Combinatorial bases for representations. of the Lie superalgebra gl m n

Transcription

1 Combinatorial bases for representations of the Lie superalgebra gl m n Alexander Molev University of Sydney

2 Gelfand Tsetlin bases for gln

3 Gelfand Tsetlin bases for gl n Finite-dimensional irreducible representations L(λ) of gl n are in a one-to-one correspondence with n-tuples of complex numbers λ = (λ 1,..., λ n ) such that λ i λ i+1 Z + for i = 1,..., n 1.

4 Gelfand Tsetlin bases for gl n Finite-dimensional irreducible representations L(λ) of gl n are in a one-to-one correspondence with n-tuples of complex numbers λ = (λ 1,..., λ n ) such that λ i λ i+1 Z + for i = 1,..., n 1. L(λ) contains a highest vector ζ 0 such that E i i ζ = λ i ζ for i = 1,..., n and E i j ζ = 0 for 1 i < j n.

5 Suppose that λ is a partition, λ 1 λ n 0.

6 Suppose that λ is a partition, λ 1 λ n 0. Depict it as a Young diagram. Example. The diagram λ = (5, 5, 3, 0, 0) is l(λ) = 3

7 Suppose that λ is a partition, λ 1 λ n 0. Depict it as a Young diagram. Example. The diagram λ = (5, 5, 3, 0, 0) is l(λ) = 3 The number of nonzero rows is the length of λ, denoted l(λ).

8 Given a diagram λ, a column-strict λ-tableau T is obtained by filling in the boxes of λ with the numbers 1, 2,..., n in such a way that the entries weakly increase along the rows and strictly increase down the columns.

9 Given a diagram λ, a column-strict λ-tableau T is obtained by filling in the boxes of λ with the numbers 1, 2,..., n in such a way that the entries weakly increase along the rows and strictly increase down the columns. Example. A column-strict λ-tableau for λ = (5, 5, 3, 0, 0):

10 Theorem (Gelfand and Tsetlin, 1950). L(λ) admits a basis ζ T parameterized by all column-strict λ-tableaux T such that the action of generators of gl n is given by the formulas E ss ζ T = ω s ζ T, E s,s+1 ζ T = T c TT ζ T, E s+1,s ζ T = T d TT ζ T.

11 Theorem (Gelfand and Tsetlin, 1950). L(λ) admits a basis ζ T parameterized by all column-strict λ-tableaux T such that the action of generators of gl n is given by the formulas E ss ζ T = ω s ζ T, E s,s+1 ζ T = T c TT ζ T, E s+1,s ζ T = T d TT ζ T. Here ω s is the number of entries in T equal to s, and the sums are taken over column-strict tableaux T obtained from T respectively by replacing an entry s + 1 by s and s by s + 1.

12 For any 1 j s n denote by λ s j the number of entries in row j which do not exceed s and set l s j = λ s j j + 1.

13 For any 1 j s n denote by λ s j the number of entries in row j which do not exceed s and set l s j = λ s j j + 1. Then c TT = (l s i l s+1,1 )... (l s i l s+1, s+1 ) (l s i l s1 ) (l s i l s s ), d TT = (l s i l s 1,1 )... (l s i l s 1, s 1 ) (l s i l s 1 ) (l s i l s s ), if the replacement occurs in row i.

14 Equivalent parametrization of the basis vectors by the Gelfand Tsetlin patterns:

15 Equivalent parametrization of the basis vectors by the Gelfand Tsetlin patterns: λ n 1 λ n 2 λ n n λ n 1,1 λ n 1,n 1 T λ 2 1 λ 2 2 λ 1 1

16 Equivalent parametrization of the basis vectors by the Gelfand Tsetlin patterns: λ n 1 λ n 2 λ n n λ n 1,1 λ n 1,n 1 T λ 2 1 λ 2 2 λ 1 1 The top row coincides with λ and the entries satisfy the betweenness conditions λ k i λ k 1, i λ k, i+1.

