Gerrit van Dijk, Yurii Neretin (1), Yurii Neretin (2), Katsunori Kawamura (1), Hisayosi Matumoto, Pavle Pandzic (1), Christof Geiss, Katsunori Kawamura (2), Jorge Vargas, Pavle Pandzic (2), Kyo Nishiyama, Genkai Zhang, Andreas Nilsson (1), Andreas Nilsson (2), Soo Teck Lee, Raphael Rouquier, Eric Sommers, Munibur R. Chowdhury
Date:  September 16 (Tue), 2003, 16:3017:30 
Room:  RIMS Room 402 
Speaker:  Gerrit van Dijk (Leiden University) 
Title:  Generalized Gelfand Pairs: A Survey 
Abstract:  The group G = SL(2,R) of 2 × 2 matrices with determinant 1, acts on the complex upper half plane by fractional linear transformations. The measure dxdy/y^{2} is a Ginvariant measure on the upper half plane, and the group action extends easily to the L^{2}space of the invariant measure. It is wellknown that this action is multiplicity free: the L^{2} space decomposes multiplicity free as a direct integral of irreducible spaces. This property was studied and extended by Gelfand a.o. to pairs (G,K), where G is a Lie group and K a compact subgroup. The equivalent of the upper half plane is the space G/K. Pairs (G,K) such that L^{2}(G/K) splits multiplicity free are called Gelfand pairs. The most wellknown examples are given by pairs (G,K) where G is a semisimple Lie group and K a maximal compact subgroup. We shall discuss an extension of the notion of Gelfand pair for pairs (G,K) where K is a closed, nonnecessarily compact subgroup of G. Several examples of (generalized) Gelfand pairs will be given. 
Date:  September 19 (Fri), 2003, 10:3011:30 
Room:  RIMS Room 402 
Speaker:  Yurii Neretin (ITEP) 
Title:  Inverse limits of unitary groups, matrix betaintegrals, and harmonic analysis on infinitedimensional symmetric spaces 
Abstract:  There exists a natural distinguished map from a larger unitary group
U(n) to the smaller group U(n1). (Firstly, this map apeared in works of M. S. Livsic on spectral theory of nonselfadjoint operators in 1940s, but the construction was not well known in the representation theory.) This map is consistent with Haar measures.
The inverse limit of the spaces U(n) is equipped with 2parametric family of canonical U(∞) × U(∞)quasiinvariant measures. It will be given a selfconsistent description of this construction and also a nonformal introduction to the Plancherel formula, which was recently obtained by G. Olshanski and A. Borodin. 
Date:  September 29 (Mon), 2003, 10:3011:30 
Room:  RIMS Room 402 
Speaker:  Yurii Neretin (ITEP) 
Title:  Structures of boson Fock space in the space of symmetric functions 
Abstract: [pdf] 
We give explicit realization of Weil representation of
infinitedimensional (FriedrichsShale) symplectic group in the space
Λ of symmetric functions in infinite number of variables
For each operator Λ → Λ we associate a formal series K(x_{1}, x_{2}, …; y_{1}, y_{2}, …) (bisymmetric kernel of operator) symmetric with respect to x_{j} and with respect to y_{j}. Our representations is realized by operators corresponding to kernels of the form We also show, that the set of all operators having such kernels is closed with respect to multiplication and describe this semigroup 
Date:  October 14 (Tue), 2003, 13:0014:00 
Room:  RIMS 402 
Speaker:  Katsunori Kawamura (쑺I) (RIMS) 
Title:  An introduction to representation theory of the Cuntz algebras 
Abstract:  The Cuntz algebra is a noncommutative simple infinite dimensional example of C*algebra which is generated by a family of isometries with some relations. We introduce a class of representations of the Cuntz algebra and characterizeexistence, irreducibility, unitary equivalence, uniqueness, complete reducibility and canonical complete orthonormal basis. By using these properties, we show two branching laws of these representations, 1) restricted on UHF subalgebra which is the inductive limit of matrix algebras, 2) restricted on the image of an endomorphism. Next we construct a representations of the Cuntz algebra on a measure space by branching function system on it. As applications, we construct a complete orthonormal basis of selfsimilar set. At last, we show application of representation theory of the Cuntz algebra to string theory. 
