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When one tries to read about automorphic representation few terms come up more than others namely,

1.Cuspidal

2.Square Integrable

3.Absolutely Cuspidal

4.Super Cuspidal

My understanding about them is Cuspidal and Square Integrable Representation are the same. Older authors used the term Square Integrable, where as now days people use Cuspidal.

Similarly Absolutely Cuspidal and Super Cuspidal Representation are same. Older authors used the term Absolutely Cuspidal, but now days people use the term Super Cuspidal to mean the same thing.

What is the reason and history behind this change of terminology? Or am I completely wrong and each of them refer to different objects?

Also what is the relationship between Cuspidal and Super Cuspidal Representation?

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  • $\begingroup$ It seems that you are talking about representations of p-adic groups. Why do you mention automorphic in the title? $\endgroup$
    – MBN
    Commented Mar 31, 2010 at 1:42
  • $\begingroup$ Martin, at least as an etymological question, one has to know about automorphic forms to know why we might call a representation of a $p$-adic group supercuspidal. (I agree that it's not clear whether Dipramit wants to know about representations of groups over local or global fields, though.) $\endgroup$
    – LSpice
    Commented Mar 31, 2010 at 1:51
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    $\begingroup$ My 2 cents: my understanding was that cuspidal typically applied to automorphic representations, global objects, and supercuspidal to representations of p-adic groups, i.e. local objects, the logic being that if a global object was supercuspidal at one place then it was forced to be cuspidal---the local condition implies the global. I guess this follows from Langlands' theory of Eisenstein series: if you're not cuspidal you're "globally induced" and hence locally induced. $\endgroup$
    – Kevin Buzzard
    Commented Mar 31, 2010 at 6:46
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    $\begingroup$ Oh: and I guess the Steinberg representation of GL_2(Q_p) is square integrable but not supercuspidal. $\endgroup$
    – Kevin Buzzard
    Commented Mar 31, 2010 at 9:23
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    $\begingroup$ Kevin, indeed, any reductive group over a $p$-adic or finite field (and probably even in more generality) admits a Steinberg representation, and all such are in the discrete series but not (super)cuspidal. $\endgroup$
    – LSpice
    Commented Mar 31, 2010 at 19:02

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‘Square-integrable’ and ‘cuspidal’ are definitely not equivalent; the former latter representations are among the latter former, but do not exhaust them. To the best of my knowledge, ‘supercuspidal’ and ‘cuspidal’ are just the same concept with different etymologies behind them; and ‘absolutely cuspidal’ is meant to refer to representations that remain cuspidal upon extension of the ground field, hence is only interesting for representations in non-algebraically-closed fields. (For $p$-adic groups, the terminology ‘supercuspidal’ is used almost exclusively. I think one sees ‘cuspidal’ more in the global- or finite-field setting.)

I find it amusing that ‘very cuspidal’ (as used by Carayol, for example) is more restrictive than ‘supercuspidal’!

In the automorphic-forms settings, discrete-series representations are those that appear as direct summands of $L^2$ of our symmetric space. In this sense, they should be viewed as subrepresentations of the part of $L^2$ that decomposes ‘discretely’, as a direct sum, rather than continuously, as a direct integral. (In a measure-theoretic sense, the discrete series contains the atoms for the Plancherel measure. I think, but wouldn't swear to it, that Plancherel measure is absolutely continuous on the remainder of the tempered spectrum.) The cuspidal representations are those that appear in the space of $L^2$ functions that die off rapidly—in other words, that vanish at the cusps of a suitable compactification of the symmetric space. It is a theorem that they automatically appear as direct summands.

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  • $\begingroup$ I am quite certain that by Absolutely Cuspidal and Super Cuspidal are the same thing as far as automorphic representation are concerned. It is also used to be called Parabolic.(Reference: Definition 1.3 Cartier Notes in Corvalis titled Representation of p-adic Groups) Also I would be really surprised if Cuspidal and Super-Cuspidal representation are same. Super cuspidal representation are not usually square integrable, where as cuspidal representations are. $\endgroup$
    – Dipramit Majumdar
    Commented Mar 31, 2010 at 1:26
  • $\begingroup$ Dipramit, certainly it would be contrary to all the usage with which I am familiar to have a non-square-integrable supercuspidal representation. Can you give a reference? If you buy that ‘absolutely cuspidal’ and ‘supercuspidal’ are the same thing, then you may want to look at Bill Casselman's notes, available at ttp://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf. Although it's probably overkill, Theorem 4.4.6 and Proposition 5.1.1 there together show that absolutely cuspidal representations are always square integrable. $\endgroup$
    – LSpice
    Commented Mar 31, 2010 at 1:34
  • $\begingroup$ To see that Casselman's definition is the same as Cartier's, look at Theorem 5.2.1 of the linked notes. $\endgroup$
    – LSpice
    Commented Mar 31, 2010 at 1:36
  • $\begingroup$ I don't see how Proposition 5.1.1 and Thm 4.4.6 imply super-cuspidal representation are square integrable. The Proposition 5.1.1 is just restatement of the definition in terms of Jacquet's fuctor. $\endgroup$
    – Dipramit Majumdar
    Commented Mar 31, 2010 at 2:00
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    $\begingroup$ When you say "the former are among the latter", do you mean the converse? (I.e. cuspidal is square integrable, but not necessarily conversely.) $\endgroup$
    – Emerton
    Commented Mar 31, 2010 at 12:49
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For global automorphic representations, square-integrable is weaker than cuspidal. According to Moeglin-Waldspurger, the square-integrable automorphic representations of $GL_n(\mathbb{A})$ are generated by residues of Eisenstein series on certain special parabolics (those of type $(p,p,p,\\dots,p)$ for some factorization $n=pm$). A trivial example: the constant function generates a square-integrable representation on $SL_2(\mathbb{Q}) \backslash SL_2(\mathbb{A})$! Cuspidal means that the constant Fourier coefficient $\int_{P(\mathbb{Q} \backslash P(\mathbb{A})}f(pg)dp$ vanishes for any proper parabolic subgroup and any $f\in \pi$.

Now, why the term "cuspidal"? Classically, modular forms are functions on quotients $\Gamma \backslash \mathfrak{H}$ of the upper half-plane. The cusps of such a quotient are those points whose $SL_2(\mathbb{R})$-orbit lies in the boundary of $\mathfrak{H}$, and the parabolic subgroups of a (semisimple) group are precisely those groups which stabilize points in the boundary of an associated symmetric space.

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Carayol's terminology 'very cuspidal' does not apply to representations of the group itself (${\rm GL}(p)$ in Carayol's paper) but to certain representations of its compact open subgroups.

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  • $\begingroup$ Paul, I haven't read Carayol for a while, so I can't speak directly to his usage; but certainly the ‘modern’ terminology uses ‘very cuspidal’ to refer to the representations of the entire group (see Section 6.3 of Adler and DeBacker's “Murnaghan–Kirillov theory …” reference-global.com/doi/pdf/10.1515/crll.2004.080; I'm sure Murnaghan uses it, too, but can't find a reference). $\endgroup$
    – LSpice
    Commented Mar 31, 2010 at 19:00
  • $\begingroup$ I thank you for the reference ! P.B. $\endgroup$
    – Paul Broussous
    Commented Mar 31, 2010 at 20:23

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