ijectures (5.1)
SOME RESULTS ON INVARIANT THEORY
;s his gratitude
sions.
Y,
BY S. HELGASON
1
Communicated by Felix Browder, April 11, 1962
1. Symmetric invariants. Let V be a finite-dimensional vector
space over R. Each XE V gives rise (by parallel translation) to a
Math. Ann. 69
vector field on V which we consider as a differential operator d(X) on
'r in groups with a
V. The mapping X-->(X) extends to an isomorphism of the complex
symmetric algebra S(V) over V onto the algebra of all differential
eiseitigen Flachen.
operators on V with constant complex coefficients. Let G be a sub-
lath. Gesellschaft
group of the general linear group GL(V). Let I(V) denote the set
of G-invariants in S(V) and let I+(V) denote the set of G-invariants
without constant term. The group G acts on the dual space V* of V by
Soc. (3) 7 (1957),
of Math. (2) 66
(g. v*)(v) = v*(g-l'v),
er. Math. Soc. 64
g E G, v
V, v*
V*
and we can consider S(V*), I(V*), I+(V*). An element p
S(V*) (a
polynomial function on V) is called G-harmonic if (J)p=0 for each
, Leipzig, 1934.
JC-I+(V). Let H(V*) denote the set of G-harmonic polynomial functions.
Let VCdenote the complexification of V. Suppose B is a nondegenerate symmetric bilinear form on VX V. If XC VC let X* denote the linear form Y--B(X, Y) on V. The mapping X--X* extends
to an isomorphism P--P* of S(V) onto S(V*). If G leaves B invari-
!erthan four, Ann.
. Math. Soc. 66
. Math. Mech. 10
ler. Math. Soc. 67
ant then I( V)*= I(V*).
i
I
i
We shall use the following notation: If E and F are linear subspaces of the associative algebra A then EF denotes the set of all
sums
i eif,, (eiGE,fi F).
THEOREM
1. Let B be a nondegenerate symmetric bilinear form on
VX V and let G be a Lie subgroupof GL( V) leaving B invariant. SupI
i
i
pose that either (1) G is compact and B positive definite or (2) G is
connected and semisimple. Then
S(V*) = I(V*)H(V*).
The case of a compact G was noted independently by B. Kostant.
It is a simple consequence of the fact that under the standard strictly
positive definite inner product on S(V*) (invariant under G), the
space H(V*) is the orthogonal complement to the ideal in S(V*)
generated by I+( V*). For the noncompact case, let g denote the complexification of the Lie algebra of G. It is not difficult to prove that
'This work was supported by the National Science Foundation, NSF G-19684.
367
21 a
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orm W of Vc
ct case can be
ing on V= Rn
I 962]
Let NG denote
. Then H(V*)
369
group U(n) becomes a subgroup G of the orthogonal group 0(2n).
Let Zk = Xk +i Yk (1 _ k _ n) be an orthonormal basis of E and let
xl, y1, · · · , xn, yn be the basis of V* dual to the basis X1, Y1, · · ,
X,, Yn of V. It is easy to show that the element
.2 and H(V*)
rem 1 reduces
n be written
is also known
is in this case
·e a, · - · , an
ig generaliza-
SOME RESULTS ON INVARIANT THEORY
n
u = E xk A yk
2
1
and its powers form a basis of J+(V*). In view of Theorem 3 each
vGA(V*) can therefore be written
v=
E uk A Pk,
k
where each Pk satisfies
p. 26]).
(u)k=0,
(compare Weil [10, Theoreme 3,
.3. Invariants of Weyl groups. Let u be an arbitrary semisimple Lie
algebra over R whose adjoint group U is compact. Let 0 be an arbitrary involutive automorphism of u and let u = f+p be the decomposiomials (X*)k,
tion of u into eigenspaces of 0 for the eigenvalue +1 and -1 respec-
armonic poly-
tively. Let K denote the analytic subgroup of U corresponding to .
s Nullstellen-
Let bpbe a maximal abelian subspace of and extend fp to a maximal
of ut. The Weyl group of is defined as the
abelian subalgebra
group of linear transformations of
in U which leave
Lively, denote
and V*. Each
3y
kA.
