Гильбертовы многообразия флагов и связанные с ними представления Текст научной статьи по специальности «Математика»

MSC 22E65, 14M15, 35Q58, 43A80, 17B65

Hilbert flag varieties and associated representations 1

© G. F. Helminck and A. G. Helminck

Korteweg-de Vries Institute, University of Amsterdam, The Netherlands North Carolina State University, USA

Let H be a complex Hilbert space. If H is finite dimensional, then one knows from the Borel-Weil theorem that the finite dimensional irreducible representations of the general linear group GL(H) can be realized geometrically as the natural action of the group GL(H) on the space of global holomorphic sections of a holomorphic line bundle over a space of HH

GL(H)

GL(H)

In this overview we will give an infinite dimensional analogue of all these

HH

of flags that can be given a Hilbert space structure. It is a homogeneous space for an analogue of the general linear group, the so-called restricted linear group. There is a proper analogue of the notion of maximal torus in this restricted linear group. Over the flag variety there exist line bundles that are similar to the finite dimensional ones. In the “dominant” case the space of global holomorphic sections of such a line bundle turns out to be nontrivial. However, the action of the restricted linear group can, in general, not be lifted to the line bundle under consideration and one has to pass to a central extension of this group. The representation space contains a unique section on which the maximal torus acts by the dominant weight. It is the generator of an irreducible highest weight module of the central extension.

Keywords: Hilbert flag varieties, holomorphic line bundles, global sections, central extension, maximal torus, dominant weight, highest weight module

§ 1. The flag variety

Let H be a separable complex Hilbert space with innerproduet < ■, ■ >, One will consider certain finite chains of subspaces in H and they will be called flags as in the finite dimensional case. First one has to specify the “size” of the components of

H

H = Hi © ... © Hm, where Hi ± Hj for i = j. (1.1)

1This work has been performed within the NWO-program Geometric Aspects of Quantum Theory and Integrable Systems with the project number 047.017.015.

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One assumes that m, = dim H, satisfies 1 ^ ^ ro and that H has a Hilbert basis

indexed by the integers.

Remark 1. A natural wav to get such a decomposition is to consider in GL(H) the maximal torus T(N) of all invertible diagonal operators that differ from the identity by a nuclear operator. Concretely, it consists of all operators of the form

diag({1 + ts}), with 1 + ts = 0 and |ts| < ro.

s£Z

In T (N) we have the dense subgroup Tf given by

Tf = {t | t = diag({1 + ts}) G T(N), ts = 0 for only finitely many s in Z}

Any analytic grouphomomorphism of Tf into C* has the form

t = diag((1 + ts)) i—^ ^(1 + ts)ms = Xm(t),

sGZ

where m = {ms}, with ms G Z for all s G Z. This character xm can be continued to

T( N)

ms, s G Z. This extension of xm is also denoted by xm and one writes T for the group of analytic characters of T(N). To each xm one associates the decomposition

ms

Let pi; 1 ^ i ^ m, be the orthogonal projection of H onto H Then we will use throughout this paper the following

Notation 1. If g belongs to B(H), the space of bounded linear operators from H to H, then g = (gij), 1 ^ i ^ m and 1 ^ j ^ m, denotes the decomposition of g with respect to the {H, | 1 ^ i ^ m}. That is to say gij- = p o g | Hj,

To the decomposition (1) one associates the basic flag F(0) given by

r

0 C Hi C ... C @ Hj C ... C H. j=1

In H one considers flags F = {F(0),...,F(m)}, that is to say chains of closed H

{0} = F(0) C F(1) C ... C F(m) = H, that are of the same “size” as the basic flag F(0), i.e. for all i, 1 ^ i ^ m,

dim(F(i)/F(i — 1)) = dim H,.

FH

H = F1 © ... © Fm , where F, = F(i) n F(i — 1)x.

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One will denote such a flag F by F = {F(0),..., F(m)} as well as F = {F1,..., Fm}, The class of flags one obtains in this way is still too wide and it will be required that our flags do not differ too much from the basic flag.

