# Institut de Recherche Mathématique Avancée, UMR 7501

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Accueil > Agenda > Colloques et rencontres > Archives > Agenda 2006 > Théorie des représentations et espaces préhomogènes

## Théorie des représentations et espaces préhomogènes

IRMA, Strasbourg, 11-14 Septembre 2006

Organisateurs : Yumiko HIRONAKA (Tokyo), Iris MULLER (Strasbourg), Hiroyuki OCHIAI (Nagoya),
Hubert RUBENTHALER (Strasbourg), Fumihiro SATO (Tokyo)

La conférence traitera des développements récents des théories des représentations et des espaces préhomogènes, et de leurs interactions.

Conférenciers :

• Leticia BARCHINI (Oklahoma State),
• Jean-Louis CLERC (Nancy)
• Raf CLUCKERS (ENS, Paris)
• Jacques FARAUT (Paris 6),
• Hideyuki ISHI (Yokohama City),
• Toshiyuki KOBAYASHI (RIMS, Kyoto),
• Khalid KOUFANY (Nancy)
• Gen MANO (Tokyo),
• Kyo NISHIYAMA (Kyoto University),
• Toshio OSHIMA (Tokyo University),
• Michael PEVZNER (Reims)
• Fumihiro SATO (Rikkyo University, Tokyo),
• Marcus SLUPINSKI (Strasbourg),
• Robert STANTON (Ohio State),
• Akihito WACHI (Hokkaido Institute of Technology, Sapporo).
• Genkai ZHANG (Göteburg),

### RESUMES / ABSTRACTS

Leticia BARCHINI (Oklahoma State, USA) : Conformally Invariant systems of differential equations and prehomogeneous vector spaces

Abstract : Let \mathfrak{g} be a simple complex Lie algebra and let \mathfrak{q} \subset \mathfrak{g} be a Heisenberg parabolic subalgebra. Write \mathfrak{q} = \mathfrak{l} \oplus \mathfrak{n}. Then \mathfrak{n} = V^+ \oplus z(\mathfrak{n}) where z(\mathfrak{n}) is one dimensional. If G = Aut(\mathfrak{g})^o and L is the connected subgroup of G corresponding to \mathfrak{l}, then V^+ is stable under the adjoint action of L. By Vinberg’s Theorem (L, Ad, V^+) is a prehomogeneous vector space. We call these prehomogeneous vector spaces of parabolic Heisenberg type.

Let \mathcal D(\mathfrak{n}) denote the Weyl algebra of \mathfrak{n}, this is, the algebra of partial differential operators in \mathfrak{n} with polynomial coefficients. We consider a family of homomorphims of \mathfrak{g} into \mathcal D(\mathfrak{n}) parametrized by a complex number s. Naively, the family of homomorphisms that we require is the one derived from the family of unnormalized induces representations ind _{\overline {Q}}^G( \chi^{-s}).

To covariants of the prehomogenous vector space V^+ we associate systems of differential equation and determine the value of s for which the system is conformally invariant. We check that the values of s we find are roots of the b-function of the relative invariant of the prehomogeneous vector space.

This talk is based on joint work with A. Kable and R. Zierau

Jean-Louis CLERC (Nancy, FRANCE) : Geometry of the Shilov boundary

Abstract : Let \cal D be a bounded symmetric domain and let G be the neutral component of its group of holomorphic diffeomorphisms. Let S be the Shilov boundary of \cal D. Then G acts transitively on S. An invariant for the action of G on S\times S\times S, named the (generalized) Maslov triple index is constructed. The action of G on S\times S\times S is further studied : it has a finite number of orbits if (and only if) the domain \cal D is of tube type. We give a complete classification of the orbits in this case, which uses the Maslov index.

Raf CLUCKERS (ENS, Paris) : Igusa’s conjecture on p-adic exponential sums : some recent attacks

Abstract : By looking at prehomogeneous vector spaces, J. Igusa discovered many phenomena of p-adic integrals, exponential sums, monodromy, etc., when the integrands, arguments of the character, and so on, are relative invariants. He often conjectured these phenomena to be true for all homogeneous polynomials, instead of only for relative invariants. Many of these conjectures are proven now, by Igusa, Denef, and many others, but some of these conjectures only have a substantial amount of evidence and remain open. So is Igusa’s conjecture on exponential sums, stated in the introduction of his ’78 book, of which only special cases are proven. Igusa himself treated the projectively smooth case. Denef and Sperber studied the nondegenerate case under extra conditions. Recently, we obtained more partial results towards this conjecture. We proved the modulo p and modulo p^2 cases for homogeneous polynomials. We generalized the Denef - Sperber results to the nondegenerate case by removing their extra conditions.

