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On the theory of b-functions. (English) Zbl 0389.32005

##### MSC:
 32A45 Hyperfunctions 46F15 Hyperfunctions, analytic functionals
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 [1] , [30], [34] and [37]. Historical remarks around the equation (1) is as follows. I. M. Gelfancl conjectured in Amsterdam Congress that the analytic properties of f+ could be well investigated by use of the desingulariza- tion theorem. In fact, I. N. Bernstein-S. I. Gelfand [39] and M. F. Atiyah [38] proved the meromorphic dependence of /+ in s and discribed its poles by the resolution theorem of H. Hironaka. In 1961, M. Sato initiated a theory of ^-functions for relative in- variants on prehomogeneous vector spaces, in connection with the Fourier transforms and C-functions associated with these spaces [23], [26]. On the other hand, I. N. Bernstein independently took the equation (1) and proved an existence theorem of such ^-functions which does not vanish identically when / is a polynomial [7]. J. E. Bjork succeedingly generalized Bernstein’s result for analytic functions [8]. Since then much effort has been forcused on the general theory of ^-functions [12], [19], [27]. The author’s contribution has been done since this stage. B. Malgrange pointed out a close connection between ^-functions of f and the local monodromy of jf ”*(()) [16], He proved that, when f has an isolated singularity, the eigenvalues of local monodromy are just exp(27T -la) for roots a of ^-function [17]. After- wards, M. Kashiwara proved the rationality of the roots of ^-function for general f in a completely different way [14]. As for the analytic property of fa, we note also that an important result that fa satisfies a holonomic system was proved by I. N. Bernstein [2] in a special case and by M. Kashiwara-T. Kawai [10] for any /. More generally, analytic property of fau for holonomic u is studied in [3] in a special case and in the author’s subsequent paper [32] for general cases. The ^-functions associated with prehomogeneous vector spaces are well-investigated and they are determined by many people. The micro- local calculus finds its good application in the area of ^-function theory, and that theme will be fully treated in M. Kashiwara-T. Kimura-M. Muro [41]. The author is grateful to Dr. M. Minami and Professor T. Kawai for their critical reading of the manuscripts. He would like to express his hearty gratitude to Professor M. Sato and Professor M. Kashiwara for their fruitful advices, enlightening discussions and constant en- couragement. Chapter I. Gen.erari.tles In this chapter, we study the basic features of general 3)_t9s Modules and ^-functions associated with them, which are indispensable to later chapters. The author develop the general theory of such b- functions and Modules in [32] . §1. 3) [t, s] -Modules and 6-Functions Let C\t, s ] be the associative algebra over C with generators s and t and defining relation (1) ts-sl = t. Set 3)\t9sḡ)\textregistered C\t9s. A ^-Module Jtt is called a 3)[s -Module c (respectively 3)\t, s] -Module) , if 3ttl)s3tt (respectively 3tt Ds3tt, JUl^tJH) holds. In this chapter, all Modules are 31 [*, s] -Modules unless otherwise stated. Since tvs - (s-v)tv in view of (1), Ker T, Coker tv and Im tv are 3) [t, s] -Modules along with a given £D[tys  Module. Definition 1.1. Let X be a 2) _s Module. If stE&ut^X) has the non-zero minimal polynomial, zue denote it by dj?