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Author: Grafiati

Published: 16 November 2022

Last updated: 16 December 2022

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1

Sorlin, Karine. "Représentations de Gelfand–Graev pour les groupes réductifs non connexes." Comptes Rendus Mathematique 334, no. 3 (February 2002): 179–84. http://dx.doi.org/10.1016/s1631-073x(02)02239-2.

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2

Sorlin, Karine. "Éléments réguliers et représentations de Gelfand-Graev des groupes réductifs non connexes." Bulletin de la Société mathématique de France 132, no. 2 (2004): 157–99. http://dx.doi.org/10.24033/bsmf.2463.

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3

Bonnafé, Cédric, and Raphaël Rouquier. "Coxeter Orbits and Modular Representations." Nagoya Mathematical Journal 183 (2006): 1–34. http://dx.doi.org/10.1017/s0027763000009259.

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AbstractWe study the modular representations of finite groups of Lie type arising in the cohomology of certain quotients of Deligne-Lusztig varieties associated with Coxeter elements. These quotients are related to Gelfand-Graev representations and we present a conjecture on the Deligne-Lusztig restriction of Gelfand-Graev representations. We prove the conjecture for restriction to a Coxeter torus. We deduce a proof of Brouée’s conjecture on equivalences of derived categories arising from Deligne-Lusztig varieties, for a split group of type An and a Coxeter element. Our study is based on Lusztig’s work in characteristic 0 [Lu2].

4

Mishra, Manish, and Basudev Pattanayak. "Principal series component of Gelfand-Graev representation." Proceedings of the American Mathematical Society 149, no. 11 (August 5, 2021): 4955–62. http://dx.doi.org/10.1090/proc/15642.

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5

Geck, Meinolf. "Character Sheaves and Generalized Gelfand-Graev Characters." Proceedings of the London Mathematical Society 78, no. 1 (January 1999): 139–66. http://dx.doi.org/10.1112/s0024611599001641.

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6

TAYLOR, JAY. "GENERALIZED GELFAND–GRAEV REPRESENTATIONS IN SMALL CHARACTERISTICS." Nagoya Mathematical Journal 224, no. 1 (September 9, 2016): 93–167. http://dx.doi.org/10.1017/nmj.2016.33.

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Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_{p}}$ of the finite field of prime order $p$ and let $F:\mathbf{G}\rightarrow \mathbf{G}$ be a Frobenius endomorphism with $G=\mathbf{G}^{F}$ the corresponding $\mathbb{F}_{q}$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig (Adv. Math. 94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math. 94(2) (1992), 139–179) is only valid under the assumption that $p$ is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that $p$ is an acceptable prime for $\mathbf{G}$ ($p$ very good is sufficient but not necessary). As an application we show that every irreducible character of $G$, respectively, character sheaf of $\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever $p$ is good for $\mathbf{G}$.

7

Letellier, Emmanuel. "Deligne–Lusztig restriction of Gelfand–Graev characters." Journal of Algebra 294, no. 1 (December 2005): 239–54. http://dx.doi.org/10.1016/j.jalgebra.2005.05.031.

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8

Chan, Kei Yuen, and Gordan Savin. "Iwahori component of the Gelfand–Graev representation." Mathematische Zeitschrift 288, no. 1-2 (March 23, 2017): 125–33. http://dx.doi.org/10.1007/s00209-017-1882-3.

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9

RAINBOLT, JULIANNE G. "WEYL GROUPS AND BASIS ELEMENTS OF HECKE ALGEBRAS OF GELFAND–GRAEV REPRESENTATIONS." Journal of Algebra and Its Applications 10, no. 05 (October 2011): 849–64. http://dx.doi.org/10.1142/s0219498811005002.

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The initial section of this article provides illustrative examples on two ways to construct the Weyl group of a finite group of Lie type. These examples provide the background for a comparison of the elements in the Weyl groups of GL(n, q) and U(n, q) that are used in the construction of the standard bases of the Hecke algebras of the Gelfand–Graev representations of GL(n, q) and U(n, q). Using a theorem of Steinberg, a connection between a theoretic description of bases of these Hecke algebras and a combinatorial description of these bases is provided. This leads to an algorithmic method for generating bases of the Hecke algebras of the Gelfand–Graev representations of GL(n, q) and U(n, q).

10

Dudas, Olivier. "Deligne-Lusztig restriction of a Gelfand-Graev module." Annales scientifiques de l'École normale supérieure 42, no. 4 (2009): 653–74. http://dx.doi.org/10.24033/asens.2105.

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11

Breeding-Allison, Jeffery, and Julianne Rainbolt. "The Gelfand–Graev representation of GSp(4,𝔽q)." Communications in Algebra 47, no. 2 (January 17, 2019): 560–84. http://dx.doi.org/10.1080/00927872.2018.1485228.

