数理解析研究所講究録
第 855 巻 1993 年 1-10

1Highest weight modules and b‐functions of semi-invariants.
AKIHIKO GYOJA(行者 明彦)Department of Fundamental Sciences, Faculty of Integrated Human Studies,
Kyoto University, Kyoto 606-01, Japan.
0. In [31], S.Suga observed a relation between the irreducibility of certain highest

weight modules and the b-functions of certain prehomogeneous vector spaces. The purpose ofthis note is to summarize [13] and [14], in which we have studied how the observation of Sugashould be generalized.

Convention. The complex number field (resp. the rational integer ring) is denoted
by C (resp. Z).

\S 1. b-Functions
Let $f_{i}(x_{1}, \cdots x_{n})(1\leq i\leq l)$ be analytic functions, $D$ the ring of analytic linear

differential operators, $0\leq k<l,$ $s=$$(s_{k+1}, \cdots , s_{l})$independent variables. We know thefollowing.
For any$b(s)\in \mathbb{C}[s]\backslash \{0\}$(1)$\lambda_{1},$$\cdots\lambda_{k},$$\mu_{k+1},$ $\cdots\mu_{l}\in Z_{\geq 0_{Z}}$there exists $Q(s)\in D[s]=D\otimes_{C}\mathbb{C}[s]$ andsuch that$Q(s)(f_{1}^{\lambda_{1}}\cdots f_{k}^{\lambda_{k}}f_{k+1}^{s_{k+1}+\mu_{k+1}} ...f_{l}^{s\iota+\mu_{l}})=b(s)(f_{1}^{\lambda_{1}}\cdots f_{k}^{\lambda_{k}}f_{k+1}^{s_{h+1}}\cdots f_{l}^{s_{l}})$(2) $b(s)= \prod_{i}(a_{i,k+1^{S}k+1}+\cdots+a_{i,l}s_{l}+\alpha_{i})$ with somean,$(a_{i,k+1}, \cdots a_{i,l})\in(Z_{\geq 0})^{l-k}\backslash \{0\}$$d$(3) with some$\alpha_{i}\in \mathbb{Q}_{>0}$.

This type of theorem is first obtained by M. Sato [27] for relative invariants of aprehomogeneous vector spaces. When $k=0$ and $l=1,$ (1) is obtained by I.N.Bernsteim [3] fora polynomial , and by J.E.Bj\"ork in general; (2) $+(3)$ is obtained by M.Kashiwara [18]. When$f$2
$k=0$ and $l>1,$ (1) and (2) are obtained by C.Sabbah [25].

See [11] for (3). Because of thepositivity, we get the general assertion from the case where $k=0$ by a specialization$s_{j}arrow\lambda_{j}$for some $j’ s$ with $\mu_{j}=0$ .

We call a polynomial $b(s)$ appearing in (1) a b-function. The totality of such $b(s)s$ (notnecessarily satisfying (2) or (3)) forms an ideal of$\mathbb{C}[s]$. The author does not know whether,in general, this is a principal ideal or not. (Cf. [12, 6.4].) If it is a principal ideal, we call its
generator the b-function.
If we have the b-function, we get$a$b-function by a specialization as above, but theresulting polynomial is not necessarily the b-function. In fact, this difference is one of our mainconcern.

\S 2. Generalized Verma modules.
Let
$p$

$G$

be a simply connected complex simple Lie group,$P$

a parabolic subgroup,

their Lie algebras, and $U(-)$ the enveloping algebra. A g-module

dimensional irreducible -module
$p$

where

$p$

$E$

$U(\mathfrak{g})\otimes_{U(p)}E$$\mathfrak{g}$andwith a finite

is called a generalized Verma module. In the special caseis a Borel subalgebra, such a g-module is called a Verma module.
The Verma modules are first introduced by D.N.Verma [32]. Since then a considerable

progress has been made [4], [5], [6], [17], [20], [7], [1],$\cdots$, and our present knowledge is fairly

satisfactory.

