SPECTRAL FUNCTIONS OF INVARIANT OPERATORS ON SKEW MULTIPLICITY FREE SPACES

Transcription

1 SPECTRAL FUNCTIONS OF INVARIANT OPERATORS ON SKEW MULTIPLICITY FREE SPACES BY MICHAEL WEINGART A dissertation submitted to the Graduate School New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of Friedrich Knop and approved by New Brunswick, New Jersey May, 2007

2 ABSTRACT OF THE DISSERTATION Spectral Functions of Invariant Operators on Skew Multiplicity Free Spaces by Michael Weingart Dissertation Director: Friedrich Knop This thesis extends results on spectral functions of invariant differential operators on multiplicity free spaces to the setting of skew multiplicity free spaces, which are representations of a reductive group whose exterior algebra decomposes into a direct sum of pairwise nonisomorphic irreducibles. We prove in the general skew multiplicity free case that the spectral functions satisfy a vanishing property and a transposition formula which are formally identical to those satisfied by their multiplicity free analogues. We investigate two special cases, the GL n C modules S 2 C n and 2 C n, for which the spectral functions of invariant operators form a family of supersymmetric functions which can be identified with the factorial Schur Q functions. From this equivalence we deduce several properties of each family, giving the spectral functions a combinatorial interpretation and the factorial Schur Q functions a new representation theoretic one. ii

3 Acknowledgements My advisor Friedrich Knop, der Meisterlehrer von Nürnberg, for setting the problem, guiding me through each stage of its solution, and demonstrating exceptional patience all through our years of work together. Without his constant, expert, and generous guidance this thesis could not possibly have been completed. Siddhartha Sahi and Roe Goodman for many graduate courses, independent studies, conversations, answers to questions both long and short, and words of encouragement. Hieu Nguyen for his kind willingness to serve on the defense committee. Doron Zeilberger, for teaching me to program in MAPLE, an indispensible skill in performing this research, and for his unwavering and contagious optimism. Amy Cohen and Stephen Greenfield, for their ongoing encouragement and advice throughout my years of graduate study. Aaron Lauve, David Nacin, and Waldeck Schutzer, my brothers in matters symmetric and skew symmetric, for many talks and conversations. iii

4 Dedication To the blessed memory of my parents, who always valued knowledge as an end in itself. iv

5 Table of Contents Abstract ii Acknowledgements iii Dedication iv 1. Introduction and Overview The Transposition Formula for General Skew Multiplicity Free Spaces Skew Capelli Polynomials and Operators The Transposition Operator The Transposition Formula Spectral functions for S 2 C n and 2 C n Combinatorics of highest weights Factorial Schur Q functions Characterization Theorem Equivalence with factorial Schur Q functions, and consequences Dimension formulas Computed examples Skew Capelli operators Spectral polynomials Factorial Schur Q polynomials Values of spectral functions Transposition Dimension Polynomials v

6 6. Topics for future research References Vita vi

7 1 Chapter 1 Introduction and Overview Let G be a connected reductive group. A G-module W is said to be multiplicity free (MF) if no two irreducible submodules of its symmetric algebra S W are isomorphic. A G-module V is said to be skew multiplicity free (SMF) if no two irreducible submodules of its exterior algebra V are isomorphic. The primary objects of interest in this study are G-invariant polynomial coefficient differential operators on V where V is an SMF space, and our viewpoint will be that any such invariant operator can be viewed as a complex valued function on the set of highest weights of V as a G-module. The goal is to demonstrate certain general properties of these functions, and to give complete characterizations of them in two special cases. To justify this viewpoint, let V denote any SMF G-module, and Λ the set of highest weights occurring in V. Then we can write the decomposition of V into irreducible G-submodules as (1.1) V = λ Λ M λ Denoting the polynomial coefficient differential operators on V by PD(V ), we consider the natural G-module isomorphisms (1.2) PD(V ) = Cliff(V V ) = V V = (V V ) Least familiar among these isomorphisms is Cliff(V V ) = V V ; see [FH, Lemma 20.9] for a proof and discussion. We have

8 2 (1.3) PD(V ) = V V = λ,µ M λ M µ Taking G-invariants gives (1.4) PD(V ) G = (M λ Mµ) G λ,µ But by Schur s Lemma, (1.5) PD(V ) G = (M λ Mλ )G λ Thus for each λ Λ there is a 1-dimensional space (M λ Mλ )G of G-invariant polynomial coefficient differential operators, hence, up to scalars, a unique invariant operator D λ associated with each λ Λ. Moreover, these operators {D λ, λ Λ} form a basis for PD(V ) G. The analogous objects in the MF case are, after a suitable normalization, the famous Capelli operators, so that our {D λ, λ Λ} may reasonably be termed the skew Capelli operators. Now each such D λ maps V to itself in a G equivariant way, so that by another application of Schur s lemma, any nonzero image of the restriction of D λ to a particular M µ must lie entirely in M µ, and its action on M µ must simply be multiplication by a scalar c λ (µ). In this sense we may interpret D λ as a complex-valued function c λ ( ) on Λ, referred to henceforth as the spectral function of D λ. The properties of spectral functions in the symmetric case, i.e. for invariant differential operators on MF spaces, have been well investigated by Knop, Sahi, Okounkov, Olshanskii, Benson and Ratcliff, and others. The development of this subject in the skew symmetric case, i.e. that of SMF spaces, follows that of the symmetric case closely. Among the properties which have SMF analogues are the existence of transposition formulas, in the following sense.

9 3 The action of G on V induces an action of the universal enveloping algebra U(g) as differential operators on V, and thus of its center Z(g) as invariant differential operators on V. There is an antiautomorphism of U(g), which we call transposition by analogy to the work of Knop [K1], whose induced effect on differential operators is to reverse the order of multiplication; (1.6) x x,, x x, x x Thus for each of our invariant operators D λ as described above, there exists a transposed operator D t λ, hence there also exist the corresponding spectral functions c λ and c t λ, respectively. Chapter 2 derives the following transposition formula, which expresses the value of c t λ at a given highest weight as a linear combination of values of several c µ: (1.7) c t λ (ν) = c λ(χ w 0 ν) = ( 1) λ d λ c µ (λ)c µ (ν) d µ µ λ where w 0 is the longest element of the Weyl group of G, χ is the sum of the weights of V, and d ν is the superdimension of the irreducible submodule M ν, whose elements have homogeneous degree ν as tensors. This is a general result on skew multiplicity free spaces which is completely analogous to a corresponding result proved by Knop in the multiplicity free case [K1]. Another property of spectral functions of invariant operators on MF spaces, investigated by several authors beginning with Sahi [S2] and proven in full generality by Knop [K1], is the vanishing condition, c λ (µ) = 0 if µ λ, µ λ and c λ (λ) 0. This condition, together with simple conditions on symmetry and degree, suffice to determine uniquely a polynomial (with ρ-shifted arguments) which interpolates c λ at all {µ Λ}. Most remarkably, the spectral functions vanish at many more points than those indicated in the vanishing condition, and indeed satisfy the extra vanishing condition that for all λ, µ Λ, c λ (µ) = 0 if λ µ, in settings in which λ and µ may be interpreted as partitions. Other examples of families of functions satisfying the vanishing and extra

