Timeline for the character tables of irreducible representations of $SL(3,Z_q)$
Current License: CC BY-SA 3.0
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| Mar 11, 2017 at 21:14 | comment | added | A Stasinski | I think it is misleading to say that the answer is certainly "no". See my comment to the OP. I don't think we currently know the character table of any $\mathrm{SL}_3(\mathbb{Z}/p^r)$, $r\geq 2$, but for $p$ large enough all the dimensions of the irreps and their numbers are known. A construction of the irreps can be approached via a Kirillov orbit method. | |
| Mar 11, 2017 at 8:04 | comment | added | Uri Bader | There is also this paper by Uri Onn, which classifies all irreps of $\text{GL}_2$ over a discrete valuation ring, and has a nice introduction which regards the history of the problem, as well as a discussion of the higher rank case: arxiv.org/pdf/math/0611383.pdf. | |
| Mar 10, 2017 at 20:52 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| Mar 10, 2017 at 14:13 | comment | added | Jim Humphreys | @Denis: Thanks for the reminder of earlier work. It's not easy to give a complete survey, but I wanted to emphasize that even in rank 1 there is a continuing issue about how much can be written down explicitly. The underlying question is certainly natural in the context of $p$-adic groups. | |
| Mar 10, 2017 at 6:08 | comment | added | Denis Chaperon de Lauzières | Kloosterman studied the case of $SL_2(\mathbf{Z}/p^n\mathbf{Z})$ in "The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups I, II," Ann. of Math. 47 (1946), 317–447; see also the account of Springer in his survey of Kloosterman's work ams.org/notices/200008/fea-springer.pdf | |
| Mar 10, 2017 at 1:33 | history | answered | Jim Humphreys | CC BY-SA 3.0 |