Haar expectations of ratios of random characteristic polynomials

We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups K = O(N), SO(N), and USp(N). To that end, we start from the Clifford-Weyl algebera in its canonical realization on the complex of holomorphic differential forms for a C-vector space V. From it we construct the Fock representation of an orthosymplectic Lie superalgebra osp associated to V. Particular attention is paid to defining Howe's oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of sp in osp. In the process, by pushing the semigroup representation to its boundary and arguing by continuity, we provide a construction of the Shale-Weil-Segal representation of the metaplectic group. To deal with a product of n ratios of characteristic polynomials, we let V = C^n \otimes C^N where C^N is equipped with its standard K-representation, and focus on the subspace of K-equivariant forms. By Howe duality, this is a highest-weight irreducible representation of the centralizer g of Lie(K) in osp. We identify the K-Haar expectation of n ratios with the character of this g-representation, which we show to be uniquely determined by analyticity, Weyl group invariance, certain weight constraints and a system of differential equations coming from the Laplace-Casimir invariants of g. We find an explicit solution to the problem posed by all these conditions. In this way we prove that the said Haar expectations are expressed by a Weyl-type character formula for all integers N \ge 1. This completes earlier work by Conrey, Farmer, and Zirnbauer for the case of U(N).


Introduction
In this article we derive an explicit formula for the average where K is one of the classical compact Lie groups O N , SO N , or USp N equipped with Haar measure dk of unit mass K dk = 1 and depends on a set of complex parameters t := (e iψ 1 , . . . , e iψ n , e φ 1 , . . . , e φ n ), which satisfy Re φ j > 0 for all j = 1, . . ., n . The case of K = U N is handled in [4]. Note that . This means that Z(t, k) is a product of ratios of characteristic polynomials, which explains the title of the article.
The Haar average I(t) can be regarded as the (numerical part of the) character of an irreducible representation of a Lie supergroup (g , G) restricted to a suitable subset of a maximal torus of G. The Lie superalgebra g is the Howe dual partner of the compact group K in an orthosymplectic Lie superalgebra osp . It is naturally represented on a certain infinite-dimensional spinor-oscillator module a(V ) -more concretely, the complex of holomorphic differential forms on the vector space C n ⊗ C N -and the irreducible representation is that on the subspace a(V ) K of K-equivariant forms.
To even define the character, we must exponentiate the representation of the Lie algebra part of osp on a(V ). This requires going to a completion A V of a(V ), and can only be done partially. Nevertheless, the represented semigroup contains enough structure to derive Laplace-Casimir differential equations for its character.
Our explicit formula for I(t) looks exactly like a classical Weyl formula and is derived in terms of the roots of the Lie superalgebra g and the Weyl group W. Let us state this formula for K = O N , USp N without going into the details of the λ -positive even and odd roots ∆ + λ ,0 and ∆ + λ ,1 and the Weyl group W (see §4. 3.2 for precise formulas). If W λ is the isotropy subgroup of W fixing the highest weight λ = λ N , then (1 − e −w(α) ) (lnt) .
( 1.1) To prove this formula we establish certain properties of I(t) which uniquely characterize it and are satisfied by the right-hand side. These are a weight expansion of I(t) (see Corollary 4.1), restrictions on the set of weights (see Corollary 2.3), and the fact that I(t) is annihilated by certain invariant differential operators (see Corollary 4.2). As was stated above, the case K = U N is treated in [4]. Here, we restrict to the compact groups K = O N , K = USp N , and K = SO N with t ′ = (e −iψ 1 , e iψ 2 , . . ., e iψ n , e φ 1 , . . . , e φ n ), since Det(k) Z(t, k) = (−1) N Z(t ′ , k).

Comparison with results of other approaches. -
To facilitate the comparison with related work, we now present our final results in the following explicit form. Let x j := e −iψ j and y l := e −φ l . Consider first the case of the unitary symplectic group K = USp N (where N ∈ 2N). Then for any pair of non-negative integers p, q in the range q − p ≤ N + 1 one directly infers from (1.1) the formula .
The sum on the right-hand side is over sign configurations ε ≡ (ε 1 , . . . , ε p ) ∈ {±1} p . The proof proceeds by induction in p , starting from the result (1.1) for p = q and sending x p → 0 to pass from p to p − 1. In published recent work [5,3] the same formula was derived under the more restrictive condition q ≤ N/2 . In [3] this unwanted restriction on the parameter range came about because the numerator and denominator on the left-hand side were expanded separately, ignoring the supersymmetric Howe duality (see §2 of the present paper) of the problem at hand. For K = SO N the same induction process starting from (1.1) yields the result as long as q − p ≤ N − 1 . Please note that this includes even the case of the trivial group K = SO 1 = {Id} with any p = q > 0 . For K = O N one has an analogous result where the sum on the right-hand side is over ε with an even number of sign reversals. The very same formulas for SO N and O N were derived in the recent literature [5,3] but, again, only in the much narrower range q ≤ Int[N/2].