17 Example. The column-strict tableau with λ = (5, 5, 3, 0, 0)

18 Example. The column-strict tableau with λ = (5, 5, 3, 0, 0) corresponds to the pattern

19 Given ω = (ω 1,..., ω n ) Z n +, consider the weight subspace L(λ) ω = {η L(λ) E ss η = ω s η for all s}.

20 Given ω = (ω 1,..., ω n ) Z n +, consider the weight subspace L(λ) ω = {η L(λ) E ss η = ω s η for all s}. The character of L(λ) is the polynomial in variables x 1,..., x n defined by ch L(λ) = ω dim L(λ) ω x ω x ωn n.

21 Given ω = (ω 1,..., ω n ) Z n +, consider the weight subspace L(λ) ω = {η L(λ) E ss η = ω s η for all s}. The character of L(λ) is the polynomial in variables x 1,..., x n defined by ch L(λ) = ω dim L(λ) ω x ω x ωn n. Corollary. ch L(λ) = s λ (x 1,..., x n ), the Schur polynomial.

22 Lie superalgebra glm n

23 Lie superalgebra gl m n Basis elements of gl m n are E i j with 1 i, j m + n.

24 Lie superalgebra gl m n Basis elements of gl m n are E i j with 1 i, j m + n. The Z 2 -degree (or parity) is given by deg(e i j ) = ī + j, where ī = 0 for 1 i m and ī = 1 for m + 1 i m + n.

25 Lie superalgebra gl m n Basis elements of gl m n are E i j with 1 i, j m + n. The Z 2 -degree (or parity) is given by deg(e i j ) = ī + j, where ī = 0 for 1 i m and ī = 1 for m + 1 i m + n. The commutation relations in gl m n have the form [E i j, E k l ] = δ k j E i l δ i l E k j ( 1) (ī+ j)( k+ l), where the square brackets denote the super-commutator.

26 The span of {E i j 1 i, j m} is a Lie subalgebra isomorphic to gl m,

27 The span of {E i j 1 i, j m} is a Lie subalgebra isomorphic to gl m, the span of {E i j m + 1 i, j m + n} is a Lie subalgebra of isomorphic to gl n,

28 The span of {E i j 1 i, j m} is a Lie subalgebra isomorphic to gl m, the span of {E i j m + 1 i, j m + n} is a Lie subalgebra of isomorphic to gl n, the Lie subalgebra of even elements of gl m n is isomorphic to gl m gl n.

29 Finite-dimensional irreducible representations of gl m n are parameterized by their highest weights λ of the form λ = (λ 1,..., λ m λ m+1,..., λ m+n ),

30 Finite-dimensional irreducible representations of gl m n are parameterized by their highest weights λ of the form λ = (λ 1,..., λ m λ m+1,..., λ m+n ), where λ i λ i+1 Z +, for i = 1,..., m + n 1, i m.

31 Finite-dimensional irreducible representations of gl m n are parameterized by their highest weights λ of the form λ = (λ 1,..., λ m λ m+1,..., λ m+n ), where λ i λ i+1 Z +, for i = 1,..., m + n 1, i m. The corresponding representation L(λ) contains a highest vector ζ 0 such that E i i ζ = λ i ζ for i = 1,..., m + n and E i j ζ = 0 for 1 i < j m + n.

32 Covariant representations L(λ)

33 Covariant representations L(λ) These are the irreducible components of the representations C m n... C m n }{{} k

34 Covariant representations L(λ) These are the irreducible components of the representations C m n... C m n }{{} k They are distinguished by the conditions: all components λ 1,..., λ m+n of λ are nonnegative integers;

35 Covariant representations L(λ) These are the irreducible components of the representations C m n... C m n }{{} k They are distinguished by the conditions: all components λ 1,..., λ m+n of λ are nonnegative integers; the number l of nonzero components among λ m+1,..., λ m+n is at most λ m.