A x,)
induced by the set of elements
invariant; the Weyl group of fp is defined as the
group of linear transformations of fp induced by the set of elements
in K which leave bp invariant. Let these groups be denoted by
W(b))and W(bp) and let I(b*) and I(tb*) denote the corresponding
sets of invariant polynomial functions. It is known that W(bp) can
be described as the group of linear transformations of bpinduced by
those members of W(b) which leave b%invariant. Consequently, if
the restriction to bpof a function f on
is denoted by , the mapping
,8(X) extends
all endomorLet J(V) and
f--f mapsI(1*)intoI(p).
, respectively,
rant term. An
and 0 any involutive automorphism of u. Then the restriction mapping
Lch JGJ+(V).
'THEOREM
4. (i) Suppose u is a classicalcompact simple Lie algebra
f-f maps1(*) ontoI(*).
(ii) Part (i) does not hold in general for the exceptional simple Lie
algebras u = e, e7, e.
Then
(iii) Let Q(b*) and Q(b*), respectively, denote the set of invariant
rational functions on
and bp. Under the restriction mapping f--f,
Q(b*) is mapped onto Q(*p).
over C. Con-
R the unitary
+ip is the most general semisimple Lie algebra over R. Parts (i) and (ii) above therefore express
IREMARKS.
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NOSVDYIH 'S
O04
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yebras over R,
t (i) is proved
'oot structures
.ps W(Ib) and
and I(D*) are
,enerators can
nd that if the
the ring I(t)
respectively,
371
SOME RESULTS ON INVARIANT THEORY
I962]
for q= 1 because 0(1, 1) is not semisimple and the theorem fails to
shows.
hold for (p, q)=(1, 1) as the example f(xi, x2) =cosh-'(lxjl)
The case (p, q)= (1, 2) was settled by Loewner [8] using special fea-
tures of the Lobatchefsky plane.
3. By a method of descent the remaining cases can be reduced to
the case x +x2-A3 = 1 (which differs radically from the case x --x
-x 3 = 1 by the noncompactness of the isotropy group). Here one can
make use of the special property of the identity component of the
group 0(2, 1), namely that every representation of it extends to a
Part (iii) had
representation of the corresponding complex subgroup of GL(3, C),
(see Harish-Chandra [5]).
a topological
4. From Theorem 1 it is clear that the polynomial P can be taken
to be an 0(p, q)-harmonic polynomial, that is a polynomial satisfying the equation
s gH with the
ts function on
/ r2
.ven by fz(gH)
zl (Cartan
[3,
the functions
d2
1...-[.
Ox1
042
2
02
Ox 2+ 1
\
P=0.
.
X
It follows that the function f is necessarily a sum of eigenfunctions
of the Laplace-Beltrami operator on C,,q (formed by means of the
indefinite Riemannian metric on C.,,,, [7]).
(P
3
~! 1 q -z= ).
of RP+ q leav-
n C,q and the
to (p-1, q)
BIBLIOGRAPHY
1. . Cartan, Sur certaines formes riemanniennes remarquables des gomtries a
groupefondamental simple, Ann. Sci. Acole Norm. Sup. 44 (1927), 345-467.
2.
, Lemons sur la gomdtrie projective complexe, Gauthier-Villars, Paris,
1931.
, Sur la determination d'un systeme orthogonalcomplet dans un espace de
3.
Riemann symdtrique dos, Rend. Circ. Mat. Palermo 53 (1929), 217-252.
4. C. Chevalley, Invariants of finite groups generatedby reflections,Amer. J. Math.
)mial then the
:he other hand
Cp,,q. Assume
Xi,
'' , Xp+q)
on C,q.
q) is compact)
77 (1955), 778-782.
5. Harish-Chandra, Lie algebrasand the Tannaka duality theorem,Ann. of Math.
51 (1950), 299-330.
6. E. Hecke, Uber orthogonalinvarianteIntegralgleichungen,Math. Ann. 78 (1918),
398-404.
7. S. Helgason, Some remarks on the exponential mapping for an affine connection,
Math. Scand. 9 (1961), 129-146.
8. C. Loewner, On some transformation semigroups invariant under Euclidean and
non-Euclidean isometries,J. Math. Mech. 8 (1950), 393-409.
9. H. Maass, Zur Theorie der harmonischen Formen, Math. Ann. 137 (1959) 142149.
10. A. Weil, Varitts Kahleriennes, Hermann, Paris, 1958.
MASSACHUSETTS
d by use of a
_ie algebra
of
procedure fails
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