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[[back to conference page>ochiai/pv]] - Hideyuki Ishi (Nagoya) --Analysis on regular convex cones associated to decomposable graphs --Regular convex cones of positive definite real symmetric matrices with prescribed zero entries have been studied intensively in multivariate statistics. It turned out that analysis on the cone is quite feasible if the zero pattern corresponds to a decomposable graph. Indeed, an explicit formula is known for the Fourier-Laplace transform of a product of powers of minors over the cone. Inspired by these statistic works, we develop analysis on the cone in a similar way to theory of homogeneous cones. In particular, we consider Riesz distributions on the cone and associated b-functions. -Yumiko Hironaka (Waseda) --Spherical functions on certain $p$-adic homogeneous spaces, and some relation to PV-theory --First I want to introduce a typical spherical function on certain homogeneous space $X$, and give its expression formula by using functional equations of sph. f's and data of the group. In this talk everything is assumed to be defined over a $\mathfrak{p}$-adic field $k$. Typical sph. f's are obtained by Poisson transform from relative $P$-invariant on $X$, where $P$ is a minimal parabolic that has Zarisky open orbit in $X$ over $\overline{k}$, and their functional equations are often reduced to those for certain limited type of prehomogeneous vector spaces. Then I want to discuss about some spaces of sesquiliear forms, and give explicit formulas of sph.f's by using specialized Hall-Littlewood polynomials associated to the root system, parametrization of all the sph.f's, and Plancherel formula. (The latter half is joint work with Y. Komori.) -Koichi Takase (Miyagi) --On spherical functions of supercuspidal representations of $GL_n(F)$ and prehomogeneous vector spaces over finite fields --We will consider the spherical function of a square-integrable irreducible representation of $G$ with respect to a $K$-type with a compact subgroup $K$ of $G$, and want to determine the non-zero set of the Fourier transform of the spherical function. In the case of the holomorphic discrete series, there exists a close relationship between the non-zero set and a pre homogeneous vector space of parabolic type associated with a boundary component of $G/K$. We can consider a parallel problem for $p$-adic reductive group. In this talk, I will report on the result of an experiment for the simplest case of an irreducible level-zero super cuspidal representation of $GL_n(F)$. -Jean-Louis Clerc (Lorraine) --Conformally invariant trilinear forms on the sphere --Given three scalar principal series representations of the conformal group of the sphere, I will first recall the construction of the (essentially unique) invariant trilinear form in the generic case. I will then discuss the singular cases. -Salah Mehdi (Lorraine) --Representation theoretic differential operators --We will discuss several results on representations of Lie groups related to invariant differential operators on homogeneous spaces, with an emphasis on Dirac operators. If time allows, we will also present some connections between Dirac operators and coherent families of modules. -Pascale Harinck (CNRS) --Fourier transform of the Schwartz space of a $p$-adic reductive symmetric space --Let $X=H\backslash G$ be a $p$-adic reductive symmetric space over a non archimedean local field $\mathbb F$ of characteristic different from $2$. An explicit Plancherel formula for $L^2(X)$ (spectral decomposition) was recently described when $G$ is split and $\mathbb F$ of characteristic zero by Y.Sakellaridis and A.Venkatesh, and for general $G$ and $\mathbb F$ of characteristic different from $2$ by P. Delorme. In this talk, I will explain a joint work with Y.Sakellaridis and P.Delorme in which we describe the Fourier transform on the Harish-Chandra Schwartz space of $X$. We obtain a spectral decomposition of this space. Our proof uses the strong version of the Plancherel formula and properties of Eisenstein integrals and their weak constant term. -Robert J. Stanton (Ohio) --Extensions on real bounded symmetric domains --The real bounded symmetric domains were classified by H. Jaffee as the fixed point sets of anti-holomorphic involutions of bounded symmetric domains in $\Bbb C^n$ (there are now several alternative descriptions). B. Krötz and I showed how the harmonic analysis on the real domain has a holomorphic continuation and determined the extent of this. In this talk we will re-visit this setting from the point of view of split complex structures. We will show the existence of a split holomorphic domain containing the real domain and we will show the existence of split holomorphic extensions of the harmonic analysis to this domain. This is joint work with G.