On this page we will announce solutions to the problems posed in the book. There will be a short description in html and a link to a postscript file containing more detailed information. If you wish to place an announcement or a postscript file on this page you can send TeX or postscript files to Joachim Hilgert (E-Mail: hilgert@math.tu-clausthal.de)
September 26, 1999
Wolfgang Bertram (Nancy) announces progress concerning his article ``Jordan algebras and conformal geometry.''
The main progress concerns Problem C: it is indeed possible to find a common framework for the Liouville theorem and the fundamental theorem of projective geometry, at least from a differential geometric point of view. This result is contained in Bertram's paper Fundamental theorem for a class of symmetric spaces which is going to appear in Math. Zeitschrift. A preliminary version of it can be found at the
Innsbruck Jordan preprint server
This result, among many others, can also be found in the paper The Geometry of Jordan and Lie structures on
The problems D, E, F mentioned in the article remain in general open. However, there is an additional reference concerning Problem D: O. Loos, Charakterisierung symmetrischer R-Räume durch ihre Einheitsgitter, Math. Z. 189 (1985), 211 - 226. This paper offers an answer to Problem D in the compact case by means of root systems and unit lattices.
February 21, 1999 (with an addition from June 16, 1999)
Yuri Neretin (Moscow) announces that he has realized part of the program of investigation of kernel representations proposed in his article ''Boundary values of holomorphic functions and some spectral problems for unitary representations''.
1. Plancherel formula. Consider the group O(p,q) with p less or equal than q and the symmetric space Bp,q=O(p,q)|O(p) x O(q). We realize this space as a space of real p x q matrices with a norm less than 1.
Consider a Hilbert space Hs of functions on Bp,q defined by a positive definite kernel
K(z,u)=det(1-zut)-s,
where s=0,1,...,p-1 or s>p-1. Let Rs be the natural representation of O(p,q) in Hs.
Theorem. Let 2s>(p+q)-2$. Then a density of the Plancherel measure for the representation Rs can be given explicitly by a formula involving products of Gamma functions.
Analogous results are obtained for all series of classical symmetric spaces.
2. Separation of spectra. Let Cr be a set of real p x q matrices z of norm 1 for which (1-zzt) has rank r.
Theorem.
a) Let 2s<(q-p)+r+1. Then operator of restricting a function f in Hs to Cr is well defined.
b) Let 2(s+k)<(q-p)+r+1. Then the restrictions of all partial derivatives of order k to Cr are well-defined.
This theorem gives a natural O(p,q)-invariant filtration in the space Hs. It provides a possibility to separate spectra of kernel repesentation to pieces of different nature.
A postscript version of the announcement which contains the explicit Plancherel formula is available.
Added on June 16, 1999: The postscript files of the papers mentioned above can be obtained here
February 19, 1999
Dragomir Djokovic (Waterloo) announces the solution of Problem 6.3 of his article ''Problems on the exponential function of Lie groups'' with K.H. Hofmann which was to prove or disprove the statement
A connected Lie group is exponential if and only if its minimal parabolic subgroups are exponential.
As a test case, in the article it is proposed (Problem 6.3a) to decide
whether or not the minimal parabolics of
SLn(H)
are exponential. (The groups SLn(H)
themselves
are exponential, and there is no harm in replacing
SLn(H) by
GLn(H))
He has shown that the answer to the test question is negative
for n larger than 7 and affirmative for n less than 7
(the case n=7 remains open but is probably affirmative).
As a consequence, he can deduce that the minimal parabolics
of the exponential groups Sp(p,q) are not exponential
for p greater equal q and both at least 8, and the
same is true for the groups
SO*(2n) for n at least 15. On the other hand,
he can show that, for the exponential groups U(p,q) and
SO(p,1)0, the minimal parabolic subgroups are
exponential.
His article will appear in
Journal of Lie Theory
July 14, 1998
Mick McCrudden and Seth Walker (Manchester) announce the solution of Problem 3 of McCruddens article ``An introduction to the embedding problem for probabilities on locally compact groups.''
They obtain a classification of infinitely divisible probabilities on linear p-adic groups, extending the work of Riddhi Shah on the same problem for p-adic algebraic groups. Their article is entitled ``Infinitely divisible probabilities on linear p-adic groups'' and can be downloaded as a postscript file:
July 10, 1998
Karl Heinrich Hofmann (Darmstadt) provides an addition to his article ``Problems on the exponential function of Lie groups'' with D. Dokovic in which he recalls a result of Karl-Hermann Neeb (J. Algebra 179 (1996), 331-361) on the characterization of Cartan subgroups for arbitrary connected Lie groups.
To download this addition click:
May 26, 1998
Bernhard Krötz (Clausthal) announced a solution to Problem 7(1) of G. Ólafsson's article ``Open Problems in Harmonic Analysis on Ordered Symmetric Spaces'':
Let G/H be a compactly causal symmetric space. For any H-spherical unitary highest weight representation of G the character of this representation can be identified with an H-biinvariant holomorphic function on a certain complex semigroup S which has G as its Shilov boundary.
On the other hand on the c-dual space of G/H one has a theory of spherical functions. It was known that the character of an H-spherical unitary highest weight representation of G can be viewed as a multiple of a certain spherical function on the c-dual space. The problem posed by G. Ólafsson's was to determine this multiple. The answer is that this multple is given explicitly in terms of the c-function on the non-compactly causal c-dual space of G/H. The solution was obtained by a result which relates the H-spherical vector of the representation with an H-integral over the highest weight vector.
A more detailed version of this announcement and the full paper Formal dimesnion of semisimple symmetric spaces by Bernhard Krötz which contains the stated result as Theorem V.3 are available as postscript files. To download them click as indicated below:
postscript version: announcement