%Paper: hep-th/9405122
%From: akrol@gnumath.rutgers.edu (A.Kazarnovski-Krol)
%Date: Wed, 18 May 94 23:43:55 EDT
%Date (revised): Mon, 6 Mar 1995 02:15:56 -0500
%Date (revised): Thu, 9 Mar 1995 22:39:58 -0500
%Date (revised): Tue, 21 Mar 1995 22:00:15 -0500

% hepth 9405122
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\topmatter
\author  A. Kazarnovski-Krol               \endauthor
\title Value of generalized hypergeometric function at unity \endtitle
\address{
  Department of Mathematics
  Rutgers University
  New Brunswick, NJ 08854, USA}


\vskip 10mm
%\date { May 18, 1994 }
\vskip 10mm

\abstract{
Value of generalized hypergeometric function at
a special point is calculated. More precisely, value of certain
multiple integral over vanishing cycle (all arguments collapse to unity)
is calculated. The answer is expressed in terms of
$\Gamma$-functions. The constant is relevant to the part of $\rho$
in the Gindikin-Karpelevich formula for c-function of Harish-Chandra.
Calculation is an adaptation of classical calculations of Gelfand and Naimark
(1950) to the Heckman-Opdam hypergeometric functions in the case of root system
 of type $A_{n-1}$.
 }
\endtopmatter

\document
\head {0 . Introduction}\endhead

In this paper value of certain multiple integral over vanishing cycle
(all arguments collapse to unity)
is calculated. This is a normalization constant. It is related to
the part of $\rho$ in the Gindikin-Karpelevich formula for
$c$-function of Harish-Chandra.

The cycle is a
distinguished one in the theory of zonal spherical functions. It
appeared for the first time in [1] and its role is clarified in [21].

Technically our calculations is an adaptation of formulas of [1]
(elliptic coordinates)
for the case of generic $k$, i.e. for the Heckman-Opdam hypergeometric
functions of type $A_{n-1}$.

The integral may be regarded as a variation on a theme of Selberg
integral
[19],
and recently integrals of this type have drawn much attention because
of applications to conformal field theory.

The paper has continuation in [20,21].


\vskip 5mm
\head { 1. Theorem}\endhead
Let $z_l,\; l=1,...,n;\; t_{i,j},\;i=1,...,j,\;j=1,...,n-1$ be the set of
real variables. It is convenient to organize these variables in the form
of a pattern cf. fig.1.

\midinsert
$$
\matrix
z_{1}&&z_{2}&&\ldots&&\ldots&&z_{n}\\\\
&t_{1,n-1}&&t_{2,n-1}&&\ldots&&t_{n-1,n-1}&\\\\
&&\ldots&&\ldots&&\ldots&&\\\\
&&&t_{1,2}&&t_{2,2}&&&\\\\
&&&&t_{1,1}&&&&
\endmatrix
$$
\botcaption{Figure 1}
 Variables organized in a pattern
\endcaption
\endinsert

\vskip 0.5cm

\remark{Remark 1.1 } This type of variables reflects the flag
structure ($i$th row corresponds to $i$-dimensional plane in a flag)
,
cf.[1,2]. One should notice that this type of variables is used in [10]
for the isomorphism of Heckman-Opdam hypergeometric system with  a
particular case of trigonometric version of
Knizhnik-Zamolodchikov equation.
\endremark

\definition{Definition 1.2} Consider the following multivalued form
$\omega_{\Delta}:$


$$
\align
\omega_{\Delta}:= &\prod_{i=1}^{n} z_i^{a_i}
 \prod_{i_1 > i_2}{(z_{i_1}-z_{i_2})^{1-2k}}\\
 &\times\prod_{i<l}{(z_l-t_{i,{n-1}})}^{k-1}\prod_{i \ge
l}{(t_{i,{n-1}}-z_l)}^{k-1}  \\
 &\times\prod_{j=1}^{n-1} \{
 \prod_{i_1 \ge i_2} ( t_{i_{1},j} - t_{i_2,{j+1}} )^{k-1}
\prod_{i_1 > i_2} ( t_{i_{1},j+1 } - t_{i_2,j} )^{k-1}\\
 &\times\prod_{i_1>i_2} {(t_{i_1,j}-t_{i_2,j})^{2-2k} }\\
 &\times\prod_{i=1}^{j} {t_{ij}^{a_{ij} }} \}\quad
 {dt_{11} dt_{12} dt_{22} \ldots dt_{n-1,n-1}  }
\endalign
$$

