The Kontsevich' Formality theorem implies not only the existence of the Duflo map but shows that it is canonical. We shall describe this construction and indicate how does this construction extend to the whole Poisson cohomology of an arbitrary finite-dimensional real Lie algebra.

In the present talk we discuss two different ring structures on the set of holomorphic discrete series. First one comes from the convolution of functions on the symmetric cone underlying the Hermitian symmetric space of tube type G/K.

The second one is non-commutative and is induced by the composition of operators whose symbols belong to the discrete series representations of the causal symmetric space of Cayley type G/H.

We also discuss the relationship that exists between these ring structures, intertwining operators for tensor products of holomorphic discrete series representations and the so-called Rankin-Cohen brackets.

The pseudo-Riemannian structure of S(q,n) allows us to give an interpretation of the "infinitesimal cross ratio", which is a complex valued 2-form on the two point configuraion space of a knot K, K × K-(diagonal): The real part of it can be interpreted as an area element of a surface in S(q,n).

I want to consider several examples, starting of SL(2,R), and to discuss hoe realistic looks today this Gelfand's project.

In this talk, we show that the Cayley transform image is convex if and only if the domain is symmetric and the parameter is a specific one so that the Cayley transform coincides with the Korányi-Wolf Cayley transform.

Z(f,\pi,s)=\int_{G/H} f( . g) |P( . g)|^s \pi( .g) u d ( .g).

For a class of symmetric spaces we will make this definition precise in the case where \pi belongs to the spherical minimal series, and we will prove a functional equation.

例えば，誘導表現や部分群へ制限された表現について，既約分解し，繋絡 作用素を構成し，また関連する不変微分作用素環を調べたい．このような 状況に直面すると，たとえ指数型可解リー群の場合ですら，我々はごく僅 かなことしか知らない．より多くの道具を手にできるのは冪零リー群に対 してのみである．リー群の表現論は半単純リー群と可解リー群の間でかな り異なった様相を見せている. 半単純リー群の豊富な代数構造は多くの研 究材料と結果を提供し，可解リー群の貧弱な構造は帰納法を唯一の有効な 手段としている．いずれにしても可解リー群のユニタリ表現論において軌 道の方法が非常に実り多いことは疑いのないところである．既約ユニタリ 表現に余随伴軌道を対応させるという Kirillov の革新的なアイデアはその 価値ある成果の数々を誇っているように見える．それは Mackey 理論の可 解リー群への見事な応用であるが，ひとたびこの枠組みが採用されると， 解析学における多くの対象物を余随伴軌道の代数的または幾何的性質を用 いて研究することができる．

冪零リー群の場合を中心に指数型可解リー群に対する軌道の方法の現状を 紹介する．

In the past few years, Barbasch and Barbasch-Ciubotaru have given a striking description of the spherical unitary dual of G. The point of this talk is to recall their results and then show how they can be used to give very precise information about the unitary dual of the double cover of G.

The relationship between unitary representations of G and its double cover is the Shimura correspondence of the title. This talk represents joint work with Jeffrey Adams, Dan Barbasch, Annegret Paul, and David Vogan.