## Compatible intertwiners for representations of finite nilpotent groups

HTML articles powered by AMS MathViewer

- by Masoud Kamgarpour and Teruji Thomas PDF
- Represent. Theory
**15**(2011), 407-432 Request permission

## 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

- D. Boyarchenko and V. Drinfeld. A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic.
*monospaceitalicarXiv:math/0609769v1, 2006.* *Jean Barge, Jean Lannes, FranĂ§ois Latour, and Pierre Vogel,**$\Lambda$-sphĂšres*, Ann. Sci. Ăcole Norm. Sup. (4)**7**(1974), 463â505 (1975) (French). MR**377939**- P. Deligne,
*Le dĂ©terminant de la cohomologie*, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp.Â 93â177 (French). MR**902592**, DOI 10.1090/conm/067/902592 - D. Gaitsgory,
*On de Jongâs conjecture*, Israel J. Math.**157**(2007), 155â191. MR**2342444**, DOI 10.1007/s11856-006-0006-2 - Paul GĂ©rardin,
*Weil representations associated to finite fields*, J. Algebra**46**(1977), no.Â 1, 54â101. MR**460477**, DOI 10.1016/0021-8693(77)90394-5 - Shamgar Gurevich and Ronny Hadani,
*The geometric Weil representation*, Selecta Math. (N.S.)**13**(2007), no.Â 3, 465â481. MR**2383602**, DOI 10.1007/s00029-007-0047-3 - Shamgar Gurevich and Ronny Hadani,
*Quantization of symplectic vector spaces over finite fields*, J. Symplectic Geom.**7**(2009), no.Â 4, 475â502. MR**2552002**, DOI 10.4310/JSG.2009.v7.n4.a4 - Roger E. Howe,
*On the character of Weilâs representation*, Trans. Amer. Math. Soc.**177**(1973), 287â298. MR**316633**, DOI 10.1090/S0002-9947-1973-0316633-5 - Roger E. Howe,
*On representations of discrete, finitely generated, torsion-free, nilpotent groups*, Pacific J. Math.**73**(1977), no.Â 2, 281â305. MR**499004**, DOI 10.2140/pjm.1977.73.281 - M. Kamgarpour. Weil representations over finite fields.
*www.masoudkamgarpour.com/**Media Files/masterthesis.pdf*, 2005. - D. Kazhdan,
*Proof of Springerâs hypothesis*, Israel J. Math.**28**(1977), no.Â 4, 272â286. MR**486181**, DOI 10.1007/BF02760635 - A. A. Kirillov,
*Unitary representations of nilpotent Lie groups*, Uspehi Mat. Nauk**17**(1962), no.Â 4 (106), 57â110 (Russian). MR**0142001** - A. A. Kirillov,
*Lectures on the orbit method*, Graduate Studies in Mathematics, vol. 64, American Mathematical Society, Providence, RI, 2004. MR**2069175**, DOI 10.1090/gsm/064 - T. Y. Lam,
*Introduction to quadratic forms over fields*, Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005. MR**2104929**, DOI 10.1090/gsm/067 - Serge Lang,
*Algebra*, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR**1878556**, DOI 10.1007/978-1-4613-0041-0 - Michel Lazard,
*Sur les groupes nilpotents et les anneaux de Lie*, Ann. Sci. Ecole Norm. Sup. (3)**71**(1954), 101â190 (French). MR**0088496**, DOI 10.24033/asens.1021 - GĂ©rard Lion and Patrice Perrin,
*Extension des reprĂ©sentations de groupes unipotents $p$-adiques. Calculs dâobstructions*, Noncommutative harmonic analysis and Lie groups (Marseille, 1980) Lecture Notes in Math., vol. 880, Springer, Berlin-New York, 1981, pp.Â 337â356 (French). MR**644839** - GĂ©rard Lion and MichĂšle Vergne,
*The Weil representation, Maslov index and theta series*, Progress in Mathematics, vol. 6, BirkhĂ€user, Boston, Mass., 1980. MR**573448**, DOI 10.1016/0012-365x(79)90168-7 - Patrice Perrin,
*ReprĂ©sentations de SchrĂ¶dinger, indice de Maslov et groupe metaplectique*, Noncommutative harmonic analysis and Lie groups (Marseille, 1980) Lecture Notes in Math., vol. 880, Springer, Berlin-New York, 1981, pp.Â 370â407 (French). MR**644841** - R. Parimala, R. Preeti, and R. Sridharan,
*Maslov index and a central extension of the symplectic group*, $K$-Theory**19**(2000), no.Â 1, 29â45. MR**1740881**, DOI 10.1023/A:1007775418690 - Daniel Quillen,
*Higher algebraic $K$-theory. I*, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp.Â 85â147. MR**0338129** - Z. Reichstein and B. Youssin,
*A birational invariant for algebraic group actions*, Pacific J. Math.**204**(2002), no.Â 1, 223â246. MR**1905199**, DOI 10.2140/pjm.2002.204.223 - Teruji Thomas,
*The Maslov index as a quadratic space*, Math. Res. Lett.**13**(2006), no.Â 5-6, 985â999. MR**2280792**, DOI 10.4310/MRL.2006.v13.n6.a13 - Teruji Thomas,
*The character of the Weil representation*, J. Lond. Math. Soc. (2)**77**(2008), no.Â 1, 221â239. MR**2389926**, DOI 10.1112/jlms/jdm098 - AndrĂ© Weil,
*Sur certains groupes dâopĂ©rateurs unitaires*, Acta Math.**111**(1964), 143â211 (French). MR**165033**, DOI 10.1007/BF02391012

## 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
- Received by editor(s): October 29, 2009
- Received by editor(s) in revised form: August 16, 2010
- Published electronically: 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 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**15**(2011), 407-432 - MSC (2010): Primary 20C15
- DOI: https://doi.org/10.1090/S1088-4165-2011-00395-2
- MathSciNet review: 2801175