Definition 1. Let F be the collection of flags F = {F1,...,Fm}, satisfying dim F = dim H,, and for all i and j with j = i, the orthogonal projection pj : F, ^ Hj is a Hilbert-Schmidt operator. One calls F the flag variety corresponding

(1)

F

certain unitary group. Let F = {F1,..., Fm} belong to F- From the definition of F one knows that there is for each i, 1 ^ i ^ m, an isometry u, between H^d F,. If one puts u = u1 © ... © um, then u belongs to the group of unitary transformations, U(H), of H and for each i, 1 ^ i ^ m, one has

In other words, the flag F is the image under u of the basic flag. The condition defining F implies that u = (uj) satisfies: Uj is a Hilbert-Schmidt operator for

i = j, This brings us to the introduction of the following group.

Definition 2. The restricted unitary group, Ures(H), consists of all u = (uj) in U (H) such th at uj is a Hilbert-Schmidt ope rator if i = j.

Clearly the stabilizer of F(0) in Ures(H) is equal to H U(Hj) and therefore one

For several reasons, like the description of the manifold structure on F and the consideration of non-unitarv flows on F it is convenient to have a description of F as the homogeneous space of a larger group of automorphisms of H, The Banach structure of this group follows directly from that of its Lie algebra. Therefore one starts with the analogue of the Lie algebra of the general linear group.

Definition 3. A restricted endomorphism, of H is a u = (uj ) in B(H ) such th at Uj is a Hilbert-Schmidt operator for all i = j. One denotes the space of all restricted endomorphisms of H by Bres(H),

The space Bres(H) is a subalgebra of B(H) since the collection of Hilbert-Schmidt operators is closed under left and right multiplication with bounded operators. Hence it is also a Lie subalgebra of the Lie algebra B(H). On Bres(H) one will introduce a norm. The algebra Bres(H) becomes a Banach algebra if one equips it with the norm

m

can identify F with the homogeneous space

m

Ures(H)/n U(Hj).

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defined by

Iu || 2 = ||u|| + ||uj ||hS .

2

Since the adjoint of a Hilbert-Sehmidt operator is again Hilbert-Sehmidt, it is clear that Bres(H) is stable under “taking adjoints”, If GL(H) denotes the group of invertible elements in B(H), then one considers

Definition 4. The restricted linear group GLres(H) consists of g such that g belongs to GL(H) nBres(H).

One easily verifies that GLres(H) consists of the invertible elements in Bres(H), Thus one can identify the tangent space at any point of GLres(H) with Bres(H), With each g in GLres(H) one can associate the flag

0 c gHi c g(Hi © H2) c ... g(Hi © ... © H*) c ... c H.

From the definition of GLres(H) one sees directly that this flag belongs to F- The stabilizer in GLres(H) of the basic flag is the “parabolic subgroup” P consisting of upper triangular matrices g G GLres(H):

/ gii .............. gim \

0

V 0 ... 0 gmm )

g

with ga G GLres(H), 1 ^ i ^ m. Thus one can identifv F also with the homogeneous space GLres(H)/^^et t : GLres(H) ^ F be the projection t(g) = g ■ F(0^^n F one puts the quotient topology that makes t into an open continuous map,

F

the set Q in GLres(H) consisting of g such that matrices

g11 . . . g1,

g,1 . . . g,,

belong to GLres(H1 © ... © Hi) for all i ^ m. The set Q is open and, as in the finite dimensional case, can be decomposed. For, let U_ be the subgroup of GLres(H) consisting of g = (gj) such that g,, = IdHi for all i and gj = 0 for j > i. The group U_ is given induced by that of GLres(H), The set Q has a similar

description as in the finite dimensional situation:

Lemma 1 The 'map (u,p) up from U- x P ^ GLres(H) determines a homeomor-

phism between U- x P and Q.

