## Lie Groups and Representation Theory Seminar 2016

List of speakers:
Winter School 2016, Piotr Pragacz, Hidenori Fujiwara #1, Hidenori Fujiwara #2,
 Winter School 2016 in Representation Theory of Reductive Groups Date: January 22-27, 2016 Place: Graduate School of Mathematical Sciences, the University of Tokyo Date: April 12 (Tue), 2016, 17:00-18:30 Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Piotr Pragacz (Institute of Mathematics, Polish Academy of Sciences) Title: Universal Gysin formulas for flag bundles Abstract: [ pdf ] We give generalizations of the formula for the push-forward of a power of the hyperplane class in a projective bundle to flag bundles of type A, B, C, D. The formulas (and also the proofs) involve only the Segre classes of the original vector bundles and characteristic classes of universal bundles. This is a joint work with Lionel Darondeau. Wu (ȊwʍuII) Date: June 6 (Mon)-10 (Fri), 2016, 15:00-17:00 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Hidenori Fujiwara (p) (Kinki University) Title: w^[Q̒a Abstract: Ƃ̖ڕWF[Q̕\_ɂĊ{IȋO̕@𗝉A̓IȖ \͂gB uTvF[Q̕\_͔P[Qɑ΂cȌƎقɂĂB RpNgQ⃋[gnȂǂ̓ĂɌ[Q̌ɂĂ͐w IA[@B̎@łB̎@̖͂ɑ΂Ă͂ƂĂɗLł ʂ񋟂Ă邪ǍʂɎߒ͈ł̒łB\Iȋc_ ɂ͍ޗsȂ̂łB @ɂĂ[Q̕\_ɂĂA.A. Kirillov nn̋O̕@ ȂB񃆃j^\̓lނƗ]\̋OƂP΂PɑΉƂ ̂ł邪A̍Ȃ̂͌Q̔ς̗U\ɑ΂Mackey _łBU \̗_͗LQɑ΂Frobenius ̌Ɏn܂邪AqoĔςJ ԂēAEPAȉ[Q̕\_ɂƂMackey _͊{łB [QAp냊[QA[QƐELƂAꂼ̃JeS[ ̃Mbv͑傫BX͙p냊[QƉ[Q̊ԂɎw^[Q̃NX 悤BɋǏRpNgȈʑQɑ΂Q̃j^\̗U\ Å{IqׂBXɁAQ̃j^\Q̃j^\ ̗U\ɓlł邽߂̏Mackey ɏ]Ę_悤BQ̂P j^\Uꂽ\P\ƂACӂ̊񃆃j^\P \ɓlłƂǍQ͒PłƂB́Aw^[Q͒Pł 邪Aʂ̉[Q͒Pł͂ȂAɉAuslander-Kostant Pukanszky ɂO̕@͈ʉĂB ŌɎ̒ႢT^ƂāAHeisenberg Qip냊[QjAax+b QiS[ QjAGrélaud Qiw^[QjグA̓Iȉ͂sAꂼ̒iKŐ錻ۂmFƂƂɁǍQ̃j^o΂肷B ѕ]@F|[goɂB 66() ͑S̓IȘbƂĉL̍usCiw^[Q̗U^dx P\j7() self-contained ȍu`sD Date: June 6 (Mon), 2016, 15:00-17:00 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Hidenori Fujiwara (p) (Kinki University) Title: Monomial representations with multiplicities of discrete type of an exponential solvable Lie group (w^[Q̗U^dxP\) Abstract: [ pdf ] Let $G=\exp {\mathfrak {g}}$ be a connected and simply connected exponential solvable Lie group with Lie algebra ${\mathfrak {g}}$. By the orbit method the unitary dual $\hat G$ is realized as the orbit space ${\mathfrak {g}}^{\ast}/G$ of the coadjoint representation of $G$. In this framework we study some monomial representations of $G$. Let $H$ be a closed connected subgroup of $G$ and $\chi$ a unitary character of $H$. We consider the induced representation $\tau=\operatorname{ind}_H^G \chi$ of $G$. When the irreducible decomposition of $\tau$ has multiplicities of discrete type, we describe explicitly the Penneyfs Plancherel formula for $\tau$ and show the commutativity of the algebra $D_{\tau}(G/H)$ of the $G$-invariant differential operators on the line bundle over the base space $G/H$ associated with data $(H,\chi)$. We also give an example which replies negatively to a related Duflo's question. $G= \exp {\mathfrak{g}}$ [ ${\mathfrak {g}}$ APAȎw^[QƂBO̕@ɂA$G$ ̃j^o $\hat G$ $G$ ̗]\̋O ${\mathfrak {g}}^{\ast}/G$ ɂB̘gg݂̒ $G$ ̂̒P\𒲂ׂ悤B $H$ $G$ ̘AQA$\chi$ $H$ ̃j^wWƂ $G$ ̗U\ $\tau = \operatorname{ind}_H^G \chi$ lB$\tau$ ̊񕪉U^dxƂA $\tau$ ɑ΂PenneyPlancherel̓IɋLqAf[^ $(H,\chi)$ ɔ $G/H$  $G$-sϔpf $D_{\tau}(G/H)$ ̉B܂֘ADuflo̖ɔےIȗ^悤B