(s)9 and say ”d_r(i) exists.” ’* b-f unctions” for a S)[t,s Module 3?, are defined by bm> v (s) = dm/t*m (s ), V = 1, 2 ●- ●. Usually, £yu is abbreviated as b^. As is easily seen, bjitV exist if and only if bm exists. It should be remarked that if X is a holonomic 3) [£, s -Module dx(s) exists, since &tdg)(£) x (x^X) is finite dimensional and <£#4>(_£) is coherent [13] . Standard example of 3) [£, s ] -Module is constructed as follows. Let / be a holomorphic function on UdX, let X be a coherent < £D-Module and let u be its section over U. We denote the annihilator of u by S* that is; J = {Qzi £)\QU=0} . Define the ideal ^($$)<Z.2)[>] by the condition that P(s, .r, \textsterling >) e^O) if and only if , x, D + - grad/Wc|>](x)c5, for some w. We denote by Jl the Module 3)\sg(s) and by /'M the class (1 mod /(*)). Jl=S)\_s]fsu is a .$$[>, s] -Module with actions of t and ^ given by, The map £is injective in 37. In fact, if P (5 + 1)/ e ^ (j) then for some m and Qj^S, The left-hand side equals to and the right-hand side can be rewritten in the form for some R^J. Therefore, - 4 - g r a d / ) = which implies P ( j ) e ^ ( 5 ) . The ID-Module S)fsu is coherent, and if u is a holonomic section, S)fsu is subholonomic (see [32]). Definition 1.2. With a non-zero polynomial p(s), ive associate a number zu(p)^N0 in the folio-wing manner (zv(p) is called the width of />.) i) // P(s) eC* then <w(p) -0, ii) // p(s) = iii) // p(s) has the form k P(s) - XI Pi (-0 > where each p3 (s) is of the form in ii) , i = l ^a^ mod Z 0>/) ; = max ze; Theorem 1. 3, // dx(s) exists, then tw(d^X = Q. Furthermore if -we assume that t is injective or surjective, then X = 0. Proof, we have and by virtue of (1), 0 - t^-^dx 0) X = dx (s + w (djr^ ) t It follows from the definition of w (dx) that g.c.d. (dx (s ,dx(s + w (dx) ) = 1 . Hence the assertion follows. When t is injective or surjective, it is obvious that _£= 0. Q.E.D, A coherent < 2)-Module X is called holonomic (resp. sub-holonomic) if Z*tg)(jL., £D) -0 for i<^n (resp. i<^n - 1) . This condition is equivalent NX NX to codim SS(£)  >_n (resp. codim SS (X) >72 - 1) . X is called purely subholonomic if £*Jg(X9 ^)) =0 for i=£n - l. It is known that for any coherent ^-Module, Z*t$$(X, 2)$$ (resp. ^-’(X, 5))) is holonomic (resp. sub-holonomic) and £*J$$(X9 3)) =0, z'>?2. Let T^ be an irreducible NX component of SS(X). Then the multiplicity of X at a generic point NX .r0 of an irreducible component of SS (X) can be defined (which is denoted by ?nXo(X))9 and has the additivity, that is, if o<-j:1<-.cI<-.\textsterling .<-o, is an exact sequence of coherent ^-Modules, mXo(Xz) =niXQ(Xi) Corollary Ie4e Let 31 be a sub-holonomic S)\_t,s~\-Module such that t:Jl-^c3l is injective. Then, 3? is purely sub-holonomic. Proof. Consider the exact sequence Set {\mathtt<\hskip-.5e<}C==<\textsterling W\textsterling (32, 3)). Then X is holonomic and the long exact sequence of &*t gives us the surjection X- >J?-^0. Therefore J? - 0 by virtue of Theorem 1. 3. Q.E.D. Proposition 1. 5. Upon the conditions in Corollary 1. 4, bm exists. N/ Proof. Consider an irreducible component W of SSffl.). Since t is injective, the multiplicity of Jl/tJl at a generic point of W vanishes. Therefore codim SS(Jl/tJl) >_n which implies that Jl/tJl is holonomic. Thus bm exists (and so does bjiiV, by the argument after Definition 1.1). Q.E.D. The conditions in Corollary 1.4 are satisfied for Jl ~S)\s\fsu, if one of the following two conditions holds. i) f is arbitrary holomorphic function, u = l. ii) f is quasi-homogeneous, 3)u is holonomic. In the present paper, we restrict ourselves to case i) . We investigate case ii) in [32], where the detailed structure of b-jiiV(s) and the relation between Jla and S)fau (aeC) are