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An easy eonseqenee is that the flagvariety F is a Hilbert manifold based on the Lie algebra E of U—

2. The connected components of F

Let g = (gj) be an element of GLres(H) and put g 1 = (hj), Then one has, by definition for all i, 1 ^ i ^ m,

giihii = IdH — ^ gifc hfcj.

fc=i

This implies that each g^ is a Fredholm operator, that is to say it has a finite dimensional kernel and eokernel. The collection of Fredholm operators on a Hilbert space K is denoted by $(K) and it is an open part of the space B(K), Its connected components are given by the index, which is defined as

ind(B) = dimker(B) — dimcoker(B), for B G $(K).

Since all off-diagonal operators are Hilbert-Schmidt and hence compact, the operator

/ gii 0 \

Y 0 gmm J

where g = (gj) G GLres(H), is a Fredholm operator of index zero. Hence the indices of the g^, 1 ^ i ^ m, satisfy

m

^^ind(gii) = 0 and ind(gkk) = 0 if mk < w.

i=1

These relations lead to the introduction of the subgroup Z of Zm consisting of vectors z = (z1,..., zm), zi G Z, such that

zi = 0, and zk = 0 if mk < w.

i=1

The standard properties of the index imply that the map i : GLres(H) ^ Z,

g ^ (md(gn),... ,ind(g mm)) ,

is a continuous grouphomomorphism. Hence the subsets GL^H) of GLres(H), consisting of g such that i(g) = z with z G Z are open. In fact, they are exactly the connected components of GLres(H), for one can show

Proposition 1 For each z G Z, the set GL^Z^H) is non-empty and connected.

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Since the parabolic group P is connected, one concludes for the flag variety

F

F(z) = {g.F(0) I g G GL«(H)} .

Remark 2, A holomorphie line bundle L over F consists simply of a collection of holomorphie line bundles {Lz M F(z) | z G Z}, Therefore one restricts one’s attention to holomorphie line bundles over F(0) in the next section.

§ 3. The holomorphie line bundles over F(0)

Each F in F(0) is equal to g.F(0) with g G GLres(H) of the form

(a) For each i, 1 ^ i ^ m, gii = IdHi + a “finite-size” operator,

(b) For all md j, i < j, gij is a “finite-size” operator,

(c) For all md j, j < i, gij belongs to HS(Hj, Hi).

Note that for all the operators gii from (a) one can speak of det(gii). Since one is working in an analytic setting it is convenient to consider a somewhat wider class of operators such that on one hand the framework is complete and on the other one can take determinants of certain minors. Recall that the determinant is defined for each operator of the form “identity + a nuclear operator”. Therefore one introduces B2(H) consisting of g G Bres(H) such that gii — IdHi G N(H) and gij G HS(Hj, H^ for i = j. On B2(H) one puts a different topology than the one induced by Bres(H), For, let 2 ^e the subspace of Bres(H) consisting of b G Bres(H) such th at bii G N (Hi)

and bij G HS(Hj, Hi) for i = j. Then 2 is a Banach space if it is equipped it with

the norm || • ||z given by

m

Ilb|z = ^2 |bij IIhS + ^2 |bii lltr.

i=j i=1

The collection B2(H) is nothing but 2 shifted by the identity and one transfers the Banach structure on 2 to B2(H) by means of the map g g + Id, Since the product of two Hilbert-Sehmidt operators is nuclear, one sees that B2(H) is closed under multiplication. Moreover the multiplication with an element of B2(H) is an analytic map from B2(H) to itself. In B2 (H) one has the subgroup U- and its “adjoint” the group

U+ = {u* | u G U-} .

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Consider an element b in B2(H). Since Ьц has the form “IdHi + a nuclear operator”, one can find an operator Ьц in GL(Hj) such that Ьц - IdHi and (b^)-1 - IdHi both belong to N(Hj). Now one defines u = (u^) in U- and v = (v^) in U+ by

uii = vjj = Id#*, ujj = -bjj (bjj )-1 if i > j and Ujj = 0 if j > i,

Vjj = -(bii)-1bij if i < j and Vjj = 0 if i > j.

