Lie Group and Representation Theory Seminar, Kyoto 2003
Lie Group and Representation Theory Seminar, Kyoto 2003 | Date: | December 26 (Fri), 2003, 16:00--17:00 |
| 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_cD^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 Seegeev - Heinzner. |
| Date: | December 26 (Fri), 2003, 17:30--18:30 |
| 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: |
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 Bernstein-Sato type formula. Some application to Macdonald polynomials will be mentioned if time permits. |
| Date: | December 19 (Fri), 2003, 15:30--16:30 |
| 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 x 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 (finite-dimensional) representation E of H, we first calculate the K-types of \Theta(E) using some ``see-saw" techniques and branching laws. Then we identify \Theta(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 11:00-12:00 at 402. |
| Date: | December 19 (Fri), 2003, 17:00--18:00 |
| Room: | RIMS 402 |
| Speaker: | 西山 享 （Nishiyama Kyo) 氏 (Kyoto Univ.) |
| Title: | Equivariant smooth completion of spherical nilpotent orbits |
| Abstract: [pdf] |
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: | November 28 (Fri) , 2003, 15:00--16:00 |
| 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 $H \cap K$ is a maximal compact subgroup of $H.$ We assume that the group $G $ has a nonempty Discrete Series. Let $(\pi, V)$ be a square integrable irreducible representation of $G$, and let $(\tau, W)$ be its lowest $K-$type. Let $E:=G \times_K W \longrightarrow G/K $ be the $G-$homogeneous, Hermitian, smooth vector bundle attached to the representation $\tau .$ 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 $F :=H \times_{H\cap K}W \longrightarrow H/H\cap K .$ 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, $rr^\star .$ We also analyze $(p,p)-$continuity of the Berezin transform. Later on, we will suppose $ (G,H) $ is a generalized symmetric pair. We write, $ \mathfrak{g}= \mathfrak{k} \oplus\mathfrak{s} = \mathfrak{h} \oplus \mathfrak{q}. $ For each nonnegative integer $m,$ let $S^m( \mathfrak{q} \cap \mathfrak{s})$ denote the $m^{th}-$symmetric power of $\mathfrak{q} \cap \mathfrak{s}. $ Thus, $S^m(\mathfrak{q} \cap \mathfrak{s}) \otimes W$ is an $H \cap K-$module. A basic idea in branching theory is to consider normal derivatives corresponding to the immersion $H/H\cap K \rightarrow G/K. $ Using this we may show that if $(\rho, Z) $ is an $H-$irreducible discrete factor of $(\pi, V).$ Then, there exists $ m \geq 0$ and an injective, continuous, linear $H-$map from $V$ into $ L^2(H \times_{H\cap K} (S^m(\mathfrak s \cap \mathfrak q) \otimes W)). $ That is, the discrete spectrum of the restriction of $\pi $ to $H,$ up to multiplicities, is contained in the discrete spectrum of $L^2(H \times_{H \cap K} (S(\mathfrak s \cap \mathfrak q) \otimes W)) .$ The above facts together with some consequences are joint work with Bent \O{}rsted. |
| Date: | November 19 (Wed) , 2003, 18:00-19:00 |
| Room: | RIMS 402 |
| Speaker: | 川村 勝紀 (Katsunori Kawamura) (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 N-additive operation when there is an embedding of the Cuntz algebra O_N into A. The set BSpec(A) of unitary equivalence classes of unital *-representations of A is an abelian semigroup and it is a right Sect(A)-module. BSpec(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 BSpec(A), and fusion rules as algebraic operations in Sect(A). |
| Date: | November 12 (Wed) , 2003, 18:00-19:00 |