Howe duality and weight expansion. -
To find an explicit expression for the integral I(t), we first of all observe that the integrand Z(t, k) is the supertrace of a representation ρ of a semigroup (T 1 × T + ) × K on the spinor-oscillator module a(V ) (cf. Lemma 4.5). More precisely, we start with the standard K-representation space C N , the Z 2 -graded vector space U = U 0 ⊕U 1 with U s ≃ C n , and the abelian semigroup T 1 × T + := {(diag(e iψ 1 , . . ., e iψ n ), diag(e φ 1 , . . ., e φ n )) | Re φ j > 0 , j = 1, . . . , n} of diagonal transformations in GL(U 1 ) × GL(U 0 ). We then consider the vector space V := U ⊗ C N which is Z 2 -graded by V s = U s ⊗ C N , the infinite-dimensional spinoroscillator module a(V ) := ∧(V * 1 ) ⊗ S(V * 0 ), and a representation ρ of (T 1 × T + ) × K on a(V ). We also let V ⊕V * =: W = W 0 ⊕W 1 (not the Weyl group).
Averaging the product of ratios Z(t, k) with respect to the compact group K corresponds to the projection from a(V ) onto the vector space a(V ) K of K-invariants (Corollary 4.1). Now, Howe duality (Proposition 2.2) implies that a(V ) K is the representation space for an irreducible highest-weight representation ρ * of the Howe dual partner g of K in the orthosymplectic Lie superalgebra osp(W ). This representation ρ * is constructed by realizing g ⊂ osp(W ) as a subalgebra in the space of degree-two elements of the Clifford-Weyl algebra q(W ). Precise definitions of these objects, their relationships, and the Howe duality statement can be found in §2.
Using the decomposition a(V ) K = ⊕ γ∈Γ V γ into weight spaces, Howe duality leads to the weight expansion I(e H ) = STr a(V ) K e ρ * (H) = ∑ γ∈Γ B γ e γ(H) for t = e H ∈ T 1 × T + . Here STr denotes the supertrace. There are strong restrictions on the set of weights Γ. Namely, if γ ∈ Γ, then γ = ∑ n j=1 (im j ψ j − n j φ j ) and − N 2 ≤ m j ≤ N 2 ≤ n j for all j . The coefficients B γ = STr V γ (Id) are the dimensions of the weight spaces (multiplied by parity). Note that the set of weights of the representation ρ * of g on a(V ) K is infinite. 1. 3. Group representation and differential equations. -Before outlining the strategy for computing our character in the infinite-dimensional setting of representations of Lie superalgebras and groups, we recall the classical situation where ρ * is an irreducible finite-dimensional representation of a reductive Lie algebra g and ρ is the corresponding Lie group representation of the complex reductive group G. In that case the character χ of ρ, which automatically exists, is the trace Tr ρ , which is a radial eigenfunction of every Laplace-Casimir operator. These differential equations can be completely understood by their behavior on a maximal torus of G.
In our case we must consider the infinite-dimensional irreducible representation ρ * of the Lie superalgebra g = osp on a(V ) K . Casimir elements, Laplace-Casimir operators of osp, and their radial parts have been described by Berezin [1]. In the situation U 0 ≃ U 1 at hand we have the additional feature that every osp-Casimir element I can be expressed as a bracket I = [∂ , F] where ∂ is the holomorphic exterior derivative when we view a(V ) K as the complex of K-equivariant holomorphic differential forms on V 0 .
To benefit from Berezin's theory of radial parts, we construct a radial superfunction χ which is defined on an open set containing the torus T 1 × T + such that its numerical part satisfies numχ(t) = I(t) for all t ∈ T 1 × T + . If we had a representation (ρ * , ρ) of a Lie supergroup (osp, G) at our disposal, we could define χ to be its character, i.e. χ(g) ? = STr a(V ) K ρ(g) e ∑ j ξ j ρ * (Ξ j ) (see §4. 1.4). Since we don't have such a representation, our idea is to define χ as a character on a totally real submanifold M of maximal dimension which contains a real form of T 1 × T + and is invariant with respect to conjugation by a real form G R of G , and then to extend χ by analytic continuation.
Thinking classically we consider the even part of the Lie superalgebra osp(W 0 ⊕W 1 ), which is the Lie algebra o(W 1 )⊕sp(W 0 ). The real structures at the Lie supergroup level come from a real form W R of W . The associated real forms of o(W 1 ) and sp(W 0 ) are the real orthogonal Lie algebra o(W 1,R ) and the real symplectic Lie algebra sp(W 0,R ). These are defined in such a way that the elements in o(W 1,R ) ⊕ sp(W 0,R ) and iW R are mapped as elements of the Clifford-Weyl algebra via the spinor-oscillator representation to anti-Hermitian operators on a(V ) with respect to a compatibly defined unitary structure. In this context we frequently use the unitary representation of the real Heisenberg group exp(iW 0,R ) × U 1 on the completion A V of the module a(V ).
Since ∧(V * 1 ) has finite dimension, exponentiating the spinor representation of o(W 1,R ) causes no difficulties. This results in the spinor representation of Spin(W 1,R ), a 2:1 covering of the compact group SO(W 1,R ). So in this case one easily constructs a representation R 1 : Spin(W 1,R ) → U(a(V )) which is compatible with ρ * | o(W 1,R ) .
Exponentiating the oscillator representation of sp(W 0,R ) on the infinite-dimensional vector space S(V * 0 ) requires more effort. In §3.4, following Howe [8], we construct the Shale-Weil-Segal representation R ′ : Mp(W 0,R ) → U(A V ) of the metaplectic group Mp(W 0,R ) which is the 2:1 covering group of the real symplectic group Sp(W 0,R ). This is compatible with ρ * | sp(W 0,R ) . Altogether we see that the even part of the Lie superalgebra representation integrates to G R = Spin(W 1,R ) × Z 2 Mp(W 0,R ) .
The construction of R ′ uses a limiting process coming from the oscillator semigroup H(W s 0 ), which is the double covering of the contraction semigroup H(W s 0 ) ⊂ Sp(W 0 ) and has Mp(W 0,R ) in its boundary. Furthermore, we have H(W s 0 ) = Mp(W 0,R ) × M where M is an analytic totally real submanifold of maximal dimension which contains a real form of the torus T + (see §3.2). The representation R 0 : H(W s 0 ) → End(A V ) constructed in §3. 3 facilitates the definition of the representation R ′ and of the character χ in §4. 2. It should be underlined that Proposition 3.24 ensures convergence of the superfunction χ(h), which is defined as a supertrace and exists for all h ∈ H(W s 0 ). On that basis, the key idea of our approach is to exploit the fact that every Casimir invariant I ∈ U(g) is exact in the sense that I = [∂ , F]. By a standard argument, this exactness property implies that every such invariant I vanishes in the spinor-oscillator representation. This result in turn implies for our character χ the differential equations D(I)χ = 0 where D(I) is the Laplace-Casimir operator representing I. By drawing on Berezin's theory of radial parts, we derive a system of differential equations which in combination with certain other properties ultimately determines χ.
In the case of K = O N the Lie group associated to the even part of the real form of the Howe partner g is embedded in a simple way in the full group G R described above. It is itself just a lower-dimensional group of the same form. In the case of K = USp N a sort of reversing procedure takes places and the analogous real form is USp 2n × SO * 2n . Nevertheless, the precise data which are used as input into the series developments, the uniqueness theorem and the final calculations of χ are essentially the same in the two cases. Therefore there is no difficulty handling them simultaneously.

Howe dual pairs in the orthosymplectic Lie superalgebra
In this chapter we collect some foundational information from representation theory. Basic to our work is the orthosymplectic Lie superalgebra, osp , in its realization as the space spanned by supersymmetrized terms of degree two in the Clifford-Weyl algebra. Representing the latter by its fundamental representation on the spinor-oscillator module, one gets a representation of osp and of all Howe dual pairs inside of osp . Roots and weights of the relevant representations are described in detail.

Notion of Lie superalgebra. -
A Z 2 -grading of a vector space V over K = R or C is a decomposition V = V 0 ⊕V 1 of V into the direct sum of two K-vector spaces V 0 and V 1 . The elements in (V 0 ∪V 1 ) \ {0} are called homogeneous. The parity function | | : (V 0 ∪V 1 ) \ {0} → Z 2 , v ∈ V s → |v| = s , assigns to a homogeneous element its parity. We write V ≃ K p|q if dim K V 0 = p and dim K V 1 = q .
gives End(V ) the structure of a Lie superalgebra, namely gl(V ). The Jacobi identity in this case is a direct consequence of the associativity (XY )Z = X (Y Z) and the definition of [X ,Y ].
In fact, for every Z 2 -graded associative algebra A the bracket which is the same as The set of fixed points Fix(σ ) = {X ∈ gl(V ) | σ (X ) = X } is a real Lie superalgebra, a real form of gl(V ) called the unitary Lie superalgebra u := u(V ). Note that u 0 is the compact real form u(V 0 ) ⊕ u(V 1 ) of the complex Lie algebra gl(V 0 ) ⊕ gl(V 1 ).
Then consider the canonical alternating bilinear form A on W 0 , and the canonical symmetric bilinear form S on W 1 , The orthosymplectic form of W is the non-degenerate bilinear form Q : W × W → K defined as the orthogonal sum Q = A + S : Note the exchange symmetry Q(w, w ′ ) = −(−1) |w||w ′ | Q(w ′ , w) for w, w ′ ∈ W 0 ∪W 1 . Given Q , define a complex linear bijection τ : End(W ) → End(W ) by the equation for all w, w ′ ∈ W 0 ∪W 1 . It is easy to check that τ has the property which implies that τ is an involutory automorphism of the Lie superalgebra gl(W ) For X ∈ End(V ), we denote by X = ∑ s,t X ts the corresponding decomposition of an operator. The supertrace on V is the linear function STr : End(V ) → K , X → TrX 00 − Tr X 11 = ∑ s (−1) s Tr X ss .
(If dimV = ∞ , then usually the domain of definition of STr must be restricted.) An ad-invariant bilinear form on a Lie superalgebra g = g 0 ⊕ g 1 over K is a bilinear mapping B : g × g → K with the properties 1. g 0 and g 1 are B-orthogonal to each other ; 2. B is symmetric on g 0 and skew on g 1 ; for all X ,Y, Z ∈ g . We will repeatedly use the following direct consequences of the definition of STr.
Recalling the setting of Example 2.3, note that the supertrace for W = V ⊕V * is odd under the gl-automorphism τ fixing osp(W ), i.e., STr W • τ = −STr W . It follows that STr W X = 0 for any X ∈ osp(W ). Moreover, STr W (X 1 X 2 · · · X 2n+1 ) = 0 for any product of an odd number of osp-elements. The universal enveloping algebra U(g) is defined as the quotient of the tensor algebra T(g) = ⊕ ∞ n=0 T n (g) by the two-sided ideal J(g) generated by all combinations for homogeneous X ,Y ∈ T 1 (g) ≡ g . If U n (g) is the image of T n (g) := ⊕ n k=0 T k (g) under the projection T(g) → U(g) = T(g)/J(g), the algebra U(g) is filtered by U(g) = ∪ ∞ n=0 U n (g). The Z 2 -grading g = g 0 ⊕ g 1 gives rise to a Z 2 -grading of T(g) by , and this in turn induces a canonical Z 2 -grading of U(g).
One might imagine introducing various bracket operations on T(g) and/or U(g). However, in view of the canonical Z 2 -grading, the natural bracket operation to use is the supercommutator, which is the bilinear map T(g) × T(g) → T(g) defined by {a, b} := ab −(−1) |a||b| ba for homogeneous elements a, b ∈ T(g). (For the time being, we use a different symbol { , } for better distinction from the bracket [ , ] on g .) Since by the definition of J(g) one has the supercommutator descends to a well-defined map { , } : U(g) × U(g) → U(g).
Proof. -Compatibility with the Z 2 -grading, skew symmetry, and Jacobi identity are properties of { , } that are immediate at the level of the tensor algebra T(g). They descend to the corresponding properties at the level of U(g) by the definition of the two-sided ideal J(g). Thus U(g) with the bracket { , } is a Lie superalgebra.
To see that {U n (g), U n ′ (g)} is contained in U n+n ′ −1 (g), notice that this property holds true for n = n ′ = 1 by the defining relations J(g) ≡ 0 of U(g). Then use the associative law for U(g) to verify the formula for homogeneous a, b, c ∈ U(g). The claim now follows by induction on the degree of the filtration U(g) = ∪ ∞ n=0 U n (g). By definition, the supercommutator of U(g) and the bracket of g agree at the linear level: It is therefore reasonable to drop the distinction in notation and simply write [ , ] for both of these product operations. This we now do.
For future use, note the following variant of the preceding formula: if Y 1 , . . .,Y k , X are any homogeneous elements of g , then which expresses the supercommutator in U(g) by the bracket in g .

