Dear colleagues,

I'm writing to advertise a lecture series on the mixed quantum group given by Quan Situ at POSTECH on November 19 and 21. The mixed quantum group is a version of the quantum group at a root of unity interpolating between Lustzig's version and the De Concini-Kac version; its structure and category (O) of representations have recently been of interest in representation theory (connections with Lustzig's quantum group, the small quantum group, KL theory and geometry) as well as in number theory (fundamental local equivalence, metaplectic local Langlands etc.).

The lectures will be on zoom. See below for details. I would be grateful if you can forward this message to anyone who might be interested in the talks.

If you have any questions about the lecture series please let me know.

With best wishes,
Valentin

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Lecture Series on the hybrid (mixed) quantum group
Title: Hybrid quantum group and its representation theory
Speaker: Quan Situ
Affiliation: Université Clermont Auvergne
Dates: November 19 and November 21
Time: 6pm-8pm/8:30pm South Korea time
Website (has all the info, zoom links etc. ): buciumas.github.io/postech-representation-theory/index
Zoom link I (November 19): us06web.zoom.us/j/9450725725?pwd=MldrQkg3TURkOTEwQy92U1gzOHVUdz09&omn=89681414519
Zoom link II (November 21): us06web.zoom.us/j/9450725725?pwd=MldrQkg3TURkOTEwQy92U1gzOHVUdz09&omn=81861224988
Abstract: The hybrid (or mixed) quantum group is a quantum algebra with triangular decomposition whose positive part is given by the one of Lusztig’s quantum group, and whose negative part is given by the one of De Concini-Kac quantum group. Its category O can be viewed as a quantum analogue of the BGG category O for semi-simple Lie algebra. In the first talk, I will start by recalling basic results on structures and representations of Lusztig’s quantum group, De Concini-Kac quantum group and small quantum group. Then I will introduce the hybrid quantum group and its relation with the quantum algebras above. I will also introduce its category O and discuss some fundamental structures, e.g. block decomposition, linkage principle, BGG reciprocity and translation functors. In the second talk, I will focus on the principal block and the Steinberg block of this category O. The main result is equivalences from these blocks to coherent-sheaf-theoretic incarnations of the (singular) affine Hecke category. As an application, the principal block is a categorification of the periodic Hecke module, and in particular the (graded) multiplicity of simple module inside Verma module is given by the “generic Kazhdan-Lusztig polynomial”.