Action of the “nilpotent torus” on representations

ADVAN(‘ES

IN M4THEMATI(‘S

Action

80.

261-768

( 1990)

of the “Nilpotent RANEE

I.

Torus”

KATHRYN

on Representations

BRYI.INXI

INTRODUCTION

Let g be a complex semisimple Lie algebra of rank 1. The nilpotent elements c E g such that the Lie centralizer 8” has dimension I are called pri@~~l nilpotent; they form a single adjoint orbt. Each such g” is, in a precise sense, a limit of Cartan subalgebras of R; in particular, each g’ is an abelian subalgebra of the nilpotent radical of the unique Bore1 subalgebra containing t’. We regard these subalgebras g” as “nilpotent tori.” In the case g=&, g” is spanned by the positive powers (as an endomorphism) of c. Kostant [K] first studied the g”-invariants in irreducible linear representations V of the adjoint group G of g. He proved, in particular, that dim( V’F’) = dim( I’!), for _r a Cartan subalgebra of g. In [Br], we have shown that, for any finite-dimensional irreducible g-representation k: I’,@ carries information on the q-atzrr1og.s of the weight multiplicities of V. Not surprisingly, Vs’ is rather difficult to analyze using usual machinery such as weight space decompositions and character theory. We conjecture that dim( I”“) is equal to the dimension of the largest weight space of I’; we have proven [Br, Proposition 2.41 it is at least that large. In this note, we study the action of the “nilpotent torus” on representations in the case g = &. Our method is to use a concrete realization for the representations, in which the limit (as X- + 8%) of the universal enveloping algebra &(g”) becomes a familiar object. In particular. we deduce results for & from results for ,gJ, We find some surprismg behavior on the generation of I’ over ‘ll(g”). Call two principal nilpotents e and ,f: or their centralizers, oppositr iff the unique Bore1 subalgebras containing e and ,f‘ are opposite. Then, we find (for generic I’) I’= &( g’ ). Vs’, for suitably chosen opposite nilpotent tori g’ and g”. This seems to be a curious analogy to the basic fact from highest weight-theory: I’=&(r/r ). I”“‘, where g=ez + @f@yr is a triangular decomposition. Let us point out, though,-that c and f‘do not turn out to lie in the same TDS of g. 261 (K)01-x70x/90$7.50 (‘ryjrzght s 1940 hy Acodem,c Prc,, Inc .\I1 rightr d r~pd~m~ In :,“, kJrm rocr,d

262

RANEEKATHRYN

BRYLINSKI

We now state two main results. The rlegrrr of an irreducible .&-representation is the smallest positive integer Y such that V occurs in (C”)@‘. THEOREM 1.1. Suppose V is u finite-diirnemionul irreducible u’egree r representation of‘ g = & , and k 3 r. Then, given an?’ principal nilpotent f in g, there exists a pj.incipal nilpotent e, opposite to ,f; SIKII thut V is generuted as a J’( g ’ )-module by the spuce V y’. Al.vo, VF’ has the same Dimensions u.s the lurg&t weight space of‘ V.

Let L be an infinite-dimensional complex vector space with countable basis It,,, t, , ... j. For any x E End,. L. let x”” be the pth power of s as an endomorphism, p 3 1. Given a sequence7 = (7,) yZ, ... ) of scalars, define linear “shift” operators 6, and e;. on L by &( t,) = ;‘, + , t, + , , and e;,(t, + , ) = ;j, + , t,, i 3 0, ei.(to) = 0. Let g:T and CJ.Tbe, respectively, the linear spans in End,. L of ie7, e:,“. ... ) and (.f;. f I,?‘,:.. ). For any partition X, let (see Section 2) E, be the corresponding irreducible complex linear symmetric group representation, and S’ the corresponding Schur functor on vector spaces. THEOREM 1.2. Suppose y UFU~ ci ure sequencesI$ non-zero scalars, ti.ith y, 6,= i, Jbr all i> 1. Let 3 he u purtitiorl of‘r. Then S”(L) is u free module qf‘rarzk dim( E,) oiler ‘)/(g,: ) = C[,f,, f:,“, .... ,flr’], \cith hasis Sa(L)“,:.

After a talk of the author at MIT on this work and initial distribution of the preprint, Victor Kac kindly informed us that Theorem 1.2 can also be derived from the results in [Ka-P]. The approach in [Ka-P] is particularly interesting, for Kac and Peterson consider representations of a natural Heisenberg algebra made up of shift operators on an infinitedimensional space with basis i..., t ] , I,,, t , , )

2. ACTION OF g/,

ON POLYNOMIAL

RINGS

Let gl, be the infinite-dimensional Lie algebra of all infinite matrices (QL,)l.,2” such that every row and column contains only finite many nonzero entries. The bracket is given, as usual, by the anti-commutator for matrix multiplication. Let Xi,, denote the (i, j)-elementary matrix. For all k 3 1, g_lk(&) embeds naturally in s_lX as the space of (traceless) matrices (a,.,L.i,o satisfying a,, , = 0 unless O 1, we form the k th truncation ~=zJoc., ~,.,X,.,Eglr uikl=

c O
a,. I

/XL

, E &hi.