10 4 vanishing conditions, or suitable analogues thereof, are the shifted MacDonald functions introduced by Okounkov [O2], and even certain non symmetric polynomials introduced by Knop [K5]. Chapter 2 demonstrates that spectral functions of invariant operators on SMF spaces satisfy the same basic vanishing property, c λ (µ) = δ λµ for all µ λ. Since V has only finitely many highest weights for each SMF space V, the interesting cases to investigate are those in which infinite families occur. Howe [H] has classified irreducible SMF spaces for simple groups (possibly augmented by C ), and apart from the defining representations of the classical groups, the only infinite classes of such spaces are the GL n modules S 2 C n, n 2 and 2 C n, n 2. Chapter 3 investigates the spectral functions of invariant operators on S 2 C n and 2 C n for n 2, and proves a characterization theorem analogous to that of Knop in the multiplicity free case: For each highest weight λ of V there exists a polynomial p λ on Span C Λ which satisfies the following properties. CT1a. p λ (µ + ρ) = c λ (µ) for every µ Λ n. CT1b. p λ (µ + ρ) = 0 whenever µ λ unless µ = λ, and p λ (λ + ρ) = 1. CT2. p λ C[p 1, p 3, p 5,...], where p 1, p 3,.. are the odd degree power sum polynomials. CT3. p λ has degree λ 2. Furthermore, p λ is uniquely determined by CT1b, CT2, and CT3. Moreover, c λ satisfies the extra vanishing condition that c λ (µ) = 0 if λ µ.

11 5 A symmetric polynomial p(z 1, z 2,.., z n ) is said to be supersymmetric if for any integers 1 i < j n, p(z 1,..., z i 1, t, z i+1,..., z j 1, t, z j+1,..., z n ) does not depend on t. Pragacz (cf. [P]) shows that the odd degree power sum polynomials p 1, p 3,... generate the algebra of supersymmetric functions. Henceforth we will use the term supersymmetric with this interpretation in mind; we call p supersymmetric if p C[p 1, p 3, p 5,...]. Condition CT2 of the characterization theorem just stated can thus be reformulated; p λ is supersymmetric. The factorial Schur Q-functions Q λ are a family of symmetric functions defined by Okounkov and investigated by Ivanov in [I1] and[i2]. Like the classical Schur Q-functions of which they are analogues, the Q λ are supersymmetric and indexed by strict partitions. After replacing each Q λ by a suitably rescaled polynomial function q λ, the family {q λ : λ a strict partition} are characterized by the same vanishing, supersymmetry, and degree conditions as our spectral polynomials, except that strict partitions are the defining set of evaluation points instead of highest weights of V. The argument in chapter 3 associates to each highest weight λ a unique strict partition ˇλ, constructs an automorphism φ of the algebra of supersymmetric functions such that φ(p λ ) = qˇλ, and concludes that c λ (µ) = qˇλ(ˇµ) for all highest weights λ and µ of V. By virtue of this correspondence between the two families of supersymmetric functions, the spectral functions inherit several important properties of the factorial Schur Q functions, as well as the combinatorial interpretation of Q µ(λ) as a term in a simple formula which counts shifted skew tableaux of shape λ/µ. Conversely, the factorial Schur Q functions acquire a new representation theoretic interpretation as spectral functions of invariant operators, and satisfy a suitable reformulation of the transposition formula. Chapter 4 employs the Weyl Dimension Formula together with the peculiar combinatorics of the highest weights occurring in V, V = S 2 C n or 2 C n, to obtain explicit dimension formulas for irreducible submodules of V in terms of the top-row Frobenius coordinates of their highest weights. For each such irreducible M λ, dimm λ can be expressed explicitly as a polynomial in n. This makes possible the derivation of an

12 6 explicit formula for the leading coefficients of the spectral functions, the factorial Schur Q-functions, and the classical Schur Q-functions. Chapter 5 presents computational examples. Among these are explicitly written operators D λ, spectral functions, factorial Schur Q-functions, tables of values of these functions, and dimension polynomials. Chapter 6 sets out several topics for further research which arise from, or are natural extensions of, the current project.

13 7 Chapter 2 The Transposition Formula for General Skew Multiplicity Free Spaces In the following discussion, multiplications are understood to be skew multiplications, i.e. multiplication inside an exterior algebra. Other notational conventions are as follows. G denotes a connected reductive group. V denotes any skew multiplicity free G-module. PD(V ) denotes the algebra of polynomial coefficient differential operators on V, and PD(V ) G the algebra of G-invariant such operators. For a specific V understood from context, Λ denotes the set of highest weights of V as a G-module, and the elements of Λ are denoted by lower case Greek letters λ, µ, ν,... For λ Λ, M λ denotes the irreducible G-module of highest weight λ. d λ = ( 1) λ dimm λ denotes the superdimension of M λ. Sections 2.1 and 2.2 are largely inspired by Benson and Ratcliff s treatment of the subject in the symmetric case in [BR, 43-57], and closely mimic much of the pattern of their argument. 2.1 Skew Capelli Polynomials and Operators Assume that V has finite dimension n. To define a pairing V V C, let {z i } be a basis of V and { z i } a basis of V, such that < z i, z j >= δ ij and < z i, z j >= δ ij. We define i and i by i (z j ) =< z i, z j >= δ ij, and i ( z j ) =< z i, z j >= δ ij. The G-module isomorphisms PD(V ) = (V V ) = V V permit us to adopt

14 8 the useful viewpoint that the canonical invariant operator D λ considered in the introduction can be regarded as a polynomial P λ in skew symmetric variables, which can in turn be viewed as a tensor. We may move freely between these interpretations. Beginning with the tensor viewpoint, we consider the basis-independent element (2.1.1) d λ v i vi (M λ Mλ )G ( V V ) G i=1 where {v i } is any basis for M λ and {v i } is its dual basis, in the sense that < v i, v j >= δ ij and < v i, v j >= δ ij. We can define the canonical skew invariant d λ (2.1.2) Pλ = v i vi, i=1 and the normalized invariant (2.1.3) P λ = 1 d λ Pλ. By the natural isomorphism (2.1.4) V V = (V V ) we may regard P λ as a skew-polynomial in the variables z 1,..., z n, z 1,..., z n, recalling that z 1,..., z n V and z 1,..., z n V. With this interpretation we write d λ (2.1.5) Pλ (z, z) = v i (z)vi ( z) i and refer to P λ (z, z) as the skew Capelli polynomial determined by λ. Finally, consider the isomorphism of G-modules (2.1.6) π : (V V ) PD(V )

