
Affiliation:
The University of British Columbia, Vancouver, Canada V6T 1Z2
Email:
masoud@math.ubc.ca
Teruji
Thomas
Affiliation:
The University of Edinburgh, Edinburgh, United Kingdom EH9 3JZ
Email:
t.thomas@ed.ac.uk
DOI:
10.1090/S1088-4165-2011-00395-2
PII:
S 1088-4165(2011)00395-2
Keywords:
Orbit method,
$p$-groups,
neighboring polarizations,
Lie rings,
intertwiners,
Weil representation,
determinant,
orientation,
Maslov index,
determinant of abelian groups
Received by editor(s):
October 29, 2009
Received by editor(s) in revised form:
August 16, 2010
Posted:
May 16, 2011
Additional Notes:
The first author was supported by NSERC PDF grant. The second author was supported by a JRF at Merton College, Oxford and a Seggie Brown Fellowship at Edinburgh.
Copyright of article:
Copyright
2011,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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