\enddefinition
\smallskip

\remark{Remark 1.3} For the applications one should let
$a_{ij}= {\lambda_{n-j+1}-
 \lambda_{n-j} -k}$ and
$a_i={\lambda_1 +{k {(n-1)}\over 2}}$,

but we formulate the theorem in this more general form
(with $a_{ij}$ and $a_i$ unspecified).
\endremark
\smallskip

\definition{Definition 1.4}
Set
$$
\Phi(z_1,\ldots,z_n)= \int_{\Delta} \omega_{\Delta}
, $$
where $\omega_{\Delta}$ is defined above and ,
assuming that $z_1, z_2, \ldots , z_{n}$ are real and satisfy
$0 < z_1 <z_2 < \ldots <z_{n}$,
define cycle $\Delta$ by the following inequalities:
$t_{i,j+1} \le t_{ij} \le t_{i+1, j+1}$ and
$z_i \le t_{i,n-1} \le z_{i+1}$ .

We assume that phases
of the factors of the form $\omega_{\Delta}$ are equal to zero
provided $k$ , $a_{ij}$, and $a_i$ are real.
\enddefinition

\smallskip

\proclaim{Theorem 1.5}
The limit of $\Phi(z_1,\ldots,z_n)$ as all $z_i$
approach $1$, while $z_1<z_2<\ldots<z_n$, is equal to:
$$
{\Phi(1,\ldots,1):={\lim\limits_{all\; {z_i}\rightarrow
1}}\Phi(z_1,\ldots,z_n)}=
 {
 {\Gamma(k)\Gamma(k)^2\ldots\Gamma(k)^n}
 \over
 {\Gamma(k)\Gamma(2k) \ldots\Gamma(nk)}
 }\quad .
$$
\endproclaim

