72
RICE UNIVERSITY STUDIES
nation of all admissible representations. This has been carried out for
simple groups of type А-D. In this manner, such classical invariants as the
“Pfaffian,” the “fundamental cubic form in triality,” etc., and some new
invariants appear in the picture.
Actually, it is expected that the theorem is true for any connected,
semisimple group G defined over ∕<. In fact, it should be possible to prove it
directly without using the classification. At any rate, using the classification
Stephen J. Haris (of our department) is examining the absolutely admissible
representations of exceptional simple groups.
3. Going back to the rational representation ρ: G → Aut(X) of a con-
nected, semisimple group G in a vector space X, all defined over k, we take
a minimal set of G-invariant, homogeneous polynomials, say∕1(x),∕2(x), ,
∕λ(x), in ∕<[X] satisfying
Ω[X]c = Ω[∕1,Λ,-,∕w].
We also fix a non trivial character χ of the adele group kA which takes the
value 1 on k. Furthermore, we shall denote by ∣dx∣4 the Haar measure
on Xa normalized by the condition that XA/Xk is of measure 1. Then,
for any function Φ in the Schwartz-Bruhat space У(ХА), we put
∫' / n ∖
Φ(χ)∙z ∑A(χ)ia* ∙∣⅛.
Xa \я = 1 /
This is called the Eisenstein-Siegel series associated with the morphism
f:X → I(X). We observe that, if this series is absolutely and uniformly
convergent for every Φ in any compact subset of ∙X'(Xa) , the correspondence
Φ → E,(Φ) defines a tempered distribution E' on Xa. Furthermore, this
distribution is intrinsic in the sense that it does not depend on the choice
°f ∕ι >A> "'>‰ ∙
On the other hand, choose a subset X' of X and denote by (X')k the
intersection of X' and Xk. Then, denoting by dμ(g) the Haar measure
on Gλ normalized by the condition that GA/Gk is of measure 1, we put
f'(Φ)= f (' Σ Φ(p(g)'ζ)∖ ' dμ(g),
Jga∣G∣< ∖ξe(x-)lc J
in which Φ is in S^(Xa). If p is admissible over k, the correspondence
Φ → f'(Φ) defines a tempered, positive measure Γ on X. Since it is expected
that, for any absolutely admissible representation ρ, G operates very nicely
on X, we will take as X' the union of all G(i). With this understanding,
we shall propose the following conjecture: