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### Perspectives on representations of semi-simple Lie algebras

In honor of Gregg Zuckerman on the occasion of his 70 th birthday

Bochum 22-24 May 2019

IA 1/53

This event is funded by CRC/TRR 191 "Symplectic Structures in Geometry, Algebra and Dynamics" of the Deutsche Forschungsgemeinschaft and the DFG Project PE 980/7-1.

Siddhartha Sahi (Rutgers),
Vera Serganova (Berkeley),
Catharina Stroppel (Bonn),
Valdemar Tsanov (Bochum)
David Vogan (MIT)
Joseph A. Wolf (Berkeley)

## Direction & Schedule

Talks will take place at the Ruhr-Universität Bochum, Faculty of Mathematics.

Every participant of the workshop is welcome to join us for the conference dinner on Thursday evening.

Wednesday 22nd Thursday 23rd Friday 24th
10:00 - 11:00 Adams 10:00 - 11:00 Stroppel 9:30 - 10:30 Lian
11:00 - 11:30 Coffee break 11:00 - 11:30 Coffee break 10:30 - 11:00 Coffee break
11:30 - 12:30 Sahi 11:30 - 12:30 Tsanov 11:00 - 12:00 Petukhov
12:30 - 14:00 Lunch 12:30 - 14:00 Lunch 12:00 - 13:30 Lunch
14:00 - 15:00 Cupit-Foutou 14:00 - 15:00 Wolf 13:30 - 14:30 Penkov
15:00 - 15:30 Coffee break 15:00 - 15:30 Coffee break 14:30 - 15:00 Coffee break
15:30 - 16:30 Boe 15:30 - 16:30 Vogan 15:00 - 16:00 Serganova
by Gregg Zuckerman

## Titles and abstracts

Wednesday 22nd
10:00 - 11:00 Adams - "Subgroups of Reductive Groups"

The study of reductive subgroups of reductive group has a long history, as does the study of finite subgroups. In this talk I'll discuss some interesting subgroups which arise as the normalizer of a reductive subgroup. These play a role in questions in geometry and representation theory.

11:00 - 11:30 Coffee break

11:30 -12:30 Sahi - "Metaplectic Representations of Hecke algebras"

We construct certain representations of affine Hecke algebras and Weyl groups, which depend on several auxiliary parameters. We refer to these as "metaplectic" representations, and as a direct consequence we obtain a family of "metaplectic" polynomials, which generalizes the well-known Macdonald polynomials.
Our terminology is motivated by the fact that if the parameters are specialized to certain Gauss sums, then our construction recovers the Kazhdan-Patterson action on metaplectic forms for GL(n); more generally it recovers the Chinta-Gunnells action on p-parts of Weyl group multiple Dirichlet series.
This is joint work with Jasper Stokman and Vidya Venkateswaran.

12:30 - 14:00 Lunch

14:00 - 15:00 Cupit-Foutou - "Momentum polytopes of Kähler compact multiplicity-free manifolds"

We shall present some results about the momentum polytopes of the multiplicity-free Hamiltonian compact manifolds acted on by a compact group which are Kählerizable. Our mail goal will be to give a characterization of these polytopes, explain how much they determine these manifolds and sketch some applications of this characterization -- some of these results have been obtained jointly with G. Pezzini and B. Van Steirteghem.

15:00 - 15:30 Coffee break

15:30 -16:30 Boe - "Perspectives on support varieties and tensor triangular geometry"

I will describe settings, some "classical" and some more recent, in which the geometry of support varieties gives algebraic information about individual representations, and where the closely-related notion of tensor triangular geometry sheds light on the structure of whole categories of representations.

Thursday 23rd

10:00 - 11:00 Stroppel - Fusion rings from quantum groups and DAHA actions

In this talk I will give a short overview about fusion rings arising from quantum groups at odd and even roots of unity. These are Grothendieck rings of certain semisimple tensor categories. Then I will study these rings in more detail. The main focus of the talk will be an expectation by Cherednik that there is a certain DAHA action on these rings which can be used to describe the multiplication and semisimplicity of these rings. As a result we present a theorem which makes Cherednik's expectation rigorous.

11:00 - 11:30 Coffee break

11:30 - 12:30 Tsanov - "Geometric Invariant Theory on Flag Varieties"

We study the action of reductive subgroups on flag varieties of semisimple complex groups in the context of Geometric Invariant Theory. We derive a closed formula for the unstable locus of an arbitrary line bundle and use it to determine the GIT-classes in the ample cone. The result are applied to study properties of GIT-quotients, the structure of the generalized Littlewood-Richardson cone, branching laws and multiplicities of components, In particular, we show that a generic quotient is a Mori Dream Space, the Mori structure of whose Picard group is isomorphic to the GIT-structure in the Picard group of the flag variety. Such a quotient is shown to contain global information on all invariants of the subgroup in irreducible modules of the ambient group. In the process we derive new numerical invariants of semisimple subgroups of semisimple groups. This is joint work with Henrik Seppänen.

12:30 - 14:00 Lunch

14:00 - 15:00 Wolf - "A nilmanifold version of flag domains"

The usual flag domains lead to geometric construction of discrete series representations of semisimple Lie groups. With some care, this leads to construction of enough representations to support the Plancherel formula. I'll describe an analog of these constructions for a class of nilpotent Lie groups and weakly symmetric pseudo-riemannian nilmanifolds.

