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

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?

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Presumably you meant to put an adjective in front of irreps. "Finite-dimensional" perhaps? – Ben Webster Dec 8 '10 at 19:13
No, I thought in this "easy" case, there could be a description of all irreps. Maybe that was stupid. – Jan Weidner Dec 9 '10 at 15:17
@Jan: I too was taking for granted that you intended finite dimensional here. – Jim Humphreys Dec 9 '10 at 19:17
Jan- I wouldn't say stupid; just leading into the weakness of the term "classification." When you classify something finite, then you can answer concrete questions about it, like "how many of them are there?" or "how many satisfying property X?" Of course, I won't claim that you can't classify things which there are infinitely many of, but that it's easy for such an exercise to become meaningless at times. – Ben Webster Dec 9 '10 at 20:28

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.

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Why fool's errand? It is only your judgement... – Bugs Bunny Dec 9 '10 at 14:16

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.

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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.

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Maybe I am confused. The universal enveloping algebra is the W-algebra of the $0$-orbit, and the minimal orbit is minimal in the set of all orbits greater then $0$ right? – Jan Weidner Dec 9 '10 at 15:11
Both "minimal" and "classified" get fuzzy at some points here. Block dealt with the ordinary universal enveloping algebra in rank one and produced a sort-of-classification (probably not usable in practice) to refute Dixmier's earlier assertion that a classification would be impossible. But the minimal nilpotent orbit is not the zero orbit, rather the unique one just above it in the closure ordering. Here the $W$-algebra picture is unrelated to what Block did. (Indeed, in rank one the minimal nilpotent orbit is the regular one.) – Jim Humphreys Dec 9 '10 at 19:23
Jim is right. I am fixing my answer. – Bugs Bunny Dec 9 '10 at 21:49