Structure of osp(W
Similarly, the symplectic Lie algebra sp(W 0 ) is the Lie algebra of the automorphism group Sp(W 0 ) of W 0 equipped with the non-degenerate alternating bilinear form A : For the next statement, recall the definition of the orthosymplectic Lie superalgebra osp(W ) and the decomposition osp(W ) = osp(W ) 0 ⊕ osp(W ) 1 .
Since both S and A are non-degenerate, the component X 01 is determined by the component X 10 , and one therefore has osp(W ) 1 ≃ Hom(W 0 ,W 1 ) ≃ W 1 ⊗W * 0 .
We now review how sp(W 0 ) and o(W 1 ) decompose for our case W s = V s ⊕ V * s . For that purpose, if U is a vector space with dual vector space U * , let Sym(U,U * ) and Alt(U,U * ) denote the symmetric resp. alternating linear maps from U to U * . 1. Let s = 1 and write the corresponding decomposition of X ∈ End(W 1 ) as

Lemma 2.4. -As vector spaces,
for all v, v ′ ∈ V 1 and ϕ, ϕ ′ ∈ V * 1 . Thus D = −A t , and the maps B, C are alternating. This already proves the statement for the case of o(W 1 ).
The situation for sp(W 0 ) is identical but for a sign change: the symmetric form S is replaced by the alternating form A , and this causes the parity of B, C to be reversed.
By adding up dimensions, Lemmas 2.3 and 2.4 entail the following consequence.
There exists another way of thinking about osp(W ), which will play a key role in the sequel. To define it and keep the sign factors consistent and transparent, we need to be meticulous about our ordering conventions. Hence, if v ∈ V is a vector and ϕ ∈ V * is a linear function, we write the value of ϕ on v as Based on this notational convention, if V is a Z 2 -graded vector space and X ∈ End(V ) is a homogeneous operator, we define the supertranspose This definition differs from the usual transpose by a change of sign in the case when X has a component in Hom(V 1 ,V 0 ). From it, it follows directly that the negative supertranspose gl(V ) → gl(V * ), X → −X st is an isomorphism of Lie superalgebras: The modified notion of transpose goes hand in hand with a modified notion of what it means for an operator in Hom(V,V * ) or Hom(V * ,V ) to be symmetric. Thus, define the subspace Sym(V * ,V ) ⊂ Hom(V * ,V ) to consist of the elements, say B, which are symmetric in the Z 2 -graded sense: By the same principle, define Sym(V,V * ) ⊂ Hom(V,V * ) as the set of solutions C of To make the connection with the decomposition of Lemma 2.3 and 2.4, notice that and similar for the corresponding intersections involving Sym(V * ,V ).
Next, expressing the orthosymplectic form Q of W = V ⊕V * as and writing out the conditions resulting from Q(X w, w ′ ) + (−1) |X||w| Q(w, X w ′ ) = 0 for the case of X ≡ B ∈ Hom(V * ,V ) and X ≡ C ∈ Hom(V,V * ), one sees that This situation is summarized in the next statement.

Lemma 2.5. -
The orthosymplectic Lie superalgebra of W = V ⊕V * decomposes as where g (+2) := Sym(V,V * ), and g (−2) := Sym(V * ,V ), and The decomposition of Lemma 2.5 can be regarded as a Z-grading of osp(W ). By the 'block' structure inherited from W = V ⊕ V * , this decomposition is compatible with the bracket [ , ] : [ 0}. It follows that each of the three subspaces g (+2) , g (−2) , and g (0) is a Lie superalgebra, the first two with vanishing bracket.
Proof. -Since the negative supertranspose A → −A st is a homomorphism of Lie superalgebras, so is our embedding A → A ⊕ (−A st ). This map is clearly injective. To see that it is surjective, consider any homogeneous X = A ⊕ D ∈ End(V ) ⊕ End(V * ) viewed as an operator in End(W ). The condition for X to be in osp(W ) is (2.1). To get a non-trivial condition, choose (w, w ′ ) = (v, ϕ) or (w, w ′ ) = (ϕ, v). The first choice gives Valid for all v ∈ V 0 ∪ V 1 and ϕ ∈ V * , this implies that D = −A st . The second choice leads to the same conclusion. Thus X = A⊕D is in osp(W ) if and only if D = −A st .
In the following subsections we will often write osp(W ) ≡ osp for short.

Roots and root spaces.
-A Cartan subalgebra of a Lie algebra g 0 is a maximal commutative subalgebra h ⊂ g 0 such that g 0 (or its complexification if g 0 is a real Lie algebra) has a basis consisting of eigenvectors of ad(H) for all H ∈ h . Recall that |[X ,Y ]| = |X | + |Y | for homogeneous elements X ,Y of a Lie superalgebra g . From the vantage point of decomposing g by eigenvectors or root spaces, it is therefore reasonable to call a Cartan subalgebra of g 0 a Cartan subalgebra of g . We will see that X ∈ osp 1 and [X , H] = 0 for all H ∈ h ⊂ osp 0 imply X = 0 , i.e., there exists no commutative subalgebra of osp that properly contains a Cartan subalgebra. Lie superalgebras with this property are called of type I in [1]. Let us determine a Cartan subalgebra and the corresponding root space decomposition of osp . For s,t = 0, 1 choose bases {e s,1 , . . ., e s, d s } of V s and associated dual bases Then for j = 1, . . ., d s and k = 1, . . . , d t define rank-one operators E s, j ;t, k by the equation E s, j ;t, k (e u,l ) = e s, j δ t,u δ k,l for all u = 0, 1 and l = 1, . . . , d u . These form a basis of End(V ), and by Lemma 2.6 the operators form a basis of g (0) . Similarly, let bases of Hom(V * ,V ) and Hom(V,V * ) be defined by F s, j ;t, k ( f u,l ) = e s, j δ t,u δ k,l ,F s, j ;t, k (e u,l ) = f s, j δ t,u δ k,l , for index pairs in the appropriate range. Then by Lemma 2.5 and equations (2.3, 2.4) the subalgebras g (−2) and g (2) are generated by the sets of operators X (−2) s j,tk := F s, j ;t, k + F t, k ; s, j (−1) |s||t| , X (2) s j,tk :=F s, j ;t, k +F t, k ; s, j (−1) |s||t|+|s|+|t| .  A root of a Lie superalgebra g is called even if its root space is in g 0 , it is called odd if its root space is in g 1 . We denote by ∆ 0 and ∆ 1 the set of even roots and the set of odd roots, respectively. For g = osp we have