\subhead{Proof} \endsubhead
  Following the classical work of I.M. Gelfand and
M.A. Naimark cf. [1]
 ,  let
$$
\tau_{ij}=
{
 {\prod\limits_{i_1=1}^{j-1}(t_{i_1,j-1}-t_{ij})}
 \over
 {\prod\limits_{i_1\ne i}(t_{i_1,j}-t_{ij})}
}\;\; ,
$$
$i=1,\ldots,j$
$j=1,\ldots,n-1$.
Note that ${\sum\limits_{i=1}^j}\tau_{ij}=1$, and
$$
{
 {D(\tau_{1,j},\ldots,\tau_{j-1,j})}
 \over
 {D(t_{1,j-1},\ldots,t_{j-1,j-1})}
}
=
{
 {{\prod\limits_{1\le i<k\le j-1}}(t_{i,j-1}-t_{k,j-1})}
 \over
 {{\prod\limits_{1\le i<p\le j}}(t_{ij}-t_{pj})}
}
$$
[see \cite{1} for the details].
Let also
$$
\tau_{in}={
           {{\prod\limits_{i=1}^{n-1}}(t_{i_1,n-1}-z_i)}
           \over
           {{\prod\limits_{i_1\ne i}}(z_{i_1}-z_i)}
          }
\;\;\;\;\;\;\;\;\;\;\;  i=1,\ldots,n.
$$
One has ${\sum\limits_{i=1}^n}\tau_{in}=1$ and
$$
{
 {D(\tau_{1,n},\ldots,\tau_{n-1,n})}
 \over
 {D(t_{1,n-1},\ldots,t_{n-1,n-1})}
}
=
{
 {{\prod\limits_{1\le i<k\le n-1}}(t_{i,n-1}-t_{k,n-1})}
 \over
 {{\prod\limits_{1\le i<p\le n}}(z_i-z_p)}
}
$$
In the variables $\tau_{ij}\;\;\; i=1,\ldots,j-1,\;\; j=1,\ldots,n$
integral for $\Phi(z_1,\ldots,z_n)$ is written as:
$$
\align
&\Phi(z_1,\ldots,z_n)
=
\prod_{i=1}^n{z_i^{a_i}}\int{\prod\limits_{j=1}^{n-1} \;
              \prod\limits_{i=1}^j
               }
    (t_{ij})^{a_{ij}}\\
&\times
{
{\prod\limits_{j=1}^n}
 (
  (\tau_{1j}\tau_{2j}\ldots\tau_{j-1,j})
  (1-\tau_{1j}-\ldots -\tau_{j-1,j})
 )^{k-1}
}
d\tau_{12}d\tau_{13}d\tau_{23}\ldots d\tau_{1n}\ldots
d\tau_{n-1,n}\;\; .
\endalign
$$
Integration is taken over
$$
\tau_{ij}>0,\;\;\;\;\;\;\;\; {\sum\limits_{i=1}^{j-1}}\tau_{ij}<1.
$$
As all $z_i$, $i=1,\ldots,n$ approach $1$, all $t_{ij}$ also approach
$1$, so
$$
\align
\Phi(1,\ldots,1)=&
\int 1 \times
{
 {\prod\limits_{j=1}^n}
                            (
                              (\tau_{1j}\tau_{2j}\ldots\tau_{j-1,j})
                              (1-\tau_{1j}-\ldots -\tau_{j-1,j})
                            )^{k-1}
}\\
&\times d\tau_{12}d\tau_{13}d\tau_{23}\ldots d\tau_{1n}\ldots
d\tau_{n-1,n}.
\endalign
$$
So using Dirichlet's formula one gets
$$
{\Phi(1,\ldots,1)}=
{
 {\Gamma(k)\Gamma(k)^2\ldots\Gamma(k)^n}
 \over
 {\Gamma(k)\Gamma(2k) \ldots\Gamma(nk)}
}
$$
\remark{ Remark 1.6} The same calculation as in theorem 1.5. shows
that
 $\Phi(z_1,z_2,\ldots,z_n)$ is equal to the same constant as in
theorem
provided that all  $a_{ij}=0$ and $a_i=0$. Then this constant is
interpreted as the volume of a maximal compact subgroup
(under certain normalizations)cf. [1,13,14] in the case of generic $k$, i.e.
when
there is no group and no subgroup at all.
The fact that this constant does not depend on $z_i$ also implies the
monodromy properties of cycle $\Delta$, cf. [21].
\endremark
\subhead Acknowledgments \endsubhead I would like to express my gratitude to
I. M. Gelfand for many stimulating discussions on the representation
theory and the theory of hypergeometric functions. I would like to
thank S. Shatashvili for helpful discussions.

\Refs

\ref
\no1
\by  Gelfand I.M., Naimark M.A
\paper  Unitary representations of classical groups
\jour   Tr. Mat. Inst. Steklova
\vol 36
\yr 1950
\pages 1-288
\paperinfo [in Russian,
 see also extract in English  in Collected Papers
 of I.Gelfand, vol. 2, pp.182-211]
\endref

\ref
\by  \quad \quad Gelfand I.M., Naimark M.A
\paper  Unitare Darstellungen der
Klassischen Gruppen
\paperinfo  Akademie Verlag ( German translation)
\yr 1957
\endref

\ref
\no2
\by  Gelfand I.M., Tsetlin M.L.
\paper  Finite-dimensional
representations of the group of unimodular matrices.
\jour  Dokl. Akad. Nauk
SSSR
\vol 71
\yr 1950
\pages 825-828.
\endref

\ref
\no3
\by  Gelfand I.M.
\paper   Spherical functions on symmetric
Rimannian spaces.
\jour Dokl. Akad. Nauk SSSR
\vol 70
\year 1950
\pages 5-8
\endref


\ref
\no4
\by Gelfand I.M.,  Berezin F.A.
\paper   Some remarks on the
theory of spherical functions on  symmetric Rimannian manifolds.
\jour Tr.
Mosk. Mat. O.-va
\vol 5
\yr  1956
\pages 311-351
\paperinfo [Transl., II.Ser.,Am.Math.Soc.
21(1962) 193-238]
\endref

