GENERALIZATION OF AN INTEGRAL OF HUA LOO-KENG

GUY ROOS

In his book “Harmonic Analysis in Classical Domains”, Hua Loo-keng has computed integrals of the following type

∫_{ℜ_{I}} det(I − Z\overline{Z}′)^{λ} Z^{.},

where

ℜ_{I} = \{Z ∈ M_{m,n}(ℂ)|I_{m} − Z^{t}\overline{Z} >> 0\}

is the “generalized unit ball” of m × n complex matrices. Analogous integrals have been computed by Hua for all classical bounded symmetric domains. For classical real simple Lie groups, similar integrals have been computed recently by Yu. Neretin.

If Ω is any bounded irreducible complex symmetric domain (including the two exceptional cases), the true generalization of these integrals appears to be

J(s) = ∫_{Ω} N(x, x)^{s}α^{n},

where N(x, x) denotes the generic norm of Ω and α is a suitable volume form . We will call J(s) the Hua integral. Using an integral of Selberg and the presentation of bounded symmetric domains via Jordan triple systems, we show that J(s) is, as a function of s, the inverse of polynomial χ(s), with integer or half-integer coefficients, whose degree is the complex dimension of Ω.

The main points of the theory of Jordan triple systems will be recalled.

We are asking about an interpretation of the polynomial χ, which is related with restricted root systems of type BC_{r}, and seems to carry all the information about the corresponding symmetric domain.

191024 S. Petersburg, Nevski pr. 113/4-53 Russia

   E-mail address: roos@mx.amss.ac.cn

Date: February 2003.

1991 Mathematics Subject Classification. 32H10.

Key words and phrases. Cartan domain, Bergman kernel, holomorphic automorphism group, bounded symmetric domain.