Concerning the generalized Verma modules, their significance was first recognized by
J.Lepowsky. He showed in [22] that every irreducible Harish-Chandra module can be obtained

as a subquotient of a (non-unitary) principal series representation, which can be constructedfrom a generalized Verma module. (Cf. [9, Chapter 9].) This result was largely improved by

W.Casselman, and then by A.Beilinson and J.Bernstein [2]. Casselman showed that ‘subquotient’ may be replaced with ‘submodule’. (Cf. [29, Introduction].)Concerning the properties of the generalized Verma modules, we have [24] (generalization of [4] and [6]), [16] (irreducibility criterion) and [17, 2.25] (translation principle).

At present, our knowledge about the homomorphisms between generalized Verma modules

3
is very poor, although some results are obtained by J.Lepowsky, B.D.Boe, D.H.Collingwood,

R.S.Irving, Hisayosi Matumoto,$\cdots$\S 3. Observation of Suga.
Let

$L$

be a Levi subgoup ofcommutative. Then

$P$

, and

the nilpotent radical of . Assume that$p$$\iota\iota$

$u$

is

has an open $ad(L)$ -orbit, i.e., ( , ad, u) is a prehomogeneous vector
$L$

$n$

space. Assume further that there exists a relatively $ad(L)$ -invariant non-constant irreducible
polynomial function$f_{0}$

on . Then$\mathfrak{u}$$p$is a maximal parabolic subalgebra corresponding to oneof the following diagrams.

$(A_{2p-1},p)$..

$(B_{p}, 1)$$\circ-\bullet$$(C_{p},p)$$\bullet-\bullet-\cdots-\bullet\Leftarrow 0$$(D_{p}, 1)$$0-\bullet-\cdots-$$(D_{2p},2p)$$\bullet-\bullet$–.

–.

$–\bullet\Rightarrow\bullet$..

I–$\bullet$$-\bullet-\bullet o|$$(E_{7},7)$$\bullet-\bullet-$

Let $b(s)$ be the b-function of$b_{0}(s)f^{8}$$f$I$-\bullet-\bullet-0$, i.e., the minimal polynomial such that $Q(s)f_{0}^{s+1}=$

with some $Q(s)\in D[s]$ , whose explicit form is given by$(A_{2p-1},p)$$b_{0}(s)=(s+1)(s+2)\cdots(s+p)$$(B_{p}, 1)$$b_{0}(s)=(s+1)(s+ \frac{2p-1}{2})$$(C_{p},p)$$b_{0}(s)=(s+1)(s+ \frac{3}{2})(s+\frac{4}{2})\cdots(s+L_{2}^{+\underline{1}})$$(D_{p}, 1)$$b_{0}(s)=(s+1)(s+ \frac{2p-2}{2})$4$(D_{2p},2p)$$b_{0}(s)=(s+1)(s+3)\cdots(s+2p-1)$$(E_{7},7)$$b_{0}(s)=(s+1)(s+5)(s+9)$

See [21], also [23] and [15]. (Cf. [10, 2.5].)
Let cv be the fundamental weight corresponding to the white node of the above diagram.

Then$\varpi$can be extended to a Lie algebra character of .
$p$

Suga Observation. [31]. For
g-module$\lambda\in \mathbb{C}$, the following $condi$ tion are $eq$ uivalent. (1) The$s$

is irredu cible. (2) $b_{0}(\lambda-j)\neq 0$ for any $j=1,2,$$U(\mathfrak{g})\otimes_{U(\mathfrak{p}),\lambda\varpi}\mathbb{C}$$\cdots$.

This observation can be checked by determining the irreducibility using the criterionof Jantzen [16] and by comparing with the explicit form of$b_{0}(s)$.

\S 4. Roughly speaking( , adjoint action,

(4.1)

$L$

At one hand, we have the$\mathcal{D}$-module$\mathcal{D}f_{0}^{\lambda}$$\mathfrak{u}$)$\fallingdotseq$( , left action, $G/P$ ).
$L$, which is related to the left hand side. (Heredenotes the sheaf of differential operators.) We can show that$b(\lambda-j)\neq 0$$\mathcal{M}(\lambda)$

if$M(\lambda)$

is simple if and only if

for any $j\in$ Z. On the other hand, we can expect that we get a D-module, say, on $G/P$ by ‘localizing’ the generalized Verma module[1]. Then$\mathcal{D}f_{0}^{\lambda}$$\mathcal{D}$$\mathcal{M}(\lambda)$$M(\lambda)=U(\mathfrak{g})\otimes_{U(\mathfrak{p}),\lambda\varpi}\mathbb{C}$as in, which is related to the right hand side of (4.1), would be simple if and onlyis simple. Hence, by showing that$\mathcal{M}(\lambda)\fallingdotseq Df_{0}^{\lambda}$, we would be able to explain theobservation of Suga to some extent.

