数理解析研究所講究録
第 816 巻 1992 年 1-21

1Some Aspects of Representations and Algebraic Geometry of Lie Algebras

finiteness criteria for the restriction of-modulesand applications to Harish-Chandra modules$U(\mathfrak{g})$京大 理 山下 博
Hiroshi YAMASHITA
Department of Mathematics, Kyoto University

Introduction and main results.
Let be a finite-dimensional complex Lie algebra, andbe the universal enof
definesveloping algebra of . The natural increasing filtration, which i\‘o isomorphic to the
a commutative graded ring gr
Poincar\’e-Birkhoff-Wittsymmetric algebraof by thetheorem. The identification
allows us to relate various objects in (non-commutative) enveloping altetry for
gebra theory with those in commutative algebra and algebraicand, the dual space of (see [2], [4], [14], [17, 18]).
For instance, if $H$ is a,-module generated by a finite-dimensional subspace
we can associate to the pair $(H, H_{0})$ a graded-module of finite type by $gr(H;H_{0})$ $:=$
with. The annihilator $J(H;H_{0})$ of $gr(H;H_{0})$ in
defines$H$
the associated variety
as he set of common
of , independent ofzeros of all the elements of $J(H;H_{0})$ . The celebrated Hilbert-Serre heorem in commutative ring theory says that this variety-modulesupports well the graded$gr(H;H_{0})$ (see Theorem 1.1).
In this paper, we give useful criteria for finitely generated U(g)-modules $H$ to remain
finite under the restriction to subalgebras of, by means of the ssociated varieties. Applying the criteria, we specify among other things, a large class of Liesubalgebras of a semisimple Lie algebra on which all the $Harish- Ci$ } $andra$ modules areof finite type. This extends a result of Casselmann-Osborne [8] and.Joseph [13] on the
restriciton of admissible modules to nilpotent Lie subalgebras
ing in the Iwasawa$re(,iprocity$
decomposition. Moreover we develop, with the help of Frobenius, the finite
multiplicity theorems for induced representations of a semisimple Lie group, obtained inour earlier work [20].
$U(\mathfrak{g})$$\mathfrak{g}$$(U_{k}(\mathfrak{g}))_{k=}$$\mathfrak{g}$$U(\mathfrak{g})$}$U(\mathfrak{g})=\oplus_{k}U_{k}(\mathfrak{g})/U_{k-1}(\mathfrak{g})$$S(\mathfrak{g})$$\mathfrak{g}$$S(\mathfrak{g})=grU(\mathfrak{g})$$S(\mathfrak{g})$$geo\iota’$$\mathfrak{g}^{*}=SpecS(\mathfrak{g})$$\mathfrak{g}$$H_{0}$$U(\mathfrak{g})$$S(\mathfrak{g})$$\oplus_{k}H_{k}/H_{k-1}$$H_{k}=U_{k}(\mathfrak{g})H_{0}$$S(\mathfrak{g})$$H_{0},$$\mathcal{V}(\mathfrak{g};H)\subset \mathfrak{g}^{*}$$\mathfrak{t}$$V(\mathfrak{g};H)$$S(\mathfrak{g})$$U(\mathfrak{g})$$\dot{(}\iota$$\mathcal{V}(\mathfrak{g};H)$$appea_{t_{-}}^{\tau}$

Let us now explain our basic ideas and the principal results of this article.
A. For a subalgebra ofcontaining the identity element, let denote the assowith $A_{k}=$. We sayciated graded subalgebra gr $:=\oplus_{k\geq 0}A_{k}/A_{k-1}$ of$H$that a finitely generated
if there exists a-modulehas the good restriction$H$$M$$:=gr(H:-ff_{0})$of for which theis of finite type-modulegenerating subspaceover . It is standard to verify that the original $H$ is finitely ger
over
if itsrestriction to is good.
We can characterize the U(g)-modules $H$ having the good $res$ riction to a given, by using the associated varieties. To be specific, we first obse;that the graded$A$$\overline{h}^{1}$$U(\mathfrak{g})$$A$$H_{0}$$‘\cap U_{k}(\mathfrak{g})$$S(\mathfrak{g})$$t()$$U(\mathfrak{g})$$A$$S(\mathfrak{g})$$-\neg rated$

$R$

$A$

$A$

$\cdot re$$A$2-module $M=gr(H;H_{0})$ is finitely generated over if and only if the quotient $S(g)-$module $M/R_{+}M$ is of finite-dimension, where $R+denotes$ the ma cimal graded idealof . Secondly, the Hilbert-Serre theorem (or Hilbert’s Nullsteller
tells us that$\dim M/R_{+}M<\infty$ whenever$S(g)$$R$$R$$\cup^{\backslash atz)}$(VHRO)$\mathcal{V}(g;H)\cap R_{+}^{\#}=(0)$holds, where
denotes the algebraic variety ofdetermined by $R+as$ the set ofcommon zero points. Furthermore, it is shown that the converse is also true provided
that $R$ is Noetherian. (See Proposition 2.1.)
Summing up the above discussion, we obtain the first main result of this paper, as
follows.
$R_{+}^{\#}$$g^{*}$

Theorem I. (see Theorems 2.1 and $2.2(1)$ ) (1) A finitely generated $U(g)$ -module $H$ has
the good restriction to a subalgebra
whenever (VHRO) is fulfilled for $R=$ gr A. The
converse is also true if the ring is Noetherian.
(2) The condition (VHRO) guarantees that $H$ is of finite type over .
$A$

$R$

$A$

If $A=U(n)$ for a Lie subalgebra of , then the corresponding $gr_{:},ded$ ringis Noetherian andof in . $Accordi_{x}1gly$ , one sees fromequals the orthogonal$H$$U(n)$. In
if and only if
Theorem I thathas the good restriction tothis case, we find that, besides the finiteness, $H$ preserves some other invariants under
the restriction to $U(n)$ :$\mathfrak{n}$$R=S(\mathfrak{n})$$g$$\mathfrak{n}^{\perp}$$R_{+}^{\#}$$\mathfrak{n}$$g^{*}$$\mathcal{V}(g;H)\cap \mathfrak{n}^{\perp}=(O)$Theorem II. (see Theorem $2.2(2)$ ) If the restriction of an $H$ tois good, the Gelfand(see 1.2 for
Kirillov dimension
and the Bernstein degree$H$the definition) of as a-module coincide respectively with $thos\epsilon(f(g;H)$ and $c(g;H)$$U(g)$
by the reas a-module. Furthermore, the variety
is carried into
striction of linear forms on to .
$U(\mathfrak{n}^{I}$$d(\mathfrak{n};H)$$:=\dim \mathcal{V}(\mathfrak{n};H)$$-\vee(\mathfrak{n};H)$$U(\mathfrak{n})$$\mathcal{V}(g;H)$$g$$\mathcal{V}(\mathfrak{n};H)$$\mathfrak{n}$B. The general results given in $A$ , have remarkable applications
Harish-Chandra
modules of a semisimple Lie algebra.
be a real semisimple Lie algebra, andNow letbe Cartan decomposition of . We denote by the complexified Lie algebra of , and , he complexification, cor ventionally. By aof a real vector subspace
will be denoted byofon which
Harish-Chandra $(g, t)$ -module is meant a finitely generated
the subalgebra
acts locally finitely, where
denotes the center of $U(g)$ . Weofregard the variety
by identifying
as a subset
with $g|$ hrough the Killing
form of .
The following two facts are essential for our applications to Harish-Chandra modules.
of a Harish-Chandra $(g, f)$ -module $H$ is contained(1) The associated varietyof all the nilpotent elements in (Lemma 3.1).
in the set(2) There exists a Harish-Chandra module
for whichcoincides with the(Proposition 3.2).
whole
These facts together with Theorem I yield the following$vo$$Bo=t_{0}\oplus \mathfrak{p}_{0}$$g_{0}$$g$$g_{0}$$\mathfrak{h}_{0}$$\iota$$\uparrow$$g_{0}$$\mathfrak{h}(\subset g)$$g_{0}$$U(g)- 11^{\backslash \backslash }duleH$$U(f)\mathcal{Z}(g)$$\mathcal{Z}(g)$$\mathcal{V}(g;H)$$g^{*}$$g$$g$$\mathcal{V}(g;H)$$\mathcal{N}(\mathfrak{p})$$\mathfrak{p}$$\tilde{H}$$\mathcal{N}(\mathfrak{p})$$\mathcal{V}(g;\tilde{H})$3
Theorem III. (see Theorem 3.1) All the Harish-Chandra
restriction to a subalgebraof $U(g)$ ifconverse is also true when is Noetherian.
$A$-modules have the goodholds for $R=$ gr A. The$(g, \epsilon)$$\mathcal{N}(\mathfrak{p})\cap R_{+}^{\#}=(0)$$R$