36 To each highest weight λ satisfying these conditions, associate the Young diagram Γ λ containing λ λ m+n boxes.

37 To each highest weight λ satisfying these conditions, associate the Young diagram Γ λ containing λ λ m+n boxes. It is determined by the conditions that the first m rows of Γ λ are λ 1,..., λ m while the first l columns are λ m+1 + m,..., λ m+l + m.

38 To each highest weight λ satisfying these conditions, associate the Young diagram Γ λ containing λ λ m+n boxes. It is determined by the conditions that the first m rows of Γ λ are λ 1,..., λ m while the first l columns are λ m+1 + m,..., λ m+l + m. The condition l λ m ensures that Γ λ is the diagram of a partition.

39 Example. The following is the diagram Γ λ associated with the highest weight λ = (10, 7, 4, 3 3, 1, 0, 0, 0) of gl 4 5 :

40 A supertableau Λ of shape Γ λ is obtained by filling in the boxes of the diagram Γ λ with the numbers 1,..., m + n in such a way that

41 A supertableau Λ of shape Γ λ is obtained by filling in the boxes of the diagram Γ λ with the numbers 1,..., m + n in such a way that the entries weakly increase from left to right along each row and down each column;

42 A supertableau Λ of shape Γ λ is obtained by filling in the boxes of the diagram Γ λ with the numbers 1,..., m + n in such a way that the entries weakly increase from left to right along each row and down each column; the entries in {1,..., m} strictly increase down each column;

43 A supertableau Λ of shape Γ λ is obtained by filling in the boxes of the diagram Γ λ with the numbers 1,..., m + n in such a way that the entries weakly increase from left to right along each row and down each column; the entries in {1,..., m} strictly increase down each column; the entries in {m + 1,..., m + n} strictly increase from left to right along each row.

44 Example. The following is a supertableau of shape Γ λ associated with the highest weight λ = (10, 7, 4, 3 3, 1, 0, 0, 0) of gl 4 5 :

45 Theorem. The covariant representation L(λ) of gl m n admits a basis ζ Λ parameterized by all supertableaux Λ of shape Γ λ.

46 Theorem. The covariant representation L(λ) of gl m n admits a basis ζ Λ parameterized by all supertableaux Λ of shape Γ λ. The action of the generators of the Lie superalgebra gl m n in this basis is given by the formulas E ss ζ Λ = ω s ζ Λ, E s,s+1 ζ Λ = Λ c ΛΛ ζ Λ, E s+1,s ζ Λ = Λ d ΛΛ ζ Λ.

47 Theorem. The covariant representation L(λ) of gl m n admits a basis ζ Λ parameterized by all supertableaux Λ of shape Γ λ. The action of the generators of the Lie superalgebra gl m n in this basis is given by the formulas E ss ζ Λ = ω s ζ Λ, E s,s+1 ζ Λ = Λ c ΛΛ ζ Λ, E s+1,s ζ Λ = Λ d ΛΛ ζ Λ. The sums are over supertableaux Λ obtained from Λ by replacing an entry s + 1 by s and an entry s by s + 1, resp.

48 Here ω s denotes the number of entries in Λ equal to s.

49 Here ω s denotes the number of entries in Λ equal to s. Corollary (Sergeev 1985, Berele and Regev 1987). The character ch L(λ) coincides with the supersymmetric Schur polynomial s Γλ (x 1,..., x m x m+1,..., x m+n ) associated with the Young diagram Γ λ.

50 Given such a supertableau Λ, for any 1 i s m denote by λ s i the number of entries in row i which do not exceed s.

51 Given such a supertableau Λ, for any 1 i s m denote by λ s i the number of entries in row i which do not exceed s. Set r = λ m1 and for any 0 p n and 1 j r + p denote by λ r+p, j the number of entries in column j which do not exceed m + p.