Casimir elements.
-As before, let g = g 0 ⊕ g 1 be a Lie superalgebra, and let U(g) = ∪ ∞ n=0 U n (g) be its universal enveloping algebra. Denote the symmetric algebra of g 0 by S(g 0 ) and the exterior algebra of g 1 by ∧(g 1 ). The Poincaré-Birkhoff-Witt theorem for Lie superalgebras states that for each n there is a bijective correspondence The collection of inverse maps lift to a vector-space isomorphism, called the super-symmetrization mapping. In other words, given a homogeneous basis {e 1 , . . . , e d } of g , each element x ∈ U(g) can be uniquely represented in the form x = ∑ n ∑ i 1 ,..., i n x i 1 ,..., i n e i 1 · · · e i n with super-symmetrized coefficients, i.e., The isomorphism ∧(g 1 ) ⊗ S(g 0 ) ≃ U(g) gives U(g) a Z-grading (by the degree n). Now recall that U(g) comes with a canonical bracket operation, the supercommutator [ , ] : U(g) × U(g) → U(g). An element X ∈ U(g) is said to lie in the center of U(g), and is called a Casimir element, iff [X ,Y ] = 0 for all Y ∈ U(g). By the formula (2.2), a necessary and sufficient condition for that is [X ,Y ] = 0 for all Y ∈ g .
In the case of g = osp , for every ℓ ∈ N there is a Casimir element I ℓ of degree 2ℓ, which is constructed as follows. Consider the bilinear form B : osp × osp → K given by the supertrace (in some representation), B(X ,Y ) := STr (XY ). Recall that this form is ad-invariant, which is to say that Taking the supertrace in the fundamental representation of osp , the form B is nondegenerate, and therefore, if e 1 , . . . , e d is a homogeneous basis of osp , there is another homogeneous basis e 1 , . . . , e d of osp so that B( e i , e j ) = δ i j . Note | e i | = |e i | and put Since the supertrace of any bracket vanishes, one concludes that [I ℓ , X ] = 0. The other statement, |I ℓ | = 0 , follows from | e i | = |e i |, the additivity of the Z 2 -degree and the fact that STr (a) = 0 for any odd element a ∈ U(g).
We now describe a useful property enjoyed by the Casimir elements I ℓ of osp(V ⊕V * ) in the special case of isomorphic components V 0 ≃ V 1 . Recalling the notation of §2.2.1, let ∂ := ∑ j X By the same argument, [ ∂ , Λ] = 0 . One also sees that Λ 2 = Id . Now define F ℓ to be the following odd element of U(osp): e i 1 · · · e i 2ℓ STr (e i 2ℓ · · · e i 1 ∂ Λ) .

Lemma 2.9.
-Let osp(V ⊕ V * ) be the orthosymplectic Lie superalgebra for a Z 2graded vector space V with isomorphic components V 0 ≃ V 1 . Then for all ℓ ∈ N the Casimir element I ℓ is expressible as a bracket: Proof. -By the same argument as in the proof of Lemma 2.8, Using the relations [∂ , Λ] = 0 and Λ 2 = Id , one has for any a ∈ U(osp) that where the second equality sign is from STr The statement of the lemma now follows on setting a = e i 2ℓ · · · e i 1 .
As we shall see, Lemma 2.9 has the drastic consequence that all osp-Casimir elements I ℓ are zero in the spinor-oscillator representation of osp(V ⊕V * ) for V 0 ≃ V 1 . osp. -In the present context, a pair (h, h ′ ) of subalgebras h, h ′ ⊂ g of a Lie superalgebra g is called a dual pair whenever h ′ is the centralizer of h in g and vice versa. In this subsection, let K = C . Given a Z 2 -graded complex vector space U = U 0 ⊕U 1 we let V := U ⊗ C N , where C N is equipped with the standard representation of GL(C N ), O(C N ) , or Sp(C N ) , as the case may be. As a result, the Lie algebra k of whichever group is represented on C N is embedded in osp(V ⊕V * ). We will now describe the dual pairs (h, k) in osp(W ) for W = V ⊕V * . These are known as dual pairs in the sense of R. Howe. Let us begin by recalling that for any representation ρ : K → GL(E) of a group K on a vector space E , the dual representation ρ * : K → GL(E * ) on the linear forms on E is given by (ρ(k)ϕ)(x) = ϕ(ρ(k) −1 x). By this token, every representation ρ :

Howe pairs in
preserves the canonical pairing V ⊗V * → C . Since the orthosymplectic form Q : W ×W → C uses nothing but that pairing, it follows that Passing to the Lie algebra level one obtains ρ W * (k) ⊂ osp(W ). The operator ρ W (k) preserves the Z 2 -grading of W ; therefore one actually has ρ W * (k) ⊂ osp(W ) 0 .
Let us now assume that the complex Lie group K is defined by a non-degenerate bilinear form B : C N × C N → C in the sense that We then have a canonical isomorphism ψ : C N → (C N ) * by z → B(z , ), and an iso- Proof. -If u ∈ U , ϕ ∈ U * , and z ∈ C N , then by the definition of Ψ and ρ W (k), Since B is K-invariant, one has ψ(z)k −1 = ψ(kz), and therefore Let us now examine what happens to the orthosymplectic form Q on W when it is pulled back by the isomorphism Ψ to a bilinear form Ψ * Q on (U ⊕U * ) ⊗ C N : By definition, ψ(z)(z ′ ) = B(z , z ′ ), and writing B(z , z ′ ) = (−1) δ B(z ′ , z) where δ = 0 if B is symmetric and δ = 1 if B is alternating, we obtain In view of this, let U denote the vector space U = U 0 ⊕U 1 with the twisted Z 2 -grading, i.e. U s := U s+1 (s ∈ Z 2 ). Moreover, notice that Ψ determines an embedding In the following we often write O(C N ) ≡ O N and Sp(C N ) ≡ Sp N for short.
Proof. -For K = O N the bilinear form B of C N is symmetric and the bilinear form Q of W pulls back -see (2.6) -to the standard orthosymplectic form of U ⊕U * . For K = Sp N , the form B is alternating. Its pullback, the orthosymplectic form of U ⊕U * twisted by the sign factor (−1) δ , is restored to standard form by switching to the Z 2 -graded vector space U ⊕ U * with the twisted Z 2 -grading.
To go further, we need a statement concerning Hom G (V 1 ,V 2 ), the space of G-equivariant homomorphisms between two modules V 1 and V 2 for a group G .