AC‘TION

OF THE

NILPOTENT

TORUS

263

Let A, := C[t,, .... r,] be the polynomial ring on r indeterminates, any r 3 1. The symmetric group S, on Y letters acts on A, by permutations of the variables. S, also acts on any r-fold tensor product I’@’ by permutation of tensor position. We now observe that <@, acts on ,4, “by acting on exponents.” For all i, j, r, let p!. ,cr be the linear operator on ‘4, which acts on monomials by the rule:

Let A,:, be the linear span in A, of all monomials with all exponents less than or equal to li - 1. LEMMA 2.2:. For all r > 1, the litwur ttzap p,: @, + End, A,. giuen II), p,(X,, ,) = p ,,,, r, is u Lie al‘yehra representation. Fur~hertwre, (C[r])@” utd A, ure isomorphic (viu rhe natural mup) us representations of’gl, x S,; Arih and (C”)@’ ure isotnorphic as rc~prcJ.set1tation.F of‘ Elk. x S,. Prooj p, is clearly the natural embedding of @~, into End,. C[t] relative to its basis by powers of t. Each matrix a ~~9, corresponds to an operator ci on C[r]; j’,,,(c,t”+ ... +co)=c,t’. So p, is just the natural induced action of gl, on C[r]@‘. The natural isomorphism C[r]@‘>A, carries (C,‘~ ‘[t])@’ onto A,:,. Recall that, for each irreducible representation E of S,, the Schur filnctor S” on vector spaces is defined as the space of invariants S”( V) = (E@ I/“‘)“r. For 1’ finite-dimensional, S”( I’) is an irreducible representations of g/( b’), and rJBr = @ 6,,,,dE@ S”( b’). These two facts also hold for I’= C[f]. The latter is clear, as it is just a statement about the action of S,. The former is easily checked, by considering the exhaustive filtration of S”(C[t]) by its subspaces(E@ A,,,)“‘2 S,(C/‘). The following construction of the irreducible symmetric group representations is well known. Let x = (r,,. 1 , . ...) be a partition of r, written x t r, with 6 the conjugate purfilion, so that the parts 2, of 5 are the column lengths of (the diagram of) a. So 2, is equal to the length I(r) of J. Put n, = C,,,, ice,. Let ~1~4,~ .... 14J = n,, , (24, - 14,), and form the product of Vandermonde determinants:

I’,= IIt,. ...?t,,,ll. IIt,,,+,. .... ti,r+i,)l Then the subspace M, of Ayl spanned by the S,-translates of 11, is irreducible as an S,-representation; call this representation E,. Moreover, E, occurs exactly once in 24rs”X,so that M, is the unique lowest degree

264

RANEE KATHRYNBRYLINSKI

copy of E, in A,. The set {EJsr +--r> is a complete set of inequivalent irreducible complex linear representations of S,. (In fact, we have obtained the Z-forms, in this way.) Let us fix some choices for &: ik will be the Cartan subalgebra of diagonal matrices, and hk will be the Bore1 subalgebra of upper triangular matrices. Let n_z,and of ~2~ be, respectively, the spacesof strictly upper and strictly lower triangular matrices. Let vi, i= 0, .... k - I, be the linear functional on _tk given by the ith coordinate entry. For each partition x = (c(,, c(,, ...) and k > 1, define a weight w,:~ = tlO~~O + ... + txk , vk ,. The partial ordering induced by h, on the weights of fk then coincides with the one given by (up to a shift) by the dominance partial order on sequences. The finite-dimensional complex linear representations of d, are then classified (via the theory of CartanWeyl) by their highest weights relative to @aI h, 1. PROPOSITION 2.3. Let r he u partition of r, and let k he any positive integer. Then ever)) non-zero vector v in the lolvest degree copy’ M, of E, in A, is u highest \r.eight vector jbr &. The &,-representation generated by v is isomorphic to S “o.- .,‘I “(C”); so it is isomorphic to Sz(Ck), ifk 3 I(u).