15:00 - 15:30 Coffee break

15:30 - 16:30 Vogan - "The local Langlands conjecture for finite groups of Lie type"

Suppose G is a reductive group over a local field. The local Langlands conjecture posits a partition of irreducible representations of G into finite "L-packets" indexed by some number-theoretic parameters. The conjecture was established by Langlands over the real numbers; in the p-adic case it has been proven by Taylor, Harris, Arthur, and others for most classical groups. The big question raised by the conjecture is how to index the representations within each packet. Knapp and Zuckerman solved this problem over the real numbers, and gave examples illustrating some very surprising behavior in the p-adic case. Their results admit nice formulations in terms of "lowest K-types." In the 1970s, Macdonald formulated and proved an analogue of Langlands' conjecture for the finite group GL(n,F_q). I will explain how one can (conjecturally) extend Macdonald's formulation to any finite group of Lie type; what results of Lusztig and Shoji offer toward proof of this extension; and how these questions may be related to "lowest K-types" for p-adic fields.

Friday 24th

9:30 - 10:30 Lian - "From cyclic covers to mirror symmetry"

We will consider a class of Calabi-Yau varieties given by cyclic branched covers of a xed semi Fano manifold. The rst prototype example goes back to Euler, Gauss and Legendre, who considered 2-fold covers of P1 branched over 4 points. Two-fold covers of P2 branched over 6 lines have been studied more recently by many authors, including Matsumoto, Sasaki, Yoshida and others, mainly from the viewpoint of their moduli spaces and their comparisons. I will outline a higher dimensional generalization from the viewpoint of mirror symmetry. We will introduce a new compacti cation of the moduli space cyclic covers, using the idea of abelian gauge xing' and fractional complete intersections'. This produces a moduli problem that is amenable to tools in toric geometry, particularly those that we have developed jointly in the mid-90's with S. Hosono and S.-T. Yau in our study of toric Calabi-Yau complete intersections. In dimension 2, this construction gives rise to new and interesting identities of modular forms and mirror maps associated to certain K3 surfaces. We also present an essentially complete mirror theory in dimension 3, and discuss generalization to higher dimensions. The lecture is based on on-going joint work with S. Hosono, T.-J. Lee, H. Takagi, S.-T. Yau.

10:30 - 11:00 Coffee break

11:00 - 12:00 Petukov - "Minimal nilpotent W-algebras and Zigzag algebras"

In my talk I would like to establish an equivalence between blocks of the category of finite-dimensional representations of a minimal nilpotent W-algebra and the category of finitely generated representations of an appropriate Zigzag algebra.
Let $G$ be a simple Lie group over $\mathbb C$ and $\frak g$ be its Lie algebra. Let $e\in\frak g$ be a highest weight vector of $\frak g$. The orbit $Ge$ is the minimal (by dimension) nilpotent $G$-orbit and we denote it $\mathcal O_{min}$. We attach to $e\in\frak g$ an associative algebra ${\rm U}(\frak g, e)$ and we call it {\it $W$-algebra} (in our case $e\in\mathcal O_{min}$ and hence we call ${\rm U}(\frak g, e)$ the minimal nilpotent $W$-algebra).
It is well known that simple finite-dimensional modules of $W$-algebras correspond to primitive ideals of ${\rm U}(\frak g)$ (i.e. the annihilators of simple modules); isomorphism classes of simple finite-dimensional ${\rm U}(\frak g, e)$-modules are in bijection with primitive ideals of ${\rm U}(\frak g)$ whose associated variety equals the closure of $Ge=\mathcal O_{min}$. It is natural to expect that the finite-dimensional modules of $W$-algebras are closely connected to (not necessarily primitive) two-sided ideals of ${\rm U}(\frak g)$ whose associated variety equals to the closure of $\mathcal O_{min}$.
The main result is an explicit description of the basic algebra of every block of finite-dimensional ${\rm U}(\frak g, e)$-modules (everything works well in types other than $A$ and it is easy to see that it can't work well in type A). This basic algebra is a Zigzag algebra.
The main tool of the article is the categorical action of the Weyl group of $\frak g$ on the 0-block of $\frak g$-modules of ${\rm U}(\frak g)$, and this approach was suggested by Gregg Zuckerman back in 1982.

12:00 - 13:30 Lunch

13:30 - 14:30 Penkov - "Primitive ideals of U(sl(infty)), U(o(infty)), U(sp(infty))"

I will present the classification of such primitive ideals. In the case of sl(infty), this is joint work with A. Petukhov. In the other two cases it is recent work of my student A. Fadeev. The answer is by far not a straightforward analog of the finite-dimensional case.

14:30 - 15:00 Coffee break

15:00 - 16:00 Serganova - "On the abelian envelope of the Deligne category GL(t)."

The category GL(t) for an arbitrary complex t is a "free" symmetric monoidal rigid Karoubian category generated by an object of dimension t. If t is not an integer, GL(t) is abelian and semisimple. A few years ago in collaboration with Entova-Aizenbud and Hinich we constructed an abelian tensor category V(t) for any integer t which extends GL(t) and satisfies nice universal properties. In this talk I plan to present several recent results about V(t), in particular, define an action of sl(\infty) on the Grothendieck group K[V(t)] and compute the radical filtration of the corresponding sl(\infty)-module. Connections with symmetric and supersymmetric polynomial rings will be also explored.

16:00 - 16:30 Concluding comments by Gregg Zuckerman

There is no registration fee nor formal registration but for organizational matters, please send an email to Mrs. Gabriele Koenig if you plan to attend the workshop.

For hotel recommendations, you can contact Mrs. Gabriele Koenig.

Organizers: Peter Heinzner (RUB), Alan Huckleberry (Bochum and JU Bremen) and Ivan Penkov (JU Bremen), for contact email Peter Heinzner.