In [13], we have tried to realize this idea and get [13, 9.13]. In the case considered in\S 3, this result asserts the following:(A) Assume that

(4.2)$\lambda\in \mathbb{C}$satisfies$\langle\lambda\varpi+\rho, \alpha^{\vee}\rangle\neq 0,$$-1,$ $-2,$$\cdots$

for any positive coroot$\alpha^{\vee}$.

5Then the following $condi$ tions are equivalent.
(4.3)$U(\mathfrak{g})\otimes_{U(\mathfrak{p}),\lambda\varpi}\mathbb{C}$

(4.4)

Here

$b_{0}(\lambda-j)\neq 0$$\rho$

is irred ucible.

for any $j\in Z$ .

denotes the half of the sum of the positive roots.

Remark. For instance, in the case $(A_{2p-1},p)$ ,$-2,$ $-1$$(4.3)\Leftrightarrow\lambda\neq-p+1,$$-p+2,$ $-p+3,$$(4.4)\Leftrightarrow\lambda\not\in Z$

In general, the totality of the parameterof arithmetic series,$(4.2)\Leftrightarrow\lambda\neq\cdots-3,$$\lambda_{j}+Z_{\geq 0}$$\lambda$$\cdots$,.
for which$U(\mathfrak{g})\otimes_{U(\mathfrak{p}),\lambda\varpi}\mathbb{C}$is reducible is a finite union. But as in the case $(A_{2p-1},p)$ , first several terms of these arithmetic

series are veiled by the assumption (4.2). By this reason, the above result is unsatisfactory. In

our proof, we needed this undesirable assumption in order to use the generality concerning the
localization of g-modules [1].

\S 5. Semi-invariants.
Although [13, 9.13] is unsatisfactory, it gives us an insight. So let us give a slightly

more detailed explanation.
Let

$T$

be a maximal torus of $G,$ $B=B_{+}$ a Borel subgroup, and $B$-the Borel subgroupsuch that $B_{+}\cap B_{-}=T$ . Since$\varpi$$G$

is assumed to be simply connected, any integral weight, i.e., a Z-linear combination of the fundamental weights

character of , which we shall denote by the same letter$T$the compositions of the projections $B\pmarrow T$ and
Let

on

$G$

$\varpi\in\sum_{i=1}^{l}Z_{\geq 0}\varpi_{\dot{*}}$$\varpi$$\varpi$$\varpi_{1},$$\cdots\varpi_{l}$, can be integrated to a

. We also denote by the same letter.

. It is known that there exists a unique holomorphic function

such that$f^{\varpi}(e)=1$

and $f^{\varpi}(b’gb)=\varpi(b’)f^{\varpi}(g)\varpi(b)$ .

$f^{\varpi}$

6
for any $b’\in B_{-},$ $g\in G$ , and $b\in B$ . This function is called semi-invariant associated to

Now let us explain [13, 9.13] in the case where$p$$\varpi$.

is a maximal parabolic subalgebra

whose nilpotent radical is not necessarily commutative.
(B) Let$\varpi$be the unique fun damental weight which can be exten $ded$ to a Lie algebra

character of , and $b(s)$ the b-function of$p$$f^{\varpi}$.

Under the condition (4.2), the conditions (4.3)

an $d(4.4)$ are $equi$ valen .
$t$Remark. (1) Strictly speaking, we need to assume some other conditions. But theauthor conjectures that the remaining conditions are automatically satisfied, and has proved itin several cases.
(2) In the case considered in \S 4, we can show that $b_{0}=b$ [ $14$ , proof of (4.2.1)]. Therefore(B) is a generalization of (A).

\S 6. What is important concerning (B) is that it gives us an insight into the significanceof the b-functions of semi-invariants. In fact, we conjecture the following.

