The center of small quantum groups II: singular blocks

Abstract

We generalize to the case of singular blocks the result in Bezrukavnikov and Lachowska [Quantum groups, Contemporary Mathematics 433 (American Mathematical Society, Providence, RI, 2007) 89-101] that describes the center of the principal block of a small quantum group in terms of sheaf cohomology over the Springer resolution. Then using the method developed in Lachowska and Qi [Int. Math. Res. Not., Preprint, 2017, arXiv:1604.07380], we present a linear algebrogeometric approach to compute the dimensions of the singular blocks and of the entire center of the small quantum group associated with a complex semisimple Lie algebra. A conjectural formula for the dimension of the center of the small quantum group at an lth root of unity is formulated in type A.

Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants

“…For example, X 2 = C 2 and, in a similar way, one can identify X p for other roots of unity. We expect X p 's to be close cousins of the spaces [82,83] that appear in the context of the small quantum group. While the explicit description of X p is very desirable (and we hope to report on it in the future), it will not be necessary for what follows.…”

“…After this paper appeared, further evidence for (4.24) was presented in[86] 22. Part of the difficulty is that the detailed structure of projective modules was understood only recently using the tools of geometric representation theory[82,83], and mostly in the context of the small quantum…”

The Center of Small Quantum Groups I: The Principal Block in Type A

“…In the sequel [LQ17] to this paper, we consider the same problem for singular blocks of small quantum groups. We will first formulate a generalization of the result in [BL07] relating singular block centers to the zeroth Hochschild cohomology of parabolic Springer varieties, and then compute the center of the singular blocks for g = sl 3 together with some other examples via the method developed in this paper.…”

“…The following conjecture comes from the explicit computations of several examples in type A, as well as some singular block computations which will appear in a sequel [LQ17].…”

“…The result can be computed explicitly using the algorithm developed in Section 2. A proof of a more general statement valid for any N P := T * (G/P ), where G is a simple complex Lie group and P is a parabolic subgroup, will be given in the sequel [LQ17], via some basic properties of (semi)stable vector bundles. To compute the remaining entries, we use the following three steps:…”