Lemma 2.12.
-Let X 1 , X 2 ,Y 1 ,Y 2 be finite-dimensional vector spaces all of which are representation spaces for a group G . If the G-action on X 1 and X 2 is trivial, then Proof. -Hom(X 1 ⊗Y 1 , X 2 ⊗Y 2 ) is canonically isomorphic to X * 1 ⊗Y * 1 ⊗ X 2 ⊗Y 2 as a G-representation space, with G-equivariant maps corresponding to G-invariant tensors.
Since the G-action on X * 1 ⊗ X 2 is trivial, one sees that Hom G (X 1 ⊗Y 1 , X 2 ⊗Y 2 ) is isomorphic to the tensor product of X * 1 ⊗X 2 ≃ Hom(X 1 , X 2 ) with the space of G-invariants in Y * 1 ⊗Y 2 . The latter in turn is isomorphic to Hom G (Y 1 ,Y 2 ). Proposition 2. 1 Proof. -Here we calculate the centralizer of k in osp(W ) for each of the three cases k = gl N , o N , sp N and refer the reader to [7] for the remaining details.
Since both V ⊂ W and V * ⊂ W are K-invariant subspaces, End K (W ) decomposes as By Schur's lemma, End K (C N ) ≃ C , and therefore Lemma 2.12 implies By the same reasoning, End K (V * ) = End(U * ). Applying Lemma 2.12 to the two remaining summands, we obtain plus the same statement where each vector space is replaced by its dual.
for W = U ⊗ C N ⊕U * ⊗ C N * is an isomorphism. This means that the centralizer of gl N in osp(W ) is the intersection Φ(End(U ) ⊕End(U * )) ∩osp(W ), which can be identified with End(U ) = gl(U ). Thus we have the first dual pair, (gl(U ), gl N ).
In the case of K = O N , the discussion is shortened by recalling Lemma 2.11 and the K-equivariant isomorphism Ψ : (U ⊕U * ) ⊗ C N → W . By Schur's lemma, these imply End K (W ) ≃ End(U ⊕ U * ). From Corollary 2.2 it then follows that the intersection osp(W ) ∩ End K (W ) is isomorphic as a Lie superalgebra to osp(U ⊕ U * ). Passing to the Lie algebra level for K, we get the second dual pair, (osp(U ⊕U * ), o N ).
Finally, if K = Sp N , the situation is identical except that Corollary 2.2 compels us to switch to the Z 2 -twisted structure of orthosymplectic Lie superalgebra in End K (W ) ≃ End(U ⊕U * ). This gives us the third dual pair, (osp( U ⊕ U * ), sp N ).

Clifford-
The universal enveloping algebra of the Jordan-Heisenberg Lie superalgebra is called the Clifford-Weyl algebra (or quantum algebra). We denote it by q(W ) ≡ U( W ).
Equivalently, one defines the Clifford-Weyl algebra q(W ) as the associative algebra generated by W = W ⊕ K subject to the following relations for all w, w ′ ∈ W 0 ∪W 1 : In particular, w 0 w 1 = w 1 w 0 for all w 0 ∈ W 0 and w 1 ∈ W 1 . Reordering by this commutation relation defines an isomorphism of associative algebras q(W ) ≃ c(W 1 ) ⊗ w(W 0 ), where the Clifford algebra c(W 1 ) is generated by W 1 ⊕K with the relations ww ′ +w ′ w = S(w, w ′ ) for w, w ′ ∈ W 1 , and the Weyl algebra w(W 0 ) is generated by As a universal enveloping algebra the Clifford-Weyl algebra q(W ) is filtered, and it inherits from the Jordan-Heisenberg algebra W a canonical Z 2 -grading and a canonical structure of Lie superalgebra by the supercommutator -see §2. 1.2 for the definitions. The next statement is a sharpened version of Lemma 2.2.
for the general case of a Lie superalgebra g with bracket [g, g] ⊂ g . For the specific case at hand, where the fundamental bracket [W,W ] ⊂ K has zero component in W , the degree n +n ′ −1 is lowered to n +n ′ −2 by the very argument proving that lemma.
It now follows that each of the subspaces q n (W ) for n ≤ 2 is a Lie superalgebra. Since [q 2 (W ), q 1 (W )] ⊂ q 1 (W ), the quotient space q 2 (W )/q 1 (W ) is also a Lie superalgebra. By the Poincaré-Birkhoff-Witt theorem, there exists a vector-space isomorphism sending q 2 (W )/q 1 (W ) to s , the space of super-symmetrized degree-two elements in q 2 (W ). Hence q 2 (W ) has a direct-sum decomposition q 2 (W ) = q 1 (W ) ⊕ s . If {e i } is a homogeneous basis of W , every a ∈ s is uniquely expressed as (2.7) By adding and subtracting terms, one sees that the product ww ′ for w, w ′ ∈ W has scalar part (ww By the linearity of the supercommutator, it suffices to consider a single term of the sum (2.7). Thus we put a = ww ′ + (−1) |w||w ′ | w ′ w, and have Now we compute the scalar part of the right-hand side. Using the Jacobi identity for the Lie superalgebra q(W ) we obtain The last expression vanishes because [w, w ′ ] ⊂ K lies in the center of q(W ).

osp(W
, s is a Lie superalgebra. Now from Lemma 2.13 and the Jacobi identity for q(W ), one sees that s ⊂ q 2 (W ) acts on each of the quotient spaces q n (W )/q n−1 (W ) for n ≥ 1 by a → [a, ] . In particular, s acts on q 1 (W )/q 0 (W ) = W by a → [a, w], which defines a homomorphism of Lie superalgebras The mapping τ is actually into and since [a, [w, w ′ ]] = 0 , this vanishes by the Jacobi identity. Using that a ∈ s has a uniquely determined expansion a = ∑ a i j e i e j with supersymmetric coefficients These agree with those of osp(W ) as recorded in Corollary 2.1. Hence our injective linear map τ : s → osp(W ) is in fact a bijection.
Let us conclude this subsection by writing down an explicit formula for τ −1 . To do so, let {e i } and { e j } be homogeneous bases of W with Q(e i , e j ) = δ i j as before. For X ∈ osp(W ) notice that the coefficients a i j := Q(e i , X e j )(−1) |e j | are supersymmetric: where the last equality sign uses (2.8) To verify this formula, one calculates the double supercommutator and shows that the result is equal to [e i , X e j ] = Q(e i , X e j ), which is precisely what is required from the definition of τ by [τ −1 (X ), e j ] = X e j .

Spinor-oscillator representation. -As before, starting from a
Consider now the following tensor product of exterior and symmetric algebras: . Following R. Howe we call it the spinor-oscillator module of q(W ). Notice that a(V ) can be identified with the graded-commutative subalgebra in q(W ) which is generated by V ⊕ K . As such, a(V ) comes with a canonical Z 2 -grading and its space of endomorphisms carries a structure of Lie superalgebra, gl(a(V )) ≡ End(a(V )).
The algebra a(V ) now is to become a representation space for q(W ). Four operations are needed for this: the operator ε(ϕ 1 ) : The operators ε and ι obey the canonical anti-commutation relations (CAR), which is to say that ε(ϕ) and ε(ϕ ′ ) anti-commute, ι(v) and ι(v ′ ) do as well, and one has . The operators µ and δ obey the canonical commutation relations (CCR), i.e., µ(ϕ) and µ(ϕ ′ ) commute, so do δ (v) and δ (v ′ ), and one has . Given all these operations, one defines a linear mapping q : W → End(a(V )) by , and δ (v 0 ), µ(ϕ 0 ) on the second factor. Of course the two sets ε, ι and µ, δ commute with each other. In terms of q , the relations CAR and CCR are succinctly summarized as where [ , ] denotes the usual supercommutator of the Lie superalgebra gl(a(V )). By the relation (2.9) the linear map q extends to a representation of the Jordan-Heisenberg Lie superalgebra W = W ⊕ K , with the constants of W acting as multiples of Id a(V ) . Moreover, being a representation of W , the map q yields a representation of the universal enveloping algebra U( W ) ≡ q(W ). This representation is referred to as the spinor-oscillator representation of q(W ). In the sequel we will be interested in the osp(W )-representation induced from it by the isomorphism τ −1 .
There is a natural Z-grading a(V ) = m≥0 a m (V ) , Thus Λ ∈ osp is represented on the spinor-oscillator module a(V ) by the degree. §2.3, and we require U 0 and U 1 to be isomorphic with dimension dimU 0 = dimU 1 = n . Recall that are Howe dual pairs in osp(W ) which we denote by (g, k). There is a decomposition in both cases. The notation highlights the fact that the operators in g (m) ֒→ osp(V ⊕V * ) change the degree of elements in a(V ) by m . Note that the Cartan subalgebra h of diagonal operators in g is contained in g (0) but h = g (0) .
Since the Lie algebra k is defined on C N , the k-action on a(V ) preserves the degree. This action exponentiates to an action of the complex Lie group K on a(V ).
Proof. -This is a restatement of Theorems 8 and 9 of [7].
Proposition 2.2 has immediate consequences for the weights of the g-representation on a(V ) K . Using the notation of §2.2.1, let {H s j } be a standard basis of h and {ϑ s j } the corresponding dual basis. We now write ϑ 0 j =: φ j and ϑ 1 j =: iψ j ( j = 1, . . ., n).
Proof. -Recall from §2.3 the embedding of osp(U ⊕U * ) and osp( U ⊕ U * ) in osp(W ), and from Lemma 2.15 the isomorphism τ −1 : osp(W ) → s where s is the Lie superalgebra of supersymmetrized degree-two elements in q(W ). Specializing formula (2.8) to the case of a Cartan algebra generator H s j ∈ h ⊂ g one gets . From Lemma 2.7 the roots α corresponding to root spaces g α ⊂ g (2) are of the form where the indices j, j ′ are subject to restrictions that depend on g being osp(U ⊕U * ) or osp( U ⊕ U * ). From a(V ) K = g (2) .1 one then has m j ≤ N 2 ≤ n j for every weight Proof. -Since the K-action on a(V ) preserves the degree, the subalgebra a(V ) K is still Z-graded by the same degree. Summing the above expressions for (q • τ −1 )(H s j ) over s, j and using CAR and CCR to combine terms, we obtain which is in fact the operator for the degree of the Z-graded module a(V ) K .