52 Example. The supertableau with λ = (7, 5, 2 2, 1)

53 Example. The supertableau with λ = (7, 5, 2 2, 1) corresponds to the patterns U and V:

54 Set l i = λ i i + 1, l s i = λ s i i + 1, l r+p, j = λ r+p, j j + 1.

55 Set l i = λ i i + 1, l s i = λ s i i + 1, l r+p, j = λ r+p, j j + 1. The coefficients in the expansions of E s,s+1 ζ Λ and E s+1,s ζ Λ are given by

56 Set l i = λ i i + 1, l s i = λ s i i + 1, l r+p, j = λ r+p, j j + 1. The coefficients in the expansions of E s,s+1 ζ Λ and E s+1,s ζ Λ are given by c ΛΛ = (l s i l s+1,1 )... (l s i l s+1, s+1 ) (l s i l s 1 ) (l s i l s s ), d ΛΛ = (l s i l s 1,1 )... (l s i l s 1, s 1 ) (l s i l s 1 ) (l s i l s s ), if 1 s m 1 and the replacement occurs in row i,

57 and by c ΛΛ = (l r+p, j l r+p+1, 1 )... (l r+p, j l r+p+1, r+p+1 ) (l r+p, j l r+p, 1 ) (l r+p, j l r+p, r+p ), d ΛΛ = (l r+p, j l r+p 1, 1 )... (l r+p, j l r+p 1, r+p 1 ) (l r+p, j l r+p, 1 ) (l r+p, j l r+p, r+p ), if s = m + p for 1 p n 1 and the replacement occurs in column j.

58 Formulas for the expansions of E m,m+1 ζ Λ and E m+1,m ζ Λ are also available.

59 Formulas for the expansions of E m,m+1 ζ Λ and E m+1,m ζ Λ are also available. Example (Palev 1989). The basis ζ Λ of the gl m 1 -module L(λ 1,..., λ m λ m+1 ) is parameterized by the patterns λ m 1 λ m 2 λ m m λ m 1,1 λ m 1,m 1 U = λ 2 1 λ 2 2 λ 1 1

60 The top row runs over partitions (λ m 1,..., λ m m ) such that either λ m j = λ j or λ m j = λ j 1 for each j = 1,..., m.

61 The top row runs over partitions (λ m 1,..., λ m m ) such that either λ m j = λ j or λ m j = λ j 1 for each j = 1,..., m. E m,m+1 ζ U = m (l mi + λ m+1 + m) i=1 i 1 ( 1) λ j λ m j j=1 l m i l j l m i l m j m j=i+1 λ j λ m j =1 l m i l m j + 1 l m i l j + 1 ζ U+δ m i, E m+1,m ζ U = m (l m i l m 1,1 )... (l m i l m 1, m 1 ) i=1 (l m i l m 1 ) (l m i l m m ) i 1 ( 1) λ j λ m j l m i l m j 1 l m i l j 1 j=1 i 1 j=1 λ j λ m j =1 l m i l m j l m i l j ζ U δm i.

62 Example. The basis ζ Λ of the gl 1 n -module L(λ 1 λ 2,..., λ n+1 ) is parameterized by the trapezium patterns V = λ r+n, 1 λ r+n, 2 λ r+n, r+n λ r+1, 1 λ r+1, 2 λ r+1, r

63 Example. The basis ζ Λ of the gl 1 n -module L(λ 1 λ 2,..., λ n+1 ) is parameterized by the trapezium patterns V = λ r+n, 1 λ r+n, 2 λ r+n, r+n λ r+1, 1 λ r+1, 2 λ r+1, r The number r of 1 s in the bottom row is nonnegative and varies between λ 1 n and λ 1. The top row coincides with (λ 1,..., λ p, 0,..., 0), where p = λ 1.

64 Yangian Y(gl n )

65 Yangian Y(gl n ) The Yangian Y(gl n ) is a unital associative algebra with generators t (1) i j, t (2) i j,... where i and j run over the set {1,..., n}.

