Terminology occuring in automorphic representation and relationship between them 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?
 A: ‘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.
A: 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.
A: 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.