Date:  October 24 (Fri), 2003, 15:0016:00 
Room:  RIMS 402 
Speaker:  Hisayosi Matumoto ({v`) (Univ. Tokyo) 
Title:  The homomorphisms between scalar generalized Verma modules associated to maximal parabolic subalgebras 
Abstract: [pdf] 
Let g be a reductive Lie algbebra over the complex
number field. A gmodule induced from a onedimenssional
module of a parabolic subalgebra is called a scalar generalized Verma
module (SGVM). It is known that the homomorphisms between SGVMs
correspond to the equivariant differntial operators between equivariant
line bundles on generalized flag varieties. Our main results are:
(1) We give a classification of the homomorphisms between SGVMs associated to maximal parabolic subalgebras. (2) Using a comparison theorem, we construct a homomorphism between SGVMs associated to (not necessarily maximal) parabolic subalgebra from a homomorphism between SGVMs associated to a maximal parabolic subalgebra. Prior to the seminar, Matumoto will give an introductory seminar 13:0014:30 at 402. 
Date:  October 31 (Fri), 2003, 15:0016:00 
Room:  RIMS 402 
Speaker:  Pavle Pandzic (Zagreb & RIMS) 
Title:  Dirac operators and applications to representation theory 
Abstract:  In the 1970s Parthasarathy has started the construction of discrete
series representations of semisimple Lie groups using an analogue of the
Dirac operator. The final form of the construction was given by Atiyah
and Schmid. In the 1990s Vogan has studied an algebraic version of Parthasarathy's Dirac operator. He conjectured that if the Dirac operator has nonzero kernel for a unitary (g,K)module X, then this kernel (explicitly) determines the infinitesimal character of X. This conjecture was recently proved by J.S. Huang and myself. In the meantime, Kostant has defined a more general, ''cubic'' Dirac operator, attached to other subalgebras than k. In the case of a Levi subalgebra, this Dirac operator is closely related to the corresponding ucohomology. In a recent preprint with J.S. Huang and D. Renard, we study this relationship in detail in certain special situations. 
Date:  November 12 (Wed), 2003, 18:0019:00 
Room:  RIMS 402 
Speaker:  Christof Geiss (Ciudad Universitaria, Mexico) 
Title:  Semicanonical bases and preprojective algebras 
Abstract: [pdf] 
This is a report on joint work with J. Schröer and B.
Leclerc.
Let g be a simple Lie algebra of type A, D, E and n a maximal nilpotent subalgebra of g. Moreover, let N be a maximal unipotent subgroup of a simple Lie group with Lie algebra g. Finally, let Π denote the corresponding preprojective algebra. Lusztig's semicanonical basis S of U(n) is parametrized by irreducible components of the corresponding nilpotent varieties mod(Π,d). The dual S^{*} is a basis of C[N]. We can show: For two elements of S^{*} holds b_{C} • b_{D} ∈ S^{*} if for the corresponding irreducible components holds ext_{Π}^{1}(C,D)=0. On the other hand, the dual canonical basis B_{q}^{*} of U_{q}(n) specializes for q=1 to a basis B. We can show, that S and B have many elements in common, but B and S coincide only if Π is representation finite, i.e. in the cases A_{2,3,4}. This explains the multiplicative properties of the dual canonical basis observed previously in these cases. On the other hand it gives us a good control over the dual semicanonical basis in the cases A_{5} and D_{4}, i.e. when Π is tame, since we have in this case a precise combinatorial description of the irreducible components of mod(Π,d) in terms of indecomposable components. This is closely related to an elliptic root system of type E_{8}^{(1,1)} resp. E_{6}^{(1,1)}. 
Date:  November 19 (Wed), 2003, 18:0019:00 
Room:  RIMS 402 
Speaker:  Katsunori Kawamura (쑺I) (RIMS) 
Title:  Algebras of sectors and their spectrum modules 
Abstract: [pdf] 
The quotient space Sect(A) of unital *endomorphisms of a unital *algebra A by the inner automorphism group of A is called the sector of A. Sect(A) is a non abelian semigroup with unit and it is an algebra with Nadditive operation when there is an embedding of the Cuntz algebra O_{N} into A. The set B Spec(A) of unitary equivalence classes of unital *representations of A is an abelian semigroup and it is a right Sect(A)module. B Spec(A) is called the spectrum module of Sect(A). By these general tools, we explain branching laws of representations of A by endomorphisms as (admissible) submodules of B Spec(A), and fusion rules as algebraic operations in Sect(A). 
