Junko Inoue, Richard Stanley, Hidenori Fujiwara, Ali Baklouti,
| Joint with Tuesday Seminar on Topology | |
| Date: | Nov 12 (Tue), 2024, 17:30-18:30 |
| Speaker: | Junko Inoue (井上順子) (Tottori University) |
| Title: | Holomorphically induced representations of some solvable Lie groups |
| Abstract: [ pdf ] |
From a viewpoint of the orbit method, holomorphic induction is
originally based on the idea of realizing an irreducible unitary
representation of a Lie group $G$ in an $L^2$-space of some holomorphic
sections of some line bundle over a $G$-homogeneous space associated with
a polarization for a linear form of the Lie algebra of $G$. It is a
generalization of ordinary induction from a unitary character; Through
this process, Auslander-Kostant constructed the irreducible unitary
representations of type 1, connected, simply connected solvable Lie
groups.
In this talk, focusing on the class of exponential solvable Lie groups,
we are concerned with holomorphically induced representations $\rho$ in some
general settings. We would like to discuss the following problems:
(1) conditions of non-vanishing of $\rho$, |
| Date: | Nov 13 (Wed), 2024, 17:30-18:30 |
| Speaker: | Richard Stanley (MIT/University of Miami) |
| Title: | Some combinatorial aspects of cyclotomic polynomials |
| Abstract: [ pdf ] | Euler showed that the number of partitions of n into distinct parts is equal to the number of partitions of n into odd parts. MacMahon showed that the number of partitions of n for which no part occurs exactly once is equal to the number of partitions of n into parts divisible by 2 or 3. Both these results are instances of a general phenomenon based on the fact that certain polynomials are the product of cyclotomic polynomials. After discussing this assertion, we explain how it can be extended to such topics as counting certain polynomials over finite fields and obtaining Dirichlet series generating functions for certain classes of integers. |
| Date: | Nov 27 (Wed), 2024, 13:30-14:30 |
| Speaker: | Hidenori Fujiwara (藤原英徳) (OCAMI/Kindai University) |
| Title: | Inductions and restrictions of unitary representations for exponential solvable Lie groups |
| Abstract: [ pdf ] | Let $G=\exp{\mathfrak{g}}$ be a connected and simply connected real nilpotent Lie group with Lie algebra ${\mathfrak{g}}$, $H=\exp{\mathfrak{h}}$ an analytic subgroup of $G$ with Lie algebra ${\mathfrak{h}}$, $\chi$ a unitary character of $H$ and $\tau=\operatorname{ind}_H^G \chi$ the monomial representation of $G$ induced from $\chi$. Let $D_{\tau}(G/H)$ be the algebra of the $G$-invariant differential operators on the line bundle over $G/H$ associated to the data $(H,\chi)$. We denote by $C_{\tau}$ the center of $D_{\tau}(G/H)$. We know that $\chi$ is written as $\chi_f$, where $\chi_f(\exp X)=e^{if(X)}$ $(X∈{\mathfrak{h}})$ with a certain $f∈{\mathfrak{g}}^{\ast}$ verifying $f([{\mathfrak{h}}, {\mathfrak{h}}])=\{0\}$. Let $S({\mathfrak{g}})$ be the symmetric algebra of ${\mathfrak{g}}$ and ${\mathfrak{a}}_{\tau}=\{X+\sqrt{-1} f(X) ; X∈{\mathfrak{h}}\}$. We regard $S({\mathfrak{g}})$ as the algebra of polynomial functions on ${\mathfrak{g}}^{\ast}$ by $X(\ell)=\sqrt{-1} \ell(X)$ for $X∈{\mathfrak{g}}$, $\ell ∈{\mathfrak{g}}^{\ast}$. Now, $S({\mathfrak{g}})$ possesses the Poisson structure $\{,\}$ well determined by the equality $\{X,Y\}=[X,Y]$ if $X$,$Y$ are in ${\mathfrak{g}}$. Let us consider the algebra $(S({\mathfrak{g}})/S({\mathfrak{g}})\overline{{\mathfrak{a}}_{\tau}})^H$ of the $H$-invariant polynomial functions on the affine subspace $\Gamma_{\tau}=\{\ell \in {\mathfrak{g}}^{\ast}:\ell(X)=f(X), X \in {\mathfrak{h}}\}$ of ${\mathfrak{g}}^{\ast}$. This inherits the Poisson structure from $S({\mathfrak{g}})$. We denote by $Z_{\tau}$ its Poisson center. Michel Duflo asked: the two algebras $C_{\tau}$ and $Z_{\tau}$, are they isomorphic? Here we provide a positive answer to this question. |
| Date: | Nov 27 (Wed), 2024, 14:30-15:30 |
| Speaker: | Ali Baklouti (University of Sfax) |
| Title: | A proof of the Zariski closure conjecture for coadjoint orbits of exponential Lie groups |
| Abstract: [ pdf ] | I will begin by defining the Zariski Closure Conjecture for coadjoint orbits of exponential solvable Lie groups, examining some cases that have been solved, and addressing the ongoing challenges in resolving the conjecture fully. I will then introduce new approaches to explore the relationship between generating families of primitive ideals and the set of polynomials that vanish on the associated coadjoint orbits, ultimately aiming to advance toward a solution to the conjecture. |
© Toshiyuki Kobayashi