C. Suggested by this theorem, we say that a Lie subalgebra
exists an inner automorphism of such that$x$$\mathfrak{n}_{0}$is large in$g_{0}$

if there$g_{0}$$(x\cdot \mathfrak{n})^{\perp}\cap \mathcal{N}(\mathfrak{p})=(0)$,or equivalently, each Harish-Chandra $(g, t)$ -module has the good restriction to.
We can specify many of large Lie subalgebras of . At first, the maximal nilpotent
Lie subalgebras and also the symmetrizing Lie subalgebras ofare proved to be largein (Propositions 4.1 and 4.2). Theorems I and II applied to the foI nler example cover
results of Casselmann-Osborne [8, Th.2.3] and Joseph [13, II, 5.6]. Secondly, it is shownthat the largeness of a Lie subalgebra is preserved by the parabolic induction (see 4.2).
This means that, if
is a large Lie subalgebra of the Levi component of a parabolic
subalgebra $q_{0}=r_{0}+u_{0}$ , the semidirect product Lie subalgebrais large in 90.
is the nilradical of .
Here
Thirdly, we say that a Lie subalgebra
is quasi-spherical if there exists aofminimal parabolic subalgebra. Such Lie subalgebrasofsuch thattothegive risehomogeneous spaces of a semisimple Lie group on which each minimal
parabolic subgroup admits an open orbit (see e.g., [3], [5, 6], [15], [16]).
$U(x\cdot \mathfrak{n})$$g_{0}$$g_{0}$$g_{0}$$\mathfrak{l}_{0}$$\mathfrak{h}_{0}$$\mathfrak{h}_{0}-\vdash u_{0}$$u_{0}$$q_{0}$$\mathfrak{n}_{0}$$q_{m,0}$$g_{0}$$90=\mathfrak{n}_{0}+q_{m,0}$$g_{0}$

Theorem IV. (see Theorem 4.1) Any quasi-spherical Lie subalgebra is large in$g_{0}$.

D. Let be a connected semisimple Lie group with finite center, ar $dK$ be a maximal
compact subgroup of . We denote the corresponding Lie algebras
and , respec$Grepresent_{c’t^{4}}$tively. By Harish-Chandra, the admissible Hilbert spaceions correspond to
K-action, by
Harish-Chandra ( , K)-modules, i.e., such $(g, g)$ -modules with comppassing to the K-finite part. On the other side, if $(\eta, E)$ is a smooth Fr\’echet representation of a closed subgroup $N$ of , the spaceof real analytic sections of associatedvector bundle $G\cross NE$ , has a natural structure of compatible $(G, U(g))\cdot$-module (see 5.1).
With the aid of Frobenius reciprocity (cf. Proposition 5.1), Theorems I and III on
the restriction of $U(g)$ -modules, allow us to give useful finite multiplicity criteria for
analytically induced modules $A(G;\eta)$ (Theorems 5.3-5.5).
Among other things, we establish the following
$G$

$G$

$1_{)}yg_{0}$$f_{0}$$j^{1_{J}ible}$

$g$

$G$

$\mathcal{A}(G;\eta)$

Theorem V. (see Theorem 5.5) Let $N$ be a closed subgroup of whose Lie algebra islarge in , and take an $x\in G$ for which. Then the intertwining
number $\dim Hom_{U(9)}(H, \mathcal{A}(G;\eta))$ is finite for every Harish-Chandra $(g, K)$ -module $H$ , if
the restriction of to compact subgroup $N\cap x^{-1}Kx$ has the finite multiplicity property.
$G$$\mathfrak{n}_{0}$$(Ad(x)\mathfrak{n})^{\perp}\cap \mathcal{N}(\mathfrak{p})=(0)$$g_{0}$$\eta$

This theorem extends one of the principal results in our previous wc rk [20, , Th.2.12],
where we studied the case of certain semidirect product large Lieebras , through$(K,
N)$-spherical functions.
the theory of$I$$sub_{c\lambda}^{t}\prime g$$\mathfrak{n}_{0}$4The organization of this paper is as follows. We begin with preparing in \S 1 thenotions and fundamental facts which we need throughout this art de. \S 2 gives thetheoretical basis of this work. We develop the general theory on $1\vee>striction$ of $U(g)-$modules to subalgebras by using the associated varieties. The criteria for good restriction
to subalgebras, are established in various situations in 2.1 and 2.2, and we clarify in 2.3
and 2.4 some important properties of U(g)-modules having the good restriction.
Ne characterize, in
In \S 3, applying the results of \S 2 to semisimple Lie algebras$U(g)$, subalgebras ofrelation with the nilpotent variety
to which all the Harishult of \S 3, Theorem
Chandra $(g, f)$ -modules have the good restriction. The principal
3.1, is presented in much more general setting. \S 4 is devoted to the specification of large
Lie subalgebras of a real semisimple Lie algebra. The last \S 5 develops finite multiplicity
criteria for analytically induced representations of a (semisimple) Lie group, by making
use of the results of \S \S 2-4 and a reciprocity of Frobenius type.
An enlarged version of this article, with complete proofs, will appear elsewhere.
$i$

$g,$

$\mathcal{N}(\mathfrak{p})$$re_{\backslash J}^{\tau}$

1.

Associated varieties for finitely generated U(g)-modules.

At first, we equip ourselves with some fundamental facts in comrrl ltative algebra andalgebraic geometry, and introduce three important invariants: the associated variety, theBernstein degree and the Gelfand-Kirillov dimension, of finitely generated modules overa complex Lie algebra.
1.1. The Hilbert-Serre theorem. Let
be a finite-dimensional complex vector
the symmetric algebra of , where $S^{k}(V)$ is thespace. We denote bysubspace of $S(V)$ consisting of all homogeneous elements of degree . Let
be a finitely generated, non-zero, graded $S(V)$ -module, on which $S(V)$ acts in such a
way as $S^{k}(V)M_{k’}\subset M_{k+k’}(k, k’\geq 0)$ . Then it is easy to see that each homogeneouscomponent
is finite-dimensional. Set
$V$

$V$

$S(V)=\oplus_{k0}^{\infty_{=}}S^{k}(V)$$k$$M=\oplus_{k=0}^{\infty}M_{k}$$M_{k}$(1.1)$\varphi_{M}(q)$$:=\dim(M_{0}+M_{1}+\ldots+M_{q})$for each integar $q\geq 0$ .

Theorem 1.1. (Hilbert-Serre, see [22, Ch.VII, \S 12]) (1) Therea unique polynomial
in such thatfor sufficiently large .
(2) Let $(c(M)/d(M)!)q^{d(M)}$ be the leading term of. Then $c(M)\iota s$ a positive integer,
and the degree $d(M)$ of this polynomial coincides with the dimension of the associatedalgebraic variety$exi_{0}t_{5}$$\tilde{\varphi}_{M}(q)$$\varphi_{M}(q)=\tilde{\varphi}_{M}(q)$

$q$

$q$

$\tilde{\varphi}_{M}$(1.2)$\mathcal{V}(M)$

$:=$

{

$\lambda\in V^{*}|f(\lambda)=0$

for all$f\in Ann_{S(V)}\Lambda l$}.

Here, $Ann_{S(V)}M$ denotes the annihilator of $M$ in $S(V),$the dual space of$S(V)$ is identified with the polynomial ring overin the canonical way.
$V^{*}$$V^{*}$$V_{f}$

and5
Since the annihilator $Ann_{S(V)}M$ is a graded ideal contained in $S(V)_{+}$ $;=\oplus_{k>0}S^{k}(V)$ ,is an algebraic cone in . This combined with (2) of the above theorem
the variety
particular
the following corollary, which is one of the keys for studying in\S 2theingives
restriction of U(g)-modules to subalgebras.
$V^{*}$$\mathcal{V}(M)$Corollary 1.1. A finitely generated, non-zero, graded $S(V)$ -module
equals (0).
if and only if its associated variety$\lrcorner hl$

is finite-dimensional$\mathcal{V}(M)$Remark. It is not difficult to deduce this corollary directly from Hilbe 4 Nullstellensatz.
$s$1.2. Associated varieties for U(g)-modules. Let be a finite-dimensional complex
Lie algebra, and $U(g)$ be the enveloping algebra of . For each integer $k\geq 0$ , wethe subspace of $U(g)$ generated by elementswith $m\leq k$ anddenote by$X_{j}\in g(1\leq j\leq m)$ . One gets a natural increasing filtrationof $U(g)$ such
that
$g$

$g$

$U_{k}(g)$$X_{1}\ldots\lambda_{?’b}^{r}$$(U_{k}(g))_{k\geq 0}$$U_{k}(g)U_{m}(g)=U_{k+m}(g),$$U( g)=\bigcup_{k=0}^{\infty}U_{k}(g),$$[U_{k}(g), U_{m}(g)]\subset U_{k+m-1}(g)$.

The associated graded commutative algebra gr $U(g)$ $:=\oplus_{k>0}U_{k}(g)/\iota^{\tau}J_{k-1}(g)(U_{-1}(g)$ $:=$
of in the canonical way.
(0) ) is isomorphic to the symmetric algebra
We will identify these two algebras with each other.
Now let $H$ be a finitely generated, non-zero $U(g)$ -module. Take finite-dimensionalof $H:H=U(g)H_{0}$ . Setting $H_{k}=U_{k}(g)H_{0}$ for $k=1,2,$ ;generating subspace$H_{-1}=(0)$ , one obtains an increasing filtrationof $H$ such that
$S(g)=\oplus_{k\geq 0}S^{k}\overline{(}g)$$g$$\hat{C}$$H_{0}$$\ldots$$(H_{k})_{k}$(1.3)$H= \bigcup_{k=0}^{\infty}H_{k}$,$U_{m}(g)H_{k}=H_{k+m}$.

