Построение многочленов с локальными функциональными уравнениями, не являющихся преоднородными, с помощью представлений клиффордовых алгебр

Сато Фумихиро; Когисо Такуйоши

MSG 15A66, 11S90

Construction of non-prehomogeneous polynomials with local functional equations from representations

of Clifford algebras

Rikkvo University, Tokyo and Josai University, Saitama,

Japan

We give a new systematic construction of polynomials satisfying a functional equation, which are not relative invariants of prehomogeneous vector spaces. The construction depends on the pull back theorem of local functional equations

Keywords: homogeneous polynomials, prehomogeneous vector spaces, invariants, Euclidean (formally real) Jordan algebras, Clifford algebras, representations

1. Introduction

Let P and P* be homogeneous polynomials in n variables of degree d with real coefficients. It is an interesting problem both in Analysis and in Number theory to P P*

speaking, of the form

the Fourier transform of |P(x)|s = Gamma factor x |P*(y)|-n/d-s. (1)

A beautiful answer to this problem is given by the theory of prehomogeneous vector

P P*

prehomogeneous vector space and its dual, respectively, and if the character x and X* corresponding to P and P*, respectively, satisfy the relation xx* = 1 then, P P*

satisfactorily and it might give an impression that prehomogeneous vector spaces are the final answer to the problem.

Meanwhile, in [2], Faraut and Koranvi developed a method of constructing polynomials with property (1), starting from representations of Euclidean (formally real) Jordan algebras. What is remarkable in their result is that, from representations of simple Jordan algebras of rank 2, one can obtain a series of polynomials satisfying (1), which are not covered by the theory of prehomogeneous vector spaces. The result was later generalized by Clerc [1]. Thus we got to know that the class of

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polynomials with the property (1) is broader than the class of relative invariants of regular prehomogeneous vector spaces.

We want to get more polynomials with property (1), which are not relative invariants of prehomogeneous vector spaces. In this note, we give a new systematic construction of polynomials with property (1), which includes the results of Faraut and Koranvi as a special case. The construction depends on the pull back theorem of local functional equations, which we explain in the next paragraph.

§ 2. Pull back of local functional equations

Let V be a real vector space with dimension n and W a real vector space with dimension m with m > n. We put V* (resp. W*) be the vector space dual to

V (resp, W), Suppose that we are given quadratic mappings Q : W —y V and Q* : W* —y V*. The mappings Bq : W x W —y V and Bq* : W* x W* —y V*

defined by

Bq(Wi,W2) := Q(wi + W2) - Q(wi) - Q(W2),

Bq* (w*,w*) := Q*(w* + w*) - Q*(w*) - Q*(w*)

are bilinear. For given v G V and v* G V*, the mappings Qv* : W —y R and QV : W —y R defined by

Qv* (w) = (Q(w),v*), QV (w*) = (v,Q*(w*))

are quadratic forms on W and W*, respectively. Denote by SQ(v*) (resp. Sq* (v)) the symmetric matrix associated with the quadratic form Qv*(w) (resp. QV(w*)). Let P (resp. P*) be a R-irreducible homogeneous polvnomial on V (resp. V*), We put Q := {v G V | P(v) = 0} and Q* := {v* G V* | P*(v*) = 0}, We assume that there exists a biregular rational mapping 0 : Q —y Q* defined over R and that Q and Q* are nondegenerate and dual to each other with respect to 0, This means that Q and Q* satisfy the following:

(Duality) Sq(0(v)) = Sq* (v)-1,

(Nondegeneracy) rank (Jac(Q)(w)) = rank (Jac(Q*)(w*)) = n, where v G Q, w G Q := Q-1 (Q) w* G Q* := Q*-1(Q*),

Let Q = Q1 U ... U Q^, Q* = Q1 U ... U QV be the decomposition into connected components of Q and Q*. Notice that the existence of 0 implies th at Q and Q* have the same number of connected components, and we may assume that Q* = 0(Qj), j = 1,... ,p. We put P := P o Q, P* := P * o Q^d Qi := Q-1(Qi), Q * := Q*-1(Q*). For s G C with Re s > 0 we define a continuous function |P(v)|^ on V by

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|P (v) is = \ |P (v)|s, v G Qj ,

1 ( ? y 0, v G Qj.

sGC

meromorphicallv. We can define |P(w)|?. similarly. Then the following theorem holds.

