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126†Žº**

ŽžŠÔF **‰Î—j“ú 16:30 - 18:00**

¢˜blF ‘å“‡—˜—Yi‚‚“‚ˆ‚‰‚‚—akagi.ms.u|tokyo.ac.jpjC¬—Ñrsi‚”o‚“‚ˆ‚‰—ms.u|tokyo.ac.jpj

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We consider certain invariant integral operators on a smooth affine variety M carrying the action of a reductive algebraic group G, and assume that G acts on M with an open orbit. Then M is isomorphic to a homogeneous vector bundle, and can locally be described via the theory of prehomogenous vector spaces. We then study the Schwartz kernels of the considered operators, and give a description of their singularities using the calculus of b-pseudodifferential operators developed by Melrose. In particular, the restrictions of the kernels to the diagonal can be described in terms of local zeta functions.

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The particular geometric structure of causal symmetric spaces permits the definition of a covariant quantization of these homogeneous manifolds. Composition formulae (#-products) of quantizad operators give rise to a new interpretation of Rankin-Cohen brackets and allow to connect them with the branching laws of tensor products of holomorphic discrete series representations.

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The goal of this series of lectures will be to describe and compare two intimately related but nevertheless fundamentally different methods of quantization of symmetric spaces : on the one hand deformation quantization and symbolic calculus on the other hand. We shall also discuss interesting connections with the representation theory of semi-simple Lie groups.

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Let ${\mathfrak g}$ be a complex semisimple Lie algebra. We call a parabolic subalgebra ${\mathfrak p}$ of ${\mathfrak g}$ normal, if any parabolic subalgebra which has a common Levi part with ${\mathfrak p}$ is conjugate to ${\mathfrak p}$ under an inner automorphism of ${\mathfrak g}$. For a normal parabolic subalgebra, we have a good notion of the restricted root system or the little Weyl group. We have a comparison result on the Bruhat order on the Weyl group for ${\mathfrak g}$ and the little Weyl group. We also apply this result to the existence problem of the homomorphisms between scalar generalized Verma modules.

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A unitary representation of a reductive Lie group can decompose when restricted to a subgroup which is a symmetric pair with finite or infinite multiplicity. On the other hand, T. Kobayashi proved that an irreducible unitary highest weight representation of scalar type decomposes with multiplicity-free when restricted to any subgroup which is a semisimple symmetric pair, and R. Howe proved that the Weil representation decomposes with multiplicity-free when restricted to any subgroup which is a dual pair.

In this talk, with respect to the minimal representation of the indefinite orthogonal group which was constructed by Kazhdan, Kostant, Binegar-Zierau and Kobayashi-Ørsted, we will show that the multiplicity-free theorem holds when restricted to more subgroups than symmetric subgroups.

‰—F‹x‰É‘ºŽ]Šò ŒÜF‘ä

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Letabe a hyperbolic element in a semisimple Lie algebra over the real number field. LetKbe the complexification of a maximal compact subgroup of the corresponding real adjoint group. We study the asymptotic cone of the semisimple orbit through a under the adjoint action byK. The resulting asymptotic cone is the associated variety of a degenerate principal series representation induced from the parabolic associated toa.

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This is a joint work with Jean Ludwig (University of Metz). Let G be an exponential solvable Lie group, and ƒÎ be an irreducible unitary representation of G. By induction from a character on a connected subgroup H, ƒÎ is realized on a Hilbert space of L^{2}-functions on a homogeneous space G/H. We investigate a subspaceSEof C^{‡}-vectors of ƒÎ consisting of functions with a certain property of rapidly decreasing at infinity. We give a description ofSEas the space of C^{‡}-vectors of an extension of ƒÎ to an exponential solvable group containing G.

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The principal series representations of a semisimple Lie group play an important role in the representation theory. We study the principal series representation of a complex semisimple Lie group and determine when there exists a nonzero homomorphism between the representations.

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Let M_{n,k}denote the vector space of real n~k matrices. The matrix motion group is the semidirect product (O(n)~O(k)) \ltimes M_{n,k}, and is the Cartan motion group associated with the real Grassmannian G_{n,n+k}. The matrix Radon transform is an integral transform associated with a double fibration involving homogeneous spaces of this group. We provide a set of algebraically independent generators of the subalgebra of its universal enveloping algebra invariant under the Adjoint representation. One of the elements of this set characterizes the range of the matrix Radon transform.

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We determine the irreducible complex symmetric spaces on which SL(2,R) acts properly. We use the T. Kobayashi's criterion for the proper actions. Also we use the symmetry or unsymmetry of the weighted Dynkin diagram of the theory of nilpotent orbits.

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The classification of Fuchsian equations without accessory parameters was formulated as Deligne-Simpson problem, which was solved by Katz and they are studied by Haraoka and Yokoyama. If the number of singular points of such equations is three, they have no geometric moduli. We give a unified connection formula for such differential equations as a conjecture and show that it is true for the equations whose local monodromy at a singular point has distinct eigenvalues. Other Fuchsian differential equations with accessory parameters and hypergeometric functions with multi-variables are also discussed.

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