| 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. Schr\"oer and B. Leclerc. Let $\mathfrak{g}$ be a simple Lie algebra of type $\mathsf{A,D,E}$ and $\mathfrak{n}$ a maximal nilpotent subalgebra of $\mathfrak{g}$. Moreover, let $N$ be a maximal unipotent subgroup of a simple Lie group with Lie algebra $\mathfrak{g}$. Finally, let $\Pi$ denote the corresponding preprojective algebra. Lusztig's semicanonical basis $\mathcal{S}$ of $U(\mathfrak{n})$ is parametrized by irreducible components of the corresponding nilpotent varieties $\operatorname{mod}(\Pi,\mathbf{d})$. The dual $\mathcal{S}^*$ is a basis of $\mathbb{C}[N]$. We can show: For two elements of $\mathcal{S}^*$ holds $b_{C}\cdot b_{D}\in \mathcal{S}^*$ if for the corresponding irreducible components holds $\operatorname{ext}_{\Pi}^1(C,D)=0$. On the other hand, the dual canonical basis $\mathcal{B}_q^*$ of $U_q(\mathfrak{n})$ specializes for $q=1$ to a basis $\mathcal{B}$. We can show, that $\mathcal{S}$ and $\mathcal{B}$ have many elements in common, but $\mathcal{B} $ and $\mathcal{S}$ coincide only if $\Pi$ is representation finite, i.e. in the cases $\mathsf{A}_{2,3,4}$. 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 $\mathsf{A}_5$ and $\mathsf{D}_4$, i.e. when $\Pi$ is tame, since we have in this case a precise combinatorial description of the irreducible components of $\operatorname{mod}(\Pi,\mathbf{d})$ in terms of indecomposable components. This is closely related to an elliptic root system of type $\mathsf{E}_8^{(1,1)}$ resp. $\mathsf{E}_6^{(1,1)}$. |
| Date: | October 31 (Fri), 2003, 15:00--16:00 |
| 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 u-cohomology. In a recent preprint with J.-S. Huang and D. Renard, we study this relationship in detail in certain special situations. |
| Date: | October 24 (Fri), 2003, 15:00--16:00 |
| Room: | RIMS 402 |
| Speaker: | 松本久義 Hisayosi Matumoto (Univ. Tokyo) |
| Title: | The homomorphisms between scalar generalized Verma modules associated to maximal parabolic subalgebras |
| Abstract: [pdf] |
Let ${\mathfrak g}$ be a reductive Lie algbebra over the complex number field.
A ${\mathfrak g}$-module induced from a one-dimenssional 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.
|
| Date: | October 14 (Tue), 2003, 13:00--14:00 |
| Room: | RIMS 402 |
| Speaker: | 川村勝紀 (Katsunori Kawamura) (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 self-similar set. At last, we show application of representation theory of the Cuntz algebra to string theory. |
| Date: | September 29 (Mon), 2003, 10:30--11:30 |
| 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 infinite-dimensional (Friedrichs--Shale) symplectic group
in the space $\Lambda$ of symmetric functions in infinite number of variables
For each operator $\Lambda\to\Lambda$ we associate a formal series $K(x_1, x_2, \dots; y_1, y_2, \dots)$ (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
$$
K(x,y)= We also show, that the set of all operators having such kernels is closed with respect to multiplication and describe this semigroup |
| Date: | September 19 (Fri), 2003, 10:30--11:30 |
| Room: | RIMS Room 402 |
| Speaker: | Yurii Neretin (ITEP) |
| Title: | Inverse limits of unitary groups, matrix beta-integrals, and harmonic analysis on infinite-dimensional symmetric spaces |
| Abstract: |
There exists a natural distinguished map from a
larger unitary group $U(n)$ to the smaller group $U(n-1)$.
(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 2-parametric family of canonical $U(\infty)\times U(\infty)$-quasiinvariant measures. It will be given a self-consistent 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 16 (Tue), 2003, 16:30--17:30 |
| Room: | RIMS Room 402 |
| Speaker: | Gerrit van Dijk (Leiden University) |
| Title: | Generalized Gelfand Pairs: A Survey |
| Abstract: |
The group G=SL(2,R) of 2x2 matrices with determinant 1, acts on the complex
upper half plane by fractional linear transformations. The measure dxdy/y^2 is
a G-invariant measure on the upper half plane, and the group action extends
easily to the L^2-space of the invariant measure.
It is well-known 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 well-known examples are given by pairs
(G,K) where G is a semi-simple 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, non-necessarily compact subgroup of G. Several
examples of (generalized) Gelfand pairs will be given.
van Dijk氏は９月１３日から２０日まで京都に滞在されます。 当日夕方にvan Dijk氏の歓迎会をかねて、食事に行きたいと思っています。 詳しくは当日相談します。 |