Correspondingly, we have a graded $S(g)$ -module(1.4)$M= \bigoplus_{k}M_{k}$with $M_{k}=H_{k}/H_{k-1}$ ,which will be denoted by $gr(H;H_{0})$ because the above filtration of if is determined by. Since $M_{k}=S^{k}(g)M_{0},$ $M$ is finitely generated over $S(g)$ . So we can define for this
$M$ the variety. It is easy
, the integers $c(M)$ and $d(M)$ as in The
to see that these quantities are independent of the choice of a gener iting subspace.
$c(g;H)$ ,Hereafter, we will denote these three invariants of $H$ respectively
and by $d(g;H)$ , emphasizing that $H$ is being considered as a $U(g)- mc_{t}dule$ .
$H_{0}$$\mathcal{V}(M)\subset g^{*}$$o_{i^{\backslash }}em1.1$$H_{0}$$b_{V^{-}}\mathcal{V}(g;H),$[4, III], [17, 18]). For a finitely generated non-zet $JU(g)$ -module $H$ ,$V(g;H),$ $c(g;H)$ , and $d(g;H)$ ( $=\dim \mathcal{V}(g;H)$ by Theorem 1.1 (2)) are called respectively
the associated variety, the Bernstein degree and the Gelfand-Kirillo ? dimension of $H$ .
Definition(cf.

6Restriction of $U(g)$ -modules to subalgebras.

2.

be a subalgebra of $U(g)$ containing the identitiy element 1
Denote by
gr $A=\oplus_{k\geq 0}A_{k}/A_{k-1}$ with $A_{k}=A\cap U_{k}(g)$ , the graded subalgebra of $S(g)=$ gr $U(g)$associated to . We say that a finitely generated $U(g)$ -module $H$ has the good restriction
if there exists a finite-dimensional generating subspaceof $H$ for which thetoassociated graded $S(g)$ -module $gr(H;H_{0})$ is finitely generated over gr .
This section characterizes, by means of the associated varieties, U(g)-modules $H$(Theorem 2.1). We show that sllch $H’ s$ are finitely
having the good restriction to(Theorem $2.2(1)$ ). Some more properties of these modules $H$ are
generated over
specified in 2.3.
Let

$A$

$’\vee^{-U(g)}$

$A$

$A$

$H_{0}$

$A$

$A$

$A$

2.1. Restriction of $S(V)$ -modules to graded subalgebras. We first discuss theis any complex vector space of finite direstriction of graded $S(V)$ -modules, wheremension. Let $R=\oplus_{k\geq 0}R_{k},$ $R_{k}\subset S^{k}(V)$ , be a graded subalgebra of $S(V)$ containing
the identity element $1\in S(V)$ . $R+=\oplus_{k>0}R_{k}$ denotes the maximal ltomogeneous idealof without constant term. We set for any subset of $S(V)$ ,
$V$

$R$

$Q$

(2.1)

$Q^{\#}$$:=${$\lambda\in V^{*}|$$f(\lambda)=0$

for all $f\in Q$ }.

Let $M$ be, as in 1.1, a finitely generated, non-zero, graded $S(V)$ -rnodule. We consider
the following four conditions on $M$ in relation with :$R$$:=(R_{+})\#$ , and(0) where(a)$M$
associated variety ofdefined in (1.2).
(b) The ideal $Ann_{S(V)}M+R_{+}S(V)$ is of finite codimension in $S[V$ ).
(c) The $S(V)$ -submodule $R_{+}M$ is of finite codimension in $M$ .
(d) $M$ is finitely generated as an R-module.
$\mathcal{V}(M)\cap R_{+}^{\#}=$$R_{+}^{\#}$$\mathcal{V}(M)=(_{d}^{\underline{t}}\backslash nn_{S(V)}M)\#$$,$

Then we get the following proposition on the relation among the$Prop_{Q}sition2.1$

Moreover, (a)(2) If the ringother.

. (1) The condition (a) (resp.

conditions.

) is equivalent to (b) (resp.

.
implies (c)is Noetherian, the four conditions $(a)-(d)$ are$(\Leftrightarrow(b))$

$R$

$(c)$

$e$

is the$(d)$).

$(\Leftrightarrow(d))$,$\prime^{3}$

tuivalent with each

Corollary 2.1. For a vector subspace $W$ of , set $W^{\perp}=\{\lambda\in V^{*}|$
$w>=$
$M,$
$w\in
W$
$S(V)$, is of finite type-module} . finitely generated gradedfor allover the subalgebra $S(W)$ if and only if $\mathcal{V}(M)\cap W^{\perp}=(O)$ .
$V$

$0$

$<\lambda,$$A$$\neq(\cup)$

2.2. Good restriction of U(g)-modules. Now, let be any complex Lie algebra,
and $H$ be a finitely generated, non-zero (g)-module. Proposition 2.1 gives the following
criterion for $H$ to have the good restriction to a subalgebra of $U(g)$ .
$g$

$U$

7
Theorem 2.1. Let be a subalgebra of $U(g)$ containing the identity element $1\in U(g)$ .
is good whenever the condition(1) The restriction of $H$ to
$A$

$A$

V $(g;H)\cap R_{+}^{\#}=(0)$(2.2)

is satisfied. Here $V(g;H)$ is the associated variety of $H$
on algebraic varieties indefined in 1.2, and $R=gr$ A denotes the graded subalgebm of $S(g)$ a ssociated to .
to , one(2) Conversely, if
is Noetherian and if $H$ admits the goodnecessarily has (2.2).
$g^{*}$$A$$r\epsilon.;triction$

$R$

$A$

Remark. The condition (2.2) guarantees that the graded $S(g)$ -module $gr(H;H_{0})$ isfinitely generated over $R=gr$ for every generating subspaceof $H$ .
$A$$H_{0}$

Let be a Lie subalgebra of . Applying Theorem 2.1 to the case $A=U(\mathfrak{n})(R=S(\mathfrak{n})$is obviously Noetherian), we obtain immediately the following$g$$\mathfrak{n}$Corollary 2.2.
finitely generated $U(g)$ -module $H,$ $\neq(0)$ , has the good restriction toholds.
if and only if$A$$\mathcal{V}(g;H)\cap \mathfrak{n}^{\perp}=(O)$$U(\mathfrak{n})$For later applications in \S 3, we give here another consequence of Theorem 2.1. Let, be a subalgebra of $U(g)$ , and let $C(B)$ denote the category of finitely generated
U(g)-modules $H$ on which $B$ acts locally finitely:$B,$$\ni 1$$\dim Bv<\infty$

for all $v\in H$.

We can (and do) take, for such an $H$ , a finite-dimensional B-stable generating subspace. Set $Q=grB$ . Then it is easily verified that the $correspoll(ling$ graded $S(g)-$module $M=gr(H;H_{0})$ is annihilated by the maximal graded ideal ($\prime l+ofQ$ . Hence one
gets
.
(2.3)
$H_{0}\subset H$$\mathcal{V}(g;H)\subset Q_{+}^{\#}$

Definition. We say that a subalgebra of $U(g)$ is large relative tomodule $H$ in the category $C(B)$ have the good restriction to .
$A$

$B$

if all the $U(g)-$$A$

From (2.3) combined with Theorem 2.1, we conclude, be subalgebra of $U(g)$Proposition 2.2. Let $B,$ $Q=grB$ be as above, and $A,$
for which $R=gr$ is Noetherian. Then is large relative to if and only if$\ni 1$

$A$

$B$

$A$

(2.4)

$\iota$$\mathcal{V}_{B}\cap R_{+}^{\#}=(0)$holds for the subset$C(B)$ . In particular, so is the case if$\mathcal{V}_{B}$$:= \bigcup_{H}\mathcal{V}(g;H)$of$Q_{+f}^{\#}$

where $H$ runs over th $\rho U(g)$ -modules in$Q_{+}^{\#}\cap R_{+}^{\#}=(0)$.

8. We will show thatof
Remark. It can be interesting to describe the subvariety
holds for the category $C(B)$ of Harish-Chandra modules of a semisimple LieCorollary 3.1).
(seealgebra$Q_{+}^{\#}$$\mathcal{V}_{B}$$\mathcal{V}_{B}=Q_{+}^{\#}$$g$

Now define the double regular representation of $U(g)\otimes U(g)$ on$(D_{1}\otimes D_{2})v=D_{1}v{}^{t}D_{2}$for$D_{1},$$D_{2}\in U(g)$and$:=U(g)$ by$i1$$v\in \mathcal{U}$.