Many of Igusa’s conjectures turned out to be true for polynomials in general, and not only for homogeneous ones. For the conjecture on exponential sums, it is not even clear what a more general conjecture would be.

Jacques FARAUT (Paris 6, FRANCE) : Berezin kernels and analysis on Makarevich spaces

Abstract : Makarevich symmetric spaces are realized as domains in the conformal compactification of a simple real Jordan algebra.

By using harmonic analysis associated to a Riemannian Makarevich symmetric space X of non compact type, it is shown that the Berezin kernel satisfies a Bernstein identity.

This identity permits the computation of the Fourier expansion of the Berezin kernel on the compact dual Y of X.

(joint work with Michael Pevzner)

Hideyuki ISHI (Yokohama City, JAPAN) : An elementary approach to relative invariants

Abstract : It is known that the fundamental relative invariants of a prehomogeneous vector space (G, V) are easily computed if \dim G = \dim V. Indeed, they are irreducible factors of a determinant of a certain operator-valued function on V in this case. We consider variations of the method to obtain relative invariants associated to a homogeneous cone.

Toshiyuki KOBAYASHI (RIMS, Kyoto, JAPAN) : Visible actions on complex manifolds

Abstract :

Motivated by the notion of visible actions on complex manifolds’’, we raise a question whether or not the multiplication of three subgroups L, G' and H surjects a Lie group G in the setting that G/H carries a complex structure and contains G'/G' \cap H as a totally real submanifold.

Particularly important cases are when G/L and G/H are generalized flag varieties, and we classify pairs of Levi subgroups (L, H) such that LG'H = G, or equivalently, the real generalized flag variety G'/H \cap G' meets every L-orbit on the complex generalized flag variety G/H in the setting that (G, G') = (U(n), O(n)). For such pairs (L, H), we introduce a herringbone stitch method to find a generalized Cartan decomposition for the double coset space L \backslash G/H, for which there has been no general theory in the non-symmetric case.

Our geometric results provide a unified proof of various multiplicity-free theorems in representation theory of general linear groups.

Khalid KOUFANY ( Nancy, FRANCE) : Hua operators for bounded symmetric domains

Abstract : Following Hua, we consider the Poisson transform \mathcal{P}_s for hyperfunctions on the Shilov boundary of a bounded symmetric domain. For all s satisfying a non-integral condition, we characterize the image of \mathcal{P}_s by a covariant differential operator.

Gen MANO (RIMS, Kyoto, JAPAN) : The unitary inversion operator for the minimal representation of O(p,q)

Abstract : The L^2-model of the minimal representation of the indefinite orthogonal group O(p,q) (p+q : even, greater than 4) was established by Kobayashi-Orsted (2003). In this talk, we present an explicit formula for the unitary inversion operator, which plays a key role in this L^2-model. Our proof uses the analysis on the Radon transform of distributions supported on the light cone.

Kyo NISHIYAMA (Kyoto University, JAPAN) : Resolution of the null fiber and conormal bundles of closed orbits on a flag variety

Abstract : We give an explicit realization of the resolution of singularities of the null fiber of standard contraction map of U \otimes V + U \times V^{\ast} by the action of GL(V) . Then, the categorical quotient by O(U) \times O(U) of the resolution turns out to be a conormal bundle of a certain closed GL(V) -orbit in the Lagrangean flag variety. The moment map image of the conormal bundle is the closure of a nilpotent orbit, which is the theta lift of the trivial nilpotent orbit for a certain indefinite orthogonal group. We will explain the construction in detail, and relationships with representation theory and prehomogeneous spaces.

Toshio OSHIMA (Tokyo University, JAPAN) : Generalized flag manifolds, hypergeometric functions and prehomogeneous vector spaces

Abstract : A generalized flag manifold is a homogeneous space of a real reductive Lie group whose isotropy group with respect to a point is a parabolic subgroup. We will describe a good generator system of the annihilator in the universal enveloping algebra acting on the sections of a line bundle over a generalized flag manifold. Then our hypergeometric functions are defined by prehomogeneous actions of subgroups on the flag manifolds. Our functions are defined by the solutions of a system of differential equations and have integral representations. Zonal spherical functions, Whittaker vectors and Gelfand’s hypergeometric functions etc. are in our functions.