15 9 defined by (2.1.7) π(z i ) = z i, π( z i ) = i, i = 1, 2,..., n We define the skew Capelli operator determined by λ by (2.1.8) D λ = π( P λ (z, z)) and write D λ = P λ (z, ). We define a G-invariant form <, > on (V V ) by (2.1.9) < p, q >= p(, )(q(z, z)) z= z=0 The form has the properties (2.1.10) < ξz i, >=< ξ, i ( ) > < ξ z i, >=< ξ, i ( ) > < 1, 1 >= 1 < 1, ξ >= 0 if ξ is not a constant. In the first two statements in (2.1.10) the right side has strictly lower degree than the left, so that the four given statements show that such an inner product on (V V ) is unique. Note also that this form is supersymmetric in the sense that (2.1.11) < ξ, η >= ( 1) ξ η < η, ξ >. Computation of the inner product on concrete elements of (V V ) entails replacing

16 10 the variables of the first argument by suitable differentiation operators. Examples: 1) < z 1 z 2, z 1 z 2 >=< z 1, 2 (z 1 z 2 ) >=< z 1, 0 >= 0 2) < z 1 z 2, z 1 z 2 >=< z 1, 2 ( z 1 z 2 ) >=< z 1, z 1 2 ( z 2 ) >=< z 1, z 1 >= 1 Indeed, we can write the following formula for the inner product on straightened monomials: (2.1.12) < z a z b, z c z d >= δ ad δ bc ( 1) ( a +1 2 )+( b 2 ) where z a = n i=1 z a i i, with each a i = 0 or 1; similarly z b = n i=1 z b i i, etc. Thus if ξ and η are monomials then < ξ, η >= 0 unless ξ equals, up to sign, a permutation of the factors of η Proposition. If M λ V and M µ V, then the pairing <, > restricted to M λ M µ is zero unless λ = µ, in which case it is non-degenerate. Proof: This follows from the G-invariance of the bilinear form <, >. The following computation is central to the overall argument: Proposition. < P λ, P µ >= d λ δ λµ Proof: Let {v i } and {w j } be bases for the irreducible submodules M λ and M µ, respectively, and let {vi } and {w j } be the corresponding dual bases. In keeping with our conventions this means precisely that < v i, v j >= δ ij, < v i, v j >= δ ij, < w i, w j >= δ ij, and < w i, w j >= δ ij (2.1.13) < P λ, P µ >=< P λ (z, z), P µ (z, z) > = P λ (, )( P µ (z, z))

17 11 = i,j v i ( )v i ( )w j (z)w j ( z) Since v i M λ, w i M µ, Proposition implies that vi ( )w j(z) =< vi, w j >= δ λµ. Thus if λ µ we have < P λ, P µ >= 0. If λ = µ, so that the {v i } and {w j } are bases of the same space, we may assume by the basis independence of P λ that v i = w i for each i. Thus we may replace vi ( )w j(z) by δ λµ δ ij, and obtain d λ (2.1.14) < P λ, P µ >= δ λµ v i ( )v i ( z) d λ = δ λµ ( 1) λ since deg v i = λ i = ( 1) λ d λ δ λµ = d λ δ λµ i The argument in the next section will require the following lemma. Let Q = z 1 z z n z n Lemma. λ =k P λ = Qk k! Proof: The left side is the sum of all skew Capelli polynomials of specified degree, which together form a basis of M λ. Recalling (2.1.5) we can express each skew Capelli λ =k polynomial in a basis independent way by d λ (2.1.15) Pλ = v i (z)vi ( z) i=1 Consider the monomial basis of M λ, namely {z α : α = k}, where α = (α 1,..., α n ) λ =k is a multiindex of 0s and 1s, and z α = z α 1 1 zαn n. By the basis independence of the sum of skew Capelli polynomials of a given degree, we have that

18 12 (2.1.16) λ =k P λ = α =k z α (z α ). (z α ), the element of V dual to z α, is determined by formula (2.1.12), which implies that (2.1.17) < z α, z α >= ( 1) ( α 2 ) It follows that (2.1.18) (z α ) = ( 1) ( α 2 ) z α so that (2.1.19) λ =k P λ = α =k = ( 1) (k 2) = ( 1)(k 2) k! = ( 1)(k 2) k! = Qk k! α =k α =k z α (z α ) z α z α k!z α z α 1 i 1 <...<i k n k!( 1) (k 2) zi1 z i1 z ik z ik Corollary. P λ = e Q. λ Λ

19 The Transposition Operator Let := n n. Recall from the beginning of this chapter that by definition, i (z j ) =< z i, z j >= δ ij and i ( z j ) =< z i, z j >= δ ij. Define an operator T : (V V ) (V V ) by (2.2.1) (T P )(z, z) = (e P )(z, z) = e (P (z, z)). Define an operator M by (2.2.2) (MP )(z, z) = P (z, z). Then Proposition. T = M e = e M = T 1. Proof: Observe first that (2.2.3) k k! = i1 i1... ik ik 1 i 1 <i 2 <...<i k n In the expansion of the product k = ( n n ) k, the only terms which do not contain repeated factors and consequently vanish are those of the form i1 i1... ik ik. Furthermore, since there are k! reorderings of each k-tuple of indices {i 1,..., i k }, we have k! copies of each term of the form i1 i1... ik ik in the expansion of k. It suffices to test the equality of M e and e M on elements of the form z 1 z 1 z 2 z 2...z s z s. On the one hand,

20 14 (2.2.4) ( 1) k k M(z 1 z 1 z 2 z 2...z s z s ) =( 1) k k k! k! (( 1)s (z 1 z 1 z 2 z 2...z s z s )) = ( 1) k+s i1 i1... ik ik (z 1 z 1 z 2 z 2...z s z s ) 1 i 1 <i 2 <...<i k n = ( 1) s 1 j 1 <j 2 <...<j s k n z j1 z j1 z j2 z j2...z js k z js k The second factor of ( 1) k, which cancels the first, arises because each i ( z i ) = 1. On the other hand, (2.2.5) M k k! (z 1 z 1 z 2 z 2 z s z s )= M(( 1) k = ( 1) s 1 j 1 <j 2 <...<j s k n 1 j 1 <j 2 <...<j s k n z j1 z j1 z j2 z j2...z js k z js k z j1 z j1 z j2 z j2...z js k z js k ) Corollary. T (P λ ) = ( 1) λ e (P λ ). Proof: Immediate from the definition and the fact that deg z P λ = λ Lemma. {T (P λ ) : λ Λ} is a vector space basis for (V V ) G. Proof: T (P λ ) (V V ) is G-invariant and is a G-invariant operator. Moreover, (2.2.6) T (P λ ) = ( 1) λ P λ + R λ where P λ (V V ) λ, λ and R λ is of strictly lower degree, due to the action of e. As the {P λ } form a basis for (V V ) G, so do the {T (P λ )}. Remark: This differs slightly from argument given in the symmetric case, in which a K-invariant form is used and the statement of the lemma pertains to K-invariants. This difference has no effect on the overall argument.

