Irreducible representations of W-algebra in case $\mathfrak sl_3$

Question

Is there a paper in which are all the irreps of the finite W-algebra with trivial action of the center are classified, in the case of $\mathfrak sl_3(\mathbb C)$ and the minimal orbit?

Answer 1

The answer may depend on what you mean by "classified", but the best results so far seem to be those of Brundan-Kleshchev and Brown-Goodwin. See for example the preprint here and its references. (However, the Brown-Goodwin results might not apply directly to your situation, due to their restrictive assumption on the nilpotent orbit.)

ADDED: I'm far from being an expert on the subject, but maybe I can focus the discussion a little more. The notion of finite $W$-algebra is tricky to define (there being at least three equivalent but different-looking definitions in the literature). Essentially it's an associative algebra attached to a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ and a nilpotent element $e \in \mathfrak{g}$. Usually the case $e=0$ is ignored, but the convention then is that the resulting finite $W$-algebra is just the universal enveloping algebra $U(\mathfrak{g})$. Kostant originally studied the opposite extreme when $e$ is regular ("principal") and the resulting algebra is isomorphic to the center of $U(\mathfrak{g})$. Premet extended the definition, using an $\mathfrak{sl}_2$-triple for (nonzero) $e$.

In any case, the finite dimensional representations are the main object of study; even their existence is problematic at first. Only in rank 1 are the results given by classical methods, while in general there are close connections with the primitive ideals of the universal enveloping algebra. Besides the work I mentioned above I should have pointed out the papers by Premet and by Ivan Losev: see for example Losev's recent ICM talk here and his earlier paper here.

Answer 2

Classifying all irreps of an algebra like a W-algebra is a fool's errand, so I'm assuming that's not what you mean.

For finite dimensional irreps, there are 2; for $\mathfrak{sl}_n$, the number is always the number of standard tableaux of the corresponding Young diagram. This is Theorem C in Brundan and Kleshchev's Representations of shifted Yangians and finite W-algebras. This theorem also gives a construction, as does B&K's "Schur-Weyl duality for higher levels".

EDIT: It occurs to me that the W-algebra in this case is actually one of the algebras considered by Musson and van der Bergh in Invariants under tori of rings of differential operators and related topics. Thus, the category of all weight modules with integral regular central character actually has a nice description; it's the quotient of the path algebra of the doubled $A_4$ quiver by the relation the loops going left and going right from either of the two middle vertices are equal. The finite dimensional reps are the two middle vertices. This stuff is discussed (admittedly, without drawing the connection to W-algebras) in our recent paper Hypertoric category $\mathcal O$, especially Example 4.12.

Answer 3

All (not only finite dimensional!!) irreducible reps of $W$-algebra of $sl_2$ at its subregular nilpotent (read $U(sl_2)$) were classified by Block. A similar classification for the minimal $W$-algebra of $sl_3$ is feasible but still awaits its hero, to the best of my knowledge.