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12

Rainbolt, Julianne G. "The Gelfand–Graev Representation of U(3,q)." Journal of Algebra 188, no. 2 (February 1997): 648–85. http://dx.doi.org/10.1006/jabr.1996.6860.

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13

SHINODA, KENICHI, and ILKNUR TULUNAY. "REPRESENTATIONS OF THE HECKE ALGEBRA FOR GL4(q)." Journal of Algebra and Its Applications 04, no. 06 (December 2005): 631–44. http://dx.doi.org/10.1142/s0219498805001459.

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In this article, we explicitly calculated the values of the representations of the Hecke algebra [Formula: see text], associated with a Gelfand–Graev character of GL 4(q), at some of the standard basis elements.

14

ACKERMANN, BERND. "THE LOEWY SERIES OF THE STEINBERG-PIM OF FINITE GENERAL LINEAR GROUPS." Proceedings of the London Mathematical Society 92, no. 1 (December 19, 2005): 62–98. http://dx.doi.org/10.1017/s0024611505015443.

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In this paper we calculate the Loewy series of the projective indecomposable module of the unipotent block contained in the Gelfand–Graev module of the finite general linear group in the case of non-describing characteristic and Abelian defect group.

15

Andrews, Scott, and Nathaniel Thiem. "The combinatorics of GL n generalized Gelfand-Graev characters." Journal of the London Mathematical Society 95, no. 2 (January 15, 2017): 475–99. http://dx.doi.org/10.1112/jlms.12023.

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16

Clarke, Matthew C. "On the endomorphism algebra of generalised Gelfand-Graev representations." Transactions of the American Mathematical Society 364, no. 10 (October 1, 2012): 5509–24. http://dx.doi.org/10.1090/s0002-9947-2012-05543-7.

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17

Krattenthaler, C., and H. Rosengren. "On a hypergeometric identity of Gelfand, Graev and Retakh." Journal of Computational and Applied Mathematics 160, no. 1-2 (November 2003): 147–58. http://dx.doi.org/10.1016/s0377-0427(03)00620-4.

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18

Panov, A. N. "Representations of Gelfand–Graev Type for the Unitriangular Group." Journal of Mathematical Sciences 206, no. 5 (March 26, 2015): 570–82. http://dx.doi.org/10.1007/s10958-015-2334-4.

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19

Kotlar, D. "On the Irreducible Components of Degenerate Gelfand-Graev Characters." Journal of Algebra 173, no. 2 (April 1995): 348–60. http://dx.doi.org/10.1006/jabr.1995.1091.

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20

Rainbolt, Julianne G. "The Generalized Gelfand–Graev Representations of U(3,q)." Journal of Algebra 202, no. 1 (April 1998): 44–71. http://dx.doi.org/10.1006/jabr.1997.7250.

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21

Bonnafé, Cédric, and Radha Kessar. "On the endomorphism algebras of modular Gelfand–Graev representations." Journal of Algebra 320, no. 7 (October 2008): 2847–70. http://dx.doi.org/10.1016/j.jalgebra.2008.05.029.

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22

Ginzburg, Victor, and David Kazhdan. "Differential operators on G/U and the Gelfand-Graev action." Advances in Mathematics 403 (July 2022): 108368. http://dx.doi.org/10.1016/j.aim.2022.108368.

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23

Pellegrini, Marco Antonio. "A Description of the Steinberg Character using Gelfand–Graev Characters." Results in Mathematics 67, no. 1-2 (June 28, 2014): 71–85. http://dx.doi.org/10.1007/s00025-014-0394-2.

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24

Curtis, Charles W., and Ken-ichi Shinoda. "Unitary Kloosterman Sums and the Gelfand–Graev Representation of GL2." Journal of Algebra 216, no. 2 (June 1999): 431–47. http://dx.doi.org/10.1006/jabr.1998.7807.

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25

Zalesski, A. E. "A remark on Gelfand–Graev characters for finite simple groups." Archiv der Mathematik 100, no. 3 (March 2013): 221–30. http://dx.doi.org/10.1007/s00013-013-0491-6.

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26

Kochubei, Anatoly N., and Yuri Kondratiev. "Representations of the infinite-dimensional p-adic affine group." Infinite Dimensional Analysis, Quantum Probability and Related Topics 23, no. 01 (March 2020): 2050002. http://dx.doi.org/10.1142/s0219025720500022.

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We introduce an infinite-dimensional [Formula: see text]-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However, it is possible to define its action on some classes of functions.

27

Dong, Junbin, and Gao Yang. "Geck's conjecture and the generalized Gelfand-Graev representations in bad characteristic." Advances in Mathematics 377 (January 2021): 107482. http://dx.doi.org/10.1016/j.aim.2020.107482.

