## Lie Groups and Representation Theory Seminar 2018

List of speakers:
Christian Ikenmeyer, Oshima Yoshiki, Paul Baum, Ali Baklouti,
 Date: March 12 (Mon), 2018, 15:00-16:30 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Christian Ikenmeyer (Max-Planck-Institut fur Informatik) Title: Plethysms and Kronecker coefficients in geometric complexity theory Abstract: [ pdf ] Research on Kronecker coefficients and plethysms gained significant momentum when the topics were connected with geometric complexity theory, an approach towards computational complexity lower bounds via algebraic geometry and representation theory. This talk is about several recent results that were obtained with geometric complexity theory as motivation, namely the NP-hardness of deciding the positivity of Kronecker coefficients and an inequality between rectangular Kronecker coefficients and plethysm coefficients. While the proof of the former statement is mainly combinatorial, the proof of the latter statement interestingly uses insights from algebraic complexity theory. As far as we know algebraic complexity theory has never been used before to prove an inequality between representation theoretic multiplicities. (集中講義) Date: June 4 (Mon)-8 (Fri), 2018 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Yoshiki Oshima (大島芳樹) (Osaka Univ.) Title: 実 Lie 群の表現と指標 Abstract: [ pdf ] (連続講義) Speaker: Paul Baum (The Pennsylvania State University) Date: Oct 22 (Mon), 2018, 15:00-16:30 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Title: K-THEORY AND THE DIRAC OPERATOR I — What is K-theory and what is it good for? Abstract: [ pdf ] This talk will consist of four points. 1. The basic definition of K-theory 2. A brief history of K-theory 3. Algebraic versus topological K-theory 4. The unity of K-theory Date: Oct 24 (Wed), 2018, 15:00-16:30 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Title: K-THEORY AND THE DIRAC OPERATOR II — The Dirac operator Abstract: [ pdf ] The Dirac operator of R^n will be defined. This is a first order elliptic differential operator with constant coefficients. Next, the class of differentiable manifolds which come equipped with an order one differential operator which (at the symbol level)is locally isomorphic to the Dirac operator of R^n will be considered. These are the Spin-c manifolds. Spin-c is slightly stronger than oriented, so Spin-c can be viewed as "oriented plus epsilon". Most of the oriented manifolds that occur in practice are Spin-c. The Dirac operator of a closed Spin-c manifold is the basic example for the Hirzebruch-Riemann-Roch theorem and the Atiyah-Singer index theorem. Date: Oct 29 (Mon), 2018, 15:00-16:30 Place: Room 117, Graduate School of Mathematical Sciences, the University of Tokyo Title: K-THEORY AND THE DIRAC OPERATOR III — The Riemann-Roch Theorem Abstract: [ pdf ] 1. Classical Riemann-Roch 2. Hirzebruch-Riemann-Roch (HRR) 3. Grothendieck-Riemann-Roch (GRR) 4. RR for possibly singular varieties (Baum-Fulton-MacPherson) Date: Oct 31 (Wed), 2018, 15:00-16:30 Place: Room 122, Graduate School of Mathematical Sciences, the University of Tokyo Title: K-THEORY AND THE DIRAC OPERATOR IV — Beyond Ellipticity or K-homology and index theory on contact manifolds Abstract: [ pdf ] K-homology is the dual theory to K-theory. The BD (Baum-Douglas) isomorphism of Atiyah-Kasparov K-homology and K-cycle K-homology provides a framework within which the Atiyah-Singer index theorem can be extended to certain differential operators which are hypoelliptic but not elliptic. This talk will consider such a class of differential operators on compact contact manifolds. These operators have been studied by a number of mathematicians (e.g. C.Epstein and R.Melrose). Operators with similar analytical properties have also been studied (e.g. by Alain Connes and Henri Moscovici --- also Michel Hilsum and Georges Skandalis). Working within the BD framework, the index problem will be solved for these differential operators on compact contact manifolds. This is joint work with Erik van Erp. Speaker: Ali Baklouti (Faculté des Sciences de Sfax) Date: Dec 3 (Mon), 2018, 17:00-18:00 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Title: Monomial representations of discrete type and differential operators Abstract: [ pdf ] Let $G$ be an exponential solvable Lie group and $\tau$ a monomial representation of $G$, an induced representation from a connected closed subgroup of $G$ of a unitary character. It is well known that $\tau$ disintegrates into irreducible factors and the multiplicities of each isotypic component are explicitly determined. In the case where $G$ is nilpotent, these multiplicities are either finite or infinite almost everywhere, with respect to the disintegration's measure. We associate to $\tau$ an algebra of differential operators and it is shown that in the nilpotent case, the commutativity of this algebra is equivalent to the finiteness of the multiplicities of $\tau$. In the exponential case, we define the notion of monomial representation of discrete type. In this case, we show that such an equivalence does not hold and this answers a question posed by M. Duflo. This is a joint work with H. Fujiwara and J. Ludwig.