Positive and simple roots.
-We here record the systems of simple positive roots that we will use later (in §4. 3.5). In the case of osp(U ⊕U * ) this will be The corresponding system of positive roots for osp In the case of osp( U ⊕ U * ) we choose the system of simple positive roots The corresponding positive root system then is In both cases the roots form a system of positive roots for gl(U ) ≃ g (0) ⊂ osp.

Unitary structure.
-We now equip the spinor-oscillator module a(V ) for V = V 0 ⊕V 1 with a unitary structure. The idea is to think of the algebra a(V ) as a subset of . For such functions a Hermitian scalar product is defined via Berezin's notion of superintegration as follows.
For present purposes, it is imperative that V be defined over R, i.e., V = V R ⊗ C, and that V be re-interpreted as a real vector space V ′ := V R ⊕ J V R with complex structure J ≃ i . Needless to say, this is done in a manner consistent with the Z 2 -grading, so that , the complex of real-analytic differential forms on V ′ 0 . Fixing some orientation of V ′ 0 , the Berezin (super-)integral for the Z 2 -graded vector space whenever the integral over V ′ 0 exists. Thus the Berezin integral is a two-step process: first the section Φ is converted into a differential form, then the form Ω[Φ] is integrated in the usual sense to produce a complex number. Of course, by the rules of integration of differential forms only the top-degree component of Ω[Φ] contributes to the integral.
The subspace V R ⊂ V ′ has played no role so far, but now we use it to decompose the complexification V ′ ⊗ C into holomorphic and anti-holomorphic parts: V ′ ⊗ C = V ⊕V and determine an operation of complex conjugation V * → V * . We also fix on V = V 0 ⊕V 1 a Hermitian scalar product (a.k.a. unitary structure) , so that V 0 ⊥ V 1 . This scalar product determines a parity-preserving complex anti-linear bijection c : To get a close-up view of γ , let {e 0, j } and {e 1, j } be orthonormal bases of V 0 resp. V 1 , and let z j = ce 0, j and ζ j = ce 1, j be the corresponding coordinate functions, with complex conjugates z j and ζ j . Viewing ζ j , ζ j as generators of ∧(V ′ We fix the normalization of γ by the condition A unitary structure on the spinor-oscillator module a(V ) is now defined as follows. Let complex conjugation V * → V * be extended to an algebra anti-homomorphism , and define their Hermitian Let us mention in passing that (2.11) coincides with the unitary structure of a(V ) used in the Hamiltonian formulation of quantum field theories and in the Fock space description of many-particle systems composed of fermions and bosons. The elements for m j ∈ {0, 1} and n j ∈ {0, 1, . . .} form an orthonormal set in a(V ), which in physics is called the occupation number basis of a(V ).
i.e., they are mutual adjoints with respect to the unitary structure of a(V ).
By the first defining property of γ in (2.10) and the fact that and passing to the Hermitian scalar product by the Berezin integral we obtain By the definition of the †-operation this means that δ (v) † = µ(cv).
In the case of v ∈ V 1 the argument is similar but for a few sign changes. Our starting relation changes to • Ω and the first term on the right-hand side Berezin-integrates to zero because ι(ṽ) lowers the degree in ∧T * V ′ 0 . Therefore, which is the statement ι(v) † = ε(cv).
By the Hermitian scalar product (2.11) and the corresponding L 2 -norm, the spinoroscillator module a(V ) is completed to a Hilbert space, A V . A nice feature here is that, as an immediate consequence of the factors 1/ n j ! in the orthonormal basis (2.12), the L 2 -condition Φ, Φ a(V ) < ∞ implies absolute convergence of the power series for In the important case of isomorphic components V 0 ≃ V 1 , we may regard A V as the Hilbert space of square-integrable holomorphic differential forms on V 0 .
Note that although δ (v) and µ(ϕ) do not exist as operators on the Hilbert space A V , they do extend to linear operators on O(V 0 , ∧V * 1 ) for all v ∈ V 0 and ϕ ∈ V * 0 .

Real structures. -
In this subsection we define a real structure for the complex vector space W = V ⊕ V * and describe, in particular, the resulting real forms of the (Z 2 -even components of the) Howe dual partners introduced above.
Recalling the map c : Note that W R can be viewed as the fixed point set W R = Fix(C) of the involution By the orthogonality assumption, The symmetric bilinear form S on W 1 = V 1 ⊕V * 1 restricts to a Euclidean structure S : The connected classical real Lie groups associated to the bilinear forms S and ω are They have Lie algebras denoted by o(W 1,R ) and sp(W 0,R ). By construction we have We know from Lemma 2.15 that τ −1 (X ) is a super-symmetrized element of degree two in the Clifford-Weyl algebra q(W ). To see the explicit form of such an element, recall the definition τ(a)w = [a, w] . Since Q = S + A , and A restricts to iω , the fundamental bracket The proposed statement X † = −X now follows under the assumption that the spinoroscillator representation maps every w ∈ W R to a self-adjoint operator in End(a(V )). But every element w ∈ W R is of the form v 1 + cv 1 + v 0 + cv 0 and this maps to the operator ι(v 1 ) + ε(cv 1 ) + δ (v 0 ) + µ(cv 0 ), which is self-adjoint by Lemma 2.16.
Given the real structure W R of W , we now ask how End(W R ) intersects with the Howe pairs (osp(U ⊕U * ), o N ) and (osp( U ⊕ U * ), sp N ) embedded in osp(W ). By the observation that Q restricted to W R is not real-valued, osp(U ⊕U * ) ∩ End(W R ) fails to be a real form of the complex Lie superalgebra osp(U ⊕ U * ), and the same goes for osp( U ⊕ U * ). Nevertheless, it is still true that the even components of these intersections are real forms of the complex Lie algebras osp(U ⊕U * ) 0 and osp( U ⊕ U * ) 0 .
The real forms of interest are best understood by expressing them via blocks with respect to the decomposition W = V ⊕ V * . Since W R = Fix(C), the complex linear endomorphisms of W stabilizing W R are given by The bar here is a short-hand notation for the complex anti-linear maps When expressed with respect to compatible bases of V and V * , these maps are just the standard operation of taking the complex conjugate of the matrices of A and B.
Here the notation still means the same, i.e., a ∈ End(U ), b ∈ Hom(U * ,U ), and so on. Let a real structure (U ⊕ U * ) R of U ⊕ U * be defined in the same way as the real Proof. -The intersection is computed by transferring the conditions D = A and C = B to the level of osp(U ⊕U * ) 0 . Of course D = A just reduces to the corresponding condition d = a . Because the isometry ψ : C N → (C N ) * in the present case is symmetric one has ψ −1 = ψ t = +ψ, so the condition C = B transfers to c = b . For the same reason, the parity of the maps b, c is identical to that of B, C, i.e., b| U * 0 →U 0 is symmetric, b| U * 1 →U 1 is skew, and similar for c . Hence, computing the intersection osp(U ⊕U * ) 0 ∩End(W R ) amounts to the same as computing osp(V ⊕V * ) 0 ∩End(W R ), and the statement follows from our previous discussion of the latter case.
In the case of the Howe pair (osp( U ⊕ U * ), sp N ) the isometry ψ : At the same time, the parity of b, c is reversed as compared to B, C: now the map b| U * 0 →U 0 is skew and b| U * 1 →U 1 is symmetric (and similar for c). Therefore, which is a compact real form usp(U 1 ⊕U * 1 ) of sp(U 1 ⊕U * 1 ); and which is a non-compact real form of o(U 0 ⊕U * 0 ) known as so * (U 0 ⊕U * 0 ). Let us summarize this result.