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28

Meinolf, Geck. "Generalized gelfand-graev characters for steinberg's triality groups and their applications." Communications in Algebra 19, no. 12 (January 1991): 3249–69. http://dx.doi.org/10.1080/00927879108824318.

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29

Paolini, Alessandro, and Iulian I. Simion. "On Refined Bruhat Decompositions and Endomorphism Algebras of Gelfand-Graev Representations." Algebras and Representation Theory 23, no. 4 (April 10, 2019): 1243–63. http://dx.doi.org/10.1007/s10468-019-09885-5.

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30

Yamashita, Hiroshi. "On Whittaker vectors for generalized Gelfand-Graev representations of semisimple Lie groups." Proceedings of the Japan Academy, Series A, Mathematical Sciences 61, no. 7 (1985): 213–16. http://dx.doi.org/10.3792/pjaa.61.213.

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31

Yamashita, Hiroshi. "On Whittaker vectors for generalized Gelfand-Graev representations of semisimple Lie groups." Journal of Mathematics of Kyoto University 26, no. 2 (1986): 263–98. http://dx.doi.org/10.1215/kjm/1250520922.

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32

Smirnov, Yu F., and Yu I. Kharitonov. "Quantum algebra U q(2, 1): q analogs of the Gelfand-Graev formulas." Physics of Atomic Nuclei 64, no. 12 (December 2001): 2167–72. http://dx.doi.org/10.1134/1.1432920.

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33

Curtis, C. W. "On the Gelfand-Graev Representations of a Reductive Group over a Finite Field." Journal of Algebra 157, no. 2 (June 1993): 517–33. http://dx.doi.org/10.1006/jabr.1993.1113.

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34

HIROSHI, ANDO. "ON THE LOCAL STRUCTURE OF THE REPRESENTATION OF A LOCAL GAUGE GROUP." Infinite Dimensional Analysis, Quantum Probability and Related Topics 13, no. 02 (June 2010): 223–42. http://dx.doi.org/10.1142/s0219025710004036.

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We discuss the local structure of the net [Formula: see text] of von Neumann algebras generated by a representation of a local gauge group [Formula: see text]. Our discussion is independent of the singularity of spectral measures, which has been discussed by many authors since the pioneering work of Gelfand–Graev–Veršic. We show that, for type (S) operators UA,b, second quantized operators with some twists, the commutativity only with those U(ψ) is sufficient for the triviality of them, where ψ belongs to an arbitrary (small) neighborhood of constant function 1. Some properties of 1-cocycles for the representation V : ψ ↦ Ad ψ are also discussed.

35

Kawanaka, N. "Generalized Gelfand-Graev representations of exceptional simple algebraic groups over a finite field I." Inventiones Mathematicae 84, no. 3 (October 1986): 575–616. http://dx.doi.org/10.1007/bf01388748.

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36

Curtis, Charles W. "On the irreducible components of a Gelfand–Graev representation of a finite Chevalley group." Pacific Journal of Mathematics 307, no. 1 (August 8, 2020): 109–19. http://dx.doi.org/10.2140/pjm.2020.307.109.

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37

Yamashita, Hiroshi. "Finite multiplicity theorems for induced representations of semisimple Lie groups and their applications togeneralized Gelfand-Graev representations." Proceedings of the Japan Academy, Series A, Mathematical Sciences 63, no. 5 (1987): 153–56. http://dx.doi.org/10.3792/pjaa.63.153.

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38

Yamashita, Hiroshi. "Finite multiplicity theorems for induced representations of semisimpmle Lie groups II, -Applications to generalized Gelfand-Graev representations-." Journal of Mathematics of Kyoto University 28, no. 3 (1988): 383–444. http://dx.doi.org/10.1215/kjm/1250520400.

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39

Niukkanen, A. V. "Transformation of the triple series of Gelfand, Graev, and Retakh into a series of the same type and related problems." Mathematical Notes 89, no. 3-4 (April 2011): 374–81. http://dx.doi.org/10.1134/s0001434611030096.

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40

Rainbolt, Julianne G. "THE IRREDUCIBLE REPRESENTATIONS OF THE HECKE ALGEBRAS CONSTRUCTED FROM THE GELFAND-GRAEV REPRESENTATIONS OF GL(3, q) AND U(3, q)." Communications in Algebra 30, no. 9 (January 11, 2002): 4085–103. http://dx.doi.org/10.1081/agb-120013305.

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41

Rainbolt, Julianne G. "Notes on the norm map between the Hecke algebras of the Gelfand–Graev representations of GL(2,q2) and U(2,q)." Journal of Algebra 320, no. 9 (November 2008): 3493–511. http://dx.doi.org/10.1016/j.jalgebra.2008.07.025.

