Annales de l’institut Fourier (AIF)

Kurth, Alexandre

${\rm SL}_2$-equivariant polynomial automorphisms of the binary forms

Chemla, Sophie

ANNALES DE L’INSTITUT FOURIER
SOPHIE CHEMLA
Extremal projectors in the semi-classical case
Annales de l’institut Fourier, tome 47, no 5 (1997), p. 1335-1343
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Ann. Inst. Fourier, Grenoble
47, 5 (1997), 1335-1343
EXTREMAL PROJECTORS
IN THE SEMI-CLASSICAL CASE
by Sophie CHEMLA
1. Introduction.
Let 0 be a complex semi-simple finite dimensional Lie algebra, () a
Cartan subalgebra of Q and A the root system associated to (). We will
write A4' (respectively A~) for the set of positive (respectively negative)
roots of A and put p = . S 7- We will denote by B == (o;i, . . . , ai) the
2-YCA+
set of simple roots. Let Q^ be the root space associated to the root 7. We
put
n = 9 0-y, b = i) On ,n-= 9 Q-^.
7GA+ 7CA+
Let R(^)) be the field of rational functions on ()*. One introduces the
algebra U ' (5) = U(o) 0 R(fy). Let us consider the generic Verma module
SW
V = U4—' Zhelobenko ([Zl]) showed that V" = ^(())1+ (where 1+ =
U'[Q)n
1 + U'^n). The decomposition V = n~V C -R((})1+ defines a projector p
onto J?(())l+ called the extremal projector. Inspired by a work of Asherova,
Smirnov and Tolstoy ([AST]), Zhelobenko ([Zl]) showed that p factorizes
into elementary projectors. Let (71,... ,7m) be a normal ordering on the
positive roots. Introduce the following notations:
00 / i\k
^ ^~1) ^ ^
^—ru^-TvT) -V
v l / rfc ekpa ~ ̂ k\ f . e-OLeoL
k=0 K ' J ^ k
fa,0 = I?
if k > 0, f^k = {ha + p(ha) + 1)... (h^ + p(M + k)
Key words: Extremal projectors - Semi-simple Lie algebras.
Math. Classification: 17B20.
1336 SOPHIE CHEMLA
(eg being the root vector associated to the root 6 and hs the coroot). We
have p = p^ .. .p^ ([Zl]). Let w = « i . . . Sj be a reduced decomposition
of w e W (with 5fc = 5^, ^ a simple root ). Put w, = 5i . . . s,. The roots
7z = w^-i(/^) (wo = 1) are pairwise distinct and
A^ = {a e A+ | w-^a) < 0} = {71,... ,^.}.
Put n^ = e 9a- In [Z2], Zhelobenko gives an explicit description of V^.o:eA-u;
We will establish similar results for the symmetric algebra (the so-called
semi-classical case).
Let us consider the analytic manifold (fl/n)*. We will endow it with
the following coordinate system ((e-a)aeA+, (^)ze[i,z]). We will call Us
the open subset of (g/n)* defined by the equation hs ^ 0. We define <^ to
be the following rational map of U^
^\^U^W)=exp(e-6we,Y\\ ^W )
where the dot denotes natural action of n on (fl/n)*. By composition, <^
defines an algebra morphism of A{Ug) which we call 71-5. We put
u^ = u^ n . . . n u^.
We will denote by P(U^) (respectively A(U^)) the set of regular functions
(respectively analytic functions) on U^ and we will write P^U^)^ (re-
spectively A^)"-) the set of invariant functions of P(U^) (respectively
A(U^)) under the action of n^. We prove the following result:
THEOREM. — The algebra morphism TT^ = TT^ o . . . o 71-̂ . does not
depend on the reduced expression of w. It establishes an isomorphism
between
C^=(feA(U^) | ^ - = . , = ^ ^ o l
I <9e-^ 9e_^ J
and A^)"-. Moreover TT^ sends C^ D P(U^) onto P^)"-.
Let N^ be the connected simply connected group whose Lie algebra
is n^. The main ingredient of the proof will be the choice of a point in each
A^-orbit lying in Uyj in accordance with the following proposition:
PROPOSITION. — Let X be in U^. The point ̂ ̂ _,... ̂ (A) is the
unique point of the orbit N^ ' \ whose coordinates e-^,..., e-^. vanish.
EXTREMAL PROJECTORS IN THE SEMI-CLASSICAL CASE 1337
In the appendix, we shall give a factorization for the extremal
projector of the Virasoro algebra in the semi-classical case. Note that the
non commutative case is still open. It is very different from the semi-simple
case because the Virasoro algebra does not admit any normal ordering.
