The name is absent

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 k_u denote the completion of к with respect to
an arbitrary valuation v on к and χ_υ the и-component of the global charac-
ter χ. Let I dx ∣_v denote a Haar measure on X_u = X_k ®_k k_υ and Φ_o a function
in the Schwartz-Bruhat space У(Х„). Consider the function F* on k_υ
defined by

= £ Φ₀(χ)∙χ_υ(∕(χ)i*)∙pχ∣_u.

The problem is, then, to find good “asymptotic estimates” of F_u*(J*)
for all V and the “asymptotic expansions to 2 terms” of F_v*(i*) for almost
all v, both as i* tends to ∞ in k_u. It is important that Φ_u is not restricted
to any subspace of У (X_u) ■ 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
SL_m for in = 2,3,∙∙∙. For the sake of simplicity, assume that G = SL_m 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 U_r denote the set of all x in X satisfying
rank(x) = 2r for 0 ≤ r ≤ n. Then U_r is a G-orbit for r ≠ n and U„_ ɪ = U(O).
In this case, we have a complete Siegel formula. Let ω_r(x) denote a gauge-
form defined over к on the homogeneous space U_r and ∣ω_r(x)∣₄ the cor-
responding measure on (U_r)_λ for r ≠ n. Then, we have

f ( Σ Φ(g<fg)} ∙ dμ(g) = Σ f Φ(x)∙z(P∕(x)<*)∙∣dx∣Λ
jG_yl∕G_k∖ξ eX_k J           i*ek JX_a

n-2 f∙

+ Σ Φ(x)∙ ∣ω_r(x)∣Λ

r = O J(U_r-)_A

for every Φ in S^(X_a) . 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 k_A (or, the “Thetaformel” in
non adelic form).

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