Theorem 1 (cf [3]) If P, P * satisfy a local functional equation

V

Fv(|P|?,)(v*) = Y, Yij(s)|P*(v*)|-i”/J>-', d = dcgP = dcgP*,

j=1 "

PP PP

V ( V / 0 N

. . ( m — 2n

fw(|P^.)(w *) = ¿ j ¿efc ■ Yik(s) ■ Ykj s + m2d2^ I p(w *)!~(m/2d) s,

j=i ^fc=i

where tk is a certain constant with, 1 FV(f) (resp. FW(f )J means the Fourier

transform of a tempered distribution f on V (resp. W).

Remark. In [3], the above theorem is described in a more general setting.

3. Quartic polynomials obtained from representations of Clifford algebras

Let W be a real vector space with dimension m and W * the vector space dual to W. Let p, q be non-negative integers, p + q = n, and consider the quadratic form

p n

P(x) = Y x2 - Y x2 i=1 j=p+1

of signature (p, q), We Wentify V = Rn with its dual vector space via the standard inner product (x,y) = x1y1 + ... + xnyn. Put Q = V\{P = 0}, We determine quadratic mappings Q : W —y V that are self-dual with respect to the biregular mapping 0 : Q —y Q defined by

0(v) := 1 grad log p(v) = p^V)^''', Vp , _Vp+i’ • • •, _Vra)-

Q

polynomial P(w) := P(Q(w)) satisfies a functional equation with explicit gamma factor.

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For a quadratic mapping Q of W = Rm to V = Rn, there exist symmetric matrices S1,..., Sn of size m such that

Q(w) = (*w S1 w,..., *w Sn w).

For v = (x1,..., xn) G Rn, we put

S (v) = X1S1 + ... + XnSn. Q0

S(v)S(0(v)) = Im, v G Q.

If we define ei to be 1 or — 1 according as i ^ p or i > p, this condition is equivalent to the polynomial identity

pn

Y Yx2s2 — Y Y XjS2 + Y Y (tjSiSj + etSjSi) = p(x)im.

i=1 j=P+1 1^i<jXn

This identity holds if and only if

Im, 1 ^ i ^ n,

J SjSi, 1 ^ i ^ p < j ^ n or 1 ^ j ^ p < i ^ n, (3)

[ —SjSi, 1 ^ i, j ^ p or p +1 ^ i, j ^ n.

This means that the mapping S : V y Symm(R) can be extended to a representation

of the tensor product of the Clifford algebra Cp of x2 + ... + xp and the Clifford

algebra Cg of x^+1 + ... + x^

Q W = Rm

quadratic space (V, P) from a representation S : Cp ® Cg —y Mat(m, R). Such a representation S is a direct sum of simple modules and a simple Cp ® C^-module is a tensor product of simple modules of Cp and Cg, Therefore, by Lemma 1 below, we can choose a basis of the representation space so that S(Rn) is contained in Symm(R) and S(ei) = Si satisfy (3). These symmetric matrices S1,..., Sn defines a

W = Rm V = Rn

Q W = Rm

(V, P) correspond to representations S of Cp ® Cg such that S(V) C Symm(R).

Lemma 1 Let P be a quadratic form on V = Rn of signedu,re (p, q), p + q = n, and let e1,..., en be a basis of V in which P is given by (2). Denote by Cp,g the Clifford algebra of the quadratic form P and Iet $ : Cp,g —y Mat(m, R) be a representation of CPyq. Then, among the equivalence class of there exists a representation with the property that $(ei) is a symmetric 'matrix for 1 ^ i ^ p and a skew-symmetric matrix for p +1 ^ i ^ n.

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Proof. By the definition of the Clifford algebra Cp>g, we have

e2 = 1 (1 ^ i ^ p), e2 = —1 (p +1 ^ i ^ n), eiej = —ejei (i = j).

Hence, the multiplicative group G generated by { —1, e1,..., en} is a finite group and $ gives a group-representation of G on Rm, Therefore, if we replace $ by an

G

represented by an orthogonal matrix. Then ei = e“1 = tei for 1 ^ i ^ p, and ei = —e“1 = — tei for p +1 ^ i ^ n. □

The construction above is a generalization of the result of Faraut-Koranvi [2] on the functional equation associated with representations of the simple Euclidean Jordan algebra of rank 2, In this case, (p, q) = (1,q) and the self-dual quadratic

(1, q)

of C1 ® Cg = Cg © Cg. A representation of C1 ® Cg can be identified with the direct sum of two Cp-modules M^d M_ . On M+ (resp. M_), e1 acts as multiplication +1 —1

one for which M_ = {0},

§ 4. Prehomogeneous or non-prehomogeneous?

