A COMBINATORIAL CONJECTURE RELATED WITH COMPLEX BOUNDED SYMMETRIC DOMAINS

GUY ROOS

1. Motivations

Let Ω be a bounded irreducible symmetric domain. To each such Ω are attached numerical invariants: the multiplicities a and b, and the rank r. These three invariants characterize the domain up to isomorphism. Hereunder is the table of all possible values, corresponding to the classification of irreducible complex bounded symmetric domains:

\begin{matrix}Type & a & b & r & g & d \\ I_{m,n} (1 ≤ m ≤ n) & 2 & n − m & m & m + n & mn \\ II_{n} (n ≥ 2) & 4 & \{ \begin{matrix}0 (n = 2p) \\ 2 (n = 2p + 1) \end{matrix} & [\frac{n}{2}] & 2 (n − 1) & \frac{n(n−1)}{2} \\ III_{n} (n ≥ 1) & 1 & 0 & n & n + 1 & \frac{n(n+1)}{2} \\ IV_{n} (n > 2) & n − 2 & 0 & 2 & n & n \\ V & 6 & 4 & 2 & 12 & 16 \\ V I & 8 & 0 & 3 & 18 & 27 \end{matrix}

In the last two columns of this table

g = 2 + a(r − 1) + b

is the genus of the bounded symmetric domain Ω and

d = dim_{ℂ} Ω = r + \frac{r(r − 1)}{2}a + rb

its complex dimension.

The Hua polynomial χ of the bounded symmetric domain Ω is then defined by

(1) χ(s) = ∏_{j=1}^{r} (s + 1 + (j − 1)\frac{a}{2})_{1+b+(r−j)a} ,

where (s)_{k} denotes the raising factorial

(s)_{k} = s(s + 1) ⋯ (s + k − 1) = \frac{Γ (s + k)}{Γ (s)}.

The polynomial χ is related to the Hua integral ∫_{Ω} N(z, z)^{s}ω(z) (where N is the generic norm for Ω) by

∫_{Ω} N(z, z)^{s}ω(z) = \frac{χ(0)}{χ(s)} ∫_{Ω} ω (s > −1).

(see [1]).

One checks easily that deg χ = d. The expression of χ for the different types of bounded symmetric domains is as follows:

Date: 5th December 2004.

• Type I_{m,n} (1 ≤ m ≤ n): χ(s) = ∏_{j=1}^{m}(s + j)_{n}.

• Type II_{2p}: χ(s) = ∏_{j=1}^{p}(s + 2j − 1)_{2p−1}.

• Type II_{2p+1}: χ(s) = ∏_{j=1}^{p}(s + 2j − 1)_{2p+1}.

• Type III_{n}: χ(s) = ∏_{j=1}^{n} (s + \frac{j+1}{2})_{1+n−j} .

• Type IV_{n}: χ(s) = (s + 1)_{n−1} (s + \frac{n}{2}) .

• Type V : χ(s) = (s + 1)_{8}(s + 4)_{8}.

• Type VI : χ(s) = (s + 1)_{9}(s + 5)_{9}(s + 9)_{9}. Let μ ∈ ℝ, μ > 0. The following expansion

(2) \frac{χ(kμ)}{χ(0)} = ∑_{j=0}^{d} c_{μ,j} \frac{(k + 1)_{j}}{j!} .

and the associated rational function

(3) F_{χ,μ}(t) = ∑_{j=0}^{d} c_{μ,j} (\frac{1}{1 − t})^{j}

play a key role in the computation of the Bergman kernel of some Hartogs domains built over bounded symmetric domains (see [1], [2]).

2. The conjecture

Let

μ_{0} = \frac{g}{d + 1}.

Conjecture. The coefficients c_{μ,j} in (2):

\frac{χ(kμ)}{χ(0)} = ∑_{j=0}^{d} c_{μ,j} \frac{(k + 1)_{j}}{j!} .

are all strictly positive if and only if

μ < μ_{0}.

For μ = μ_{0}, all coefficients c_{μ,j} in (2) are strictly positive, except c_{μ,d−1} = 0 and except for the rank 1 type I_{1,n} (where c_{μ,d} = 1 and c_{μ,j} = 0 for all j < d).

We call μ_{0} the critical exponent for Ω. The values of the critical exponent are

\begin{matrix}Type & I_{m,n} & II_{n} & III_{n} & IV_{n} & V & V I \\ μ_{0} & \frac{m+n}{mn+1} & \frac{4}{n+ \frac{2}{n−1}} & \frac{2}{n+ \frac{1}{n+1}} & \frac{n}{n+1} & \frac{12}{17} & \frac{9}{14} \end{matrix}

We have always μ_{0} < 1, except in the rank 1 case I_{1,n}.

Remark 1. The conjecture has been checked with help of computer algebra software in many significant cases:

• for μ = μ_{0} and the types I_{3,3}, IV_{3}, IV_{4}, IV_{6}, V, V I;

• for type V and all values of μ.

Remark 2. The coefficients c_{μ,j} are polynomial functions of μ and linear combinations of

{χ(kμ) | k = −1, −2, …, −j − 1} .