21 Definition. The generalized skew binomial coefficients λ ν are defined for λ, ν Λ by T (P λ ) = ( 1) λ ν Λ λ ν P ν. Analogously to the symmetric case, the proof of Lemma gives the following corollary Corollary. λ λ = 1, λ ν = 0 when λ ν but λ ν. Thus (2.2.7) T (P λ ) = ν λ ( 1) λ λ ν P ν = ( 1) λ P λ + ( 1) λ λ ν ν < λ P ν Proposition. ( 1) µ λ µ µ Λ µ ν = ( 1) λ δ λν. Proof: T 2 = 1 implies that (2.2.8) P λ = T (T (P λ )) = ( 1) λ ν λ ν T (P ν ) On the other hand, (2.2.9) T (P λ ) = ( 1) λ µ λ µ P µ = ( 1) λ µ λ µ T (P ν )) (( 1) µ ν µ ν = ( 1) λ ν ( µ ( 1) µ λ µ µ )T (P ν ) ν

22 16 The proposition follows from the linear independence of the {T (P λ ) : λ Λ} Proposition. For λ Λ and k N, (2.2.10) k k! P λ = ν = λ k λ ν P ν and (2.2.11) ( 1) k k k! T (P λ) = ν = λ k λ ν T (P ν ) Proof: We have (2.2.12) ( 1) λ ν λ ν P ν = T (P λ ) = ( 1) λ e (P λ ) = ( 1) λ k k k! P λ For a given k, (2.2.13) k k! P λ = ν = λ k λ ν P ν Equating homogeneous components of degree 2( λ k) on both sides yields (2.2.10). To prove , apply the operator T to both sides of (2.2.10): (2.2.14) e M k k! P λ = ( 1) λ k e k k! P λ = λ k k ( 1) k! e P λ = ν = λ k ( 1) λ k k k! ( 1) λ T (P λ ) = λ ν T (P ν )

23 17 ( 1) k k k! T (P λ) = Lemma. Q and are adjoint. Proof: We compute (2.2.15) < z j z j ξ, η >=< z j ξ, ( 1) deg ξ j η >=< ξ, j j η >. Thus (2.2.16) < Qξ, η >=< ξq, η >=< ξ, η >. Benson and Ratcliff ([BR, p.54]) cite an unpublished Pieri formula proved by Yan, whose skew analogue is the following: Theorem. For ν Λ, k N, Q k k! P ν = λ = ν +k λ ν Pλ. Proof: The theorem states that (2.2.17) Q k k! d νp ν = λ = ν +k λ ν d λ P λ We have (2.2.18) Q k k! P ν = λ = ν +k < Qk k! P ν, P λ > < P λ, P λ > P λ since the P λ of degree ν + k form a basis for invariants of degree ν + k (2.2.19) = λ = ν +k d λ < P ν, k k! P λ > P λ

24 18 by adjointness of Q and (Lemma 2.2.8) and the fact that d λ =< P λ, P λ > (Proposition 2.1.2), so that < P λ, P λ >= 1 d λ. Thus we have (2.2.20) = d λ < P ν, λ P ν > P λ λ = ν +k µ ν = λ d λ < P ν, P ν > P λ ν λ = ν +k by the orthogonality of P λ and P ν for λ ν, and Proposition (2.2.21) = λ = ν +k d λ d ν λ ν P λ. Thus (2.2.22) i.e. Q k k! d νp ν = Q k k! P ν = λ = ν +k λ = ν +k λ d λ P λ ν λ ν Pλ Corollary. For ν Λ, e Q Pν = λ Λ λ ν Pλ. Recall from Chapter 1 that by Schur s Lemma and the skew multiplicity freeness of V, the G-invariant operator D ν = P ν (z, ) acts as a scalar on each irreducible M λ Definition. Let c ν denote the spectral function of the canonical invariant operator P ν (z, ) (M ν M ν ) G. Thus for ν, λ Λ, c ν (λ) C denotes the eigenvalue of D ν on M λ Proposition. For all λ, ν Λ, c ν (λ) = λ. ν

25 19 Proof: Note that P ν (z, ) P λ (z, z) = c ν (λ) P λ (z, z) since P λ (z, z) M λ M λ. We have (2.2.23) Pν (z, )e Q = P ν (z, z)e Q which holds in the skew case as in the symmetric. On the one hand, (2.2.24) Pν (z, z)e Q = P ν (z, z) k Q k k! = k Q k k! P ν (z, z) since Q k is homogeneous of even degree, so that Q k Pλ = P λ Q k for each k. (2.2.25) = k λ = ν +k λ ν Pλ (z, z) by Theorem = λ ν λ ν Pλ (z, z). On the other hand, using Lemma we find that (2.2.26) e Q = Q k k! k = P λ (z, z) k λ =k = P λ (z, z) λ Thus

26 20 (2.2.27) Pν (z, )e Q = λ P ν (z, ) P λ (z, z) = λ c ν (λ) P λ (z, z) Having written P ν (z, )e Q in two ways we find that (2.2.28) λ ν λ Pλ (z, z) = λ c ν (λ) P λ (z, z). For each λ, equating coefficients of P λ (z, z) gives λ ν = c ν (λ). Proposition and Corollary imply that the spectral functions satisfy the following vanishing condition: Corollary. For any λ, ν Λ, c ν (λ) = 0 if λ ν but λ ν; c ν (ν) = 1.

27 The Transposition Formula Let the transposition map τ be the antiautomorphism on skew differential operators defined for each i = 1,..., dim V by (2.3.1) τ(z i ) = z i, τ( i ) = i. It then follows that τ(z i i ) = ( 1) deg(z i)deg( i ) ( i )z i = i z i, and similarly τ( i z i ) = z i i Proposition. The action of T = e M = M e on skew polynomials corresponds to that of τ on skew differential operators in the sense that the following diagram commutes for any P (z, z) (V V ): P (z, z) π P (z, ) T (P (z, z)) τ(p (z, )) Proof: We observe that the following diagrams commute: z z (2.3.2) z z z (2.3.3) z z z z (2.3.4) z z + 1 z + 1 = z since z + z = 1. To justify that in fact e M(z z) = z z + 1, we compute

28 22 (2.3.5) e M(z z) = e ( z z) = z z + ( z z) = z z + ( z) = z z + 1 For the induction step, assume that the result holds for polynomials P (z, z) = P (z 1,.., z n 1, z 1,..., z n 1 ). We must show that it holds for P (z, z)q(z n, z n ). It suffices to check this for Q(z n, z n ) = z n, z n, or z n z n. Write (2.3.6) n 1 := n 1 n 1. Then (2.3.7) = n = n 1 + n n and in particular (2.3.8) e = e n 1 e n n Note that n 1 and n n commute, since both expressions have even degree, and therefore e n 1 and e n n commute as well. For Q(z, z) = z n, observe that e n n M(z n ) = z n. We have (2.3.9) e n 1 e n n M(P (z, z)z n ) = e n 1 (P (z, z))e n n (z n )) = (e n 1 (P (z, z)))z n Thus