Notations. — Along all this article Q will denote a complex
semi-simple finite dimensional Lie algebra and ( ) ,A ,A+,A_,n ,n_ ,
B = (ai , . . . ,c^) will be as above. Denote by W the Weyl group asso-
ciated to these choices and w its longest element. Let 7 be an element of
A"^ and let /Ly be the unique element of [0/y,fl-^] such that 7(^y) = 2.
If e/y is in 0^, then there exists a unique e_/y such that (/i^,e^,e_^) is a
sl(2)-triple. If a and {3 are two roots, we set [6^,6/3] = C^^e^^ with the
convention that Ca,p is zero if a + (3 is not a root.
The ordering (71,..., 7m) on the positive roots is normal if any
composite root is located between its components. Thus for all posi-
tive roots 7i,7j,7A;, the equality 7/c == 7^ + 7^ implies i ^ k ^ j or
j ^ k ^ i. There is a one to one correspondence between normal or-
derings and reduced expression of w ([Z2]). Let us recall it. Denote by
Si the reflexion with respect to a simple root f3i. If w = 5i . . . Sm'> then
(/3i, si(/?2), . . . , 5 i . . . s^-i(^),.... si .. . Syn-i(An)) are in normal ordering.
If V is a vector space, S(V) will be the symmetric algebra of V.
Lastly, if P is in S(V), S(V)p will be the localization of S(V) with respect
to {P71 | n € N}.
Acknowledgments. — I would like to thank M. Duflo for suggesting
to me to study the semi-classical case and for helpful discussions. I am
grateful to the referee for indicating to me that the main theorem could be
proved in a geometric way.
2. Extremal equations in (fl/n)*.
We consider (fl/n)* as an analytic manifold. We endow it with the
following coordinate system ((e-a)c^A ^^ai)i^\ii]]' If ^ is a positive
root, we will denote by Ug the open subset of (fl/n)* defined by the equation
hs 7^ 0. If U is an open subset for the Zariski topology, we will write A(U)
for the algebra of analytic functions on U and P(U) for the algebra of
regular functions on U. We will define ^s to be the following rational map
1338 SOPHIE CHEMLA
ofUg
VAe^ , ^W=exp(e--6we6}'\.
\ ^W )
By composition, <1><$ defines an algebra morphism of A{Ug) which we call
TT<$. We will denote by Xg the natural action of e<$ on A(Ug). Remark that
Xg is a derivation. If / is in P{Ug)^ we have
00 e^M ̂ =^-^^.f
fc=o °
where e_<$ denotes the multiplication by e_<$. The operator 7r<$ is the
commutative analog of the Zhelobenko's elementary projector.
Let w == 5 i . . . Sj be a reduced decomposition of w C W (with
Sk = 5/3,,, f3k C 5). Put Wz = « i . . . Si. The roots 7^ = Wz-i(^) (wo = 1)
are pairwise distinct and
A^ = { a e A + Iw-^a) < 0} = {71,.. . ,7^-}.
An ordering in A^ is called normal if it coincides with the initial segment
of some normal ordering in A4" (that is compatible with one of the reduced
expression of w). Note that (71,..., 7j) is a normal ordering of A-u;. Put
u^ = n u^
<$eA^
We have
PW=(SW} =5(0)
\ S ( Q ) n j . , \n}
We will denote by Nw the connected and simply connected group whose Lie
algebra is n^ == ® 0a. We will start by proving the following proposition.
aCAu;
PROPOSITION 2.1. — Let X be in Uw- The point <S>^^^_^... ̂ (A) is
the unique point of the orbit Nw -A whose coordinates e-^,..., e-^. vanish.
In particular <^.^._^ . . . ^ does not depend on the normal ordering on
A^.
Proof of Proposition 2.1. — Complete (71,... ,7^) into a normal or-
dering on the positive roots (71,..., 7m)- fl/n is endowed with the ba-
sis (e-^,...,e-^,/iai,...,^cj. Let (el^,... ,eL^,/i^,..., h^) be the
dual basis. We will often identify the point a^e*_^ +.. .+a^e^_ +^1^1 +
EXTREMAL PROJECTORS IN THE SEMI-CLASSICAL CASE 1339
... + bih^ with its coordinates (a^,. . . , a^, & i , . . . , bi). Let us see that
there is a unique point in Nw'\ whose coordinates e_^ , . . . , e-/y. vanish. As-
sume that there are two such points / = (0 , . . . , 0, a^ . _ ^ , . . . , a^, b\,..., bi)
and // = (0 , . . . , 0, a^ ._^, . . . , a^, b [ , . . . , b[). Then there exist complex
numbers (ti , . . . ,^j) such that exp(^ie^ + . . . + tje^) • / = / ' . One can
show easily the following equalities:
e^ ' ^--Yk = '-^,-7fce*-7z-7fc
e^ ' h^ = -h"_^{h^)e\.