Here $Darrow {}^{t}D$ denotes the principal anti-automorphism of $U(g)$ , characterized by ${}^{t}X=$-X for $X\in g$ . Identifying $U(g)\otimes U(g)$ with $U(g\oplus g)$ by the Poincar\’e-Birkhoff-Witt.
theorem, we regard as a $U(g\oplus g)$ -module generated by the identity elementwith the good(0) in Proposition 2.2 can be
The condition
restriction property of this module , as follows.
$1\in \mathcal{U}$$\mathcal{U}$$Q_{+}^{\#}\cap R_{+}^{\#}=$$re1_{\subset 1i}ed$$\mathcal{U}$Proposition 2.3. Let $A,$ $B(\ni 1)$ be two subalgebras of $U(g)$ . The restriction of $U(g\oplus$$g)- module\mathcal{U}$ to the subalgebm
is satisfied, where $R=grA$
is good if
is Noetherian.
and $Q=gr$ B. The converse is also true if$A\otimes B$$Q_{+}^{\#}\cap R_{+}^{\#}=(0)$$R\otimes Q$

2.3. Properties of U(g)-modules with good restriction. The U(g)-modules admitting the good restriction enjoy some nice properties as follows.

Theorem 2.2. Let $H$ be a finitely generated, non-zem $U(g)- mod\prime nle$ having the good
restriction to a subalgebm $A\subset U(g)$ . Then,(1) $H$ is finitely generated as an A-module.
(2) Assume thatfor some Lie subalgebm of (see Corollary 2.2). $By$
$H$
,
(1),
, and so one can define the associatcd variety
is of finite type over11
asa
-module, and Gelfand-Kirillov dimension
Bernstein degreeofas well as those as a $U(g)$ -module. These two kinds of invariants have the relations$A=U(\mathfrak{n})$$\mathfrak{n}$$g$$\mathcal{V}(\mathfrak{n};H)$$U(\mathfrak{n})$$d(\mathfrak{n};H)$$c(\mathfrak{n};H)$$c(g;H)=c(n;H)$ , $d(g;H)=d(n;H)$ ,(2.5)

and hence(2.6)$\dim \mathcal{V}(g;H)=\dim \mathcal{V}(\mathfrak{n};H)$

Moreover one has
(2.7)
where

$U(\mathfrak{n})$$p^{*}:$$g^{*}arrow \mathfrak{n}^{*}$$p^{*}\mathcal{V}(g;H)\subset \mathcal{V}(\mathfrak{n};H)$.

,

denotes the restriction map of linear forms.

The following is a direct consequence of Theorem 2.2(2).
Corollary 2.3. If a finitely generated $U(g)$ -module $H$ has the good restriction to
the Gelfand-Kirillov dimension $d(g;H)$ does not exceed.
$\dim \mathfrak{n}$We now give two more consequences of Theorems 2.1 and 2.2:

$U(\mathfrak{n})$,9Corollary 2.4. Let I be a right ideal of $U(g)$ such that $I\neq U(g)$ . For a finitely genemted$U(g)$ -module $H$ , the factor space $H/IH$ is finite-dimensional if $V(g;H)\cap(grI)\#=(0)$ ,where gr $I=\oplus_{k}I_{k}/I_{k-1}$ with $I_{k}=U_{k}(g)\cap I$ .

Corollary 2.5. Let be a Lie subalgebra $ofg$ , and $H$ be a finitely generated $U(g)$ -modulesatisfying the condition. Then, the n-homology gmups $H_{k}(n, H)(k=$$0,1,$) of $H$ (see $e.g_{f}[7]$ for the definition) are all finite-dimensional.
$\mathfrak{n}$$\mathcal{V}(g;H)\cap n^{\perp}=(O)$$\ldots$Let$I$

be a non-trivial right ideal of $U(g)$ . We denote by

in $U(g)$ :(2.8)$N_{I}=\{D\in U(g)| DI\subset I\}$

$N_{I}$

the left normalizer of$I$.

-module.
For any $U(g)$ -module $H$ , the factor space $H/IH$ becomes an$Corol1_{e}’\iotary2.4$ , as follows.
We conclude this section with an interesting generalization of$N_{I}$

Proposition 2.4. Let be a subalgebm ofcontaining the identity element. Denote
gr (resp. gr ) the gmded ideal (resp. gmded subalgebm) of $S(g)$ associated to(resp.
). For a finitely generated $U(g)$ -module $H,$ $H/IH$ is of finite type overdenotes
whenever the variety $\mathcal{V}(g;H)\cap(grI)^{\#}\cap(grB)_{+}^{\#}$ reduces to (0). Here gr
the maximal graded ideal of gr .
$B$

$by$

$N_{I}$

$B$

$I$

$B$

$B$

$I$

$($

$B)_{+}$$B$

This proposition actually includes Corollary 2.4 as a special case $B=C1$ .
An application of the proposition will be given in \S 3 for semisimple Lie algebras .
$g$3.

Nilpotent varieties in$\mathfrak{p}$and good restriction of Harish-Chandra modules.

Until the end of \S 4, let be a complex semisimple Lie algebra. In this section,
in , subalapplying the results of \S 2 we characterize, in relation with nilpotent$U(g)$$(g,e)$
to which all the Harish-Chandra-modules have the good restriction,
gebras ofwhere $g=e+p$ is a symmetric decomposition of . The main result‘ here are stated in
Theorems 3.1 and 3.2.
Although our principal interest lies in the applications to Harish. Chandra modules,
we proceed here in more general situation as much as possible.
$g$$v_{cI}^{r}rieties$$\mathfrak{p}$$g$. Let $e1$ ) any Lie subalge3.1. Associated varieties for $U(g)$ -modules in
bra of , and $Z=\mathcal{Z}(g)$ denotes the center of $U(g)$ . Set $B(e, \mathcal{Z})=U(e)\mathcal{Z}(g)$ , and we$:=C(B(e, \mathcal{Z}))$ of locally, finitely
consider as in 2.2 the category
generated U(g)-modules.
A Lie subalgebra of is said to be symmetrizing if it is the set of fixed points of an
involutive automorphism of . In this case, the U(g)-modules in $C(t, Z)$ will be called
Harish-Chandm $(g, t)$ -modules. This category of Harish-Chandra modules is enjoying anessential role in representation theory of real semisimple Lie groups (see e.g., [9, 19]).
for a Borel subalgebra , includes the highest
On the other hand, the category
weight modules (cf. [10, Chap.7]).
$C(f, \mathcal{Z})$

$e$

$g$

$C(t, \mathcal{Z})$$B(e, \mathcal{Z})- finite$$\epsilon$

$g$

$g$

$C(t, \mathcal{Z})$

$e$

10
. IdenWe now study the associated varietiesof U(g)-modules $H$ in$V(g;H)$
with through the Killing form of , we regard
tifyingas a variety in . For
a subset of , letdenote the set of nilpotent elements of contained in .
$\mathcal{V}(g;H)$$g^{*}$$g$$g$$C(e, \mathcal{Z})$

$g$

$g$

$\mathcal{N}(z)$

$g$

$s$

$g$

Lemma 3.1. (cf. [18, Cor.5.13]) Let $Q(\epsilon, z)=$ gr
be the graded subalgebm of$S(g)$ corresponding to $B(\epsilon, z)=U(t)Z(g)$ .
Then the variety $Q(e, g)+$ (see (2.1)) is
contained in, and hence, by (2.3), it holds that
$B(e, \mathcal{Z})$$\mathcal{N}(\mathfrak{p})$

(3.1)

V

$(g;H)\subset Q(f, Z)_{+}^{\#}\subset \mathcal{N}(\mathfrak{p})$. Herefor every $U(g)$ -module in the categorycomplement of in with respect to the Killing form of .
$H$

$e$

$C(t, \mathcal{Z})$$:=t^{\perp}$$\mathfrak{p}$$g$denotes the orthogonal$g$

It shoud be noted that is an (ad f)-stable subspace of .
For symmetrizing we can construct a Harish-Chandra $(g, t)$ -module
whose associated variety.
this, we need theis exactly the whole nilpotent variety$g$$\mathfrak{p}$$\tilde{H}$$\not\in$$\mathcal{V}(g;\tilde{H})$$\mathcal{N}(\mathfrak{p})$$F_{(}r$following
Proposition 3.1.
a Lie subalgebm of such that
(1) One has
as (ad e)-modules.
(2) The $U(g)$ -module$:=U(g)/U(g)(t+U(g)_{+}^{K})$(3.2)$Let\not\in be$$g$$f\cap \mathfrak{p}=(0)$for$\mathfrak{p}=e^{\perp}$.

$g=k\oplus \mathfrak{p}$$\tilde{H}$lies in the category$C(\epsilon, \mathcal{Z})$, and its associated variety is described(3.3)$\mathcal{V}(g;\tilde{H})=(S(\mathfrak{p})^{K})_{+}^{\#}\cap \mathfrak{p}$$a\vee^{-}\backslash$.

Here) denotes the set of elements $D$ in $gU(g)$ ( $lesp$ . in(resp.
that (ad $X$ ) $D=0$ for all $X\in t$ .
$U(g)_{+}^{K}$$S(\mathfrak{p})^{K}$$S(\mathfrak{p})$) suchA nilpotent element
is called normal if there exists an element T $Ee$ anda non-zero complex number such that $[T, X]=\beta X$ . Let
denote the set of
normal nilpotent elements in .
We now arrive at$X\in \mathfrak{p}$$\mathcal{N}_{nor}(\mathfrak{p}|$$\beta$$\mathfrak{p}$Proppsition 3.2. (1) Let
(3.4)

$e,$

$\mathfrak{p}=\epsilon_{J}^{\perp}$

and$\tilde{H}$be as in Proposition 3.1. Then it holds that$\mathcal{N}_{nor}(\mathfrak{p})\subset \mathcal{V}(g;\tilde{H})\subset \mathcal{N}(\mathfrak{p})$.

