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Dimensions of irreducible modules over W-algebras and Goldie ranks

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Abstract

The main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple Lie algebra. The first one, very classical, is the Goldie rank of an ideal. The second one is the dimension of an irreducible module corresponding to this ideal over an appropriate finite W-algebra. We concentrate on the integral central character case. We prove, modulo a conjecture, that in this case the two are equal. Our conjecture asserts that there is a one-dimensional module over the W-algebra with certain additional properties. The conjecture is proved for the classical types. Also, modulo the same conjecture, we compute certain scale factors introduced by Joseph, this allows to compute the Goldie ranks of the algebras of locally finite endomorphisms of simples in the BGG category \(\mathcal {O}\). This completes a program of computing Goldie ranks proposed by Joseph in the 80’s (for integral central characters and modulo our conjecture). We also provide an essentially Kazhdan–Lusztig type formula for computing the characters of the irreducibles in the Brundan–Goodwin–Kleshchev category \(\mathcal {O}\) for a W-algebra again under the assumption that the central character is integral. In particular, this allows to compute the dimensions of the finite dimensional irreducible modules. The formula is based on a certain functor from an appropriate parabolic category \(\mathcal {O}\) to the W-algebra category \(\mathcal {O}\). This functor can be regarded as a generalization of functors previously constructed by Soergel and by Brundan-Kleshchev. We prove a number of properties of this functor including the quotient property and the double centralizer property. We develop several side topics related to our generalized Soergel functor. For example, we discuss its analog for the category of Harish–Chandra modules. We also discuss generalizations to the case of categories \(\mathcal {O}\) over Dixmier algebras. The most interesting example of this situation comes from the theory of quantum groups: we prove that an algebra that is a mild quotient of Luszitg’s form of a quantum group at a root of unity is a Dixmier algebra. For this we check that the quantum Frobenius epimorphism splits.

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Notes

  1. In a recent preprint of Premet, [44], he proved that any W-algebra admits an \(A\)-stable 1-dimensional representation, but the central character is generally not integral. For example, [33] implies that, in the case of the three exceptional orbits, there are no \(A\)-stable finite dimensional representations with integral central character, so Premet’s representations automatically have non-integral central characters in these cases.

  2. The attribution to Premet is made because of his beautiful result saying that this scale factor is always integral. The first version of our proof below used that result. In fact—as was communicated to the author by Premet—one does not need that in the proof, it is enough to use the fact that the scale factor is bigger or equal than 1, which was established by the author. It is pleasant when somebody else understands your work better than you do...

  3. These are close to so called birationally rigid orbits.

  4. As I learned from Jeffrey Adams there is only one weakly rigid orbit, where this is not true: it appears in type \(E_8\).

Abbreviations

\(\mathcal {A}^{opp}\) :

The opposite algebra of \(\mathcal {A}\)

\(\widehat{\otimes }\) :

The completed tensor product of complete topological vector spaces/ modules

\((a_1,\ldots ,a_k)\) :

The two-sided ideal in an associative algebra generated by elements \(a_1,\ldots ,a_k\)

\(A^{\wedge _\chi }\) :

The completion of a commutative (or “almost commutative”) algebra \(A\) with respect to the maximal ideal of a point \(\chi \in \mathrm{Spec }(A)\)

\(\mathrm{Ann }_\mathcal {A}(\mathcal {M})\) :

The annihilator of an \(\mathcal {A}\)-module \(\mathcal {M}\) in an algebra \(\mathcal {A}\)

\(D(X)\) :

The algebra of differential operators on a smooth variety \(X\)

\(G^\circ \) :

The connected component of unit in an algebraic group \(G\)

\((G,G)\) :

The derived subgroup of a group \(G\)

\(G_x\) :

The stabilizer of \(x\) in \(G\)

\(\mathrm{Grk }(\mathcal {A})\) :

The Goldie rank of a prime Noetherian algebra \(\mathcal {A}\)

\(\mathrm{gr }\mathcal {A}\) :

The associated graded vector space of a filtered vector space \(\mathcal {A}\)

\(R_\hbar (\mathcal {A})\) :

\(:=\bigoplus _{i\in \mathbb {Z}}\hbar ^i \mathrm{F }_i\mathcal {A}\): the Rees \({\mathbb {K}}[\hbar ]\)-module of a filtered vector space \(\mathcal {A}\)

\(S(V)\) :

The symmetric algebra of a vector space \(V\)

\(\mathfrak {X}(H)\) :

The group of characters of an algebraic group \(H\)

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Acknowledgments

I would like to thank J. Adams, R. Bezrukavnikov, J. Brundan, I. Gordon, A. Joseph, G. Lusztig, V. Ostrik, A. Premet, C. Stroppel, D. Vogan and W. Wang for stimulating discussions related to various parts of this paper. Also I would like to thank the referee for many useful comments that allowed me to improve the exposition.

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Authors and Affiliations

  1. Department of Mathematics, Northeastern University, Boston, MA , 02115, USA

    Ivan Losev

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  1. Ivan Losev
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Correspondence to Ivan Losev.

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Dedication to Tony Joseph, on his 70th birthday.

Supported by the NSF grants DMS-0900907, DMS-1161584.

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Losev, I. Dimensions of irreducible modules over W-algebras and Goldie ranks. Invent. math. 200, 849–923 (2015). https://doi.org/10.1007/s00222-014-0541-0

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