Quantum Grassmannians and \(q\)-hypergeometric series.

*(English)*Zbl 0782.33015A new point of view relating quantum groups to \(q\)-special functions is presented. It is motivated by a recent work of Horikawa who showed that the contiguity relations for the general \(q\)-hypergeometric series \({}_ 2\varphi_ 1\) can be described by the quantized universal enveloping algebra \(U_ q(\text{gl}(4))\), which in turn was found by investigating \(q\)-analogues of Gelfand’s interpretation of Gauss’ hypergeometric functions by Grassmann manifold Grass(2,4). In this article, as an attempt to give an intrinsic explanation of the above relation, the case of \(\text{Grass}(2,n)\) is discussed and it is shown that a \(q\)-analogue of Lauricella’s hypergeometric series \(F_ D\) of \(n-3\) variables is such. As a consequence, the quantized universal enveloping algebra \(U_ q(\text{gl}(n))\) is obtained naturally as the algebra describing the contiguity relations of the \(q\)-hypergeometric series.

Reviewer: A.Kaneko (Komaba)

##### MSC:

33D80 | Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics |