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SOME OBSERVATIONS ON THE SIEGEL FORMULA

A General Siegel Formula. If ρ ;G → Aut(A^r) is an absolutely admissible
representation of a connected, simply connected, senιisimple group G in
a vector spade X, all defined over к, Γ and E' are both defined and they
are equal.

Actually, the assumption that G is simply connected may not be necessary.
The above conjectural formula is, so to speak, the main part of a complete
Siegel formula, and it consists of I' = E', its analogues for subrepresen-
tations of p, and of identities of lesser significance, if not trivial. As for
its proof, we can reduce the general case to the case where G is absolutely
simple. Therefore, it is enough to verify the conjecture case-by-case
using the classification. At any rate, if this is true, we will get a rather natural
domain of validity for all Siegel formulas.

On the other hand, in the case when there is only one G-invariant f(x),
we can consider the zeta-function of the morphism f. We recall that such
a zeta-function, where f{x) is a quadratic form, has been investigated in
detail by Siegel [6]. Also, Ono [d] has examined a similarly defined zeta-
function in the case where /(x) is an arbitrary polynomial. Now, suppose
that ∕^-1(i) is a G-orbit for every i ≠ O. Let dx denote the translation-
invariant gauge-form on X defined over к and put

θ(x) ≈f(_xγ^κ∙dx,

in which κ-deg(∕) = dim(X). Let A = (λ_p) denote the standard “con-
vergence factor’’ of Y = X- ∕^-1(0), i.e., λ_u = (1 — (<f_υ)^-1)~¹ with q_υ
denoting the number of elements in the residue field of к for each non archi-
median valuation v on k. Then, we get the so-called Tamagawa measure
I λθ(x) ∣₄ on Y_a . Let Φ denote an arbitrary function in -IYfX _a) and s a complex
variable. Then, the integral

z(_s,Φ) = ɛ Φ(χ)∙∣∕ω∣^sA∙μθ(x)∣_x

defines a holomorphic function in Re(s) > κ. It is expected that this function
has an analytic continuation to the entire s-plane as a meromorphic func-
tion with eventual poles of order 1, the number of which is twice the number
of G-Orbits in ≠¹(t>), and satisfies the functional equation

Z(s,Φ) = Z(κ — s, Φ*),

in which Φ* is the Fourier transform of Φ on X_λ with respect to the nor-
malized measure ∣ dx 1₄.

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