also discussed. The existence of bm(s) for ''31=- 3) \s~\f su with general f and 3)u being holonomic can be derived from that of case ii) , following the technique in {\S} 3 of [14] . (See [32]) {\S} 2. ^-Functions of Holoinorphic Functions Let X be a complex manifold of dimension n, and let f(x) be a holomorphic function. Hereafter we make vise of the notations fi = df/dxi a = S0/i, for brevity. The ^-function o f / , which we denote by \textsterling /(s), is defined by, bf (s) = bm (*) , where 31=3) |>]/f . Here, Jl is a special case of 3)[s~]fsu for u = l. We also define bfiV(s) =bjitV(i). The existence of them will be assured later by Theorem 1.8. It follows from the above definition that there are P(s) and Pv(s + v)&g)[s']9 such that (1) P(0/'+1 = */(*)/', (2) Pv(* + v)/IJ'' = */>)/', and bf(s) and bftV(s) are minimal among such polynomials in s. When we emphasize the point x^X into consideration, we use the notation bfiX(i). Furthermore given a compact set KdX, we set */.*(*) =l.c.m. bfiX(s). x^K If /(.r)^0, then -fs^=fs. Hence \textsterling ,.,(*) =1. If /(o;)=0, setting j = - l in (1) , we know (5 + 1) | bfi x (s) . If /(*)=(), grad /(*) \?=0, then \textsterling ,.,(*) = (*+!) by A/s ' l = (s + l)/s (e.g. when f, (x) Therefore, our main concern is with b f , x ( s ) at a singular point of /-'(Q). If y is in a sufficiently small neighborhood of x, bf:y(s)\bfiX(s) by (1). For g(x)e0, g(x0)^0, we have b,fil,(s) =bf,x,(s). Because, if l)grad log ff) and vice versa. Thus, \textsterling /(V) is an invariant of the hypersurface {f = 0} independent of the choice of its defining equation. For later convenience we list up basic notations in ^-function theory. Definition 1.6. i ) 3 (*') = (P W e 3) [5] | P (s)/s = 0 , ii) W= {(x, s grad log T70= {(^ f) e W|/(x) =0} U {U 0) Proposition 1.7. 3? = .0 [>] /^ (s) , Proof. The isomorphisms of 31, Jtt and 3?a are easy to verify. That of JyR is proved as follows. Let P(s) be such that P(s) (}+!)/' = QO)/!+I. Setting s = - l , we have Q(- 1) =2 ft(:c, Z>) D4. Hence, P (,) (* + I)/' = ( (S + 1) P. (5) + I] ft (X, = (5 + 1) (P. (5)/+ S ft (X, Q.E.D. If grad /(x) =^=0, /(X)=0 at xe.X, we can assume f=xt. Then (5-^A) +1] 3) \s~\D, ~3)/iz 3)D,. Therefore, =... = fn = 0}=Wr in a neighborhood of x. Since is an analytic set, we have We state the fundamental theorem of M. Kashiwara. Theorem 1,8. i) 3? is sub-holonomic and 33(71$$ = W. ii) bf(s) exists and all the roots of bf(s) =0 are strictly negative rational, For the proof of this, we refer the reader to M. Kashiwara [14], The existence of bj(s) can be derived from i) and Proposition 1. 5. See also [32]. Corollary I> 9. 3A, j/l and Jla are holonomic. More precisely, Proof. For, t gives an isomorphism on W\f l(§) in the exact sequence 0-^-43?-* JK->0, SS (JH) is contained in f l(V)^W and hence a holonomic set. Since fD.-ftsegts), SS^J^Wf] (/-●(O) U (? = 0)). (/i = 0,Vz). Q.E.D. When f is locally reduced, K. Saito proved the following: Theorem 1. 10. Q is a reflexive Ox-Module. Let Xt = Yl <Zij(x)Dj z = l, ●●- , < , be elements in Q. Then Xl9 –,Xn is a locally free basis of 3, if and only if det(a^) =g/, ge0J. Corollary 1.11. Suppose dim.X = 2. Then Q has locally free basis Xly X2 (Xi = ^aijDj) and aua22 - ci12a21 = gf, Q^O^. Converse- ly p, if i-wo vector fields Xt in Q satisfy the above formula, they form a basis of G. For the proof of these, we refer the reader to K. Saito [21]. When f is the square of the fundamental anti-invariant of a Coxeter group, considered as a function of fundamental invariants, Q is a free module. This was pointed out by K. Saito [21]. For the determination of the structure of Q and the microlocal structure of 3)\s \f we refer the reader to T. Yano [33] or T. Yano-J. Sekiguchi [35], [36]. They proved that the holonomic system 3)fa has multiplicity 1 on all the irreducible components of SS(£Dfa), and determined a basis of Q concretely. Corollary 1. 