29 23 P (z, z)z n P (z, )z n (2.3.10) (e n 1 (P (z, z)))z n τ(p (z, ))z n where e n 1 (P (z, z)) τ(p (z, )) by induction. For Q(z, z) = z n, we have e n n M( z n ) = z n, hence (2.3.11) e n 1 e n n M(P (z, z)( z n )) = e n 1 (P (z, z)))( z n ) Thus we have a commutative diagram P (z, z) z n P (z, )( n ) e n 1 (P (z, z))( z n ) τ(p (z, ))( n ) For Q(z, z) = z n z n, we have (2.3.12) e n n M(z n z n ) = e n n ( z n z n ) = z n z n + n n ( z n z n ) = z n z n + n ( z n ) = z n z n + 1 since n ( z n ) = 1, hence (2.3.13) e n 1 e n n M(P (z, z)z n z n ) = e n 1 (P (z, z))( z n z n + 1) Thus we have a commutative diagram P (z, z)z n z n P (z, )(z n n ) e n 1 (P (z, z))( z n z n + 1) τ(p (z, ))( z n n + 1) = τ(p (z, ))( n z n )

30 24 Let V be an SMF G-space of dimension n, let χ denote the sum of all weights of V, and let w 0 denote the longest element of the Weyl group of G. Consider the map (2.3.14) i V n i V n V and observe that the module M χ n V. If M λ i V then there exists M µ n i V such that multiplication is a perfect pairing, M λ M µ M χ. Then necessarily M µ = M χ M λ. Since M λ has highest weight w 0 λ it follows that the set of highest weights of V is invariant under λ χ w 0 λ. We can now formulate the following Proposition. c D t(λ) = c D (χ w 0 λ). Proof. The argument closely follows Knop s approach in the symmetric case ([K1, section 2] where the corresponding assertion is that c D t(z) = c D ( z)). Let V be any finite dimensional SMF G-space, where G has Lie algebra g. Let Z(g) denote the center of the universal enveloping algebra U(g). The action of G on V induces a homomorphism Ψ : U(g) PD(V ), whose restriction to Z(g) maps to PD(V ) G. Let e i be a basis for V consisting of weight vectors such that each e i has weight χ i. Let the operator i be defined by i (e j ) = δ ij. Write the usual decomposition g = n + h n. For η h we have Ψ(η) = χ i (η)e i i. After transposing we have (2.3.15) Ψ t (η) = χ i (η) i e i = χ i (η)(1 e i i ) = χ(η) Ψ(η). If η n ± then Ψ(η) = i j a ije i j. Since e i and j anticommute when i j, we have

31 25 Ψ t (η) = Ψ(η). Since χ is a character of g we can define an antiautomorphism τ on U(g) by τ(η) = η + χ for all η g, and thus we have shown that (2.3.16) Ψ t (ξ) = Ψ(τ(ξ)) for all ξ U(g). Let ξ Z(g) and D = Ψ(ξ). By the Poincare Birkhoff Witt theorem we can write ξ = ξ 0 + ξ 1 where ξ 0 h and ξ 1 n U(h)n +. Regarding ξ 0 as a function on h which takes the value ξ 0 (v) at v h, (2.3.17) τ(ξ 0 (v)) = ξ 0 ( v + χ). Now observe that τ(ξ) = τ(ξ 0 )+τ(ξ 1 ), where τ(ξ 0 ) U(h) and, since τ is an antiautomorphism, τ(ξ 0 ) n + U(h)n. We consider the action upon a lowest weight vector u of M λ, which has weight w 0 λ, where w 0 denotes the longest element in the Weyl group of G. Since n annihilates a lowest weight vector, Ψ(τ(ξ 1 ))u = 0, hence (2.3.18) D t u = Ψ(τ(ξ 0 ))u = τ(ξ 0 )(w 0 λ)u = ξ 0 ( w 0 λ + χ)u. Thus we have (2.3.19) c D t(λ) = ξ 0 (χ w 0 λ) for any highest weight λ. Since λ is the highest weight of M λ, we can write (2.3.20) c D (λ) = ξ 0 (λ). Since χ w 0 λ is again a weight, we may replace λ by χ w 0 λ to obtain (2.3.21) c D (χ w 0 λ) = ξ 0 (χ w 0 λ) = c D t(λ).

32 26 This result holds when D is any G-invariant differential operator arising from the center of U(g). On the assumption that D λ, for λ Λ, arises in this way, we obtain at last the desired transposition formula: Theorem. c λ (χ w 0 ν) = µ λ µ λ Equivalently, χ w 0ν = ( 1) λ d λ λ ν λ d µ µ µ ( 1) λ d λ c µ (λ)c µ (ν) for each λ, ν Λ. d µ Proof: By Definition we write in terms of unnormalized invariants that (2.3.22) T (P λ (z, z)) = ( 1) λ λ µ Λ µ P µ (z, z) In terms of normalized invariants this is written (2.3.23) T ( P λ (z, z)) d λ = ( 1) λ λ µ Λ µ P µ (z, z) d µ By Proposition we have (2.3.24) T ( P λ (z, z)) d λ = ( 1) λ c µ (λ) P µ (z, z) d µ µ Λ hence (2.3.25) T ( P λ (z, z)) = ( 1) λ d λ c µ (λ) d P µ (z, z) µ µ Λ Now we apply the map π (see (2.1.6) above) which associates to each polynomial the corresponding differential operator by mapping z z, z. Recalling that by Proposition 2.3.1, the action of T corresponds under this map to the transposition operator τ, we have

33 27 (2.3.26) τ( P λ (z, )) = ( 1) λ d λ c µ (λ) d P µ (z, ) µ µ Λ or equivalently, (2.3.27) Dλ t = ( 1) λ d λ c µ (λ)d µ d µ µ Λ When each side of this equation acts on an irreducible M ν, we replace the operator on each side by its spectral function: (2.3.28) c t λ (ν) = ( 1) λ d λ c µ (λ)c µ (ν) d µ µ Λ By Proposition this is (2.3.29) c λ (χ w 0 ν) = ( 1) λ d λ c µ (λ)c µ (ν) d µ µ Λ Theorem implies an additional symmetry which the spectral functions satisfy: Corollary. The expression ( 1) λ c λ (χ w 0 ν) d λ is symmetric in λ and ν Corollary. For each λ Λ, c λ (χ) = ( 1) λ d λ = dimm λ. Proof: By evaluating the transposition formula, Theorem 2.3.3, at µ = 0, (2.3.30) c λ (χ) = c λ (χ w 0 (0)) = ( 1) λ d λ c µ (λ)c µ (0). d µ µ λ The vanishing property, Corollary , implies that c µ (0) = 0 unless µ = (0), so we have (2.3.31) c λ (χ) = ( 1) 0 d λ d (0) c (0) (λ)c (0) (0).

34 28 Since (0) is the highest weight of the trivial representation, D (0) = 1, so that c (0) (ν) = 1 for all ν Λ, and moreover, d (0) = 1. Thus (2.3.31) becomes c λ (χ) = d λ Corollary. For each nontrivial λ Λ, ( 1) µ c µ (λ) = 0. µ λ Proof: By evaluating the transposition formula at χ, (2.3.32) c λ (χ w 0 (χ)) = c λ (0) = µ λ µ λ ( 1) λ d λ c µ (λ) d µ d µ ( 1) λ d λ c µ (λ)c µ (χ) d µ by Corollary Since λ 0 by assumption, c λ (0) = 0 by the vanishing property (Corollary ), so we have (2.3.33) 0 = ( 1) µ d λ c µ (λ) = d λ ( 1) µ c µ (λ) µ λ µ λ 0 = ( 1) µ c µ (λ). µ λ Remark: Theorem and its corollaries are analogous to results obtained by Knop in the general multiplicity free case [cf. K1].

