By Kazuhiko Aomoto (auth.), Simon Gindikin, James Lepowsky, Robert L. Wilson (eds.)

These volumes comprise eighteen invited papers by way of distinctive mathematicians in honor of the 80th birthday of Israel M. Gelfand, probably the most awesome mathematicians of our time. Gelfand has performed a vital position within the improvement of practical research over the past half-century. His paintings and his philosophy have in truth assisted in shaping our realizing of the time period 'functional analysis'. The papers in those volumes principally situation parts within which Gelfand has a really powerful curiosity at the present time, together with geometric quantum box concept, illustration thought, combinatorial constructions underlying numerous 'continuous' structures, quantum teams and geometry.

The moment of the 2 volumes comprises the a bit extra 'geometric' papers, even supposing this sort of designation is to a undeniable volume arbitrary, a result of breadth of the papers.

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**Example text**

4) it follows that ei, ei =1= 0 so that (2h i , ei , ei) is an Striple in ti. It now follows (using [B-K4, Cor. 6]) that the algebra of Ko-semiinvariants in S(tt} is a polynomial ring in s algebraically independent generators AI' ... ' As where Ai E (to is a primitive (so irreducible) homogeneous Kosemi-invariant of weight Xi and qi is the rank of the symmetric pair (ti, t~). Then P = A~' ... A~· where al, ... , as E Z+. But then we find that each ai =1= 0 since X2 = X~' ... 2) and the fact hi =1= 0 ).

10) The Jackson integral over the cycle reg [O,7]oolQ itself can be evaluated as follows. 11) Connection formulas in the q-analog de Rham cohomology 11 where C~ = L:,7=1 (aj + n - j) [,8 + (j - 1h]. On the other hand, we have the Askey-Habsieger-Kadell formula for An = L:,7=1 (aj + n - j)[l + (j - 1h], where rq(x) denotes the q-Gamma function (1- q)l-X(q)oo/(qX)oo. This observation has been done in various articles [As], [K], [H], [Ev], [T], [Ao2],etc. At first it was proved in the case where "( was a non-negative integer, but it is possible to extend it in the present form.

B-K4, Th. 4]). Let Y C ll* be the K-orbit of highest weight vectors where II ~ ll* is a simple K-module of highest weight dX. 5 (and their proofs) hold verbatim after "4" is replaced by "q" in the obvious way. 6. We now introduce the involution (J in order to compute lJz. 1) where exp : e-+ K is the exponential map. The action of (Jo defines a map (J : -+ V on each K -module V, so in particular on t, p and p*.