A direct verification shows that ubv belongs to Id + N(H). Sinee B2(H) is closed with respect to taking adjoints, there holds

Lemma 2 Every b Є B2(H) can be written in the form b = u1b1v^r b = v2b2u2, where щ and u2 belong to U— v2 and v1 belong to U+ and b1 and b2 lie іn Id+N(H).

If one takes into account that for each i, the operator Ьц can be chosen of the form bjj + fa, where fa is finite dimensional, then for all c in B2(H) sufficiently close to b one can take сц = сц + /¿¿. By using this, one shows easily that the map det : B2(H) ^ C, defined by det(b) = det(u1b1v1) = det(b^, where b = u1 b1v1 as in the lemma above, is well-defined and analytic on B2(H),

Remark 3, Since the operators in Id + N(H) lie dense in B2(H) and since det is multiplicative on Id + N(H), one gets that for each b1 and b2 in B2(H)

det(b1b2) = det(b1) det(b2)

From the fact that an operator g of the form Id + N(H) is invertible if and only if det(g) is non-zero, one sees that the invertible elements of B2(H) form a group G consisting of b G B2(H) with det(b) = 0, Clearlv G is a Banach Lie group with Lie algebra 2 and it acts analytically and transitivelv on F(0)- The stabilizer T of F(0) G

( ¿11 0

V о

0 ¿mm /

with tii G {Id + N(Hi)} n GL(Hi) and tij G HS(Hj, Hi) for j > i. Thus one can

identity F(0) with the homogeneous space G/T.

For each k = (k1,..., km) in Zm, one defines in T by

= (diag{1 + is}) = n (1+ **i)kl n (1+ **2)k2 ... n (1 + ^)km .

siGSi *2 €S2 smeSm

Clearly extends to an analytic character of T by means of the formula

(t) = det(tn)kl . . . det(*mm)km.

t

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To each 0k one can associate a holomorphic 1 ine bundle L(k) over F(0) = G/T. It is

defined as follows: consider on the space T x C the equivalence relation

(g1, A1) ~ (g2, A2) ^ g1 = g2 o t, with t G T and A2 = A10k(t).

The space T x C modulo this equivalence relation is L(k)- For each g G G and each A C (g, A) [g, A]

There is a natural projection : L(k) ^ F(0) given bv

([g A]) = g ■ F(0).

The space L(k) is a Hilbert manifold based on the Hilbert space E © C,

8 4. The central extension

There is a natural analytic action of the group G on the space L(k) by left translations

g1 ■ [g2, A] = [g1g2,A].

This is a lifting of the natural action of G on F(0) to one on L(k)- However, the natural action of GL(°S(H) can, in general, not be lifted to one on L(k)- Such an attempt may lead to nontrivial central extensions of GL(0S(H) as one will show. Note that each g in GL^H) can be written as g = dg2, with g2 G G and d belonging to the “diagonal” subgroup D of GL(0S(H) consisting of g = (gij) with gij = 0 if i = j. Clearly the group D normalizes the group G- Since the determinant of an operator of the form “identity + nuclear” is invariant under conjugation with an invertible operator, one gets that D centralizes each 0^, i.e. for each t in T and dD

0k(dtd-1) = 0k (t).

This fact permits you to lift the action of D on F(0) to one on L(k) by means of

d ■ [g, A] = [dgd-1, A].

For an element d tom D n G, this action differs by a factor 0k (d-1) from the action induced by that of G- Hence one cannot combine them to an action of GL^H) on F(0)- To overcome this problem a group extension G of GL(0S(H) will be built. It is defined by

G = |(g, d) | g G GL(0S(H), d G D and gd-1 G G j .

As one verifies directly this group acts on L(k) by means of

(g,d)[gb A1] = [gg1d-\ A1].

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GD

n : G A GL^H) be the canonical project ion, i.e. n((g,d)) = g for all (g,d) G G, For certain subgroups of GL(0S(H) there exist several wavs to embed them into G, Therefore we introduce special notations for two of them. Let i resp. j be the embedding of G resp. D into G given bv

i(g) = (g, Id) and j(d) = (d,d).