(2) Assume, and th , equalities hold insymmetrizing. Then one has$(g,t)$(3.4). Henceis a Harish-Chandm-module such that.
$tb\epsilon$$f\cap \mathfrak{p}=(0)$$\tilde{H}$$f$$\mathcal{V}(g;\tilde{H})=\mathcal{N}(\mathfrak{p})$The following is an immediate consequence of Lemma 3.1 and Proposition 3.2(2).
Corollary 3.1. (see Remark to Proposition 2.2) Assume that is symmetrizing, andletbe the subset ofdefined in Proposition 2.2, where
and $Q(gZ)=grB(t, Z)$ as before. Then one has$t$$\mathcal{V}_{B(\epsilon,z)}$(3.5)$Q(g\mathcal{Z})_{+}^{\#}$$\mathcal{V}_{B(\mathfrak{k},\mathcal{Z})}=Q(t, \mathcal{Z})_{+}^{\#}=\mathcal{N}(\mathfrak{p})$$B(\epsilon, z)=U(\epsilon)\mathcal{Z}(g)$.

11Remark. It is interesting to describe the associated varieties $V(g;H)$ for important
Harish-Chandra $(g, f)$ -modules . We can achieve this for the discrete series of a$H$semisimple Lie group by an elementary method based on Hotta-Parthasarathy’s work
[11] (see also [21]). The details will be discussed elsewhere.

. Let $A,$ , be a
3.2. Characterization of large subalgebras relative tosubalgebra of $U(g)$ , and be a Lie subalgebra of . Consider the folloviing two conditions
on in relation to .
$B(g\mathcal{Z}|$

$t$

$A$

$\ni 1$$g$$g$(NPRO), whereand $R=grA$ .
is large relative to(ALKZ), i.e., all the U(g)-modules $H$ in the category$C(f, Z)$ have the good restriction to . So, in this case, these modules $H$ have the properties specified in 2.3.
$\mathcal{N}(\mathfrak{p})\cap R_{+}^{\#}=(0)$$\mathfrak{p}=t^{\perp}$$A$$B(f, \mathcal{Z})$$A$

Getting together the results in 2.2 and 3.1, we find a close relation between these
conditions as follows, which is one of the most important results of this article.
Theorem 3.1. For
and as above, the condition (NPRO) always implies (ALKZ).
Moreover, these two conditions are equivalent with each other if $R=gr$ is Noetherian
and if is symmetrizing.
$A$

$t$

$A$

$t$

As a special case, we obtain the following criterion.

Theorem 3.2. ( : symmetrizing,) All the Harish-Chandra $(g, t)$ -module havethe good restriction to a Lie subalgebra of if and only if there doc,.: not exist any nonzero nilpotent element of orthogonal towith respect to the Ki’ling form:$e$$A=U(\mathfrak{n})$$\mathfrak{n}$

$g$

(3.6)

$g$

$t+\mathfrak{n}$$\mathcal{N}((t+\mathfrak{n})^{\perp})=n^{\perp}\cap \mathcal{N}(\mathfrak{p})=(0)$.

By applying Proposition 2.4, one gets another consequence of theas in$,:\supset ndition$(NPRO)

Proposition 3.3. Let
be as in Theorem 3.1, and let I be a
, right ideal of$U(g)$ such that $A/A\cap I$ is finite-dimensional.
,(NPRO)
isthesatisfic
factor space
If$H/IH$ is finitely genemted as a-module for every locally t-finit ‘ finitely genemted$U(g)$ -module $H$ .
$f_{f}$$A$$pr\cdot oper$$d$$\mathcal{Z}(g)$4.

Large Lie subalgebras of a real semisimple Lie algebra.

Let 90 be, throughout this section, a real semisimple Lie, andbe the Cartan decomposition of determined by an involution .
write
the complexification of a real vector subspace of by dropping tlie subscript$algebr_{\dot{c}}$$\theta$$g_{0}$$\mathfrak{h}_{0}$$g_{0}$$V_{1}’e$$g_{0}=oplus \mathfrak{p}_{0}$$\mathfrak{h}(\subset g)$

$0’$

.

for

12
A Lie subalgebra of is said to be large in if there exists an element $x\in Int(g_{0})$is large in $U(g)$ relative to $B(\not\in, Z)=U(e)\mathcal{Z}(g)$ (seefor which the subalgebra(ALKZ) in 3.2). This amounts to, thanks to Theorem 3.2, a simple geometric condition:$g_{0}$$g_{0}$$\mathfrak{n}_{0}$$U(x\cdot \mathfrak{n})$(4.1)$(x\cdot \mathfrak{n})^{\perp}\cap \mathcal{N}(\mathfrak{p})=(0)$for some $x\in Int(g_{0})$ .

Here Int
denotes the group of inner automorphisms of . Notice $|hat$ the largenessof a Lie subalgebra does not depend on the choice of a , since suct
are conjugate
with each other by inner automorphisms.
This section specifies many of large Lie subalgebras of , and we find that every
quasi-spherical Lie subalgebra (cf. [3], [15]) is large in .
$(g_{0})$$g_{0}$$t_{0}$$\dagger_{0}^{\dot{2}}s$$g_{0}$$g_{0}$

4.1. Two kinds of typical large Lie subalgebras. Let $g_{0}=t_{0}+a_{p,0}+1t_{m,0}$ be an
Iwasawa decomposition of . Here is the first important example
Lie subalgebrasof .
$of1^{\Gamma}\downarrow rge$$g_{0}$$g_{0}$

Proposition 4.1. The maximal nilpotent Lie subalgebra$u_{m,0}$is$larg\epsilon$in$g_{0}$.

The above proposition, together with Theorem 2.2, covers the results of CasselmanOsborne[8, Th.2.3] and $Joseph$ [ $13$ , II, 5.6] on the restriction of-Cahndra modules
to.
Secondly, letbe any symmetrizing Lie subalgebra ofdefined by an involutive
automorphism of 90. Then there exists an inner automorphism of 90 such that$:=yo\sigma oy^{-1}$ commutes with the Cartan involution. Let
be the$subs$
ofeigenspace decomposition of by . Take a maximal abelian} ace
and an element $X’\in a_{ps,0}$ which is regular in the sense: $\dim Ke\iota\cdot(adX’)$ is minimal
among the elements of. Then one has a Cartan decomposition $()90$ with respect to
as
,
(4.2)
$Haris\dagger_{\iota}$$u_{m}$$\mathfrak{h}_{0}$$g_{0}$$\sigma$$y$$\theta$$90=^{-}y\cdot \mathfrak{h}_{0}\oplus z_{0}$$\sigma_{y}$$g_{0}$$\sigma_{y}$$\lrcorner$$a_{ps,0}$$\mathfrak{p}_{0}\cap g_{0}$$a_{ps,0}$$y\cdot \mathfrak{h}_{0}$$g_{0}=(f_{0}+x’y\cdot \mathfrak{h}_{0})\oplus a_{ps,0}$where $x’=\exp(adX’)$ , andwith reis orthogonal toform. See [20, , Lemma 1.9] for the proof of (4.2). We thus deduce$a_{ps,0}$$t_{0}+x’y\cdot \mathfrak{h}_{0}$$\backslash \backslash ^{1}P^{ect}$to the Killing

$I$

(4.3)

because the elements ofof large Lie subalgebras.

$(x’y\cdot \mathfrak{h})^{\perp}\cap \mathcal{N}(\mathfrak{p})=\mathcal{N}(a_{ps})=(0)$$\alpha_{ps}$,are semisimple, and so this gives the second typical example

Proposition 4.2. Any symmetrizing subalgebm$\mathfrak{h}_{0}$is large in

$Bo$

.

This allows us to deduce the finite multiplicity theorem [1]the quasi-regular
representation on $L^{2}(G/H)$ , associated to a semisimple symmetric space $G/H$ .
$fo$

13
4.2. Inheritance of the largeness by parabolic induction. Let be any parabolic
subalgebra of , and $q_{0}=(_{0}+u_{0}$ with, be its Levi decomposition. Since theis reductive, one can define large Lie subalgebras
Levi component
way.
the
same
just
in
of
The largeness of Lie subalgebras is preserved by parabolic induction.
$q_{0}$$r_{0}=q_{0}\cap\theta q_{0}$$g_{0}$$(_{0=}(f\cap t_{0})+(\mathfrak{p}\cap 1_{0})$$(_{0}$

is a large Lie subalgebm of , the semidirect product Lie subalgebm
Lemma 4.1. If
is large in 90.
$t_{0}$$\mathfrak{h}_{0}$$\mathfrak{h}_{0}+u_{0}$Thanks to this lemma, we can generalize Proposition 4.2 to

Proposition 4.3. (cf. [20]) Let be a symmetrizing subalgebm of the Levi factor
a parabolic subalgebm $q_{0}=r_{0}+u_{0}$ . Thenis large in .
$\mathfrak{h}_{0}$$\mathfrak{h}_{0}+u_{0}$$\mathfrak{l}_{0}$of$g_{0}$

This proposition actually contains Proposition 4.2 as a special case $q_{0}=90$ .
Using this proposition, we can recover our finite multiplicity theorems for induced
representations of semisimple Lie groups, given in [20, I]. See 5.4 for the details.