Semigroup representation
As before, we identify the complex Lie superalgebra g := osp(W ) with the space of super-symmetrized degree-two elements in q 2 (W ), so that The adjoint representation of g on q(W ) restricts to the Lie algebra representation of g 0 = o(W 1 ) ⊕ sp(W 0 ) on W = W 1 ⊕W 0 which is just the direct sum of the fundamental representations of o(W 1 ) and sp(W 0 ). These are integrated by the fundamental representations of the complex Lie groups SO(W 1 ) and Sp(W 0 ) respectively.
Since the Clifford-Weyl algebra q(W ) is an associative algebra, one can ask if, given x ∈ g 0 ⊂ q(W ), the exponential series e x makes sense. The existence of a one-parameter group e tx for x ∈ g 0 would of course imply that Since the Clifford algebra c(W 1 ) is finite-dimensional, the series e x for x ∈ o(W 1 ) ֒→ g 0 does make immediate sense. In this way one is able to exponentiate the Lie algebra o(W 1 ) in c(W 1 ). The associated complex Lie group, which is then embedded in c(W 1 ), is Spin(W 1 ). This is a 2:1 cover of the complex orthogonal group SO(W 1 ). Its conjugation representation on W 1 as in (3.1) realizes the covering map as a homomorphism Spin(W 1 ) → SO(W 1 ).
Viewing the other summand sp(W 0 ) of g 0 as being in the infinite-dimensional Weyl algebra w(W 0 ), it is definitely not possibly to exponentiate it in such a naive way. This is in particular due to the fact that for most x ∈ sp(W 0 ) the formal series e x is not contained in any space w n (W 0 ) of the filtration of w(W 0 ).
As a first step toward remedying this situation, we consider q(W ) as a space of densely defined operators on the completion A V (cf. §2. 6.3) of the spinor-oscillator module a(V ). Since all difficulties are on the W 0 side, for the remainder of this chapter we simplify the notation by letting W := W 0 and discussing only the oscillator representation of w(W ). Recall that this representation on a(V ) is defined by multiplication µ(ϕ) for ϕ ∈ V * and the directional derivative For x ∈ w(W ) there is at least no formal obstruction to the exponential series of x existing in End(A V ). However, direct inspection shows that convergence cannot be expected unless some restrictions are imposed on x . This is done by introducing a notion of unitarity and an associated semigroup of contraction operators.
Since we have restricted our attention to the symplectic side, the vector spaces W and W R are now equipped with the standard complex symplectic structure A and real symplectic form ω = iA respectively. From here on in this chapter we abbreviate the notation by writing Sp := Sp(W ) and sp := sp(W ). Let an anti-unitary involution σ : Sp → Sp be defined by g → CgC −1 . Its fixed point group Fix(σ ) is the real form Sp(W R ) of main interest. We here denote it by Sp R and let sp R stand for its Lie algebra.
Given A and C, consider the mixed-signature Hermitian structure which we denote by A(Cw , w ′ ) =: w , w ′ s , with subscript s to distinguish it from the canonical Hermitian structure of W given by v+ϕ, v ′ +ϕ ′ : The relation between the two is Note also the relation .

Now observe that the real form Sp
, will play an important role in our considerations. It is invariant under the Sp R -action by right multiplication, π(hg −1 ) = π(h), and is equivariant with respect to the action defined by left multiplication on its domain of definition and conjugation on its image space, π(gh) = gπ(h)g −1 . Direct calculation shows that in fact the π-fibers are exactly the orbits of the Sp R -action by right multiplication. Observe that if h = exp(iX ) for X ∈ sp R , then σ (h) = h −1 and π(h) = h 2 . In particular, if t is a Cartan subalgebra of sp which is defined over R , then π| exp(it R ) is just the squaring map t → t 2 .

Actions of Sp
In other words, we may choose {e j } j=1,. ..,d to be an orthonormal basis of V and equip V * with the dual basis so that the elements t ∈ T are in diagonal form: Observe that, conversely, the elements of Sp that stabilize the decomposition W = P 1 ⊕ . . . ⊕ P d and act diagonally in the above sense, are exactly the elements of T . Moreover, exp(it R ) is the subgroup of elements t ∈ T with χ j (t) ∈ R + for all j . Note that the complex symplectic planes P j are A-orthogonal and defined over R.
We now wish to analyze H(W s ) via the map π : h → hσ (h −1 ). However, for a technical reason related to the proof of Proposition 3.1 below, we must begin with the opposite map, π ′ : h → σ (h −1 )h . Thus let M := π ′ (H(W s )) and write π ′ : H(W s ) → M . The toral semigroup T + := exp(it R ) ∩ H(W s ) consists of those elements t ∈ exp(it R ) that act as contractions on V * , i.e., 0 < χ j (t) −1 < 1 for all j . The restriction π ′ | T + = π| T + is, as indicated above, the squaring map t → t 2 ; in particular we have T + ⊂ M and In the sequel, we will often encounter the action of Sp R on T + and M by conjugation. We therefore denote this action by a special name, Int(g)t := gtg −1 .
Since m does indeed have at least one eigenvector, we have constructed a complex 2-plane Q 1 as the span of the linearly independent vectors w and Cw. The plane Q 1 is defined over R and, because 0 = w, w s = A (Cw, w), it is A-nondegenerate. Its A-orthogonal complement Q ⊥ 1 is therefore also nondegenerate and defined over R. The transformation m ∈ Sp stabilizes the decomposition W = Q 1 ⊕ Q ⊥ 1 . Hence, proceeding by induction we obtain an A-orthogonal decomposition W = Q 1 ⊕. . .⊕Q d .
Since the Q j are m-invariant symplectic planes defined over R, there exists g ∈ Sp R so that t := gmg −1 stabilizes the above T -invariant decomposition W = P 1 ⊕ . . . ⊕ P d . Exchanging w with Cw if necessary, we may assume that t acts diagonally on In other words, t ∈ T + .
If we let Sp R T + Sp R := {g 1 tg −1 2 | g 1 , g 2 ∈ Sp R , t ∈ T + } , then we now have the following analog of the KAK-decomposition.
Because Sp R and T + are connected, so is H(W s ) = Sp R T + Sp R .
The proof for the map (g, m) → mg −1 is similar, with π replacing π ′ .