Conjecture. Let$p$be a $m$ aximal araboli $csu$ balgebra,$p$$\varpi$the unique$fu$

ndament$al$weight which can be extended to a Lie algebra character of , and $b(s)$ the b-function of the$p$semi-invarian$tf^{\varpi}$. For$\lambda\in \mathbb{C}$, the following $con$ dition are equi valent. (1)irreducible. (2) $b(\lambda-j)\neq 0$ for any $j=1,2,$$s$$\cdots$$U(g)\otimes_{U(\mathfrak{p}),\lambda\varpi}\mathbb{C}$is.

Remark. In this note, we restrict ourselves to the simplest cases. See [14, \S 3 and\S 9] for our conjectures in their full generalities, where $b(s)$ is replaced with multi-variable bfunctions (cf. \S 1).

\S 7. In the individual cases, once we know the explicit form of the b-function, we cancheck the Conjecture using [16]. Let us explain what is known about the b-functions.

Except for the cases studied by Suga (cf. \S 3), our knowledge was almost nothing.
In fact, usually the determination of the b-function is very difficult and in many cases seems

actually impossible with bare hands.

7
In [26], an algorithm to calculate the b-function of a relative invariant of a prehomo-

geneous vector space is given, based on the microlocal analysis, which is an analysis on the
cotangent bundle of the base space. If the base space is a vector space, saygent bundle$T^{*}V$

is nothing but the direct product ofalso be regarded as the cotangent bundle of$V^{*}$$V$and its dual space$V$$V^{*}$, then its cotan-. Hence$T^{*}V$can. The essential part of the calculation of theb-function according to [26] is to go backward and forward between these two ways of looking.
Thus, in modifying this algorithm to handle the b-functions of the semi-invariants, ourdisadvantage mainly comes from the fact that our base space$G$

is not a vector space. On

the other hand, the following points are of our advantage. Recall that the semi-invariants are

relative invariants with respect to the natural$B_{-}\cross B$-action on$G$.

(1) The orbits of this action are the Bruhat cells, whose property is well understood.

Especially there are only a finite number of orbits, and hence the orbit decomposition gives a

Whitney stratification.
(2) The closures of each orbit is normal [8, Corollay 1 in p.85] (cf. [14, Remark following(5.11.2)]). Thus we can use the Zariski’s main theorem.
(3) As is naturally expected, we need the geometry of the flag manifold $G/B$ . Concern-

ing the flag manifold, results of R.Steinberg [30] and N.Spaltenstein [28] are$at$our disposal.

Because of these advantages, we can give an algorithm to calculate the b-functions of
the semi-invariants [14], based on the microlocal analysis as in the case of prehomogeneous
vector spaces, and we have calculated some examples by this procedure. Conjecture in \S 6 isformulated partly based on this calculation. The author hopes to discuss the other basis of ourConjecture in a different place.

\S 8. As we have explained, the b-functions of semi-invariants should control the irreducibility. Moreover, it seems that there exists an intimate relation of these b-functions tointertwining operators and unitarizability.
Before concluding this note, let us remark that the microlocal analysis of semi-invariants

is also interesting in its own sake. For example, in the case where $g=A_{1}$ and$p$

is a Borel

8subalgebra, the holonomy diagram and the W-graph of the regular representation constructedin [20] should be the same, if we assume$a$conjecture of Kazhdan-Lusztig.

References[1] A. Beilinson, J. Bernstein, Localisation de g-modules, C. R. Acad. Sci. Paris 292 (1981),

15-18.
[2] A. Beilinson, J. Bernstein, A generalization of Casselman’s submodule theorem. In: Rep-

resentation theory of reductive groups, Progress in Math. 40 (1983), 35-52, Birkh\"auser.
[3] I. N. Bernstein, The analytic continuation of generalized functions with respect to a pa-

mmeter, Funct. Anal. Appl. 6 (1972), 26-40.
[4] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, Differential operators on the base affine space

and a study of g-modules, Lie groups and their representations. Proc. summer school ingroup representations. (1975), 21-64, Halsted, New York.
[5] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, Structure of representations generated byvectors of highest weight, Funct. Anal. Appl. 5 (1971), 1-8.
[6] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, Category of g-modules, Funct. Anal. Appl.