4.3.

Quasi-spherical Lie subalgebras. Let
be a minimal
parabolic subalgebra of , where
in . We say that a Liedenotes the centralizer ofsubalgebra of is quasi-spherical if there exists a $z\in Int(g_{0})$ such that.
This is equivalent to saying that, if is a connected Lie group with Lie algebra , theanalytic subgroup of corresponding to has an open orbit on the $n\iota aximal$ flag variety$G/Q_{m}$ , where
denotes a minimal parabolic subgroup of .
It is easy to verify that the large Lie subalgebras specified in 4.1-4.2 are all quasispherical.
The following theorem is the principal result of this section.
$q_{m,0}=\mathfrak{m}_{0}+\alpha_{p,0}+u_{m,0}$$g_{0}$$\mathfrak{n}_{0}$$a_{p,0}$$\mathfrak{m}_{0}$$t_{0}$$z\cdot n_{0}+q_{m,0}=g_{0}$$g_{0}$$G$$G$$g_{0}$$\mathfrak{n}_{0}$$G$$Q_{m}$

Theorem 4.1. Quasi-spherical Lie subalgebms are always large in 90.
Remark. One can see from Theorem 3.2, coupled with a recent result of Bien-Oshima,that the converse is also true in the above theorem if
is algebraic ir , i.e.,
is the$N$
Lie algebra of an algebraic subgroupof , where is a semisim le algebraic groupwith Lie algebra .
In fact, it is easy to deduce from our Theorem 3.2 that, if is large in , the induced
representations
have the finite multiplicity property for all finite-dimensional
N-representations (see 5.4; for this,need not to be algebraic). A result of Bienare of
Oshima assures that, under the above assumption, these representat ons
is quasi-spherical.
multiplicity finite only when$g_{0}$$\mathfrak{n}_{0}$$G$$\mathfrak{n}_{0}$

$G$

$Bo$

$\mathfrak{n}_{0}$$g_{0}$$Ind_{N}^{G}(\eta)$$\mathfrak{n}_{0}$$\eta$$Ind_{N}^{G}(\eta)$$\mathfrak{n}_{0}$14Finite multiplicity theorems for induced representations.

5.

Let be any connected Lie group with Lie algebra $90$ (not necessarily semisimple),, be a subalgebra of $U(g)$ with $g=g_{0}\otimes_{R}$ C. Following the idea of induced
and $A,$representations, we can associate, to any given Fr\’echet A-module $E$ , an analytically(see 5.1).
induced G- and (g)-moduleThis section makes clear what we can know about these modules $\Gamma(G\uparrow A;E)$ byapplying our results in \S \S 2-4 (see Theorems 5.1 and 5.2). Moreover. for semisimple ,we largely develop and simplify our previous work [20] on the finiteness of multiplicities
in induced representations, by making use of the associated varieties of Harish-Chandra
modules (see Theorems 5.3-5.5).
$G$$\ni 1$$U$$\Gamma(G\uparrow A;E)$

$G$

and
. We begin with the
5.1. Analytically induced modules$E$
precise definition of our induced modules. Let be as above, and be an A-module with
Fr\’echet space structure on which the elements ofact as continuous linear operators.
We then define $\Gamma=\Gamma(G\uparrow A;E)$ to be the space of all E-valued, real analytic functions
on satisfying$R_{D}f(x)={}^{t}D\cdot f(x)$(5.1)$\Gamma(G\uparrow A;E)$$\mathcal{A}(G;\eta)$

$A$

$A$

$G$

$f$

for $D\in {}^{t}A$ and $x\in G$ . Here $Darrow {}^{t}D$ is the principal anti-automorphism of $U(g)$ (see2.2), and $Darrow R_{D}$ identifies $U(g)$ with the algebra of left invariant differential operatorson . The group acts on by left translation :
$G$

$G$

$L$

$\Gamma$(5.2)$L_{g}f(x)=f(g^{-1}x)$$(g\in G)$.

The $U(g)$ -action on , gained by differentiation, will be denoted ag ein by . We call$(L, \Gamma(G\uparrow A;E))$ the G-representation or $U(g)$ -module analytically in ‘ uced
from $E$ .
If $(\eta, E)$ is a smooth Fr\’echet representation (cf. [20, $2.1]$ ) of a closed subgroup $N$of , the real analytic functions : $Garrow E$ such that
$L$$\Gamma$

$I,$

$G$

$f$

$f(gn)=\eta(n)^{-1}f(g)$for$(n,g)\in N\cross G$,

complexified Lie. Here isform a G-submodule, say $A(G;\eta)$ , of$E$
$N$-module through differentiation. In this sense
algebra of , and is viewed as aour $\Gamma(G\uparrow A;E)s$ include the group theorical (analytically) induced modules $A(G;\eta)$ .
Now let $H$ be a $U(g)$ -module. We discuss (g)-homomorphisms from $H$ to$A;E)$ and especially the intertwining number$\Gamma(G\uparrow U(\mathfrak{n});E)$$t1^{\backslash }e$$\mathfrak{n}$$U(\mathfrak{n})$

$U$

(5.3)

$I_{U(9)}(H, \Gamma)$

When $H$ is irreducible,
Fix an element $x\in G$ . If$:=\dim Hom_{U(\mathfrak{g})}(H, \Gamma)$

(5.4)

.

gives the multiplicity of $H$ in as U(g)-submodules.
is a U(g)-homomorphism from $H$ to .

$I_{U(9)}(H, \Gamma)$$T$$\Gamma=\Gamma(G\uparrow$$\iota_{x}(T)(v)$$\Gamma$$\tau-$$:=(Tv)(x)(v\in H)$

15from $H$ to . It is easily verified thatgives rise to a linear map
with the actions of $xA:=Ad(x)A\subset U(g)$ as$E$$\iota_{x}(T)$(5.5)$\iota_{x}(T)$commutes$\iota_{x}(T)oD=(x^{-1}D)0\iota_{x}(T)$$1D=Ad(x)^{-1}D$ . Moreover, $\iota_{x}(T)=0$ implies $T=0$ , sincefor all $D\in xA$ , where.
$Tv(v\in H)$ are real analytic functions on connected
We have thus obtained a half part of the Frobenius reciprocity for induced modules,
as follows.
‘

$x$

$G$

Proposition 5.1. Let $H,$ $\Gamma=\Gamma(G\uparrow A;E)$ and $x\in G$ be as above. The assignment
defined in (5.4) gives an injective linear map$Tarrow\iota_{x}(T)$

(5.6)

where

$\iota_{x}$$E_{x}$$(e\in E)$.

:$Hom_{U(t\iota)}(H, \Gamma)^{c}arrow Hom_{xA}(H, E_{x})$,stands for the Fr\’echet space $E$ viewed as an $(xA)$ -module $oyD\cdot e=(x^{-1}D)e$This proposition allows us to give in the succeeding subsections criteria for the finiteby means of the associated varieties of $H$ andness of intertwining numbers$I_{U(\mathfrak{g})}(H, \Gamma)$$A$.

5.2. Finite multiplicity criteria, I. First, observe that the vector space $Hom_{xA}(H, E_{x})$in (5.6) is finite-dimensional if so are both A-module $E$ and factor space $H/I_{x}H$ with$:=(Ann.AE.)U(g)$ . Corollary 2.4 together with Proposition 5.1 gives the following
finiteness criterion, which is the first important result of this section.
$I_{x}$

Theorem 5.1. Let $H$ be a finitely generated $U(g)$ -module. The it tertwining number$I_{U(9)}(H, \Gamma)$
from $H$ to an analytically induced $U(g)$ -module $\Gamma=\Gamma(G\uparrow A;E)$ is finite
whenever two conditions:$\dim E<\infty$ ,
(5.7)

and
(5.8)

V $(g;H)\cap x^{-1}\cdot R_{+}^{\#}=(0)$for some $x\in G$ ,with $R=grA$ , is the algeis the associated variety
are satisfied. Herecoadjoint representation.
thethroughbmic variety of defined in 2.1, and acts on$ofH_{f}R_{+}^{\#}$$\mathcal{V}(g;H)$$G$$g^{*}$$g^{*}$, of $U(g)$ , let $C(B)$ be as in 2.2 the category of locally B-finite,
For a subalgebra $B,$
$U(g)$-modules. The above theorem together with (2.6) immediately
finitely generated
gives$\ni 1$Corollary 5.1. Let $R=grA_{f}Q=grB$ be the gmded subalgebras of $S(g)$ associatedto subalgebras $A,$ $B\subset U(g)$ respectively. If there exists an elemen $tx\in G$ such that(0), the intertwining number $I_{U(9)}(H, \Gamma(G\uparrow A;E))$ is finite for every$U(g)$ -module $H$ in $C(B)$ and for every A-moduleof finite dimension.

