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Let $k=\mathbb{F}_q$ where $q$ is a prime power of odd cardinality.

Where could I find explicit models of all irreducible cuspidal (complex) representations of $GL_n(k)$ for $n\ge 3$?

I understand that the characters of such representations was constructed by J.A. Green “The characters of the finite general linear groups,” Trans. Amer. Math. Soc.,80, No. 2, 402–447 (1955).

If $n=2$, the cuspidal representations of $GL_2(k)$ can be constructed using Weil representations, which can be found in many online notes. But I could not find any explicit construction of cuspidal representations of $GL_n(k)$ when $n\ge 3$.

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    $\begingroup$ Maybe you need to explain the term ``explicit construction''. Some works of Deligne--Lusztig and Lusztig can be used to construct the cuspidal representations. $\endgroup$ – user148212 Jan 12 '18 at 7:59
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    $\begingroup$ Your tag 'lie-groups' is not helpful here. It should be omitted (or replaced by such a tag as 'gr.group-theory' or 'algebraic-groups').. $\endgroup$ – Jim Humphreys Jan 19 '18 at 15:11
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For the finite groups GL$_n(\mathbb{F}_q)$ there is an early paper by Lusztig well worth checking out here. This predates his broader work on finite groups of Lie type with Deligne (1976), where they found a way to construct virtual characters using $\ell$-adic cohomology which in principle have all irreducible characters as constituents. Later work by Lusztig and others refines considerably how these characters can be extracted and described. But only in type $A$ is an explicit construction likely to be found as in Lusztig's earlier paper.

Historically, Frobenius and Schur were first able to treat the case $n=2$ using various character methods. For "principal" series, ordinary induction is enough to provide models for most of the irreducible representations, but for "cuspidal" (or "discrete") series, here mostly of dimension $q-1$ rather than $q+1$, a construction is quite elusive. So they worked around this obstacle using character-theoretic tricks, but the problem remained open for these and similar finite groups. (For groups not of type $A$, finding explicit models for cuspidal representations is especially difficult because there is more than one family of these.)

ADDED: The reference for the more sophisticated work by Deligne and Lusztig is here. They treat arbitrary finite groups of Lie type coming from reductive algebraic groups including general linear groups. Note that 9.9 provides a vanishing criterion for $\ell$-adic cohomology of their varieties outside one degree; this implies in particular that (up to sign) "most" of the irreducible representations can be realized on such nonvanishing cohomology modules, including most of those that are cuspidal. But of course there are exceptions for the general linear groups (and others). For example, already in rank 1 one of their virtual characters for $\mathrm{GL}_2(\mathbb{F}_q)$ has formal degree $-(q-1)$ and involves the difference of the trivial character and the Steinberg character with each of these being realized in a different cohomology degree. In higher ranks it gets much more complicated. so the main goal of subsequent work has been to extract information about irreducible characters from their virtual character construction.

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  • $\begingroup$ I could not find the book by Lusztig at this moment. The following is a description of the book taken from press.princeton.edu/titles/1464.html The book gives an explicit construction of one distinguished member, $D(V)$, of the discrete series of $GL_n (F_q)$, where $V$ is the n-dimensional $F$-vector space on which $GL_n(F_q)$ acts. This is a p-adic representation; more precisely $D(V)$ is a free module of rank $(q--1) (q^2—1)...(q^{n-1}-1)$ over the ring of Witt vectors $W_F$ of $F$. It seems that this is not complex representation I need. $\endgroup$ – Qing Zhang Jan 13 '18 at 5:17
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    $\begingroup$ @QingZhang It is a complex rep as $\bar{Q}_p$ is isomorphic to $\mathbb{C}$ as abstract fields. $\endgroup$ – user148212 Jan 13 '18 at 6:32
  • $\begingroup$ @QingZhang: This approach is admittedly indirect but does yield an explicit construction of a typical irreducible representation with a cuspidal character over $\mathbb{C}$. It might help to look at Lusztig's own comments on this paper, which is [17] on his list: arxiv.org/pdf/1707.09368.pdf (But to go further with groups of Lie type, one mainly relies on the characters.) $\endgroup$ – Jim Humphreys Jan 13 '18 at 16:12
  • $\begingroup$ Does Lusztig more regularly update the comments on his papers on his web-site, or on the arXiv? Anyway, it may be worth knowing that some set of comments is available at www-math.mit.edu/%7Egyuri/pub.html . (This is an amazing resource, wherever you find it.) $\endgroup$ – LSpice Jan 24 '18 at 16:41
  • $\begingroup$ @LSpice: As I recall, he started to keep both the publication list and the list of his own comments on some of them at his homepage, but recently the comments got transferred to the arXiv. Anyway, these comments are often quite useful for putting his papers in context. He has been remarkably productive for many decades, but sometimes his papers are hard to unpack even though they have many unexpected gems tucked away inside. $\endgroup$ – Jim Humphreys Jan 24 '18 at 20:56
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Gel′fand S.I., Representations of the general linear group over a finite field, Lie groups and their representations (Proc. Summer School on Group Representations of the Bolya: J ́anos Math. Soc., Budapest, 1971), 119–132, Halsted, New York, 1975. makes the action on the Kirilov model explicit.

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  • $\begingroup$ This comes from his work in the late 1960s, written up first in a 1970 Russian journal article with English translation: mathscinet.ams.org/mathscinet-getitem?mr=0272916. Probably I should also have cited this approach, though its influence has been very limited compared with that of Deligne-Lusztig. Apparently it's suited only to the finite general linear groups (?) and is hard to implement effectively. But it fits well into the functional analysis tradition in representation theory.. $\endgroup$ – Jim Humphreys Jan 22 '18 at 14:39
  • $\begingroup$ For those who read Russian, the 1970 paper is freely available here mathnet.ru/php/… $\endgroup$ – Jim Humphreys Jan 22 '18 at 14:44

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