10 (1976), 87-92.
[7] J. L. Brylinski, M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent.

Math. 64 (1981), 387-410.
[8] M. Demazure, D\’esingularisation des vari\’et\’es de Schubert g\’eneralisees, Ann. Sci.

\’Ec.

Norm. Sup. 7 (1974), 53-88.
[9] J. Dixmier, “Alg\‘ebres enveloppantes,” Gauthier-Villars, Paris, 1974.

[10] A. Gyoja, Theory of prehomogeneous vector spaces without regularity condition, Publ.

RIMS. Kyoto Univ. 27 (1991), 861-922.
[11] A. Gyoja, Bernstein-Sato polynomials for seveml analytic functions, to appear in J. Math.

Kyoto Univ.

$g$[12] A. Gyoja, Local $b- fi_{1}nctions$ of prehomogeneous Lagmngians, to appear in J. Math. Kyoto

Univ.
[13] A. Gyoja, Further generalization of genemlized Verma modules, Publ. RIMS, Kyoto Univ.
29 (1993), 349-395.
[14] A. Gyoja, Highest weight modules and b-functions of semi-invariants, preprint.

[15] R. Howe, T. Umeda, The Capelli identity, the double commutant $theorem_{f}$ and multiplicity

free actions, preprint.
[16] J. C. Jantzen, Kontmvariante Formen auf induzierten Darstellungen halbeinfacher Lie-

Algebren, Math. Ann. 226 (1977), 53-65.
[17] J. C. Jantzen, Moduln mit einem hochsten Gewicht, LNM 750 (1979), Springer.
[18] M. Kashiwara, b-Functions and holonomic systems, Invent. Math. 38 (1976), 33-53.

[19] M. Kashiwara, The universal Verma module and the b-function, Advanced Studies in Pure

Math. 6 (1985), 67-81.
[20] D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent.

Math. 53 (1979), 165-184.

[21] T. Kimura, The b-function and holonomy diagmms of irreducible regular prehomogenous
vector spaces, Nagoya Math. J. 85 (1982), 1-80.

[22] J. Lepowsky, Algebraic results on representations of semisimple Lie group, Trans. Amer.

Math. Soc. 176 (1973), 1-44.
[23] I. Muller, H. Rubenthaler, G. Schiffmann, Structures des espaces prehomog\‘enes associes \‘acertaines alg\‘ebres de Lie gmdu\’ees, Math. Ann. 274 (1986), 95-123.
[24] A. Rocha-T aridi, Splitting criteria for g-modules induced from a pambolic and the Bern stein-

Gelfand-Gelfand resolution of a finite dimensional irreducible g-module, Trans.

Amer.

Math. Soc. 262 (1980), 335-366.
[25] C. Sabbah, Proximit\’e e’vanescente$\Pi$

Compositio Math. 64 (1987), 213-241.

[26] M. Sato, M. Kashiwara, T. Kimura, T. Oshima, Micm-local analysis of prehomogeneous

vector spaces, Invent. Math. 62 (1980), 117-179.

10[27] M. $Sato/T$ . Shintani/M. Muro, Theory of prehomogeneous vector spaces (algebmic part)-

The English tmnslation of Sato’s lecture from Shintani’s note, Nagoya Math. J. 120(1990), 1-34.
[28] N. Spaltenstein, On the fixed point set of a unipotent element on the variety of Borel

subgmups, Topology 16 (1977), 203-204.
[29] J. T. Stafford, N. R. Wallach, The restriction of admissible modules to pambolic subalgebms,

Trans. Amer. Math. Soc. 272 (1982), 333-350.
[30] R. Steinberg, On the desingularization of the unipotent variety, Invent. Math. 36 (1976),

209-224.
[31] S. Suga, Highest weight modules associated with classical irreducible regular prehomogeneous

vector space of commutative pambolic type, Osaka J. Math. 28 (1991), 323-346.
[32] D. N. Verma, Structure of certain induced representations of complex semisimple Lie alge-bras, Bull. Amer. Math. Soc. 74 (1968), 160-166.