\ref
\no5
\by Heckman G.J., Opdam E.M.
\paper  Root systems and
Hypergeometric functions I
\jour Comp. Math.
\vol 64
\yr 1987
\pages  329-352
\endref

\ref
\no6
\by Cherednik I.V.
\paper  A unification of
Knizhnik-Zamolodchikov
and Dunkl operators via affine Hecke Algebras
\jour Invent. Math.
\vol 106
\pages 411-431
\endref

\ref
\no7
\by Varchenko, A.N.
\paper   Multidimensional Hypergeometric
 functions in Conformal Field Theory, Algebraic K-theory, Algebraic
 Geometry.
\endref

\ref
\no8
\by Gindikin S.G.,Karpelevich F.I.
\paper   Plancherel
measure for Rimannian symmetric spaces of nonpositive curvature
\jour
Dokl.Akad. Nauk SSSR
\yr 1962
\vol 145
\issue 2
\pages 252-255
\endref

\ref
\no 9
\by Schechtman,V.V., Varchenko, A.N.
\paper  Hypergeometric
solutions of Knizhnik- ~Zamolodchikov equations.
\jour Lett. Math. Phys.
\vol 20
\yr 1990
\issue 4
\pages  279-283
\endref

\ref
\no 10
\by A.Matsuo
\paper  Integrable connections related to
zonal spherical functions
\jour Invent.math.
\vol  110
\pages 95-121
\yr 1992
\endref



\ref
\no 11
\by Cherednik I.
\paper  Integral solutions of trigonometric Knizhnik
-Zamolodchikov equations and Kac-Moody algebras
\jour Publ. RIMS Kyoto
Univ.
\vol 27
\yr 1991
\pages 727-744
\endref

\ref
\no12
\by V.Knizhnik, A.Zamolodchikov
\paper  Current
algebra and Wess-Zumino models in two dimensions
\jour Nucl. Phys. B
\vol 247
\yr 1984
\pages 83-103
\endref

\ref
\no 13
\by Aomoto K.
\paper Sur les transformation
d'horisphere et les equations integrales qui s'y rattachent
\jour
J.Fac.Sci.Univ.Tokyo
\vol 14
\yr
\pages 1-23
\endref

\ref
\no14
\by I.G. Macdonald
\paper  The volume of a compact
Lie group
\jour Inv. Math.
\vol  56
\yr 1980
\pages  93-95
\endref

\ref
\no15
\by Aomoto K.
\paper Les equations aux differences
lineaires et les integrales des fonctions multiformes
\jour
J.Fac.Sci.Univ. Tokyo
\vol 22
\yr 1975
\endref

\ref
\no16
\by Etingof P.I., Kirillov A.A. Jr.
\paper
Macdonald's polynomials and representations of quantum groups
\paperinfo to appear in Math.Res.Let. , hep-th/9312103
\yr 1994
\endref

\ref
\no 17
\by Etingof P.I., Kirillov A.A. Jr.
\paper
A unified representation theoretic approach to special functions
\paperinfo
Hep-th/9312101.
\endref

\ref
\no18
\by Alexeev A.,Faddeev L., Shatashvili S.
\paper
  Quantisation of symplectic
orbits of compact Lie groups by means of functional integral
\jour Jour. of Geom. and Phys.
\vol 5
\yr 1989
\pages  391-406
\endref

\ref
\no 19
\by Selberg A.
\paper Bemerkninger om et multiplet integral
\jour Norsk. Mat. Tids.
\vol 26
\yr 1944
\pages 71-78
\endref



\ref
\no 20
\by  Kazarnovski-Krol A.
\paper Cycles for asymptotic solutions and Weyl group
\endref

\ref
\no 21
\by  Kazarnovski-Krol A.
\paper Decomposition of a cycle
\endref



\ref
\no22
\by Opdam E.
\paper An analogue of the Gauss summation formula for
hypergeometric functions related to root systems
\paperinfo preprint
\yr July  1991
\endref




\endRefs
\enddocument