66 Yangian Y(gl n ) The Yangian Y(gl n ) is a unital associative algebra with generators t (1) i j, t (2) i j,... where i and j run over the set {1,..., n}. The defining relations are given by [t (r+1) i j, t (s) k l ] [t (r) i j, t (s+1) k l ] = t (r) where r, s 0 and t (0) i j := δ i j. k j t (s) i l t (s) k j t (r) i l,

67 Using the formal generating series t i j (u) = δ i j + t (1) i j u 1 + t (2) i j u

68 Using the formal generating series t i j (u) = δ i j + t (1) i j u 1 + t (2) i j u the defining relations can be written in the equivalent form (u v) [t i j (u), t k l (v)] = t k j (u) t i l (v) t k j (v) t i l (u).

69 Using the formal generating series t i j (u) = δ i j + t (1) i j u 1 + t (2) i j u the defining relations can be written in the equivalent form (u v) [t i j (u), t k l (v)] = t k j (u) t i l (v) t k j (v) t i l (u). A natural analogue of the Poincaré Birkhoff Witt theorem holds for the Yangian Y(gl n ).

70 Every finite-dimensional irreducible representation L of Y(gl n ) contains a highest vector ζ such that t i j (u) ζ = 0 for 1 i < j n, and t i i (u) ζ = λ i (u) ζ for 1 i n,

71 Every finite-dimensional irreducible representation L of Y(gl n ) contains a highest vector ζ such that t i j (u) ζ = 0 for 1 i < j n, and t i i (u) ζ = λ i (u) ζ for 1 i n, for some formal series λ i (u) = 1 + λ (1) i u 1 + λ (2) i u , λ (r) i C.

72 Every finite-dimensional irreducible representation L of Y(gl n ) contains a highest vector ζ such that t i j (u) ζ = 0 for 1 i < j n, and t i i (u) ζ = λ i (u) ζ for 1 i n, for some formal series λ i (u) = 1 + λ (1) i u 1 + λ (2) i u , λ (r) i C. The n-tuple of formal series λ(u) = ( λ 1 (u),..., λ n (u) ) is the highest weight of L.

73 Moreover, there exist monic polynomials P 1 (u),..., P n 1 (u) in u (the Drinfeld polynomials) such that for i = 1,..., n 1. λ i (u) λ i+1 (u) = P i(u + 1) P i (u)

74 Moreover, there exist monic polynomials P 1 (u),..., P n 1 (u) in u (the Drinfeld polynomials) such that for i = 1,..., n 1. λ i (u) λ i+1 (u) = P i(u + 1) P i (u) Refs: Drinfeld (1988), Tarasov (1985).

75 For an arbitrary representation L(λ) of gl m n consider the vector space isomorphism L(λ) = µ L (µ) L(λ) + µ,

76 For an arbitrary representation L(λ) of gl m n consider the vector space isomorphism L(λ) = µ L (µ) L(λ) + µ, where L (µ) denotes the irreducible representation of the Lie algebra gl m with the highest weight µ = (µ 1,..., µ m ),

77 For an arbitrary representation L(λ) of gl m n consider the vector space isomorphism L(λ) = µ L (µ) L(λ) + µ, where L (µ) denotes the irreducible representation of the Lie algebra gl m with the highest weight µ = (µ 1,..., µ m ), and L(λ) + µ is the multiplicity space spanned by the gl m -highest vectors in L(λ) of weight µ,

78 For an arbitrary representation L(λ) of gl m n consider the vector space isomorphism L(λ) = µ L (µ) L(λ) + µ, where L (µ) denotes the irreducible representation of the Lie algebra gl m with the highest weight µ = (µ 1,..., µ m ), and L(λ) + µ is the multiplicity space spanned by the gl m -highest vectors in L(λ) of weight µ, L(λ) + µ = Hom glm (L (µ), L(λ)).