$R_{+}^{\#}\cap x\cdot Q_{+}^{\#}=$

$E$

16
Estimation of the multiplicities. Let be a Lie subalgebra of , and $H$ be a$U(g)$ -module in the category $C(B)$ with $B=U(e)$ . Take a finite-dimensional, B-stable,
is generated by 1 and as algebra, it is. By noting thatgenerating subspaceeasy to see that the subspaces $H_{k}=U_{k}(g)H_{0}(k=0,1, \ldots)$ are all 4;-stable. Hence thecorresponding graded $S(g)$ -module $M=gr(H;H_{0})=\oplus_{k\geq 0}M_{k}$ with $M_{k}=H_{k}/H_{k-1}$ ,admits a natural B-module structure. Write this B-action on $M$ by
5.3.

$t$

$g$

$B$

$H_{0}$$e$$B\cross M\ni(D, v)arrow Dov\in M$ ,in order to distinguish it from the original $S(g)$ -action. One finds from the definition,$XoDv-D(Xov)=((adX)D)v$(5.9)

for $X\in e$ and $D\in S(g)$ .
With (5.6) in mind, we can give, by using this $(S(g), B)$ -module -/7ff an upper boundof the intertwining number $I_{A}(H, E)=\dim Hom_{A}(H, E)$ as in

Proposition 5.2. Let $H,$ $B=U(e)$ be as above, andOne has for any A-module ,$A_{f}\ni 1$, be a subalgebm of $U(g)$ .

$E$(5.10)

where $R=gr$$I_{A}(H, E) \leq\sum_{k=0}^{\infty}I_{A\cap B}(M_{k}/((A\cap B)o(R_{+}M)_{k}), E)$$A$.

and $(R_{+}M)_{k}$ $:=R_{+}M\cap M_{k}$ .

This together with (5.6) immediately gives the following theorem,

Theorem 5.2. The intertwining $numberI_{U(g)}(H, \Gamma)$ from $H$ in $C(B)$ . $B=U(e)$ , to an
analytically induced $U(g)$ -module
is bounded by$\Gamma=\Gamma(G\uparrow A;E)$(5.11)$\min_{x\in G}\{\sum_{k=0}^{\infty}I_{xA\cap B}(M_{k}/((xA\cap B)o((xR_{+})M)_{k}), E_{\sigma}.)\}$.

Remarks. (1) By setting $\epsilon=(0)$ , or $B=$ Cl, one finds that this theorem recovers
Theorem 5.1. In fact, in this case (5.11) turns to be$\dim E\cross\{\min_{x\in G}(\dim(M/(xR_{+})M))\}$ ,which is finite under the assumptions (5.7) and (5.8) (see Proposition 2.1).
(2) If $A=U(n)$ for a Lie subalgebra of , one finds from the
Birkhoff-Witttheorem that $xA\cap B=U(x\cdot \mathfrak{n}\cap f)$ with $x\cdot n=Ad(x)n$ . Hence, in view of (5.9) we have$\mathfrak{n}$$g$$p_{oint^{arrow.ar\acute{e}-}}$$(xA\cap B)o((xR_{+})M)_{k}=((xR_{+})M)_{k}=((x\cdot \mathfrak{n})M)_{l}$in (5.11).

17
Finite multiplicity criteria, II: case of semisimple Lie groups. Now assume be a connected semisimple Lie group with finite center, and let $K$ be a maximal
compact subgroup of . In this subsection we apply the results of 5.2 and 5.3 to HarishChandra modules for .
with $e_{0}=Lie(K)$ . denotes a Cartan
By keeping the notation in \S 4,$H$-module (see 3.1).
decomposition of $g_{0}=Lie(G)$ . Let
be a Harish-Chandra (Assume that the compact group $K$ acts on $H$ in such a way as
5.4.

$G$

$G$

$G$

$g_{0}=g_{0}\oplus \mathfrak{p}_{0}$$g,$$\dim\{Kv\}<\infty$$\epsilon\backslash$,and$(d/dt)_{t=0}(\exp tX)v=Xv$

for $v\in H$ and $X\in f_{0}$ , where $\{Kv\}$ stands for the K-submodule of $H$ generated by. Such an $H$ is called a Harish-Chandm $(g, K)$ -module. Observe that, since our $K$ isconnected, the above two conditions assure the compatibility of and $K$ actions:
$v$

$g$

$k\cdot Xv=(k\cdot X)\cdot k^{-1}v$

for $k\in K$ and $X\in g$ , where $k\cdot X=Ad(k)X$ .
We note that, if a Harish-Chandra $(g, f)$ -module $H$ appears in some$A;E)$ as a $U(g)$ -submodule, $H$ necessarily has the $(g, K)$ -module
inheritedfrom . A fundamental theorem of Harish-Chandra says that the ( , reducible) HarishChandra ( , K)-modules correspond to the (irreducible) admissible representations of, by passing to the K-finite part (see e.g., [19, Chap.8]). From these two reasons weconcentrate on the $(g, K)$ -modules from now on.
$\Gamma=\Gamma(G\uparrow$$S\mathfrak{c}ruCture$$i$$\Gamma$$g$$G$

Definition. Let $\Gamma=\Gamma(G\uparrow A;E)$ and $A=\mathcal{A}(G;\eta)$ be the induced G- and $U(g)-$modules defined in 5.1. We say that (resp. ) has the finite multiplicity property if the(resp. $I_{U(9)}(H,$ ) is finite for every Harish-Chandra
intertwining number$(g, K)$ -module $H$ .
$\Gamma$$\mathcal{A}$$I_{U(9)}(H, \Gamma)$$\mathcal{A})$or tocommutes with theRemarks. (1) Any $U(g)$ -homomorphism from $H$ to$K$
K-actions by virtue of the connectedness of .
(2) When is a finite-dimensional unitary representation of a closed subgroup $N$ ,$\Gamma$$\mathcal{A}$$\eta$$theassignmentHarrow I_{U_{N}}g)(H,A)givesanupperboundofthemultiplicityfunctionforG- representationL^{2}- Ind(\eta)unitarilyinducedfrom\eta.HereHrursovertheHarish-$

Chandra $(g, K)$ -modules associated with irreducible unitary representations of
20, , \S 3 for the details.
$[$

$I$

$G$

. See

$]$

Here is our first application to semisimple group , of the general results in 5.2-5.3,
which follows immediately from Corollaries 3.1 and 5.1.
$G$

has the finite $7?1ltiplicity$ property
Proposition 5.3. The induced module.
for any finite-dimensional A-module , if there exists an $x\in G$ for whichis the totality of nilpctent elements in .
gr $A)_{+}^{\#}=(0)$ (cf. (NPRO) in 3.2). Here$\Gamma(G\uparrow A;E)$

$E$

$($

$\mathcal{N}(\mathfrak{p})\cap x$$\mathcal{N}(\mathfrak{p})$$\mathfrak{p}$18
As a special case, we gain
Corollary 5.2. Iftion 5.3 is true for$n_{0}$

is a large Lie subalgebm ofC.
with$A=U(\mathfrak{n})$$g_{0}$(see \S 4), the conclusion of Proposi-$\mathfrak{n}=n_{0}\otimes_{R}$In view of the large Lie subalgebras specified in \S 4, one may realize that this corollary
has numbers of applications.
(see 4.3), Bien-Osl: ima recently got a
Remark. For quasi-spherical Lie subalgebras
result similar to the above corollary. But our method here is completely different fromtheirs.
$n_{0}$

Now let $(\eta, E)$ be a smooth Fr\’echet representation of a closed subgroup $N$ of , and. For a Harish-Chandra $(g, K)$ -module $H$ , take aconsider the induced moduleof $H$ . Then the associated gradedfinite-dimensional, K-stable generating subspace$S(g)$ -module $M=gr(H;H_{0})=\oplus_{k}M_{k}$ has a natural K-module stru $(ture$ .
from $H$ to $A(G;\eta)$ :We can estimate the intertwining number$G$$\mathcal{A}(G;\eta)$$H_{0}$$I_{U(\mathfrak{g})}(H, A(G;\eta))$

Theorem 5.3. For each $x\in G$ , one has the inequality(5.12)$I_{U(9)}(H, \mathcal{A}(G;\eta))\leq\sum_{k=0}^{\infty}I_{K\cap xNx^{-1}}(M_{k}/((x\cdot n)M)_{k}, E_{x})$where $((x\cdot \mathfrak{n})M)_{k}=M_{k}\cap(x\cdot n)M$ , anddefined by $\eta_{x}(xnx^{-1})=\eta(n)(n\in N)$ .