11 was also noted by M. Sato and M. Kashiwara (not published) . Chapter II. Structure of the Ideal §(s) In this chapter, we shall restrict our attention to the structure of $$(s). First of all, we introduce a number L(/), which measures the non-quasi-homogeneity of /. We further define a class of functions called a convergent power series of simplex type, which plays an im- portant role in later applications. In the case of such a function, cor- responding$$ (s) contains a distinguished element (cf. Theorem 2.15). In §§6, 8, we shall determine the structure of $$(s) upon the following two conditions that 1^\circ L (/) <3 and 2^\circ the singularity is isolated. Section 8 is concerned with a delicate phenomenon about$$ (s), and given are counter examples against Sato-Kashiwara conjectures. § 3* Total Symbol For the later purposes it is appropriate to modify the notion of order of an element of 2)_s  by regarding s as element of order 1. To be more precise, we define Definition 2. 1. Given P(s) = X! s’Pfa, D [4] . Examples of non-isolated singularities are given in §22. § 17. Two-Dimensional Case When the space dimension is 2, we can apply Theorem 2. 24. As is shown below, we find ”explicit formulae” under some assumptions on f . Let us explain the situation. First, we assume that f is a locally reduced non-quasi-homogeneous function at O e C 2 such that (a) a:f=(x>,y*). Next, we assume that generators of $$(s) 0 ({\mathtt<\hskip-.5e<}2)s+ 3)) are given by A, (s, x, D) =xa(s- X.) + A', (x, D) , and At (5, x, D) = y ' (5 - X,) + A( (x, D) , where Xk = aklxDx-\- ak2yDy, \textsterling = 1,2, atj^Q+, and they satisfy the condition (b) the weight of A{ (x, D) (A'2 (x9 D) , respectively) is greater than that of a21a in X2 (alzb in Xi9 respectively) . Set A, = _ \;[&u ai*\;n 12\, There are two cases. La21 a22J 1^\circ rank A = l. Assume (an, a12) - c \bullet (#2i, ^22) c^Q- We write / in the following form: /* =o + 0, where f0 is the sum of monomials in f which have minimal weight, say it\;with respect to (<22i, #22) \bullet Then, } - A((x, D) (/o + sO Comparing the terms with minimal weight in these formulae, we have zv - 1 and c = I. Thus, X1 = X2 = X0 = axDx +ffyDy9 f=fQJrg, X0f0=fQ and g has the weight greater than 1 with respect to X0. This shows that, when rank A = 1, f can be considered as a higher order deformation of weighted homogeneous polynomial. Since ybA1 - xaA2 = <p(fxDy-fyDx), 0>(0)=^0, we have (2) !>(# + 1)Oi + (6 + 1)/? . 2^\circ rank A = 2. In this case, inequalities an=\textsterling a21, a12=^a22 holds in general. Then the relation xaA2 - ybA1 = <p(f3;Dy -fyDx) again shows 1 = ((2 4-1) an + (6+1) #12 = (a + 1) #21 + (6 + 1) #22 \bullet That is, A can be written in the form _ 1 A -(!-(# + : Taking the determinant of coefficients of 1 , 1 w , e have LA2J where g is the sum of monomials whose weight are strictly greater than that of xa+lyb+l with weight of X1 or X2. Moreover, we impose the condition (c) L ( / ) = 2 . Upon these conditions, we conjecture that the action of s is de- termined by a, b and A. The explicit formulae for ^-functions are given as follows. Conjecture 4.0 (EEF)0 l-a 1-/3 1^\circ rank A = 1. A = ( t f , / 9 ) . Then, /< = a /5 (l-*att)(l-i Moreover, s is semisimple. 