Ólafsson. -Marcus J. Slupinski (Strasbourg) --Symplectic goemetry of spinors in $12$-dimensions --The spinor representations of the double cover of the orthogonal group in twelve dimensions are regular prehomogeneous vector spaces and in 1970 J-I. Igusa gave a normal form for each orbit and determined the corresponding isotropy groups. These representations each carry an invariant symplectic form and their direct sum is a Clifford module. In this talk we give a new classification of the orbits and describe the geometry of each orbit (over a field of characteristic not two or three) in terms of symplectic covariants and properties with respect to Clifford multiplication. This is joint work with R.J.Stanton. -Fumihiro Sato (Rikkyo) --Automorphic pairs of distributions on prehomogeneous vector spaces and zeta functions --Let $(G,\rho,V)$ be a regular prehomogeneous vector space defined over $\mathbb Q$ and $(G,\rho^*,V^*)$ its dual. Denote by $\Omega$ and $\Omega^*$ the open orbits of $(G,\rho,V)$ and $(G,\rho^*,V^*)$, respectively. A pair of periodic distributions $T$ on $V_{\mathbb R}$ and $T^*$ on $V^*_{\mathbb R}$ is called automorphic, if $T$ and $T^*$ satisfy $T(f)=T^*(f_\infty)$ for any $f \in C^\infty_0(\Omega_{\mathbb R})$ where $f_\infty$ is defined by $f_\infty(\mathrm{grad} \log P(v)):=f(v)$ for a fixed nondegenerate relative invariant $P$. For an automorphic pair $(T,T^*)$ on a pv of commutative parabolic type, Dirichlet series with functional equation can be associated. As an application functional equations of zeta functions of certain (non-prehomogeneous) forms of degree 4 will be proved. The simplest cases of $G=GL(1)$ and $\dim V=1$ will be discussed in some detail. (This is a joint work with K.Tamura, K.Sugiyama, T.Miyazaki and T.Ueno.) -Kyo NISHIYAMA (AGU) --Robinson-Schensted-type correspondence over mirabolic double flag variety --We consider the conormal variety (or the Steinberg variety) over a mirabolic double flag variety for a symmetric pair $ (G, K) $. If the double flag variety has finitely many $ K $-orbits, the irreducible components of the conormal variety $ Y $ encode the parametrization of orbits. On the other hand, if we consider the image of a moment map called an exotic nilpotent cone, nilpotent $ K $-orbits together with its fiber (an analogue of Springer fiber) also classifies irreducible components of $ Y $. Thus we get a correspondence between geometric parametrization of orbits and the parametrization given by nilpotent orbits and its fiber. This is what we call "Robinson-Schensted-type correspondence". It turns out this whole picture is strongly related to the exotic (or enhanced) nilpotent cone, which are studied by many people including Travkin, Syu Kato, Achar-Henderson, Henderson-Trapa and Shoji-Sorlin among others. We will discuss the RS-type correspondence as well as the structure of the exotic nilpotent cone for the symmetric pair of type AIII. This is an on-going joint work with Lucas Fresse. -Takashi Taniguchi (Kobe) --Second order terms in some arithmetic functions --Using the zeta functions of PV's (prehomogeneous vector spaces), Shintani proved the second order terms in counting functions for the weighted class numbers of binary quadratic forms and for the class numbers of binary cubic forms. We discuss some further places that we can derive the second order terms, using the related PV zeta functions. -Kohji Matsumoto (Nagoya) --Zeta-functions of root systems and Poincaré polynomials --A useful way of evaluating special values of zeta-functions of root systems is to consider certain linear combinations of those zeta-functions, and express such combinations in terms of Bernoulli polynomials. By this method we can show explicit evaluation formulas for even integer points. However if we consider odd integer points, some "signature part" appears, and we have to determine when this part does not vanish. In this talk we express this part in terms of Poincare polynomilas of Weyl groups, discuss when it does not vanish, and show examples of explicit formulas for odd integer values. -Toshiyuki Kobayashi (Tokyo) --Symmetry breaking operators for rank one orthogonal groups --I give a classification of all symmetry breaing operators that intertwines two spherical principal series representations of two groups $O(n+1,1)$ to $O(n.1)$. This is a joint work with B. Speh. -Akihiko Yukie (Kyoto) --On orbits of prehomogeneous vector spaces --We consider orbits of prehomogeneous vector spaces in various situations. We first consider prehomogeneous vector spaces where the group is not necessarily split over a perfect field and show that the set of unstable points can be stratified by the convexity of GIT. Then we consider the question of orbits of prehomogeneous vector spaces over the p-adic integer ring and show in some cases orbits can be classified. -Tamotsu Ikeda (Kyoto) --PV and Siegel series --We review the theory of Siegel series, and show that the functional equation of Siegel series can be obtained from the local functional equation of PV. -Sofiane Souaifi (Strasbourg) --Paley-Wiener theorem(s) for real reductive Lie groups --In the early 80's, J. Arthur proved the Paley-Wiener theorem for real reductive Lie groups. To describe the Fourier transform of the space of compactly supported smooth functions, he uses the so-called Arthur-Campoli relations. More recently, P. Delorme, using other techniques, gave another proof of the Paley-Wiener theorem. His description of the Paley-Wiener space is now in terms of intertwining conditions. In a joint work with E. P. van den Ban, we make a detailed comparison between the two spaces, without using the proof or any validity of any of the associated Paley-Wiener theorems. This is done by use of the Hecke algebra, our techniques involving derivatives of holomorphic families of continuous representations and Harish-Chandra modules. -Gautam Chinta --Whittaker functions and Shintani zeta functions -- I will discuss some examples of coincidences between Shintani zeta functions and Whittaker functions of Eisenstein series on metaplectic double covers of linear groups. I will also describe some applications to number theory and suggest prospects for further study.