Abstract
We consider the analog of Gelfand–Graev representations for the unitriangular group. We obtain the decomposition into a sum of irreducible representations, prove that these representations are multiplicity free, and calculate the Hecke algebra.
Similar content being viewed by others
References
C. A. M. André, “Basic characters of the unitriangular group,” J. Algebra, 175, 287–319 (1995).
C. A. M. André, “On the coadjoint orbits of the unitriangular group,” J. Algebra, 180, 587–630 (1995).
C. A. M. André, “The regular character of the unitriangular group,” J. Algebra, 201, 1–52 (1998).
C. A. M. André, “Basic characters of the unitriangular group (for arbitrary primes),” Proc. Am. Math. Soc., 130, No. 7, 1943–1954 (2002).
C. A. M. André, “Hecke algebra for the basic representations of the unitriangular group,” Proc. Am. Math. Soc., 132, No. 4, 987–996 (2003).
J. Dixmier, Algèbras enveloppantes, Gauthier-Villars, Paris (1974).
I. M. Gelfand and M. I. Graev, “Categories of group representations and problem of classification of irreducible representations,” Dokl. Akad. Nauk SSSR, 146, No. 4, 757–760 (1962).
I. M. Gelfand and M. I. Graev, “Construction of irreducible representations of simple algebraic groups over a finite field,” Dokl. Akad. Nauk SSSR, 147, No. 3, 529–532 (1962).
M. V. Ignatev and A. N. Panov, “Coadjoint orbits for the group UT(7,K),” Fundam. Appl. Math., 13, No. 5, 127–159 (2007).
I. M. Isaacs and D. Karagueuzian, “Conjugacy in groups of upper triangular matrices,” J. Algebra, 202, 704–711 (1998).
D. Kazhdan, “Proof of Springer’s hypothesis,” Israel J. Math., 28, No. 4, 272–286 (1977).
A. A. Kirillov, “Unitary representations of nilpotent Lie groups,” Usp. Mat. Nauk, 17, No. 4, 57–110 (1962).
A. A. Kirillov, Lectures on the Orbit Method Grad. Stud. Math., Vol. 64, Amer. Math. Soc., Providence (2002).
A. N. Panov, The Orbit Method for Unipotent Groups over Finite Field, arXiv:1212.1980.
J. Sangroniz, “Characters of algebra groups and unitriangular groups,” in: Finite Groups, Walter de Gruyter, Berlin (2004), pp. 335–349.
R. Steinberg, Lectures on Chevalley Groups, Yale university (1968).
T. Yokonuma, “Sur la structure des anneaux de Hecke d’un groupe de Chevalley fini,” C. R. Acad. Sci. Paris, 264, 344–347 (1967).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 3, pp. 161–178, 2013.
Rights and permissions
About this article
Cite this article
Panov, A.N. Representations of Gelfand–Graev Type for the Unitriangular Group. J Math Sci 206, 570–582 (2015). https://doi.org/10.1007/s10958-015-2334-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-015-2334-4