From these equalities, one deduces easily that the term in et^ of
exp(^ie^ + ... +^e^.) • (0 , . . . ,0,a^.^,... ,a^,^i,. . . ,bi) is -ti/(/i^).
As / is in £/^, we get t\ = 0. We reproduce the same reasoning to
show that t 2 ^ 3 ? - - - ^ j are zero. So that we have proved that the two
points / and /' coincide. It is not difficult to deduce from the normal
ordering property that ^ sends the point (^,...,^^,2/1,...,^) to
a point (x^,..., x'^_^, 0, x^,..., x^, y ^ , . . . , yi) and that it sends the
point (0,.. . ,0,^,.. . ,^,2/i, . . . ,^) to a point (0, . . . ,0,a^,... ,x'^
^ / i , . . . , yi}. So that <!>^.^._^ ... ̂ ^ (A) is the unique point of Nw ' A whose
coordinates e_^ , . . . , e_^. vanish. This finishes the proof of Proposition 2.1.
As a consequence of the previous proposition, we may write ^ for
the operator ^.<1>^._^ . . . -^. The algebra homomorphism defined by ̂
on A(Uw) will be denoted by 7i\y. Using Proposition 2.1, we will give a
geometric proof of the following result.
THEOREM 2.2. — 1) Ifviw denotes the linear hull of (e-a]^^ , one
has KerTT^ === n^A(L^).
2) The operator TT^ is the projector onto AfJJw)^ with kernel
viwA{Uw) and its restriction to P{Uw) is the projector onto P^w)^ with
kernel viw'P(Uw).
3) The operator TT^ establishes an isomorphism H^ between
C^=(feA{U^) | -)j-=...-af-=o\
[ 9e-^ 9e^ }
and A^U^. Moreover 11^ sends Cw H P(U^) onto P(^)nw. If / is in
A^UU))^ , ^^l(/) is the restriction of f to the subvariety of equations
e-7i = ' " = e-^ = 0.
1340 SOPHIE CHEMLA
Proof of Theorem 2.2. — From the previous proposition, the inclusion
viw^-(Uw) C Ker7i\y is clear. Moreover, a standard reasoning shows that
A(U^} =C^en^A(U^).
Then one sees easily that KerTT^ D C^ = {0}. So that we have n^^t(L^) =
KeTTTw
Let us now show that IniTTu; == A^U^)^ and that TT^ is a projector.
Let a be in Ay;. For any / in A(Uw) and any A in £7^, we have
(X^ o TT^) (/)(A) = ̂ /(^ • • • ̂ lexP(-^)A)|^o .
But for any ^, <1>^ ... ^exp(—^ea)A is the unique point of N^ ' A whose
coordinates e_^ , . . . , e_^. vanish. So that X^OTT^ = 0. We have thus proved
the inclusion Inur^ C A^Uw)^ - Now it is clear that TT^ is a projector :
check that TT^ OTT^ = TT^ on coordinates using the formula (*). The reverse
inclusion A^U^Y^ C ImTr^ will be a consequence of the following lemma.
LEMMA 2.3. — Let k be in [l,j] and let f be in A(U^)' IfX^f = 0,
then Tr^f = /.
Proof of Lemma 2.3. — We first remark that (7r^(e-^),...,
^k^-^-^^—Yk^-rk^—rk+i^"^ ^k^—fm)^ ^i,...,^) is a coordi-
nate system in Uw Indeed, one may see by induction that for any
i ^ k — 1 (respectively i ^ k + 1), e_^ may be expressed as a
regular function of (TT^ (e-^i ) ) • . • ; Tr^ (^-^), / ^a i , . . . , /lo;;) (respectively
(7r^(e-^),...,7r^(e-^),/iai,...,^)). We put (ei , . . . ,€m+i) =
{^k ^—^l) ? • • • 5 ^fc ^—"yfe-l)? e—7fc5 ̂ fe (6—7fc+l)5 • • • 5 ^fc^—^m)? ̂ ai ?• • • I^Oil}'
f\
In these coordinates, we have X^ = h^ -—. So that if X^f = 0, then /
o^k
does not depend on e^ and it becomes clear that there exists g such that
f = TT^^g. As TT^ is a projector, we have TT^/ == TT^TT^^ = ^A;^? which
finishes the proof of the lemma.