As a group G is the semi-direct product of i(G^d j(D). We equip each GL(Hi) with the operator norm topology and we put on j (D) the product Banach Lie group structure. On i(G) we take the Banach structure based on 2, The conjugation with d D G G

the product topology of i(G) and j (D), it becomes a Banach Lie group based on

m

® B(Hi) ©2.

i=1

The group G is a fiber bundle over GL(0S(H) with fiber T n D, Next one tries to minimalize the extension of GL(0S(H) that acts on F(0) and L(k). Thereto one considers the action of the kernel of n on L(k)

(Id,d) ■ [g,A] = [gd-1,A] = [g,0k(d-1 )A].

In particular the subgroup D(k) of G consisting of (Id, d) with 0k(d) = 1 acts trivially on L(k) and one sees that it suffices to consider the extension G(k) = G/D(k) of GL^H), If the character 0k is trivial, i. e, k = 0 then G(k) is just GL<0>(H). For k = 0, one computes directly that G(k) is a central extension of GL(0S(h) with Ker(n)/D(k) = C*. '

One can describe such an extension with a Borel 2-cocvele a : GL(0)(H) x GL<2(H) A C*. It can be constructed as follows: take a section p of the fiber bunc^k G A GL^H), i. e. for each g in GL(0S(H) one has

P(g) = (g,q(g)) with q(g) G D.

By definition there holds for each g1 and g2 in GL(0S(H) that

q(g1) q(g2) q(g1 g2)-1 G D n G.

Thus one gets for the action on L(k) the relation

P(g1g2) ■ [g,A] = P(g1) ■ {p(g2) ■ [g, A0k(q(g1) q(g2) q(g1 g2)-1)]}

:= P(g1) ■ {p(g2) ■ [g, Aa(gl, g2)-1^ .

The group G(k) is ^^^n isomorphic as a group to the product space GL(0S(H) x C* with the multiplication

(g^ A1) * (g2, A2) = (g1 g2, A1 A2 a (gl, g2)).

A detailed analysis of this central extension yields the following general result:

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Theorem 1

(a) The extension G(k) is always trivial if there is at most one infinite mi.

(b) If there are at least two infinite dimensional components in the basic flag, then G(k) is trivial if and only if for all i and j,

mi = mj = to ^ ki = kj.

(c) If ki = kj for infinite dimensional Hi and Hj, iften ifte corresponding Lie algebra 2-cocycle for the extension G(k) ¿s gwen by

da(X, Y) = ki Trace te Xij Yji 'y ', Yj Xji i .

i=1 I j=i j = i J

§ 5. Holomorphic sections of L(k)

Let L(k) denote the space of global holomorphic sections of L(k) The space L(k) is given the topology of uniform convergence on compact subsets of F(0) ■ It becomes then a complete locally convex space. Let f : F(0) A L(k) belong to L(k) then it can be written as

f (g ■ F(0)) = [g,f (g)], for all g gG,

where f : G A C is a holomorphic function satisfying

f (gt) = f (g)0k(t)-1 for all g G G and all t G T. (5,2)

Thus we can identify L(k) with the space of holomorphic funotions on G that satisfy

(g, d) G

F(0) as L(k), one §ets a natural action of G on L(k) that corresponds on the functions G

^ d)(f )(g1) = f (g-1 g1 d)^ with g1 G G and (g, d) G G.

By approximating the flag variety F(0) with finite dimensional flag varieties and using the representation theory in that case, one arrives at the following results:

Theorem 2

(a) The space L(k) is non-zero if and only if k1 ^ ... ^ km.

(b) If the space L(k) is non-zero, then the space of vectors in L(k) on which T(N)

acts by 0k is one-dimensional.

(c) Let v be a nonzero in L(k) on whi ch T (N) acts b y 0k, then it is the generator of an irreducible highest weight module of G(k).

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