79 Olshanski homomorphism

80 Olshanski homomorphism Set E = [ E i j ] m i,j=1. The mapping ψ : Y(gl n) U(gl m n ) given by t (1) i j E m+i,m+j, m E m+i,k (E r 2 ) k l E l,m+j, r 2, t (r) i j k,l=1 defines an algebra homomorphism.

81 Olshanski homomorphism Set E = [ E i j ] m i,j=1. The mapping ψ : Y(gl n) U(gl m n ) given by t (1) i j E m+i,m+j, m E m+i,k (E r 2 ) k l E l,m+j, r 2, t (r) i j k,l=1 defines an algebra homomorphism. The image of ψ is contained in the centralizer U(gl m n ) gl m.

82 Theorem. The representation of Y(gl n ) in L(λ) + µ defined via the homomorphism ψ is irreducible.

83 Theorem. The representation of Y(gl n ) in L(λ) + µ defined via the homomorphism ψ is irreducible. Proof. L(λ) + µ is an irreducible representation of the centralizer U(gl m n ) gl m.

84 Theorem. The representation of Y(gl n ) in L(λ) + µ defined via the homomorphism ψ is irreducible. Proof. L(λ) + µ is an irreducible representation of the centralizer U(gl m n ) gl m. The centralizer U(gl m n ) gl m is generated by the image of the homomorphism Y(gl n ) U(gl m n ) gl m and the center of U(gl m n ).

85 Twist the Yangian action on L(λ) + µ by the automorphism t i j (u) t i j (u + m).

86 Twist the Yangian action on L(λ) + µ by the automorphism t i j (u) t i j (u + m). For each box α = (i, j) of a Young diagram define its content by c(α) = j i.

87 Twist the Yangian action on L(λ) + µ by the automorphism t i j (u) t i j (u + m). For each box α = (i, j) of a Young diagram define its content by c(α) = j i. Theorem. Suppose that L(λ) is a covariant representation. The Drinfeld polynomials for the Y(gl n )-module L(λ) + µ are given by P k (u) = α ( u c(α) ), k = 1,..., n 1, where α runs over the leftmost boxes of the rows of length k in the diagram Γ λ /µ.

88 Example. For λ = (7, 5, 2 3, 1, 0, 0) and µ = (4, 2, 1) we have

89 Example. For λ = (7, 5, 2 3, 1, 0, 0) and µ = (4, 2, 1) we have P 1 (u) = (u + 1)(u + 4)(u + 5),

90 Example. For λ = (7, 5, 2 3, 1, 0, 0) and µ = (4, 2, 1) we have P 1 (u) = (u + 1)(u + 4)(u + 5), P 2 (u) = u + 3,

91 Example. For λ = (7, 5, 2 3, 1, 0, 0) and µ = (4, 2, 1) we have P 1 (u) = (u + 1)(u + 4)(u + 5), P 2 (u) = u + 3, P 3 (u) = (u 4)(u 1).

92 Introduce parameters of the diagram conjugate to Γ λ /µ. Set r = µ 1 and let µ = (µ 1,..., µ r ) be the diagram conjugate to µ so that µ j equals the number of boxes in column j of µ.

93 Introduce parameters of the diagram conjugate to Γ λ /µ. Set r = µ 1 and let µ = (µ 1,..., µ r ) be the diagram conjugate to µ so that µ j equals the number of boxes in column j of µ. Set λ = (λ 1,..., λ r+n), where λ j equals the number of boxes in column j of the diagram Γ λ.

94 Introduce parameters of the diagram conjugate to Γ λ /µ. Set r = µ 1 and let µ = (µ 1,..., µ r ) be the diagram conjugate to µ so that µ j equals the number of boxes in column j of µ. Set λ = (λ 1,..., λ r+n), where λ j equals the number of boxes in column j of the diagram Γ λ. Corollary. The Y(gl n )-module L(λ) + µ is isomorphic to L(λ ) + µ, the skew representation associated with gl r+n -module L(λ ) and the gl r -highest weight µ.