35 29 Chapter 3 Spectral functions for S 2 C n and 2 C n In this chapter we restrict our investigation to the two special cases in which G = GL n (C) and V is the skew multiplicity free space S 2 C n or 2 C n, n 2. We let Λ n denote the set of highest weights actually occurring in V as a G-module; the subscript may be omitted when the dependence on n is irrelevant. We will regard these weights concretely as partitions of length at most n, i.e. weakly decreasing n-tuples λ consisting of l(λ) positive integers followed by n l(λ) 0s. It is primarily V whose highest weights are of interest, rather than V, though we continue to denote by χ the sum of all weights of V itself. We also continue to denote by c λ the spectral function for the action of the skew Capelli operator D λ on V, so that D λ acts on the irreducible module M µ as multiplication by the scalar c λ (µ). n p k = denotes the degree k power sum polynomial in n commuting variables, and i=1 z k i C[p 1, p 3, p 5,...] the algebra of supersymmetric polynomials in n variables on V, where n is understood from context. 3.1 Combinatorics of highest weights There are two useful characterizations of Λ n when V = S 2 C n. Consider the standard basis e 1,..., e n of C n, so that {e i e j, 1 i j n} is a basis for S 2 C n. Following Howe s approach in [H, section 4.4], put e ij = e i e j and write these basis vectors in a triangular array

36 30 e 11 e 12 e 13 e 14 e e 1n e 22 e 23 e 24 e e 2n (3.1.1) e 33 e 34 e e 3n e 44 e e 4n... e nn Now consider subsets of basis vectors with the property that if a given basis element is in the subset, then so are all basis elements above it and/or to the left of it in the array. e 11 e 12 e e 1r3... e 1r2... e 1r1 (3.1.2) e 22 e e 2r3... e 2r2 e e 3r3. Howe shows that a highest weight vector of V is obtained by taking the wedge product of all elements of such a subset, and that a highest weight vector for every irreducible submodule can be obtained in this way. Next consider the Young diagram of a highest weight λ defined by such a subset of the triangular array, and observe that the ith row of the array (3.1.3) e ii e i,i+1 e i,i+2... e i,ri 1 e i,ri contributes to the Young diagram of λ what Howe defines as an (r i +1, r i ) hook, namely the Young diagram of (r i + 1, 1 ri 1 ). Note that we must have r i n. Thus the Young diagram of λ is formed by nesting k such (r + 1, r) hooks, for a finite number k of different values of r; this property characterizes the elements of Λ n, which consists of all weights formed by nesting (r + 1, r) hooks such that r n Proposition. The set Λ n of highest weights occurring in S 2 C n consists precisely of all weights formed by nesting (r + 1, r) hooks such that r n.

37 Corollary. Λ n embeds in Λ n+1 by λ = (λ 1, λ 2,..., λ l(λ), 0 n l(λ) ) (λ 1, λ 2,..., λ l(λ), 0 n l(λ)+1 ) for each λ Λ n For example, when n = 3, the subsets of the triangular array which give rise to highest weight vectors are {}, {e 11 },{e 11, e 12 }, {e 11, e 12, e 22 }, {e 11, e 12, e 13 }, {e 11, e 12, e 13, e 22 }, {e 11, e 12, e 13, e 22 }, {e 11, e 12, e 13, e 22, e 23 }, and {e 11, e 12, e 13, e 22, e 23, e 33 }. The corresponding highest weights are (0, 0, 0), (2, 0, 0), (3, 1, 0), (3, 3, 0), (4, 1, 1), (4, 3, 1), (4, 4, 2), and (4, 4, 4). Among the Young diagrams of these weights, that of (2, 0, 0) consists of a single (2, 1) hook, that of (3, 1, 0) consists of a single (3, 2) hook, and that of (4, 1, 1) consists of a single (4, 3) hook, while the other nontrivial diagrams consist of nestings of two or three of these. An equivalent characterization, due to Knop, is based on the form of the Frobenius coordinates of elements of Λ n. If λ is any partition (λ 1, λ 2,..., λ r ), then its dual partition λ = (λ 1, λ 2,..., λ k ) is the partition whose Young diagram is obtained from that of λ by interchanging its rows with its columns. λ is said to have Frobenius coordinates α 1 α 2... α k, β 1 β 2... β k where α i = λ i i and β i = λ i i, i = 1, 2,.., k, and k is the largest index such that λ i i > 0. This description of a partition λ can be viewed as a decomposition of λ into nested hooks of shapes (α 1 + 1, 1 β 1 ), (α 2 + 1, 1 β 2 ),..., (α 1 + k, 1 β k). In private conversation, Knop reformulated Howe s characterization of Λ n by observing that weights which satisfy Howe s nested hook property are precisely those whose Frobenius coordinates are of the form (3.1.4) α 1 α 2... α k α 1 1 α α k 1

38 32 where i α i n. Since Frobenius coordinates of this form are determined by their top row alone, whose entries are by definition positive and strictly decreasing, the partitions in Λ n correspond bijectively to strict partitions of integers n. The correspondence is indeed bijective, since any strict partition of an integer n can be placed into the top row of Frobenius coordinates of the indicated form and thereby determine a highest weight of S 2 C n. For each λ Λ, denote the associated strict partition ˇλ. Observe that each ˇλ i equals half the number of boxes in the corresponding (λ i i + 1, λ i i) hook. It follows that (3.1.5) λ = 2 ˇλ. For example, when λ = (4, 3, 1), whose Young diagram is composed of a (4, 3) hook and a (2, 1) hook, we have ˇλ = (3, 1), so that indeed (4, 3, 1) = 8 = 2 (3, 1). In the case 2 C n, the same reasoning using the basis {e i e j, 1 i < j n} shows that the highest weights are those whose Young diagrams are nested (r, r + 1) hooks and whose Frobenius coordinates are therefore of the form (3.1.6) α 1 1 α α k 1 α 1 α 2... α k. Thus a bijection holds between highest weights of 2 C n and strict partitions of integers n, by associating to each highest weight its bottom row Frobenius coordinates. Moreover, the characterization by nested hooks shows that λ is a highest weight of S 2 C n if and only if its dual λ is a highest weight of 2 C n, so that λ and λ correspond to the same strict partition ˇλ. This fact will ultimately have the consequence, which is surprising a priori, that our two special cases S 2 C n and 2 C n have essentially the same spectral theory.