Date:  November 28 (Fri), 2003, 15:0016:00 
Room:  RIMS 402 
Speaker:  Jorge Vargas (FAMAF, Argentine) 
Title:  Restriction of Discrete Series representations, continuity of the Berezin transform 
Abstract: [pdf] 
Let G be a connected semisimple matrix Lie group. We fix a connected reductive subgroup H of G and a maximal compact subgroup K of G such that H ∩ K is a maximal compact subgroup of H. We assume that the group G has a nonempty Discrete Series. Let (π, V) be a square integrable irreducible representation of G, and let (τ, W) be its lowest Ktype. Let E:=G ×_{K}W → G/K be the Ghomogeneous, Hermitian, smooth vector bundle attached to the representation τ. After the work of Hotta and other authors, we may and will realize V as an eigenspace of the Casimir operator acting on L^{2}(E). Since the Casimir operator is an elliptic operator, the elements of V are smooth sections. Let F :=H ×_{H ∩ K}W → H/H ∩ K. Because of our setting, F is a subbundle of E and we may restrict the elements of V to smooth sections of F. We denote the resulting linear transformation from V into the space of smooth sections of F by r. In this talk we will show that the image of r consist of L^{2}sections of F, the (2,2)continuity of r, as well as the (2,2)continuity of Berezin transform, rr^{★}. We also analyze (p,p)continuity of the Berezin transform. Later on, we will suppose (G,H) is a generalized symmetric pair. We write, g = k ⊕ s = h ⊕ q. For each nonnegative integer m, let S^{m}(q ∩ s) denote the m^{th}symmetric power of q ∩ s. Thus, S^{m}(q ∩ s) ⊗ W is an H ∩ Kmodule. A basic idea in branching theory is to consider normal derivatives corresponding to the immersion H/H ∩ K → G/K. Using this we may show that if (ρ,Z) is an Hirreducible discrete factor of (π,V). Then, there exists m ≥ 0 and an injective, continuous, linear Hmap from V into L^{2}(H ×_{H ∩ K}(S^{m}(s ∩ q) ⊗ W)). That is, the discrete spectrum of the restriction of π to H, up to multiplicities, is contained in the discrete spectrum of L^{2}(H ×_{H ∩ K}(S(s ∩ q) ⊗ W)). The above facts together with some consequences are joint work with Bent Ørsted. 
Date:  December 19 (Fri), 2003, 15:3016:30 
Room:  RIMS 402 
Speaker:  Pavle Pandzic (Zagreb & RIMS) 
Title:  Some exceptional dual pair correspondences 
Abstract: [pdf] 
This talk describes results from a joint paper with Huang and Savin
published in Duke Math. J. in 1996.
Let G be the adjoint group of the Lie algebra of type F_{4}, E_{6}, E_{7} or E_{8} with real rank four. (For F_{4}, G is replaced with its double cover). There is a dual pair G_{2} × H in G, with G_{2} the split real group of type G_{2} and H compact. We restrict the minimal representation of G constructed by Gross and Wallach to this dual pair and obtain the explicit Howe correspondences. For an irreducible (finitedimensional) representation E of H, we first calculate the Ktypes of Θ(E) using some ''seesaw'' techniques and branching laws. Then we identify Θ(E) in the unitary dual of G_{2} as given by Vogan. Finally, we show that our results can serve as examples of Langlands correspondences. Prior to the seminar, Pandzic will give an introductory seminar from 11:0012:00 at 402. 
Date:  December 19 (Fri), 2003, 17:0018:00 
Room:  RIMS 402 
Speaker:  Kyo Nishiyama (R ) (Kyoto Univ.) 
Title:  Equivariant smooth completion of spherical nilpotent orbits 
Abstract:  I discuss an equivariant embedding of a spherical nilpotent orbit in gl(n) into a certain vector bundle over Grassmannian manifold, with an open dense image. As a consequence, we have an equivariant smooth completion of such kind of nilpotent orbits. 
Date:  December 26 (Fri), 2003, 17:3018:30 
Room:  RIMS 402 
Speaker:  Genkai Zhang (Chalmers Univ. of Tech. and Gothenburg Univ) 
Title:  Spherical transform of canoncal functions on root systems of type BC 
Abstract: [pdf] 
Consider a root system of the BC with general real positive multiplicity. We introduce the canonical functions which corresponds to the integral kernel of the canonical representations in the symmetric space case. We compute their spherical transform using the Cheredik operators and prove some BernsteinSato type formula. Some application to Macdonald polynomials will be mentioned if time permits. 
Date:  December 26 (Fri), 2003, 16:0017:00 
Room:  RIMS 402 
Speaker:  Genkai Zhang (Chalmers Univ. of Tech. and Gothenburg Univ) 
Title:  Invariant plurisubharmonc functions on extended Cartan and Siegel domains 
Abstract: [pdf] 
Let D=G/L be a bounded symmetric domain and S the corresponding Siegel domain in a vector space V. Let K be the semisimple part of L. It complexification K_{c} acts on the product V^{N} linearly and diagonally and we called the resulting domains K_{c}D^{N} and K_{c}S^{N} the extended Cartan and Siegel domains. We prove in certain cases that they are domain of holomorphy and generalize earlier results of Zhou and of SeegeevHeinzner. 
