The strange word “cuspidal”

I am currently looking at various papers (and books) about the representation theory of p-adic groups (especially GL(2,Qp)), and in particular about the so-called discrete series. I was convinced that the standard terminology for those representations (except for the special case of the Steinberg representation) was “supercuspidal”, but it turns out that various references use either “absolutely cuspidal” or simply “cuspidal”. The last is the terminology in the (outstanding) book of Bushnell and Henniart, who fortunately mention the other two possibilities, but I wonder how many outsiders have been hopelessly confused by this type of wobbling…

By a nice coincidence (though it may be showing that the Stars really suggest “cuspidate” as the right word), one of the citations for “cuspidal” in the Oxford English Dictionary is

3. Of teeth: = CUSPIDATE.
1867 BUSHNELL Mor. Uses Dark Th. 274 Cuspidal teeth.

(the reference is to the masterpiece “The complete ship-wright” of a certain Edward Bushnell in 1664).

Going further, intrepidly, we learn that “cuspidate” is an invention of a J. Hunter (“The natural history of the human teeth”, 1771–78), and that this learned man decided to call “cuspidati” what are “vulgarly called canine”. It follows that the friends of Langlands, if they moreover wish to be progressive, should speak proudly of “canine (or supercanine) representations”, of “canine forms”, and so on…

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I am a professor of mathematics at ETH Zürich since 2008.

2 thoughts on “The strange word “cuspidal””

  1. I believe this stems from Harish-Chandra originally using “supercuspidal”, but Hervé Jacquet originally using “absolutely cuspidal”. In addition to “cuspidal”, the term “parabolic” is also sometimes used (but uncommon). From his more recent papers it appears that Jacquet himself now uses “supercuspidal”, so it does appear to be the more or less standard term.

  2. I had the (faulty) memory of “supercuspidal” being the term in Jacquet-Langlands, but looking at it, you’re right that they use “absolutely cuspidal”.

    Interestingly, the first use of “cuspidal” in Math Reviews (in the context of automorphic forms) is to papers of Langlands MR0249539 and Harish-Chandra MR0232893 on analytic continuation of Eisenstein series, and for Harish-Chandra at least, “cuspidal” is used (indeed) as a synonym of parabolic, but applied to a “parabolic subgroup”!

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