Date: | October 8 (Fri), 2004, 10:30-12:00 |
Room: | RIMS 402 |
Speaker: | Bernhard Krötz (RIMS) |
Title: | Working Seminar on Integral Geometry 1 |
Abstract: [pdf] |
The objective of this seminar is to provide an introduction to some recent trends in integral geometry on real Lie groups G_{R} and, more generally, on symmetric spaces X_{R} = G_{R}/H_{R}. To be more specific, we want to discuss the possibilities for a suitable horospherical Radon transform on the aforementioned spaces. The first obstacle one encounters in this context is the right notion of horospheres on X_{R}. The naive approach, namely to define horospheres as orbits of certain unipotent subgroups of G_{R} is not fruitful -- for example they do not exist for compact spaces. It was suggested by S. Gindikin that one should consider complex horospheres without real points, i.e. orbits of appropriate complex unipotent subgroups of the complexification G_{C} of G_{R} on X_{C} = G_{C}/H_{C} which do not intersect X_{R}. Having the right notion of horospheres one then should be able to define a horospherical Cauchy-Radon transform for symmetric spaces. The final upshot would lie a new understanding of the L^{2}-spectrum of X_{R}, in particular the wave front sets of the arising series of representations should become much clearer. Literature on the subject is rather scarce. There are some notes of Gindikin on compact symmetric spaces and several articles by him for the example G_{R} = Sl(2,R). One of our main goals is to make these articles, often written in classical style, available to a wider audience. For us this will mean that after a careful and thorough discussion of the initial approach, we will comment on alternative points of view. So it will be a ''hands on'' seminar with no hiding of details and the possibility of unraveling hidden truths. We shall start with a discussion of the compact case where the picture is complete and very beautiful. [G1] S. Gindikin, Integral Geometry on Sl(2,R), Math. Res. Letters (2000), 417-432. [G2] —, An analytic separation of series of representations for Sl(2,R), Moscow Math. J. 2 (2002), no 4., 1-11. [G3] —, Horospherical Cauchy-Radon transform on compact symmetric spaces, preprint 2004. [GKO] S. Gindikin, B. Krötz and G. Ólafsson, Horospherical model for the holomorphic discrete series, in preparation. [GKO] —, Holomorphic aspects of the Radon transform on Riemannian symmetric spaces, in preparation. |
Date: | October 15 (Fri), 2004, 10:30-12:00 |
Room: | RIMS 402 |
Speaker: | Bernhard Krötz (RIMS) |
Title: | Working Seminar on Integral Geometry 2 |
Abstract: [pdf] |
In the beginning we will finish our discussion on complex horocycles on complex symmetric spaces. After that we will introduce the space of "horocycles without real points" attached to a compact symmetric space. The second part of the lecture is devoted to a thorough structural discussion of this new object. |
Date: | October 22 (Fri), 2004, 10:30-11:30 |
Room: | RIMS 402 |
Speaker: | Bernhard Krötz (RIMS) |
Title: | Working Seminar on Integral Geometry 3 |
Abstract: [pdf] |
We will finish our discussion on the space for horocycles with no real points attached to a compact symmetric space. A key fact in this context is a certain uniform boundedness result for matrix coefficients (due to Clerc) which will be explained in detail. |
Date: | October 29 (Fri), 2004, 10:30-12:00 |
Room: | RIMS 402 |
Speaker: | Bernhard Krötz (RIMS) |
Title: | Working Seminar on Integral Geometry 4 |
Abstract: [pdf] |
In this session we will complete our discussion for compact symmetric spaces. We define the horospherical Cauchy transform and explain the geometry of the underlying double fibration. We shall introduce the inverse transform and prove a geometric inversion formula. As usual, the story is rounded up with our favorite example of the sphere. |
Date: | November 5 (Fri), 2004, 10:30-12:00 |
Room: | RIMS 402 |
Speaker: | Bernhard Krötz (RIMS) |
Title: | Working Seminar on Integral Geometry 5 |
Abstract: [pdf] |
For a real symmetric space X_{R} the kernel of the real horospherical transform consist of all series of representations except the most-continuous one. In case the real horospherical transform is
faithful (for example if X_{R} is a complex group or X_{R}
is non-compact Riemannian) its inversion is essentially equivalent to
the Plancherel formula.