39 33 This bijective correspondence between highest weights and strict partitions can be realized in another way, which will be crucial to the subsequent discussion. Let ( n 1 (3.1.7) ρ =, n 3,..., n + 1 ) the half sum of the positive roots, as it were the true ρ, and, as in chapter 2, denote by χ the sum of all weights of V. Throughout this chapter we will use (3.1.8) ρ = ρ 1 2 χ When V = S 2 C n, χ = (n + 1, n + 1,..., n + 1), so that ρ = ( 1, 2, 3,..., n) and when V = 2 C n, χ = (n 1, n 1,..., n 1), so that ρ = (0, 1, 2,... n + 1). Consider any weakly decreasing m-tuple of integers (a 1,..., a m ) with at least a 1 > 0, and let r be the largest index such that a r > 0. Then let (a 1,..., a m ) + denote (a 1, a 2,..., a r ), i.e. the r-tuple obtained by discarding all 0 or negative entries of (a 1,..., a m ) Lemma. Highest weights µ Λ n correspond bijectively to strict partitions of integers n, under the map µ (µ + ρ) +. Proof: If V = S 2 C n, then the coordinates of (µ + ρ) + = (µ 1 1, µ 2 2,..., µ k k) are by definition the top-row Frobenius coordinates of µ. Similarly, if V = 2 C n, and µ has top-row Frobenius coordinates (µ 1 1, µ 2 2,..., µ k k), then its bottom-row Frobenius coordinates are (µ 1, µ 2 1,..., µ k k + 1) = (µ + ρ) Lemma. Let V = S 2 C n and ρ = ( 1, 2, 3,..., n). For any µ Λ n, the unordered set of integers { µ 1 + ρ 1, µ 2 + ρ 2,..., µ k + ρ k,..., µ n + ρ n } is precisely the set {1, 2,..., n}. In the statement of the lemma it is understood that if l(µ) < n then µ i = 0 for each

40 34 i = l(µ) + 1,..., n. A few examples are as follows: µ = (2, 0), µ + ρ = (1, 2, 3, 4,...) µ = (3, 1), µ + ρ = (2, 1, 3, 4,...) µ = (3, 3), µ + ρ = (2, 1, 3, 4,...) µ = (4, 1, 1), µ + ρ = (3, 1, 2, 4,...) µ = (4, 3, 1), µ + ρ = (3, 1, 2, 4,...) µ = (4, 4, 2), µ + ρ = (3, 2, 1, 4,...) µ = (4, 4, 4), µ + ρ = (3, 2, 1, 4,...) A more extensive list is given in Chapter 5, Section 3. Proof: We prove the result by induction on the number of nested (r+1, r) hooks making up the Young diagram of µ. If the diagram of µ consists of one hook (µ 1, µ 1 1), then µ + ρ = (µ 1 1, 1 2, 1 3,..., 1 (µ 1 1), (µ ), (µ ), (µ ),...) = (µ 1 1, 1, 2,..., 2 µ 1, µ 1, µ 1 1, µ ). The first µ 1 1 coordinates are, in absolute value, a cyclic permutation of the integers 1, 2,..., µ 1 1. The subsequent coordinates are, in absolute value, µ 1, µ 1 + 1,... Thus the result holds for µ consisting of one hook. Now suppose that the result holds for weights whose diagrams are composed of k nested (r i + 1, r i ) hooks. Consider a weight µ composed of k + 1 nested hooks. Denote by µ the partition formed by the first k of these hooks, and write ν = µ + ρ = (ν 1, ν 2,..., ν µ1 1, µ 1, µ 1 1,...), so that by induction the integers ν 1, ν 2,..., ν µ1 1 form a permutation of 1, 2,..., µ 1 1. We now consider µ + ρ, whose first k coordinates are identical to those of ν = µ + ρ. The coordinates first differ at the k + 1st position, where we add µ k+1 k. They also differ from the k + 2nd to the µ k+1 k 1st coordinates, to each of which we add 1;

41 35 after this point µ + ρ agrees with µ + ρ in all subsequent entries. We can write the entries in which they do differ as (ν k+1 + µ k+1 k, ν k+2 + 1, ν k+3 + 1,..., ν µk+1 k 1 + 1). It suffices to show that these integers, in absolute value, are a permutation of (ν k+1, ν k+2,..., ν µk+1 k 1). Note the simple but crucial fact that since the hooks are nested, µ k+1 = µ k+2 =... = µ µk+1 k 1, i.e. the length of the k + 1st nested hook must be less than that of any previous hook. Thus ν k+1 = ν k =... = ν µk+1 k 1 + µ k+1 k 2. We can be even more specific, and assert that ν k+1 = 1. After nesting k hooks, the k + 1st row has k boxes. The kth hook cannot be a (2, 1) hook, i.e. just a row with 2 boxes, or else we could not nest the k + 1st hook. So after nesting k hooks, there are exactly k boxes in the k + 1st row, and ν k+1 = 1. Our task of showing that in absolute value, (3.1.9) (ν k+1 + µ k+1 k, ν k+2 + 1, ν k+3 + 1,..., ν µk+1 k 1 + 1) is a permutation of (3.1.10) (ν k+1, ν k+2,..., ν µk+1 k 1), now reduces to showing that in absolute value, (3.1.11) ( 1 + µ k+1 k, , ,..., 1 µ k+1 + k ) = (µ k+1 k 1, 1, 2,..., µ k+1 + k + 2) is a permutation of (3.1.12) ( 1, 1 1,..., 1 µ k+1 + k + 2) = ( 1, 2, 3..., µ k+1 + k + 1). These do differ by a (cyclic) permutation, and the result follows.

42 Lemma. Let V = 2 C n and ρ = (0, 1, 2,..., n + 1). For any µ Λ n, the unordered set of integers { µ 1 + ρ 1, µ 2 + ρ 2,..., µ n + ρ n } is precisely the set {0, 1, 2,..., n 1}. P roof: The induction on the number of nested (r 1, r) hooks making up the Young diagram of µ is virtually identical to the proof of Lemma 3.1.4, but uses ρ = (0, 1, 2,..., n+ 1) instead of ( 1, 2, 3,..., n). Examples: µ = (1, 1), µ + ρ = (1, 0, 2, 3,...) µ = (2, 1, 1), µ + ρ = (2, 0, 1, 3,...) µ = (2, 2, 2), µ + ρ = (2, 1, 0, 3,...) µ = (3, 1, 1, 1), µ + ρ = (3, 0, 1, 2,...) µ = (3, 2, 2, 1), µ + ρ = (3, 1, 0, 2,...) µ = (3, 3, 2, 2), µ + ρ = (3, 2, 0, 1,...) µ = (3, 3, 3, 3), µ + ρ = (3, 2, 1, 0,...) In each of our two examples of skew multiplicity free spaces V, the number of highest weights λ of V with λ = 2d equals the number of strict partitions of d. By a celebrated result of Euler, the number of strict partitions of d equals the number of partitions of d into odd parts. Now monomials of degree d in the odd degree power sums p 1, p 3, p 5,... correspond bijectively to partitions of d into odd parts. For example, the partitions of 6 into odd parts are (1 6 ), (1 3, 3), (1, 5), and (3 2 ), and the corresponding monomials are p 6 1, p3 1 p 3, p 1 p 5, and p 2 3. Since monomials of degree d form a basis for the space of supersymmetric polynomials of degree d, we have proven the following: Proposition: For any integers n d 0, the number of highest weights λ Λ n with λ 2d equals the dimension of the space of supersymmetric polynomials of degree d.