Proof: We take the view gl, = End, C[t]. mk has a basis by homogeneous S,-invariant operators D of negative degree. So D can only carry M, to zero or to a strictly lower degree copy of E,. As the latter is impossible, D, and hence r_n,kills M,. As D, generates M,, we may now take u = I’,. The monomials are a basis of A, by weight vectors for _fkr for X,,, acts on tf’ tf by the scalar 1{sl py = i}l. But Usis a linear combination of permutations of a certain monomial, namely, f;’ ... t’;‘, where (e,, .... e,)= (&- 1, +2, .... 0, 6, - 1, i, -2, .... 0, ...). In tT’... t:, then, each i 2 0 occurs as an exponent exactly C(~times, so that X,,, acts as the scalar z, on t;’ . t:‘, and hence on c‘,. Thus v, has weight (c),:~. As o, is a highest weight vector, it generates an irreducible representation of gl,. If k > d, = I(H), then r, lies in the E,-isotypic component of A,:,, which is just a direct sum of copies of SZ(Ck). Thus v, generates a copy of Y( C”). Moreover, from what we have said so far, it follows that for any partition fi of at most h- parts, top,k is the highest weight of Sa(Ck). The Schur Jimction .s,(xI , .... x,), CI any partition, is the character of S’(C”), i.e., the trace on S’(C?) of a diagonal matrix with entries x,, .... .Y~. Then s,(x~, .... x,,) is a symmetric polynomial; if [(cc)d n, then its full symmetrization into an infinite variable set I,, x1, is the Schur function s,(x, , .Yz, ...1.

165

ACTION OF THE NILPOTENT TORUS For any graded ST-representation

A, form the generating function

g7(A) := 1 (E,. p z II We now recover the well-known COROLLARY 2.4. integer. Then

result:

Let x he u purtition

g,(A.,,)=s,(l,y g,(A,)=.F,(l,q,

>4”)q”.

of‘ r, und let k lx> un~, posithv

,..., (lb ‘)=s, ..,)=.s,

c! 1 -q” I-q

. ’

I c I-q 1

Proof Set /I = z., ix,,,. Then p(h), p = p,, is the degree operator on A,. So we can compute the generating functions in terms of traces. Letting Ix:/, be the E,-isotypic component in A,:,, we have 1 I ~JA,..A 1 = IE,I Tr,,,,,j 1,~ = IE,( Tr,,,,,,,,, I,., = Tr,,,,) S’(C)

= .sz( 1, q, .... q’

‘),

3. DIFFERENTIAL OPERATORS The Weyl ulgebra .d, is the algebra of all linear (finite-order) partial differential operators on A,. c4/, is generated by the operators t, , ...) I,., c:, , .,., c?,, where ?, := ?/?t,; these commute save for the non-trivial brackets [?i, t,] = 1. Let us call any linear operator D on A, a,formal d@fjcerentiul operutor on A, iff for each non-negative integer p, the action of D on A,<” coincides with the action of some finite order partial differential operator. Then every operator of the form D = Cr - ,:,,,- ,:,) D, 2;. D, E A,, is a formal differential operator, and every formal differential operator has a unique expression in this way. Moreover, every linear operator on A, is a formal differential operator in this sense, so that the C-algebra Pr of all formal differential operators is just another means of describing End, A,. From now on, we mainly think of PY as a Lie algebra, under the anticommutator. The linear inclusion d,: 8, ---t 2?;. given by c$,(tp(dY/dtY) = but not an algebra c:=, ty (‘:, is a Lie algebra homomorphism, homomorphism.

RANEEKATHRYN

266

BRYLINSKI

For every pair i, j of non-negative integers, detine the operators in gr:

LEMMA

3.1. For all i, j, r, P,.,:r=P,(X,.,)=

l/!i!

c yv,+ ,,,a0 .

I,,., + ,,I.I.

Proof: As the representation pr coincides with q5,, it suffices to check the Lemma for r= 1. We find pi. j:,(tp) =0 if p <,j, and otherwise,

= ” (l- l)~-y+‘-Lhp,,f~. 0 Consider the “shift operators” in g_l, : (p+i)!

xr.p+r:

%-CT r20

.f, = c xi? + I. I’ 120 p 2 1. Observe e,, and ,I, are, respectively, the pth powers of e = e, and f=fi. Let a+ and a be, respectively, the spans in gl, of the positive powers of e and f. Let i, be the identity matrix of i&trite order. Then [e, f]=i,, and gJxlr =Ci,

@a+,

gl~=Ci,CQg

Extend pr to a representation of the universal enveloping algebra. LEMMA 3.2. ( 1) C[ 2,. .... a,lsr, the s&algebra qf ~2, of symmetric constant coefficient differential operators, is equal to p,(%(a’)), and “@(a+ ) = C[e, e7, .... e,].

(2) differential

CCt1. .... rrlSr, the subalgebra operators, is equal to p,(@(a

of cdr of symmetric )), and ?/(a ) = C[f,

order zero f?, .... ,f,].