It is clear from the proof that TT^ sends Cw D P(Uw) onto ^(Uw)^ -
In particular Tr^-pm—) is the projector onto S(fy)h ...h with kernel
n-P{U^). By analogy to Asherova, Tolstoy, Smirnov and Zhelobenko's
work, we will call it the extremal projector.
Proposition 2.1 gives a geometric interpretation of the projector TT^.
EXTREMAL PROJECTORS IN THE SEMI-CLASSICAL CASE 1341
3. Appendix: Extremal projector
for the Virasoro algebra in the semi-classical case.
In this section, we shall give a factorization of the Virasoro algebra
extremal projector in the semi-classical case. Note that the non commuta-
tive case is still open. It is very different from the semi-simple case because
the Virasoro algebra does not admit any normal ordering. Recall that the
Virasoro algebra Vir is the infinite dimensional Lie algebra generated by
{ei | i C Z} U {c} with commutation rules
[̂ e,} = (j - i) e^, + (J ^ J ) <^oc, [e,, c] = 0.
Vir admits the following triangular decomposition:
Vir = Vir+ C Viro C Vir_
where
Vir+ = C Ce,, Viro = Ceo C Cc, Vir- = C Ce,.
^1 i^-l
We will also use the notation
Vir^+ = ©Ce, and Vir^_ = C Ce,.
i^r ' if^—r
Viry.,+ and Vir^_ are Lie subalgebras of Vir.
Let .R(Viro) be the field of fractions of 5'(Viro). We introduce the
algebra
5'(Vir) = ^(Vir) 0 ^(Viro) = 5" (Vir/Vir-).
5'(Viro)
There is a natural action of Vir- on 5" (Vir/Vir-). Through this action,
for any negative z, e^ defines a derivation Xi of 5' f —lr- ). Set
VVir-y
^ / ^(Vir) ^virr-
r ^'(Vi^Vir-y
The result and the proof of the following lemma is left to the reader.
LEMMA 3.1.
Tr= C -R(Viro)e^...e^-i1.
A;i,...,A^_ieN
1342 SOPHIE CHEMLA
As a consequence of Lemma 3.1, we have the following decomposition:
6" (Vir/Vir-) = Tr C Vir^+5' (Vir/Vir_).
The proof of the next lemma is an easy computation.
LEMMA 3.2. — For any i > 1, the operator
00 / i \ fc
/- _V^ \ 1 ) ^kvk.. _____^_Z±1______p y"^.w-i^i —
is an algebra morphism and satisfies the relations
X-i o TTi = 0 and TT^ o a = 0
(where e^ denotes multiplication by ei).
00
It is not hard to see that the operator Up = ]~[ TT^ is well denned.
z=r
oo
Actually n ^(e?1 • • • e^) = FI ^(e?1 • • • ̂ fc) = 0 (by Lemma 3.2).
i=r r^i^k
THEOREM 3.3. — The operator Hr satisfies the relations
Vz ^ r, X_, o life = 0, life o a = 0.
It is the projector onto Tr with kernel Vir^+5'' ( _-— ).
In particular, 111 is the extremal projector.
Proof of Theorem 3.3. — The relations of the theorem are easy to
check and they prove that IIy. is a projector. To prove that the kernel of IIy.
is Vir^+, we proceed as in the semi-simple case. The inclusion Imlly. C Tr
is a consequence of the theorem. To prove the reverse inclusion, remark
that if x is in Tr, then HrX = x, so that x is in Imll^.
EXTREMAL PROJECTORS IN THE SEMI-CLASSICAL CASE 1343
BIBLIOGRAPHY
[AST] R.M. ASHEROVA, Y.F. SMIRNOV, V.N. TOLSTOI, Description of a class of
projection operators for semi-simple complex Lie algebras, Matem. Zametki,
26, No. 1 (1979), 15-25.
[Zl] D.P. ZHELOBENKO, An introduction to the theory of <S'-algebras over reductive
Lie algebras, Representations of infinite Lie groups and algebras, Gordon and
Breach, New York, 1986.
[Z2] D.P. ZHELOBENKO, Extremal cocycles of Weyl groups, Functional analysis and
its applications, 21, No 3 (1987), 11-21.
Manuscrit recu Ie 14 octobre 1996,
revise Ie 15 juin 1997,
accepte Ie 8 septembre 1997.
Sophie CHEMLA,
Institut de Mathematiques
Universite Paris VI
Case 82
4, place Jussieu
75252 Paris Cedex 05 (France).
schemla@math.jussieu.fr

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${\rm SL}_2$ -equivariant polynomial automorphisms of the binary forms