Michael PEVZNER (Reims, FRANCE) : Projective pseudo-differential analysis and harmonic analysis

Abstract : We consider pseudo-differential operators on \mathbb{R}^{n+1} which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. Their symbols can be regarded as functions on a reduced phase space, isomorphic to the symmetric space G/H=SL(n+1,\mathbb{R})/GL(n,\mathbb{R}), and the resulting calculus is a pseudo-differential analysis of operators acting on spaces of appropriate sections of line bundles over the projective space P_n(\mathbb{R}) : these spaces are the representation spaces of the maximal degenerate series of G. This new approach to the quantization of the rank one para-Hermitian space G/H, has several advantages : as an example, it makes it possible to give a very explicit version of the continuous part from the decomposition of L^2(G/H).

Fumihiro SATO (Rikkyo University, Tokyo, JAPAN) : Quadratic mappings and Zeta functions

Abstract : By the theory of prehomogeneous vector space, the complex power of relative invariants of a regular prehomogeneous vector space satisfies a local functional equation. In their famous book "Analysis on symmetric cones", Faraut and Koranyi proved a local functional equation for certain polynomials obtained from representations of Euclidean Jordan algebrs. Their result contains a non-prehomogeneous example of local functional equations.

In this talk, we give a (slight) generalization of the result of Faraut and Koranyi to the following setting : Let Q:W \rightarrow V be a (nice) quadratic mapping and P a polynomial function on V which satisfies a local functional equation. Then, the pull back P \circ Q also satisfies a local functional equation. Based on this result we can add a few non-prehomogeneous examples of local functional equations.

Robert STANTON (Ohio State, USA) : The fine structure of symplectic PV’s, I

Abstract : \Bbb Z_5 - graded Lie algebras provide a natural class of prehomogeneous vector spaces endowed with an invariant symplectic form. That there is an interesting interplay between the symplectic PV and the graded Lie algebra is the main focus of this lecture. We present several results, algebraic in nature, to justify this viewpoint. We also illustrate this interplay in detail for the real semisimple Lie group split E_6 as we give a description of all its symmetric subgroups using a natural functorial construction originating from the symplectic perspective.

Marcus SLUPINSKI (Strasbourg, FRANCE) : The fine structure of symplectic PV’s, II

Abstract : For these prehomogeneous vector spaces the symplectic structure gives a tool - a moment map - with which to investigate the PV. Exploiting the moment map, as an alternative to the quasi-invariant, to expose the geometric structure of these PV’s will be the main topic of this lecture. Detailed information of the orbit structure of the PV as well as the symplectic geometric properties of the orbits will be presented. The relationship of the various orbit types in the PV to conjugacy classes in the transformation group will be explained as well as conjectural consequences.

Akihito WACHI (Hokkaido Institute of Technology, Sapporo, JAPAN) : Capelli identities for symmetric pairs

Abstract : We consider a see-saw pair consisting of a Hermitian symmetric pair (G_{\mathbb{R}}, K_{\mathbb{R}}) and a compact symmetric pair (M_{\mathbb{R}}, H_{\mathbb{R}}), where (G_{\mathbb{R}}, H_{\mathbb{R}}) and (K_{\mathbb{R}}, M_{\mathbb{R}}) form real reductive dual pairs in a large symplectic group. In this talk, we give Capelli identities which explicitly represent certain K_{\mathbb{C}}-invariant elements in U(Lie(G)_{\mathbb{C}}) in terms of H_{\mathbb{C}}-invariant elements in U(Lie(M)_{\mathbb{C}}). The images of these invariants coincide under the Weil (oscillator) representation. This is a joint work with Kyo Nishiyama and Soo Teck Lee.

Genkai ZHANG (Göteburg, Sweden) : Radon, cosine and sine transform on Grassmannian manifolds

Abstract : Let G_{n,r}(\mathbb{K}) be the Grasmannian manifold of r-dimensional \mathbb{K}-subspaces in \mathbb{K}^n where \mathbb{K}=\mathbb{R},\mathbb{C}, \mathbb{H} is the field of real, complex or quaternionic numbers. We consider the cosine and sine transforms, C _{r',r} and S _{r',r}, from L^2-functions on G_{n,r}(\mathbb{K}) to functions on G_{n,r'}(\mathbb{K}) for r,r'\leq n-1.

We prove two Bernstein-Sato type formulas on general root system of type BC for the sine and cosine type functions on the compact torus \mathbb{R}^r/2\pi Q^{\vee} . We find the spectral symbol of the cosine and sine transforms and we find the characterization of their images. Our results generalize those of Alesker-Bernstein and Grinberg. We prove also that the Knapp-Stein intertwining operator for certain induced representations is given by the sine transform and we give the unitary structure of the Stein’s complementary series in the compact picture. We consider also similar hyperbolic sine- and cosine-transforms on non-compact symmetric spaces. As application we generalize the results of Helgason and find certain inversion formula for the Radon transform.

Dernière mise à jour le 17-10-2006