Polynomial functions on Jordan pairs

Ottmar Loos

Institut für Mathematik, Universität Innsbruck Technikerstr. 25/7, A-6929 Innsbruck, Austria

In a recent preprint [2], F. Vandebrouck proved by a case-by-case verification that a certain rational function on hermitian Jordan triple systems, defined in terms of the generic norm, is in fact a polynomial. We give an elementary proof of this fact which is valid for all finite-dimensional Jordan pairs over arbitrary fields.

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We will need some basic facts on generic points and polynomial and rational functions on a finite-dimensional vector space, say V, over a field k. To keep the exposition simple, we fix a basis b_{1}, …, b_{n} of V and thus identify V with k^{n}. A basis-free approach may be found in [1, §18]. Let X = (ξ_{1}, …, ξ_{n}) be an n-tupel of indeterminates. The polynomial ring k[X] = k[ξ_{1}, …, ξ_{n}] in n variables may be identified with the ring of polynomial functions on V, and the n-tuple X with the generic point of V. Its quotient field k(X) is then the field of rational functions on V. By a rational map on V with values in a finite-dimensional k-vector space W we mean an element f = f(X) ∈ W ⊗_{k} k(X). Since k[X] is a factorial ring, f admits a reduced expression f(X) = g(X)/h(X) where the numerator g(X) ∈ W ⊗ k[X] and the denominator h(X) ∈ k[X], also called an exact denominator of f, have greatest common divisor 1 and are uniqe up to a nonzero scalar in k. All this has obvious extensions to the case where V is replaced by a finite direct product V_{1} × V_{2} × ⋯ with generic point (X_{1}, X_{2}, …).

Let now V = (V^{+}, V^{−}) be a finite-dimensional Jordan pair over k, with X = (ξ_{1}, …, ξ_{m}) the generic point of V^{+} and Y = (η_{1}, …, η_{n}) the generic point of V^{−} (the dimensions of V^{+} and V^{−} need not be the same). Then k(X, Y) is the field of rational functions on V^{+} × V^{−}. The pair (X, Y) is quasi-invertible in V ⊗_{k} k(X, Y) because det B(X, Y) ≠ 0 in k(X, Y). Indeed, det B(X, Y) is a polynomial and det B(0, 0) = 1. Hence (X, Y) is quasi-invertible, and the quasi-inverse X^{Y} is a rational map from V^{+} × V^{−} to V^{+}. The exact denominator N(X, Y) ∈ k[X, Y] of X^{Y}, normalized by N(0, 0) = 1, is the generic norm of V [1, 16.9].

Consider V^{+} × V^{−} × V^{+} × V^{−}, with generic point (X, Y, Z, T). We claim that

N_{4}(X, Y, Z, T) = N(X, Y)N(X^{Y} + Z, T)  (1)
= N(Z, T)N(X, Y + T^{Z})  (2)
= N(X, Y)N(Z, T)N(X^{Y} , T^{Z})  (3)

is a polynomial function on V^{+} × V^{−} × V^{+} × V^{−}.

Indeed, by the cocyle relations for the generic norm [1, 16.11.1, 16.11.2] and quasiinvertibility of (X, Y) and (Z, T) in V ⊗ k(X, Y, Z, T) we have

\begin{aligned}N(X, Y + T^{Z}) & = N(X, Y)N(X^{Y} , T^{Z}), \\ N(X^{Y} + Z, T) & = N(Z, T)N(X^{Y} , T^{Z}). \end{aligned}

By multiplying these equations with N(Z, T) and N(X, Y), respectively, we see that the right hand sides of (1)–(3) are equal. Now (1) shows that N_{4}(X, Y, Z, T) ∈ k(X, Y)[Z, T] is a polynomial in (Z, T) with coefficients in k(X, Y), while (2) shows, similarly, that N_{4}(X, Y, Z, T) ∈ k(Z, T)[X, Y]. Thus

N_{4}(X, Y, Z, T) ∈ k(X, Y)[Z, T] ∩ k(Z, T)[X, Y] = k[X, Y, Z, T]

is indeed a polynomial in all variables.

Next, consider the rational function

f(X, Y, Z, W, T) = N(X, Y)N(Z, W)N(X^{Y} + Z^{W} , T)  (4)

on V^{+}×V^{−}×V^{+}×V^{−}×V^{−} (with generic point (X, Y, Z, W, T)). This, too, is a polynomial: By symmetry in (X, Y) and (Z, W) we have

f(X, Y, Z, W, T) = N(Z, W)N_{4}(X, Y, Z^{W} , T) = N(X, Y)N_{4}(Z, W, X^{Y} , T).

Since N_{4} is a polynomial, this shows that

f(X, Y, Z, W, T) ∈ k(Z, W)[X, Y, T] ∩ k(X, Y)[Z, W, T] = k[X, Y, Z, W, T]

is a polynomial in all five variables.

Let now V be a finite-dimensional complex Jordan pair and let ¯ : V^{σ} → V^{−σ} be a an involution of the second kind, i.e., a complex antilinear map which is an involution (in the sense of [1, 1.13]) of the underlying real Jordan pair (_{ℝ}V^{+},_{ℝ}V^{−}). We recall here that Jordan pairs with involution of the second kind are in one-to-one correspondence with complex Jordan triple systems whose triple product is antilinear in the middle variable; the triple system associated to (V,¯ ) being V^{+} with triple product {xȳz}. F. Vandebrouck [2, Prop. 4] proved for semisimple Jordan triple systems by a case-by case verification that the function

|N(x, ȳ)|^{2}N ((xȳ + yx̄), z̄)  (5)

(where x, y, z ∈ V^{+}) is a polynomial on (_{ℝ}V^{+})^{3}. This is now an easy consequence of (4). Indeed, from [1, 16.10.1] and the fact that the involution is antilinear, it is easy to see that the generic norm of V satisfies \overline{N(x, ȳ)} = N(y, x̄). Hence |N(x, ȳ)|^{2} = N(x, ȳ)N(y, x̄), and thus |N(x, ȳ)|^{2}N([xȳ + yx̄], z̄) = f(x, ȳ, y, x̄, z̄) is a polynomial on (_{ℝ}V^{+})^{3}.

References

[1] O. Loos, Jordan pairs, Lecture Notes in Math., vol. 460, Springer-Verlag, 1975.

[2] F. Vandebrouck, The Poisson-Furstenberg kernel of a bounded symmetric domain, preprint, 1999.