2^\circ rank A = 2. i + l J Then, ft = 1 + - + - , a /? = ^ + (i - o (i - o (i - o (i r$$ Set d = g.c.a. (0 + 1, £+ 1). Then, –, v = l, ●●- , {\mathtt< } -1 semisimple eigenvalue of s of height tuoo. We call the formula and proviso about semisimplicity of s in 1^\circ (respectively in 2^\circ ) as ”EEF” type 1^\circ (resp. 2^\circ ). The common case where formulae type 1^\circ and 2^\circ could cover formally is the following. Z, /?) and 1 = In this case, even though these two formulae seem very different, they give the same result as directly seen. Of course, this case can never occur according to the restriction (2). We also conjecture that a second order operator showing that L (y) = 2 can be chosen in the following way Type 1^\circ (3) (s - X+c’)(s - X where each term in sA.’ - B’ has strictly positive weight with respect to XQ. Note that c’ is positive by the inequality (2) . Type 2^\circ (4) (s X^(s-X^+sA’^Bf ’, where sA’-rB’ has strictly positive weight with respect to both Xi and x2. Especially when a - b = in type 1^\circ , we can also take We add some remarks to ”EEF” type 1^\circ . According to the analysis in case 1^\circ , /=/o+ (higher weight), XQfQ=fQ. The first term of Pf(f) of type 1^\circ is the same with P/Q(t). Since there is a factor (1 - 0 in the second term, P/(f) and P/0(t) can be expanded into the fractional polynomial of the form: and Note that (5) min C0<rnm C owing to the inequality (2), because min Co = # + /9 and minC = l - act -*/9. There is a natural generalization of ”EEF” type 1^\circ to ^-dimensional case. That is, if (a), a:f=(x? ●●●, ^ {\mathtt< } ) , and first order operators associated to it are (b) n x?* (s - X0) + (higher weight) , with XQ = XI OiiXiDi, and then Moreover, 5 is semisimple. We refer this formula with proviso as ”EEF” type l(n). There are several cases where these conjectures can be verified, as we will discuss later on. Conditions (a) , (b) and (c) are essential. In fact, types Wf^q and Wi*{\mathtt< }-i in §21 satisfy (a) (with a = l9 b = 2) and (c) but violates (b) . POO is given in [32] and is different from both type 1^\circ and 2^\circ . The next example satisfies (a) and (b) but violates (c) . Example 4. 1. / - –.rn> + - (y - txm^ (y + (w2 - 1) txm^ ”’-1. \)1 J7 9 / is a non-zero parameter. We impose conditions Then a : f - Or*1”2$$^{TM}$$1”1, y) , and first order operators associated with it are the following: (w2 = 7/2 - 1) y 0 - where Q = (y + matxn^ s (Dx - 1 { (y 4 mi ni (y + Owing to the inequalit}r l/7Z2<^7721/;z1, we can check the condition (b) . However, condition (c) does not hold. In fact, 2 = 1 (f) <^L (jT) - 3 in this case. P(0 is given by the following and does not coinside with formula type 1^\circ or 2^\circ . See type X0btP in §18 and [32]. § 18. *? In this section, we study the typical example f(x, y) = -xni + - yn’-txm’ym> , < 1 ??2 where £is a parameter. We can assume \^/mi<ni - owing to Proposition 2.10. In the following, c always denotes T] - - - 1. n-i When c = 0, f is weighted homogeneous polynomial with weight j - - and hence by Theorem 3.6, we have \nl ;z2/ (7) p r o - (^-0(^-0 ●When c=^0, / is of simplex type, and when £>0, f is a /^-constant deformation of (8) J L r M + A y . Wl < 2 Therefore, the local inonodromy of / is the same as that of (8). But P (f) is not given by (7) as is shown below. When c<^0, bf(s) may have double roots. Then so do the local monodromy. In the sequel, we use the following notation. - XD x + - yDv, First of all, we determine g (s) fl (& Proposition 4. 2. a : / are § (s) fl (S)s + 3)) are given as follows. zi<Wi/2, l<wz2<;/2/2: (a;”11-1, ymz’1
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