## Lie Groups and Representation Theory Seminar at the University of Tokyo 1998

 Date: April 14 (Tue), 1998, 16:30-18:00 Speaker: Tom Roby (California State University Hayward) Title: Down-up Algebras Abstract: The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down-up algebras. We show that down-up algebras exhibit many of the important features of the universal enveloping algebra U(sl) of the Lie algebra sl including a Poincaré-Birkhoff-Witt type basis and a well-behaved representation theory. We investigate the structure and representations of down-up algebras and focus especially on Verma modules, highest weight representations, and category $\Cal O$ modules for them. We calculate the exact expressions for all the weights, since that information has proven to be particularly useful in determining structural results about posets. Date: April 21 (Tue), 1998, 16:30-18:00 Place: Room 122, Graduate School of Mathematical Sciences, Komaba Speaker: Manabu Yamaguchi (R w) (University of Tokyo) Title: Ώ̌Q̂˂Q Lie superalgebra $\frak q(n)$ duality Abstract: $k$ Ώ̌Q ${\frak S}_k$ ʏ̌QƈقȂ˂Q ${\Cal A}_k$i^Ĉӂɒ܂j, Lie superalgebra ${\frak q}(n)$ duality ̍\. ̌ʂ, Ώ̌QƈʐQ $GL(n)$ duality (Schur-Weyl duality) ̗ގł. Schur-Weyl duality ɂ, F. G. Frobenius ̑Ώ̌Q́iʏj\ wWɊւ, \̗ꂩł. Frobenius ̋LqɎ@, I. Schur ͑Ώ̌Q ˉe\̊wW̌, $Q$-֐ƌĂ΂ Ώ̊֐̊pĕ\. 񓾂ꂽ duality , Schur ̌\̌tŐ̂ł. Date: November 10 (Tue), 1998, 16:30-18:00 Speaker: Satoshi Ishikawa (ΐ N) (University of Tokyo) Title: Symmetric subvarieties in compactifications and the asymptotic properties of the Radon transform on Riemannian symmetric spaces Abstract: We study the asymptotic property of the Radon transform on rapidly decreasing function spaces on higher rank Riemannian symmetric spaces of non-compact type and generalize a part of my previous work for the totally geodesic Radon transform on real hyperbolic spaces. We prove that under certain conditions if the Radon transform of a rapidly decreasing function satisfies some strong decay condition or has a compact support, then the function satisfies some stronger decay condition or the support of the function is smaller. The conditions are given in terms of the cell decomposition of the Satake compactification etc. Date: November 24 (Tue), 1998, 16:30-18:00 Speaker: Soji Kaneyuki (ss) (Sophia University) Title: pG~[gΏ̋ԂSiegel^̈ƂĂ̎Ƃ̎ Abstract: PΏ̋ԂŐȉ~OɂȂ͕̂f\A([)G~[g ̋ԂɂȂBoȋOɂȂ̂̓pf\ApG~[g Ώ̋ԂɂȂB҂̊Ԃɂ͊􉽊wIȁA΂ގ݂B ƂāAΏ̑oȋOSiegel^̈ƂĂ̎ɂďqׂB Date: December 8 (Tue), 1998, 16:30-18:00 Speaker: J. Stembridge (University of Michigan & RIMS) Title: Computational aspects of root systems, Coxeter groups, and Weyl characters Abstract: We will discuss three fundamental computational problems that parise in working with Weyl groups and root systems: (1) the conjugacy problem (deciding when two Weyl group elements are conjugate, or producing a canonical representative of a given conjugacy class); (2) traversal (efficiently searching through the elements of a Weyl group or one of its quotients); (3) tensor product multiplicities for Weyl characters. For the classical cases, especially type A, these problems have well-known solutions that do not (easily) generalize. Here, our emphasis is on algorithms that are (largely) independent of the classification of root systems. The canonical example we always have in mind is E8. Date: December 15 (Tue), 1998, 16:30-18:00 Speaker: Kyo Nishiyama (R ) (Kyoto University) Title: Bernstein Degree and Associated Cycle of Harish-Chandra Modules Abstract: Let $G = Sp(2 n, \R)$ be symplectic group and $\cover{G} = Mp(2 n,\R)$ its metaplectic double cover. By the theory of dual pair, irreducible unitary highest weight modules of $\cover{G}$ are parametrized by irreducible finite dimensional representations of orthogonal group $O(m)$ for various $m$ (Howe correspondence, or theta correspondence). For $\sigma \in \widehat{O}(m)$, let us denote the corresponding unitary highest weight module of $\cover{G}$ by $L(\sigma)$. In this talk, I will give Bernstein degree and associated cycle of $L(\sigma) \ (\sigma \in \widehat{O}(m))$ for $m \leq n = \rank \cover{G}$ explicitly. The similar method can be applicable to other non-compact simple groups of hermitian type. I would like to comment on the pair $(U(p, q), U(m))$ and $(O^{\ast}(2 r), Sp(2 m))$. Date: January 19 (Tue), 1999, 16:30-18:00 Speaker: Yoshiaki Kakiichi (sǖ) (Toyo University) Title: Differential Geometry, Lie Algebras and Lie Superalgebras Abstract: As the tangent algebra at a base point of Lie group we obtain the Lie algebra, and also as the tangent algebra of symmetric space we have the Lie triple system. Analogously, as the tangent algebra of reductive homogeneous space, we get the Lie triple algebra (or general Lie triple system). From the view point of algebra, we obtain the standard embedding Lie algebra of the Lie triple algebra. Analogously, we get the standard embedding Lie superalgebra of Lie super triple algebra (LSTA) which is the graded generalization of Lie triple Date: January 26 (Tue), 1999, 16:30-18:00 Speaker: M. Duflo (Univ. Paris VII) Title: Transverse Poisson structure to coadjoint orbits Abstract: In the dual of a Lie algebra, the symplectic leaves are the coadjoint orbits. The transverse Poisson structure to a symplectic leaf as been defined by Weinstein. Work of F. du Cloux and M. Saint Germain show that it is an interesting object to study in relation to representation theory. Date: February 16 (Tue), 1999, 16:30-18:00 Speaker: Hiroyuki Ochiai ([V) (Kyushu University) Title: Radon-Penrose transformation and the configuration space related to hypergeometric functions Date: March 16 (Tue), 1999, 14:30-16:00 Speaker: Tibor Ódor Title: The solution of the Pompeiu problem Abstract: The longstanding open problem of integral geometry, the Pompeiu problem (posed by the Romanian mathematician, Dimitru Pompeiu in 1929), states the following. If the integral of a nonzero continuous function on the n dimensional Euclidean space vanish on every congruent copy of the domain O then the domain is a ball, provided O boundary. (Convex bodies obviously satisfies this condition.) Using the elementary ''Extremum Method'', a solution is given under the possible most general conditions. The proof extends to rank 1 symmetric spaces of non compact type. Extension to higher rank symmetric spaces is also discussed. Date: March 16 (Tue), 1999, 16:30-18:00 Speaker: Katsuhisa Mimachi (Ov) (Kyushu University) Title: A duality of the Macdonald-Koornwinder polynomials and its application to the integral representations Abstract: We give a formula representing a duality of the Macdonald-Koornwinder polynomials. Using this formula, an integral representation of the Macdonald-Koornwinder polynomials is derived, which special case is the conjectural formula in our previous work. We also present the corresponding formula to Heckman-Opdam's Jacobi polynomials of type $BC_m.$