Clearly, for each pal, a;+ ... +Sp=p,(e,,), and tp+ ... +tC The lemma now follows as the first r power-sum symmetric functions in a variable set of size r generate the full ring of symmetric functions in that variable set. Proof =p,(f,).

A(‘TION OF THE NII.PoTENT THEOREM

(1) ditnemiorz

3.3.

Let r htj a partition

For all k>r, S’(C” equal to 1E, 1.

qf’r,

For all k 3 r, S”( C/‘) is getterated

(3)

S”( C[f]

nzodzrle owr

267

and set c, = &?“‘. Then

‘[t])ci=SZ(C[t])“+,

(2)

) is a ,frw

TORUS

owr ‘//(g

and ~2/(@)

this space has

hi, its c,-itwarimrs.

), Itlith husis S’(C[t]

)“+.

Assume throughout the proof that k 2/(r); for otherwise, S’(C’) = 0. Also assumeh-3 2. Let us take L’, as in the last section, and let L,,, 3 S”(C) be the representation of g_l, generated by P,. for k finite or k = x. For any S,.-representation I,‘. write 1” for the E,-isotypic component in V. Let H, be the space of harmonic polynomials in A,, i.e., the space of polynomials annihilated by all symmetric positive degree constant coefficient differential operators. Then one knows ( [Cl, see [Bo]) that A, is a free module over 4: with basis H,. So 4: is a free module over ‘47 with basis HF. Now A: is (see Section 2) the direct sum of lE,I copies of L,, ,~. Lemma 3.2. tells us that the actions of ,4: and C[?, , .... ?,I” come from the action of +!!(gl, ), so it follows that L,: ,~ is a free module over A? with basis given by its harmonic subspace.which is just Z, := (Lx: T )“‘. Thus we get (3 ). But one also knows [C] that H, carries the regular representation of S,, so Hj! carries 1E,l copies of E,. So Z, has dimension equal to 1E,l. On the other hand, the h-th truncation e,,, is a principal nilpotent of &, and a,:,= (.Y,~,].YE~ + ) is equal to c’, For all p 3 1, the actions of e,, and (e/lb, coincide on A,:, , and hence on L,., , so that (L,,, ),r, (L,,, ),AI=(ti(c~ ).Z,),,,=‘l/(a,,,).Z,. (The restriction on li is needed just so that (Z,),, , = Z,. ) So we have (2). Proof:

COROLLARY (of Proof) 3.4. Let x hr a principal tlilpotettt itz g = si,, and let a he LI~J’ purtitiotz. Tlzctt iS’(C”)“l d lE,I. iviflt rqualit~~ if’k 2 Irl.

268

RANEE

KATHRYN

BRYLINSKI

Remark 3.5. Keeping the notation of the proof of Theorem 3.3, we may also observe that L,,, = L,:, n A,:,, for all k > l(a). For, L,., 4 L,, % n A,;,. 4 (L,,, )Cx,= L,:,. (Or, one can see this using Lemma 2.2 and Proposition 2.3.) It follows then that (Lalk)glk~=2, n Aartk. It should be possible to push our methods further to compute I(La:l.)*l~ll. We have some precise conjectures on this. The polynomial F,(q) :=zpaO (E,, H,P)q” is a q-analog of IE,I. From 2.4 and 3.3 we also obtain COROLLARY

F,(q)l[(l

3.6. [f

c1 is a partition

of r,

then sa(l/( 1 -4)) =

-q).“(l-q’)l.

Proof of Theorem 1.1. The first assertion now follows from Theorem 3.3 by conjugacy of principal nilpotents. The second follows because(in the notation of the last proof) IE,I is equal to the dimension of the largest weight space of L,:+, for all k 3 r. Proof of Theorem 1.2. For the casey = (1, 1, ...). this is Theorem 3.3(3). To get the general case,just conjugate our principal nilpotents e and f by the diagonal matrix with entries (1, y,, y,y2, yzy3, ...).

REFERENCES

lBo1 [Brl [Cl [Ka-P]

lK1

N. BOURBAKI. “Elements de Mathematiques, Groupes et algebres de Lie,” Chap. 4. 5, 6, Hermann. Paris, 1968. R. K. BRYLINSKI, Limits of weight spaces, Lusztig’s q-analogs, and iiberings of adjolnt orbits, J. Anw. Afcrth. Sot. 2 (1989). 517-533. C. CHEVALLEY. Invariants of finite groups generated by reflexions, Amer. J. Math. 77 (1955). 521-531. V. G. KAC AND D. H. PETERSON, Lectures on the infinite wedge representation and the MKP hierarchy, SGm. Moth. Sup.102 (1986), 141-186. B. KOSTANT, Lie group representations on polynomial rings, Amrr. J. Murh. 85 (1963 ), 327404.