There is a method of constructing representations of classical Lie algebras via Gelfand-Tsetlin bases. It has also been applied to Symmetric groups by Vershik and Okounkov. Does anybody know of any application of the method to complex representations of $GL_n(\mathbb F_q)$? Or, at least, any results in this directions, like what is the centralizer of $GL_{n-1}$ in $\mathbb C[GL_n]$?
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1$\begingroup$ I don't have a precise answer, but keep in mind the Schur-Weyl duality between representations of general linear and symmetric groups. Whatever is done for the latter is likely to apply to the former, even over finite fields. In another direction, Dan Rockmore and colleagues have done quite a bit of practical work on fast Fourier transforms for finite linear groups, some with connections to Gelfand-Tsetlin basis methods. $\endgroup$– Jim HumphreysCommented Nov 9, 2010 at 15:16
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2$\begingroup$ Maybe you should have a look at arXiv:0705.3605v1 [math.RT] (by Vershik and Kerov). This applies some approaches that worked successfully for the infinite symmetric group to the case of limits of finite linear groups. The constuction in the case of symmetric groups has much to do with GTs bases, as I remember. $\endgroup$– Leonid PetrovCommented Nov 9, 2010 at 20:24
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2 Answers
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My earlier comment was not at all well-focused. After more thought, I'm inclined to be pessimistic about using a Gelfand-Tsetlin approach here (even if it has some success for symmetric groups). Though of course it would be interesting to be proven wrong.
As Matt Davis reminds me, my offhand reference to Schur-Weyl duality is not helpful here since the work of Benson, Doty, and others deals mainly with the representations of various groups over fields of prime characteristic. (See especially Doty's papers on arXiv.) Irreducible representations of finite general linear groups over $\mathbb{C}$ are very difficult to construct directly and have very little in common with the finite dimensional representations of general linear groups or their Lie algebras in characteristic 0. Instead, the theory imitates more closely the infinite dimensional Harish-Chandra approach to Lie group representations in which parabolic induction is exploited together with a study of "discrete series".
J.A. Green's 1955 TAMS paper followed somewhat this pattern in developing combinatorially the character theory of finite general linear groups. But there is little insight here into constructing the elusive discrete series characters; instead orthogonality relations and the like are exploited. The best approach to an actual construction of discrete series representations was given in Lusztig's 1974 Annals of Mathematics Studies No. 81. Soon after that, Deligne and Lusztig pioneered a more sophisticated method for constructing generalized characters of arbitrary finite groups of Lie type. This has become the dominant influence in the subject, since Lusztig's earlier techniques don't go far enough beyond the finite general linear case.
answered Nov 25, 2010 at 16:14
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$\begingroup$ Thank you. I am now also pessimistic about the Gelfand-Tseitlin approach. What would be a good introduction to the Deligne-Lusztig theory? $\endgroup$ Commented Dec 21, 2010 at 17:57
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$\begingroup$ There are two detailed textbook treatments: Carter (Wiley Interscience, 1985), extensive but down-to-earth; Digne & Michel (Cambridge, 1991), more concise and sophisticated. Carter has also written useful surveys, including one translated into Russian: MR1170353 (93j:20034), Karter, R.U. [Carter, Roger William] (4-WARW), Representation theory of finite groups of Lie type over an algebraically closed field of characteristic zero. Translated from the English by N. A. Vavilov. Algebra, 9 (Russian), 5–143, 268, Itogi Nauki i Tekhniki, Vsesoyuz. $\endgroup$ Commented Dec 21, 2010 at 18:42
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Not an answer either, but in response to Jim - Schur-Weyl duality doesn't always apply over finite fields. See http://www.ams.org/mathscinet-getitem?mr=2563588 for one result and some discussion of the related issues.
