74
RICE UNIVERSITY STUDIES
4. Before we explain the example that we mentioned in the beginning,
we shall point out one of the difficulties that we have encountered in proving
the Siegel formula. For the sake of simplicity, assume that there is only
one G-invariant /(x). Let ku denote the completion of к with respect to
an arbitrary valuation v on к and χυ the и-component of the global charac-
ter χ. Let I dx ∣v denote a Haar measure on Xu = Xk ®k kυ and Φo a function
in the Schwartz-Bruhat space У(Х„). Consider the function F* on kυ
defined by
= £ Φ0(χ)∙χυ(∕(χ)i*)∙pχ∣u.
The problem is, then, to find good “asymptotic estimates” of Fu*(J*)
for all V and the “asymptotic expansions to 2 terms” of Fv*(i*) for almost
all v, both as i* tends to ∞ in ku. It is important that Φu is not restricted
to any subspace of У (Xu) ■ As we know, this becomes a problem only when
the degree of /(x) is at least 3. It appears that a solution of this problem
is the key to settling various conjectures that we have enunciated.
As we have remarked at the end of Section 1, there are essentially two
infinite sequences of new absolutely admissible representations. One of
them consists of the second fundamental representations for Worms G of
SLm for in = 2,3,∙∙∙. For the sake of simplicity, assume that G = SLm over
к and, disregarding the trivial case of an odd m, assume that m = 2n. In
this case, let X denote the vector space of alternating matrices of degree 2n
and put p(g)∙x = gx,g for g in G and x in X. Then, the Pfaffian Pf(x)
of x gives the morphism Pf : X → I(X) = Ω. Furthermore, we have U(i) =
(P∕)'^1(i) for every i ≠ 0. Let Ur denote the set of all x in X satisfying
rank(x) = 2r for 0 ≤ r ≤ n. Then Ur is a G-orbit for r ≠ n and U„_ ɪ = U(O).
In this case, we have a complete Siegel formula. Let ωr(x) denote a gauge-
form defined over к on the homogeneous space Ur and ∣ωr(x)∣4 the cor-
responding measure on (Ur)λ for r ≠ n. Then, we have
f ( Σ Φ(g<fg)} ∙ dμ(g) = Σ f Φ(x)∙z(P∕(x)<*)∙∣dx∣Λ
jGyl∕Gk∖ξ eXk J i*ek JXa
n-2 f∙
+ Σ Φ(x)∙ ∣ωr(x)∣Λ
r = O J(Ur-)A
for every Φ in S^(Xa) . It might be amusing to observe that, in the special
case when л = 1, if we identify x with its (l,2)-coefficient, this becomes
the Poisson formula for the adele group kA (or, the “Thetaformel” in
non adelic form).