$(\eta_{x}, E_{x})$,is the representatlon of $xNx^{-1}$ on$E$

This theorem enables us to deduce useful criteria for the finiteness of intertwining, which are applicable even to $infinite- dinl\langle\backslash ,Ilsional(\eta, E)s$ .
numbers
denote the set of $eq$ { ivalence classes of
To be specific, fix an $xEG$ , and let
irreducible finite-dimensional representations of $K\cap xNx^{-1}$ . Then the locally finite$(K\cap xNx^{-1})$ -module
is decomposed into a direct sumof the irreducibles as,$I_{U(g)}(H, \mathcal{A}(G;\eta))$$\Pi$$M/(x\cdot \mathfrak{n})M=\oplus_{k}M_{k}/((x\cdot \mathfrak{n})M)_{k}$$M/(x\cdot \mathfrak{n})M\simeq\oplus_{\gamma\in\Pi}[m_{\gamma}]V_{\gamma}$

where
of in
$\gamma$$V$$One_{1}$(5.13)

is an irreducible $(K\cap xNx^{-1})$ -module of class , and.
finds that (5.12) is rewritten as$\gamma$$m_{\gamma}$

denc tes the multiplicity$M/(x\cdot \mathfrak{n})M$$I_{U(\mathfrak{g})}(H, \mathcal{A}(G;\eta))\leq\sum_{\gamma\in\Pi}m_{\gamma}I_{K\cap xNx^{-1}}(V_{\gamma}, E_{x})$.

The sum in the right hand side is finite if and only if there exists a fnite subset $F$ offor whichor $I_{K\cap xNx^{-1}}(V, E_{x})=0$ for(5.14),$m_{\gamma}=0$

and(5.15)$\gamma\not\in F$.
The above discussion coupled with Corollary 2.2 leads us to the following$I_{K\cap xNx^{-1}}(V_{\gamma}, E_{x})<\infty$for$\gamma\in F$

$\Pi$

19
Theorem 5.4. Under the above notation, the intertwining numbe $rI_{U(9)}(H, \mathcal{A}(G;\eta))$takes finite value
from a Harish-Chandm module $H$ to an induced $U(g)$ -module$x\in G$ such thatexistsanthere
if$\mathcal{A}(G;r)$(5.16)$V(g;H)\cap(x\cdot n)^{\perp}=(0)$and that(5.17)

holds$I_{K\cap xNx^{-1}}(V_{\gamma}, E_{x})<\infty$

for every irreducible constituent ofdenotes the associated variety of .
$V_{\gamma}$$M/(x\cdot \mathfrak{n})M$,. Here $M=gr(H;H_{0})$ , and$\mathcal{V}(g;H)$$H$

From this theorem, we immediately deduce an interesting criterion forof multiplicity finite, as follows.

$\mathcal{A}(G;\eta)$

to be

Theorem 5.5. Let $N$ be a closed subgroup of whose Lie algebrc
is large inand take an element $x\in G$ such that
. Then, $for$. a smooth Fr\’echet
representation
has the finite multiplicity propertyof the induced module$x^{-1}Kx\cap N$ .
is
so
the
restrictiontosubgroupthecompactifof$G$$n_{0}$$g_{0f}$$(x\cdot n)^{\perp}\cap \mathcal{N}(\mathfrak{p})=(0)$$(\eta, E)$$N_{f}$$\mathcal{A}(G;\eta)$$\eta$

This theorem extends one of the principal results in our previous work, [20, , Th.2.12],
where we studied the case of semidirect product large Lie subalgebras
specified in Proposition 4.3, through the theory of $(K, N)$ -spherical fu. Interesting
applications are found in [20, II] for reduced generalized Gelfand-Gra representations.
$I$$\mathfrak{n}_{0}=\mathfrak{h}_{0}+u_{0}$$\iota_{c^{\neg}}tions$$\sim 3v$

5.5. Relation with K-harmonic polynomials on . We conc; $nde$ this article byrelating the $(K\cap xNx^{-1})$ -modulein Theorems 5.3 and 5.4, with K-harmonicpolynomials on .
As in \S 3, regard the elements ofas polynomial functions on through the Killing
form of . An elementby every $Ad(K)-$is called K-harmonic if isinvariant, constant coefficient differential operator on without constant term. Let
denote the totality of K-harmonic polynomials on . It is easily ob,served thatisa graded K-submodule of, whereis
K-stable.
A result of Kostant and Rallis (cf. [12, p.381]) says that the $mu$ ]gives a K-isomorphism$\mathfrak{p}$$M/(x\cdot \mathfrak{n})M$$\mathfrak{p}$$S(\mathfrak{p})$$g$$\mathfrak{p}$$f$$f\in S(\mathfrak{p})$$annihilat_{\sigma’}\cdot d$$\mathcal{H}(\mathfrak{p})$$\mathfrak{p}$$\mathcal{H}(\mathfrak{p})$$\mathfrak{p}$$\mathcal{H}^{k}(\mathfrak{p})=\mathcal{H}(\mathfrak{p})\cap S^{k}(\mathfrak{p})$$S(\mathfrak{p}):\mathcal{H}(\mathfrak{p})=\oplus_{k\geq 0}\mathcal{H}^{k}(\mathfrak{p})$$\}_{\mathfrak{t}}iplication(h,j)arrow$$hj(h\in \mathcal{H}(\mathfrak{p}), j\in S(\mathfrak{p})^{K})$(5.18)

where$\mathcal{H}(\mathfrak{p})\otimes S(\mathfrak{p})^{K}\simeq S(\mathfrak{p})$$S(\mathfrak{p})^{K}$(5.19),

is the algebra of $Ad(K)- fixed$ elements of$S(\mathfrak{p})$. This$i_{111}plies$$S(\mathfrak{p})=\mathcal{H}(\mathfrak{p})\oplus(\mathcal{H}(\mathfrak{p})\otimes S(\mathfrak{p})_{+}^{K})$as K-modules, withas before. The linear projection fromtoalong this decomposition will be denoted by .
$S(\mathfrak{p})_{+}^{K}=S(\mathfrak{p})^{K}\cap \mathfrak{p}S(\mathfrak{p})$$\mathcal{H}(\mathfrak{p})$$\alpha$$S(\mathfrak{p})$20
For any Harish-Chandra $(g, K)$ -module $H$ , we can and do take a finite-dimensionalof the form
generating subspace$H_{0}\subset H$(5.20)$H_{0}=\oplus_{5\in\Phi}H(\delta)$

denotes the -isotypic
for a finite subset of( $=the$ unitary dual of $K$ ), whereis stable under $K$ and $U(g)^{K}$ , one sees that thecomponent of $H$ . Noting thatassociated graded $(S(g), K)$ -module $M=gr(H;H_{0})=\oplus_{k}M_{k}$ is annihilated by and. Hence it follows from (5.19) thatand that$\Phi$$\hat{K}$$H(\delta)$$\delta$$H_{0}$$e$$M=\mathcal{H}(\mathfrak{p})M_{0}$$S(\mathfrak{p})_{+}^{K}$(5.21)$\beta$: $?\{(\mathfrak{p})\otimes M_{0}\ni h\otimes varrow hv\in M$

asgives a surjective K-homomorphism (cf. [17, Proof of Prop.5.5]). Note that
K-modules.
Now let $N$ be any closed subgroup of with complexified Lie algebra . We set$H_{0}\simeq M_{0}$$G$(5.22)$n$$\mathcal{H}(P;\mathfrak{n})=\mathcal{H}(\mathfrak{p})/\alpha(p[n]S(\mathfrak{p}))$,where $p[n]$ denotes the image of by the projection $garrow p$ along
is a $(K\cap N)$ -module.
with
as follows.
We can relate $(K\cap N)$ -module$\mathfrak{n}$$g=f\oplus \mathfrak{p}$. Note that$\mathcal{H}(\mathfrak{p};n)$$M/\mathfrak{n}M$$\mathcal{H}(\mathfrak{p};\mathfrak{n})$Proposition 5.4. (1) The K-homomorphism$(K\cap N)$ -module map(5.23)$\beta$in (5.21) natumlly nduces a surjective$\prime l$.

$\mathcal{H}(\mathfrak{p};\mathfrak{n})\otimes M_{0}arrow M/nM$(2) If$\mathcal{N}(\mathfrak{p})\cap n^{\perp}=(0)$, the space$\mathcal{H}(\mathfrak{p};\mathfrak{n})$is finite-dimensional.

This proposition, combined with Theorem 5.3, allows us to estimate the intertwining
as in
numberin (5.12) by means ofand$I_{U(\mathfrak{g})}(H, \mathcal{A}(G;\eta))$$\mathcal{H}(\mathfrak{p};\mathfrak{n})$$H_{0}\simeq M_{J}$Corollary 5.3. Let $H$ be a Harish-Chandm $(g, K)$ -module and$U(g)$ -module analytically induced from a smooth N-representation$\mathcal{A}((\urcorner\tau;\eta)$(5.24)$(?jI’\cap/)$

be the G- and. Then one has$I_{U(g)}(H, \mathcal{A}(G;\eta))\leq I_{I\backslash ’\cap xNx^{-1}}(\mathcal{H}(\mathfrak{p};x\cdot \mathfrak{n})\otimes H_{0}, E_{x})$for each $x\in G$ . Here $M=gr(H;H_{0})$ with$xnx^{-1}arrow\eta(n)(n\in N)$$H_{0}$in (5.20), and $xNx^{-1}$ acts on $E_{x}=E$ by.

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