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42

Thiem, Nathaniel, and C. Ryan Vinroot. "Gelfand–Graev Characters of the Finite Unitary Groups." Electronic Journal of Combinatorics 16, no. 1 (November 30, 2009). http://dx.doi.org/10.37236/235.

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Gelfand–Graev characters and their degenerate counterparts have an important role in the representation theory of finite groups of Lie type. Using a characteristic map to translate the character theory of the finite unitary groups into the language of symmetric functions, we study degenerate Gelfand–Graev characters of the finite unitary group from a combinatorial point of view. In particular, we give the values of Gelfand–Graev characters at arbitrary elements, recover the decomposition multiplicities of degenerate Gelfand–Graev characters in terms of tableau combinatorics, and conclude with some multiplicity consequences.

43

Andrews, Scott, and Nathaniel Thiem. "The generalized Gelfand–Graev characters of GLn(Fq)." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6406.

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International audience Introduced by Kawanaka in order to find the unipotent representations of finite groups of Lie type, gener- alized Gelfand–Graev characters have remained somewhat mysterious. Even in the case of the finite general linear groups, the combinatorics of their decompositions has not been worked out. This paper re-interprets Kawanaka's def- inition in type A in a way that gives far more flexibility in computations. We use these alternate constructions to show how to obtain generalized Gelfand–Graev representations directly from the maximal unipotent subgroups. We also explicitly decompose the corresponding generalized Gelfand–Graev characters in terms of unipotent representations, thereby recovering the Kostka–Foulkes polynomials as multiplicities.

44

GECK, MEINOLF. "GENERALISED GELFAND–GRAEV REPRESENTATIONS IN BAD CHARACTERISTIC ?" Transformation Groups, May 22, 2020. http://dx.doi.org/10.1007/s00031-020-09575-3.

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45

Breeding-Allison, Jeffery, and Julianne Rainbolt. "The Gelfand-Graev Representation of GSp(4, 𝔽q)." Communications in Algebra, October 7, 2016. http://dx.doi.org/10.1080/00927872.2016.1206341.

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46

Blagovestchenskii, A. S., and F. N. Podymaka. "On symmetry condition of images of Fourier – Gelfand – Graev integral transformation." Journal of Inverse and Ill-posed Problems 8, no. 3 (January 2000). http://dx.doi.org/10.1515/jiip.2000.8.3.233.

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47

Hetyei, Gábor. "Delannoy numbers and Legendre polytopes." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AJ,..., Proceedings (January 1, 2008). http://dx.doi.org/10.46298/dmtcs.3599.

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International audience We construct an $n$-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of $(x-1)/2$ in the $n$-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open quadrant. We thus provide a geometric interpretation of a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago. The polytopes we construct are closely related to the root polytopes introduced by Gelfand, Graev, and Postnikov. \par No construisons un polytope de dimension $n$ dont le complexe de bord est comprimé et dont les nombres de faces dans toute triangulation "en tirant des sommets'' sont les coefficients des puissances de $(x-1)/2$ dans le $n$-ième polynôme de Legendre. Nous montrons que les nombres centraux de Delannoy comptent toutes les faces dans la triangulation "en tirant des sommets'' en ordre lexicographique qui contiennent un point dans un certain quadrant ouvert. Ainsi nous produisons une interprétation géométrique d'une relation entre les nombres de Delannoy centraux et les polynômes de Legendre, notée il y a 50 ans. Nos polytopes sont reliés intimement aux polytopes de racines introduits par Gelfand, Graev, et Postnikov.

48

Savin, Gordan, and Petar Bakic. "The Gelfand-Graev representation of classical groups in terms of Hecke algebras." Canadian Journal of Mathematics, June 24, 2022, 1–26. http://dx.doi.org/10.4153/s0008414x2200030x.

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49

Matsubara-Heo, Saiei-Jaeyeong. "Global Analysis of GG Systems." International Mathematics Research Notices, June 14, 2021. http://dx.doi.org/10.1093/imrn/rnab144.

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Abstract This paper deals with some analytic aspects of GG system introduced by I.M. Gelfand and M.I. Graev: we compute the dimension of the solution space of GG system over the field of meromorphic functions periodic with respect to a lattice. We describe the monodromy invariant subspace of the solution space. We provide a connection formula between a pair of bases consisting of $\Gamma $-series solutions of GG system associated with a pair of regular triangulations adjacent to each other in the secondary fan.

50

Wang, Xiangsheng. "A new Weyl group action related to the quasi-classical Gelfand–Graev action." Selecta Mathematica 27, no. 3 (May 29, 2021). http://dx.doi.org/10.1007/s00029-021-00655-0.

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