Date:  January 9 (Fri), 2004, 16:0017:00 
Room:  RIMS 402 
Speaker:  Andreas Nilsson (Royal Institute of Technology, Stockholm, Sweden) 
Title:  Some Theorems of deLeeuw on multipliers 
Abstract: [pdf] 
To check L^{p}boundedness of multipliers it is sometimes useful to be able to restrict to simpler situations. deLeeuw has proved that if we restrict an L^{p}bounded multiplier operator to a linear subspace then the resulting multiplier operator will also be bounded. I will talk about the proof of this and some related results. In my second talk I will give some applications to this. 
Date:  January 13 (Tue), 2004, 16:0017:00 
Room:  RIMS 402 
Speaker:  Andreas Nilsson (Royal Institute of Technology, Stockholm, Sweden) 
Title:  Investigation of L^pboundedness for certain multipliers 
Abstract: [pdf] 
Together with professor Kobayashi, I have been trying to characterize multipliers by group actions. To begin with we were only concerned with the characterization and hence only asked for L^{2}boundedness. But it is natural to ask which of the multipliers that are bounded on L^{p} as well. The tools used include deLeeuw's theorem on restriction to affine subspaces and Fefferman's ball multiplier theorem. 
Date:  February 10 (Tue), 2004, 17:0018:00 
Room:  RIMS 402 
Speaker:  Soo Teck Lee (NUS, Singapore) 
Title:  A basis for the kfold tensor product algebra of GL_{n}(C) 
Abstract: [pdf] 

Date:  February 17 (Tue), 2004, 17:0018:00 
Room:  RIMS 402 
Speaker:  Raphael Rouquier (Paris VII) 
Title:  Categorification of Weyl groups and Lie algebras 
Abstract: 
It is classical that various actions of Weyl groups or Lie algebras on vector spaces come from functors acting on abelian or triangulated categories of algebraic or geometric origin, whose Grothendieck group is that space. We want to explain that the natural transformations between these functors should satisfy certain algebraic relations, leading to a better control of the triangulated categories acted on. We give two precise such frameworks. For Weyl groups, we use Soergel bimodules. In a joint work with Chuang, we explain the setting for sl2, which leads to a construction of equivalences of derived categories between blocks of Hecke algebras of type A. 
Date:  February 20 (Fri), 2004, 16:0017:00 
Room:  RIMS 402 
Speaker:  Eric Sommers (University of Massachusetts, Amherst) 
Title:  Functions on nilpotent orbits and their covers 
Abstract: 
Extending work of McGovern and Graham, we conjecture a formula for the functions on covers of nilpotent orbits and prove it in many cases. One application of this result is an explicit computation for the classical groups of a part of the LusztigVogan bijection (this is a bijection between the set of irreducible equivariant coherent sheaves on nilpotent orbits in a simple Lie algebra and the set consisting of its irreducible finitedimensional representations). 
Date:  March 9 (Tue), 2004, 16:0017:00 
Room:  RIMS 402 
Speaker:  Munibur R. Chowdhury (the University of Dhaka, Bangladesh) 
Title:  Arthur Cayley and his contribution to abstract group theory 
Abstract: [pdf] 
We critically reexamine in considerable detail Cayley's first three papers on group theory (185459), with special reference to his formulation of the (abstract) group concept. We show convincingly (we hope) that Cayley, writing his first paper on November 2, 1853, was in full and conscious possession of the abstract group concept, and that — as far as finite groups are concerned — his definition was complete and unequivocal, refuting opinion expressed by some earlier writers. Already in the first paper Cayley classified the abstract groups of orders upto 6, and suggested that there might exist composite numbers n such that the only abstract group of order n is the cyclic group of that order. We also discuss Cayley's motivation for generalizing the then current concept of a permutation group. Cayley extended the classification to groups of order 8 in the third paper. There he also initiated the study of groups in terms of generators and relations (a procedure usually attributed to Walter Dyck), and in this way constructed the abstract dihedral group of order 2n. However, these pioneering studies were swept away by the then burgeoning surge of permutation groups, and apparently went completed unheeded by his contemporaries. 
Organizer:  Toshiyuki Kobayashi 
© Toshiyuki Kobayashi