43 Factorial Schur Q functions Okounkov defines [I1, I2] the factorial Schur Q functions, a family of supersymmetric functions which, like the classical Schur Q functions of which they are an analogue, are indexed by strict partitions. Let [z k] = k (z i + 1), k = 0, 1, 2,... For any strict partition λ and n l(λ), let i=1 l(λ) (3.2.1) F λ (z 1,..., z n ) = [z i λ i ] i=1 i l(λ),i<j n z i + z j z i z j Then the factorial Schur Q polynomial indexed by λ is defined by Definition. Q λ = 2 l(λ) F λ (z (n l(λ))! w(1),..., z w(n) ) w S n From this definition it is clear that Q λ has degree λ. Ivanov demonstrates [I2] that Q λ is supersymmetric and vanishes at all strict partitions µ such that µ λ, but not at λ itself. Moreover, Q λ satisfies the extra vanishing condition that Q λ (µ) = 0 if λ and µ are strict partitions such that λ µ. Furthermore, for each n l(λ) we can express Q λ as a polynomial in n variables, but the values taken by Q λ are independent of n. Ivanov shows that Q λ (λ) = H(λ), where (3.2.2) H(λ) = l(λ) t=1 λ t! i<j λ i + λ j λ i λ j. H(λ) is indeed defined, since λ is assumed to be a strict partition. We will consider a rescaled version of the factorial Schur Q functions, defined by (3.2.3) q λ = 1 H(λ) Q λ, so that q λ (λ) = 1. Thus q λ has the same properties of supersymmetry, degree, and

44 38 independence of n as Q λ. The basic vanishing property it satisfies can be expressed as (3.2.4) q λ (µ) = δ λµ for all strict partitions µ such that µ λ. Let V = S 2 C n and ρ = ( 1, 2,..., n). Let p k denote the degree k power sum polynomial in n variables, i.e. p k = n i=1 zk i. We define an automorphism φ of C[p 1, p 3, p 5,..., p s ] in n variables, where s is the largest odd number n, by (3.2.5) φ(p k ) = 2p k + p k (ρ) for each k = 2j + 1, j = 0, 1, 2,..., s 1 2 Recall that to each λ Λ n there is associated a unique strict partition ˇλ Lemma. p k (λ + ρ) = 2p k (ˇλ) + p k (ρ) for every k = 2j + 1, j = 0, 1, 2,... and λ Λ n. Proof: By Corollary 3.1.5, the coordinates of λ + ρ are in absolute value a permutation of {1, 2,..., n}; the positive terms are precisely the top row Frobenius coordinates of λ, and are equal to the coordinates ˇλ 1,..., ˇλ l(ˇλ) of ˇλ. Thus (3.2.6) p k (λ + ρ) = ˇλ k ˇλ k l(ˇλ) j k 1 j n, j ˇλ i = 2(ˇλ k ˇλ k l(ˇλ) ) 1k 2 k... n k = 2p k (ˇλ) + p k (ρ) Proposition. For any p C[p 1, p 3, p 5,..., p s ] in n variables, p(λ + ρ) = φ(p)(ˇλ) for every λ Λ n. Proof: We can write p as a polynomial in the odd degree power sums of degree n, say p = f(p 1, p 3,...p s ). Then

45 39 (3.2.7) φ(p) = f(2p 1 + p 1 (ρ), 2p 3 + p 3 (ρ),..., 2p s + p s (ρ)) The proposition now follows directly from Lemma Lemma.: If p C[p 1, p 3,..., p s ] d vanishes at all strict partitions ˇλ such that ˇλ d then p = 0 identically. Proof: The rescaled factorial Schur Q functions {qˇµ : µ d} form a basis of C[p 1, p 3, p 5,..., p s ] d, since the vanishing property (3.2.4) which they satisfy guarantees linear independence, and we conclude from Proposition that they span. It follows that p has a unique expansion p = a 0 qˇµ0 + a 1 qˇµ a k qˇµk, where ˇµ 0,..., ˇµ k are all the strict partitions of size d, in order of weakly increasing size. But the vanishing property (3.2.4) implies that only qˇµ0 = q (0) does not vanish at (0), hence a 0 = 0. Likewise, by induction, each a i = 0, i = 1,..., k, and in fact no such supersymmetric polynomial p of degree > 0 exists Lemma.: For any integers n d > 0, if p C[p 1, p 3,..., p s ] d satisfies p(λ + ρ) = 0 on the set {λ Λ n : λ 2d}, then p = 0 identically. Proof: Suppose that there exists such a supersymmetric polynomial p of degree d > 0. Then by Proposition 3.2.3, φ(p) vanishes at all strict partitions ˇλ such that ˇλ d. Since φ is an automorphism, the result now follows from Lemma

46 Characterization Theorem Let V = S 2 C n, ρ = ( 1, 2,..., n). The goal of this section is to prove the following Characterization Theorem: Theorem. For each λ Λ n, n λ 2 there exists a polynomial p λ which satisfies the following properties. CT1a. p λ (µ + ρ) = c λ (µ) for every µ Λ n. CT1b. p λ (µ + ρ) = 0 whenever µ λ unless µ = λ, and p λ (λ + ρ) = 1. CT2. p λ is supersymmetric. CT3. p λ has degree λ 2. Furthermore, p λ is uniquely determined by CT1b, CT2, and CT3. Preliminary remark 1: Implicitly, p λ depends on n, although it will be shown in Section 3.4 that this dependence is much weaker than it appears to be. Indeed, it will be shown that for all n large enough, the values of p λ on ρ-shifted arguments are independent of n. On the one hand, since G acts on S 2 C n for each n, we can consider a fixed value of n and study all the functions p λ. On the other hand we can fix a λ and study the functions p λ as n increases, in particular for all n λ 2. The statements of the Characterization Theorem should be understood in the latter sense, with λ fixed and for all n λ 2. Preliminary remark 2: Our strategy is to prove the theorem for a certain class of invariant differential operators, namely those which arise from the center of the universal enveloping algebra of G. A combinatorial argument then shows that in fact all the

47 41 G-invariant operators on V belong to this class. Proof: The action of G on V gives rise to an action of its Lie algebra g on V as endomorphisms (3.3.1) g End(V ) hence of U(g) on V as differential operators (3.3.2) U(g) PD(V ) and thus, passing to G-invariants, of the center Z(g) of the universal enveloping algebra on V as invariant differential operators (3.3.3) Ψ : Z(g) PD(V ) G A priori it is not clear whether Ψ is surjective, but as indicated in the second preliminary remark to this proof, our strategy is to establish CT1, CT2, and CT3 for the class of operators which do arise from Z(g), and leave it to the concluding part of the proof to show that in fact every invariant operator on V arises in this way. Consider the following diagram, which will be crucial to the proof of each step of the Characterization Theorem. Z(g) C[p 1, p 2, p 3,...] (3.3.4) Ψ PD(V ) G Maps(Λ n, C)