The conjecture is equivalent to the following properties:

• the least positive root of μ ↦ c_{μ,d−1} is μ_{0};

• all positive roots of μ ↦ c_{μ,j} (0 ≤ j < d − 1) are greater than μ_{0}.

Remark 3. As the function F_{χ,μ} is related to the Bergman kernel of a family of bounded (non homogeneous) domains, it is known that all derivatives F_{χ,μ}^{(k)} (k > 0) of this function are strictly positive on [0, 1[ for all μ > 0. The conjecture says more about this function when 0 < μ < μ_{0}.

Remark 4. If the conjecture is true, it would allow to compare the Bergman metric of some Hartogs domains built over bounded symmetric domains, with the Kähler-Einstein metric of the same domains.

3. Examples

3.1. Type V (the exceptional case of dimension 16). The critical exponent is

μ_{0} = \frac{12}{17}.

The Hua polynomial χ is

χ(s) = (s + 1)_{8}(s + 4)_{8}.

A computation with Maple gives

χ (\frac{12}{17}s) = (\frac{12}{17})^{16} ∑_{j=0}^{16} c_{j}(s + 1)_{j},

with

c_{16} = 1,  c_{15} = 0,  c_{14} = \frac{595}{12},  c_{13} = \frac{4165}{6},  c_{12} = \frac{30042145}{3456},
c_{11} = \frac{14448385}{144},  c_{10} = \frac{790269316375}{746496},  c_{9} = \frac{1259425781075}{124416},

c_{8} = \frac{12447571001586875}{143327232}, c_{7} = \frac{2957566710311675}{4478976},

c_{6} = \frac{11300125622942496725}{2579890176}, c_{5} = \frac{10677213117341703625}{429981696},

c_{4} = \frac{65190770448545396318125}{557256278016}, c_{3} = \frac{10209484788366056549125}{23219011584},

c_{2} = \frac{114818904611324955416375}{92876046336}, c_{1} = \frac{35779252854815307462625}{15479341056},

c_{0} = \frac{33368892412222545303125}{15479341056}.

This shows that all coefficients c_{μ_{0},j} (0 ≤ j ≤ 16) in

\frac{χ(kμ)}{χ(0)} = ∑_{j=0}^{16} c_{μ,j} \frac{(k + 1)_{j}}{j!}

are strictly positive, except c_{μ_{0},15} = 0.

4 GUY ROOS

3.2. Type V I (the exceptional case of dimension 27). The critical exponent is

μ_{0} = \frac{9}{14}..

The Hua polynomial χ is

χ(s) = (s + 1)_{9}(s + 5)_{9}(s + 9)_{9}. A computation with Maple gives

χ (\frac{9}{14}s) = (\frac{9}{14})^{27} ∑_{j=0}^{27} c_{j}(s + 1)_{j},

with

c_{27} = 1, c_{26} = 0, c_{25} = \frac{2275}{9},
c_{24} = \frac{56875}{9},  c_{23} = \frac{38591735}{243},
c_{21} = 15425515970150/3^{11},
c_{20} = 37061881356500/3^{9},
c_{19} = 184328710104188650/3^{14},
c_{18} = 3564334218619774600/3^{14},
c_{17} = 584735324681177419750/3^{16},
c_{16} = 10020732894163060819750/3^{16},
c_{15} = 352001611351295587864253500/3^{23},
c_{14} = 586664566244061492395923000/3^{21},
c_{13} = 1988637252859632373297511212000/3^{26},
c_{12} = 25672251717038124392289396301000/3^{26},
c_{11} = 8233663487061605972803486331644375/3^{29},
c_{10} = 89384793443821000370862374382625000/3^{29},
c_{9} = 1923754293102540042201539198326959366875/3^{36},
c_{8} = 209778908005712588859591649123533801875/3^{32},
c_{7} = 399192552377373476318550395682751432975625/3^{37},
c_{6} = 2728484170046421839052459199725228012518750/3^{37},
c_{5} = 47840351197962492631409316902739852226831250/3^{38},
c_{4} = 232468257762517753158460641861539626710125000/3^{38},
c_{3} = 73037107041363504672642146434776735686797778125/3^{42},
c_{2} = 23557400955895564936769134062033297681918662500/3^{40},
c_{1} = 409456797752799914624225389536137199476376953125/3^{42},
c_{0} = 394594700340674453245747775040231797415576953125/3^{42}.

This shows again that all coefficients c_{μ_{0},j} (0 ≤ j ≤ 27) in

F_{χ,μ_{0}}(t) = ∑_{j=0}^{27} c_{μ_{0},j} (\frac{1}{1 − t})^{j}

are strictly positive, except c_{μ_{0},26} = 0.

References

[1] Yin Weiping, Lu Keping, Roos Guy, New classes of domains with explicit Bergman kernel, Science in China Ser. A Mathematics, 47(2004), 352–371.

[2] Roos, Guy, Weighted Bergman kernels and virtual Bergman kernels, Proceedings SCV2004 Beijing, to appear. Jordan archive, Preprint #169.

[3] Wang An, Yin Weiping, Zhang Liyou, Roos Guy, The Kähler–Einstein metric for some Hartogs domains over bounded symmetric domains, Preprint, 2004.

Nevski prospekt 113/4-53, 191024 St Petersburg, Russian Federation

   E-mail address: guy.roos@normalesup.org