To obtain a faithful horospherical transform Gindikin suggested to use complex horospheres in X_{C} without real points (they do not intersect X_{R}) and to replace the real horospherical transform (i.e. integration over real horospheres) with a Cauchy type transform (with singularities of the Cauchy kernel on the horospheres without real points). In our first four lectures we have seen that this method works well for compact symmetric spaces. The objective of the next two (or three) talks is to discuss affine symmetric spaces of Hermitian type. We will define a complex horospherical transform and show that it has no kernel on the holomorphic discrete series. In addition, on this part of the spectrum (i.e. hol. disc. series) there is an inversion formula in complete analogy to the compact case. Instead of giving an account of the theory in full generality we decided to discuss the most basic case X_{R} = SL(2,R)/SO(1,1) where the results are already new and interesting. We intend to provide all details for this case; of particular interest might be our approach to spherical harmonic analysis for (unitary) lowest weight modules of G = SL(2,R). The results are taken from the recently finished paper Horospherical model for the holomorphic discrete series and the horospherical Cauchy transform (authored by Gindikin, Ólafsson and myself) and my article Formal dimension for semisimple symmetric spaces, Compositio Math. 125 (2001), no. 2, 155-191. |
Date: | November 12 (Fri), 2004, 10:30-12:00 |
Room: | RIMS 402 |
Speaker: | Bernhard Krötz (RIMS) |
Title: | Working Seminar on Integral Geometry 6 |
Abstract: [pdf] |
We will finsih our discussion on the geometry of the horocycle space for a symmetric space of Hermitian type and define the horospherical Cauchy transform. After that we begin with an interlude on spherical analysis of lowest weight modules. |
Date: | December 3 (Fri), 2004, 10:30-12:00 |
Room: | RIMS 402 |
Speaker: | Bernhard Krötz (RIMS) |
Title: | Working Seminar on Integral Geometry 7 |
Abstract: [pdf] |
The topic of this talk will be analysis of spherical unitary lowest weight modules. Special emphasis will be given to the discussion of the associated c-functions and formal dimension. |
Date: | December 10 (Fri), 2004, 10:30-12:00 |
Room: | RIMS 402 |
Speaker: | Bernhard Krötz (RIMS) |
Title: | Working Seminar on Integral Geometry 8 |
Abstract: [pdf] |
The plan for the approximately next five lectures is to slowly move through Gindikin's paper "Integral geometry on SL(2,R)", Math. Res. Lett. 7 (2000), no. 4, 417-432. We begin with explanation/translation of the notations in the paper. |
Date: | December 17 (Fri), 2004, 10:30-12:00 |
Room: | RIMS 402 |
Speaker: | Bernhard Krötz (RIMS) |
Title: | Working Seminar on Integral Geometry 9 |
Abstract: [pdf] |
We will discuss the duality between horospheres and horocycles on G_{C} = Sl(2,C), both from a structural and geometrical point of view. After that we will describe G_{R} × G_{R}-invariant domains in G_{C} and explain some of their basic geometry. |
Date: | February 23 (Wed), 2005, 10:30-12:00 |
Room: | RIMS 206 |
Speaker: | Bernhard Krötz (RIMS) |
Title: | Working Seminar on Integral Geometry 10 |
Abstract: [pdf] |
We will summarize the recent overview article "Holomorphic horospherical duality 'sphere-cone'" by Simon Gindikin. Knowledge of session 1-4 of the seminar would be helpful. |
Organizer: | Toshiyuki Kobayashi |
© Toshiyuki Kobayashi