95 Construction of basis vectors

96 Construction of basis vectors produce the highest vector of the Y(gl n )-module L(λ) + µ,

97 Construction of basis vectors produce the highest vector of the Y(gl n )-module L(λ) + µ, use the isomorphism L(λ) + µ = L(λ ) + µ to get the vectors of the trapezium Gelfand Tsetlin basis of L(λ ) + µ in terms of the Yangian generators,

98 Construction of basis vectors produce the highest vector of the Y(gl n )-module L(λ) + µ, use the isomorphism L(λ) + µ = L(λ ) + µ to get the vectors of the trapezium Gelfand Tsetlin basis of L(λ ) + µ in terms of the Yangian generators, combine with the Gelfand Tsetlin basis of L (µ).

99 The extremal projector p for gl m is given by p = i<j (E j i ) k (E i j ) k ( 1) k k! (h i h j + 1)... (h i h j + k), k=0 where h i = E i i i + 1. The product is taken in a normal order.

100 The extremal projector p for gl m is given by p = i<j (E j i ) k (E i j ) k ( 1) k k! (h i h j + 1)... (h i h j + k), k=0 where h i = E i i i + 1. The product is taken in a normal order. The projector satisfies E i j p = pe j i = 0 for 1 i < j m.

101 The extremal projector p for gl m is given by p = i<j (E j i ) k (E i j ) k ( 1) k k! (h i h j + 1)... (h i h j + k), k=0 where h i = E i i i + 1. The product is taken in a normal order. The projector satisfies E i j p = pe j i = 0 for 1 i < j m. Ref: Asherova, Smirnov and Tolstoy, 1971.

102 For i = 1,..., m and a = m + 1,..., m + n set z i a = pe i a (h i h 1 )... (h i h i 1 ), z a i = pe a i (h i h i+1 )... (h i h m ).

103 For i = 1,..., m and a = m + 1,..., m + n set z i a = pe i a (h i h 1 )... (h i h i 1 ), z a i = pe a i (h i h i+1 )... (h i h m ). z i a and z a i can be regarded as elements of U(gl m n ) modulo the left ideal generated by E i j with 1 i < j m.

104 For i = 1,..., m and a = m + 1,..., m + n set z i a = pe i a (h i h 1 )... (h i h i 1 ), z a i = pe a i (h i h i+1 )... (h i h m ). z i a and z a i can be regarded as elements of U(gl m n ) modulo the left ideal generated by E i j with 1 i < j m. Example. z 1a = E 1 a, z 2 a = E 2a (h 2 h 1 ) + E 21 E 1 a, z a m = E a m, z a, m 1 = E a, m 1 (h m 1 h m ) + E m, m 1 E a m.

105 The elements z i a and z a i are odd; together with the even elements E a b with a, b {m + 1,..., m + n} they generate the Mickelsson Zhelobenko superalgebra Z(gl m n, gl m ) associated with the pair gl m gl m n.

106 The elements z i a and z a i are odd; together with the even elements E a b with a, b {m + 1,..., m + n} they generate the Mickelsson Zhelobenko superalgebra Z(gl m n, gl m ) associated with the pair gl m gl m n. The generators satisfy quadratic relations that can be written in an explicit form.

107 They preserve the subspace of gl m -highest vectors in L(λ), z i a : L(λ) + µ L(λ) + µ+δ i, z a i : L(λ) + µ L(λ) + µ δ i, where µ ± δ i is obtained from µ by replacing µ i by µ i ± 1.

108 They preserve the subspace of gl m -highest vectors in L(λ), z i a : L(λ) + µ L(λ) + µ+δ i, z a i : L(λ) + µ L(λ) + µ δ i, where µ ± δ i is obtained from µ by replacing µ i by µ i ± 1. Proposition. The element ζ µ = m j=1 ( zm+λj µ j, j... z m+2, j z ) m+1, j ζ with the product taken in the increasing order of j is the highest vector of the Y(gl n )-module L(λ) + µ.