Most of the quartic polynomials P and P* are conjectured not to be relative invariants of prehomogeneous vector spaces except for low-dimensional cases.

Theorem 1 If p + q = dim V ^ 4, then the polynomials P and P* are relative invariants of prehomogeneous vector spaces.

The prehomogeneous vector spaces appearing in the case p + q ^ 4 are given in the following table:

(p,q) prehomogeneous vector space

(1, 0) (GL(1,R) x SO(fc1,fc2), Rkl +k2)

(2, 0) (GL(1,C) x SO(k,C), Ck)

(3, 0) (GL(1,R) x SU(2) x SO *(2k), C2k)

(2,1) (GL(2,R) x SO(k1,k2), Mat(2,k. + k2; R))

(4, 0) (GL(1,H) x GL(1,H) x GL(k,H), Mat(2,k,H))

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(3,1) (GL(2, C) x SU(k1, fe), Mat(2, kx + fe; C))

(2, 2) (GL(2,R) x GL(2,R) x SL(k,R), Mat(2,k;R)®2)

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It seems that, if n ^ 5, then P and p* are relative invariants of prehomogeneous vector spaces only for few exceptional cases.

Let g be the Lie algebra of the group G = {g G GL(W) | P(gw) = P(w)} and h(= hP,q) be the Lie algebra of the group H = {h G GL(W) | Q(hw) = Q(w)}. We can prove that g contains a Lie subalgebra isomorphic to so(p, q) © h.

Conjecture 1. We ftawe

except for low-dimensional cases.

The Lie algebras ^d h depend on p, q and the choice of a representation of CP ® Cq. By the periodicity of Clifford algebras CP+8 = Mat(16, R) ® Cp, there exists a natural correspondence between representations of CP+8 ® Cq and representations of Cp ® Cq and it can be proved that the strueture of h is the same for corresponding representations. This implies the isomorphisms

hp,q = hq,p = hp+8,q = hp,q+8 = hp+4,q±4. (4)

If dim V and dim W are relatively small, then we can calculate h explicitly by using a symbolic calculation engine (such as Mathematiea and Maple) and we have the

h.

h

in the following table:

j9/q 0 1 2 3

0 gl(£, R) so(ki, £2) so(£, C) so*(2£)

2 so(£,C) so(£i, £2) gl(£, R) sp(£, R)

4 gl(£, H) sp(£i, £2) sp(£, C) sp(£, R)

6 sp(£, C) sp(£i, £2) gi(£, H) so*(2£)

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Here p = p mod 8 and q = q mod 8 and £i, £2, £3, £4 are non-negative integers determined by the multiplicities of irreducible representations in the reprentation of Cp ® Cq corresponding to the quadratic mapping Q,

Notice that, by (4), it is sufficient to give the table only for 0 ^ p ^ 7 and 0 ^ q ^ 3.

Assuming Conjectures 1 and 2, we can determine all the cases where P is prehomogeneous. For example, if p + q ^ 13, then P is non-prehomogeneous for any representation of Cp ® Cq; namely it does not come from any prehomogeneous vector space.

References

1. J.-L. Clerc, Zeta distributions associated to a representation of a Jordan algebra, Math. Z., 2002, vol. 239, 263-276.

2. J. Faraut and A. Koranvi, Analysis on symmetric cones, Clarendon Press, Oxford, 1994.

3. F. Sato. Quadratic maps and non-prehomogeneous local functional equations, Comment. Math. Univ. St. Pauli, 2007, vol. 56, No. 2.

4. F. Sato. Zeta functions in several variables associated with prehomogeneous vector spaces I: functional equations, Tohoku Math. J., 1982, vol. 34, 437-483.

5. F. Sato and T. Kogiso. Representations of Clifford algebras and quartic polynomials with local functional equations, RIMS Kokyuroku, 2008, vol. 1617, 63-82.

6. M. Sato. Theory of prehomogeneous vector spaces (notes by T. Shintani in Japanese), Sugaku no Avumi, 1970, vol. 15, 85-157.

7. M. Sato and T. Kimura. A classification of irreducible prehomogeneous vector spaces and their invariants, Nagoya Math. J., 1977, vol. 65, 1-155.

8. M. Sato and T. Shintani. On zeta functions associated with prehomogeneous spaces, Ann. of Math., 1974, vol. 100, 131-170.

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Аннотация научной статьи по математике, автор научной работы — Сато Фумихиро, Когисо Такуйоши

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