Cone realization of M.
-Let us look more carefully at M as a geometric object. First, as we have seen, the elements m of M satisfy the condition m = σ (m −1 ). We regard ψ : H(W s ) → H(W s ) , h → σ (h −1 ), as an anti-holomorphic involution and reformulate this condition as M ⊂ Fix(ψ). In the present section we are going to show that M is a closed, connected, real-analytic submanifold of H(W s ) which locally agrees with Fix(ψ). This implies in particular that M is totally real in H(W s ) with dim R M = dim C H(W s ). We will also show that the exponential map identifies M with a precisely defined open cone in isp R . We begin with the following statement.  Proof.
-We are going to use the fact that the squaring map S : Sp → Sp , g → g 2 , is a local diffeomorphism of Sp at any point t ∈ T + . To show this, we compute the differential of S at t and obtain where dL g denotes the differential of the left translation L g : Sp → Sp , g 1 → gg 1 .
The middle map Id sp + Ad(t −1 ) : sp → sp is regular because all of the eigenvalues of t ∈ T + are positive real numbers. Since dL t −1 : T t Sp → sp and dL t 2 : sp → T t 2 Sp are isomorphisms, it follows that D t S : T t Sp → T t 2 Sp is an isomorphism.
Turning to the proof of the lemma, given ξ ∈ t + we now choose n ∈ N so that 2 −n ξ is in a neighborhood of 0 ∈ sp where exp is a diffeomorphism. It follows that for U a sufficiently small neighborhood of ξ , the exponential map expressed as is a diffeomorphism of U onto its image. Now recall M = Int(Sp R )T + and consider the cone It follows from the equivariance of exp, i.e., exp(Ad(g)ξ ) = Int(g) exp(ξ ), that exp : C → Int(Sp R )T + = M is surjective. Furthermore, exp | t + : t + → T + is injective and for every ξ ∈ t + the isotropy groups of the Sp R -actions at ξ and exp(ξ ) are the same. Therefore exp : C → M is also injective.
In fact, much stronger regularity holds. For the statement of this result we recall the anti-holomorphic involution ψ : H(W s ) → H(W s ) defined by h → σ (h −1 ) and let Fix(ψ) 0 denote the connected component of Fix(ψ) that contains M. Since exp is a local diffeomorphism on U , we may also assume that U ⊂ C = Ad(Sp R )t + , and it then follows that C = Ad(Sp R )U . In particular, this shows that C is open in isp R . In summary, By the equivariance of exp , we also know that it is everywhere of maximal rank on C . Now ψ is an anti-holomorphic involution. Therefore, Fix(ψ) 0 is a totally real, halfdimensional closed submanifold of H(W s ), and since C is open in isp R , we also know that dim C C = dim R Fix(ψ) 0   Proof. -The first statement is proved by explicitly constructing a smooth inverse to each of the two maps. For this let g ′ m ′ = mg −1 = h ∈ H(W s ) and note that m = π(h) and m ′ = π ′ (h). Since the square root is a smooth map on M , a smooth inverse in the two cases is defined by The second statement follows from C ≃ M by exp and the fact that Sp R is a product of a cell and a maximal compact subgroup K . We choose K to be the unitary group U = U(V, , ) acting diagonally on W = V ⊕V * and recall that π 1 (U) ≃ Z .

3.2.1.
Lifting the semigroup. -We begin by recalling a few basic facts about covering spaces. If G is a connected Lie group, its universal covering space U carries a canonical group structure: an element u ∈ U in the fiber over g ∈ G is a homotopy class u ≡ [α g ] of paths α g : [0, 1] → G connecting g with the neutral element e ∈ G ; and an associative product U × U → U , (u 1 , u 2 ) → u 1 u 2 , is defined by taking u 1 u 2 to be the unique homotopy class which is given by pointwise multiplication of any two paths representing the homotopy classes u 1 , u 2 . The fundamental group π 1 (G) ≡ π 1 (G, e) acts on U by monodromy, i.e., if [α g ] = u ∈ U and [c] = γ ∈ π 1 (G), then one sets γ(u) := [α g * c] ∈ U where α g * c is the path from g to e which is obtained by composing the path α g with the loop c based at e . This π 1 (G)-action satisfies the compatibility condition γ 1 (u 1 )γ 2 (u 2 ) = (γ 1 γ 2 )(u 1 u 2 ) and in that sense is central. The situation for our semigroup H(W s ) is analogous except for the minor complication that the distinguished point e = Id does not lie in H(W s ) but lies in the closure of H(W s ). Hence, by the same principles, the universal cover U of H(W s ) comes with a product operation and there is a central action of π 1 (H(W s )) on U . Moreover, the product U ×U → U still is associative. To see this, first notice that the subsemigroup T + ⊂ H(W s ) is simply connected and as such is canonically embedded in U . Then for u 1 , u 2 , u 3 ∈ U observe that u 1 (u 2 u 3 ) = γ((u 1 u 2 )u 3 ) where γ ∈ π 1 (H(W s )) could theoretically depend on the u j . However, any such dependence has to be continuous and the fundamental group is discrete, so in fact γ is independent of the u j and, since γ is the identity when the u j are in T + (lifted to U ), the associativity follows.
This can be viewed as a statement of Mp-equivariance of the covering map τ H . Now, we have another real-analytic diffeomorphism Sp R ×M → H(W s ) by (g, m) → mg −1 , which transfers left translation in Sp R to right multiplication on H(W s ), and by using it we can repeat the above construction. The result is another identification H(W s ) ≃ Mp × M and another Mp-action on H(W s ). Altogether we then have two actions of Mp on H(W s ). The essence of the next statement is that they commute.

Proposition 3.5. -There is a real-analytic action
Proof. -By construction, the stated equivariance property of τ H holds for each of the two actions of Mp separately. It then follows that it holds for all (g 1 , g 2 ) ∈ Mp × Mp if the two actions commute. But by τ H ((g 1 , e) · h) = τ(g 1 )τ H (h) and τ H ((e, g 2 ) · h) = τ H (h)τ(g 2 ) −1 the commutator g := (g 1 , e)(e, g 2 )(g 1 , e) −1 (e, g 2 ) −1 acts trivially on H(W s ) by τ H , i.e., τ H (g · h) = τ H (h). Therefore g can be regarded as being in the covering group τ −1 (Id) = Z 2 of the covering τ : Mp → Sp R . Since we can connect both g 1 and g 2 to the identity e ∈ Mp by a continuous curve, it follows from the discreteness of Z 2 that g ∈ Mp × Mp acts trivially on H(W s ).

Lifting involutions.
-Let us now turn to the issue of lifting the various involutions at hand. As a first remark, we observe that any Lie group automorphism ϕ : Sp R → Sp R uniquely lifts to a Lie group automorphism ϕ of the universal covering group Sp R , and the latter induces an automorphism of the fundamental group π 1 (Sp R ) ≃ Z viewed as a subgroup of the center of Sp R . Now Aut(π 1 (Sp R )) ≃ Aut(Z) ≃ Z 2 and both elements of this automorphism group stabilize the subgroup Γ ≃ 2Z in π 1 (Sp R ). Therefore ϕ induces an automorphism of Mp = Sp R /Γ .
Since the operation h → h −1 canonically lifts from Sp R to Mp and h → (h −1 ) † is a Lie group automorphism of Sp R , it follows that Hermitian conjugation h → h † has a natural lift to Mp . The same goes for the Lie group automorphism h → shs of Sp R .

lifts to an anti-holomorphic map ψ : H(W s ) → H(W s ) which is the identity on M and
Mp × Mpequivariant in that ψ(g 1 x g −1 2 ) = g 2 ψ(x)g −1 1 for all g 1 , g 2 ∈ Mp and x ∈ H(W s ). Proof. -Recall that the simply connected space M ⊂ H(W s ) has a canonical lifting (still denoted by M) to H(W s ). Since all of our involutions stabilize M as a submanifold of H(W s ), they are canonically defined on the lifted manifold M . In particular, the involution ψ on M is the identity map, and therefore so is the lifted involution ψ.
Note furthermore that the involution defined by h → shs is holomorphic on H(W s ) and that the other two are anti-holomorphic. Now H(W s ) is connected and the lifted version of M is a totally real submanifold of H(W s ) with dim R M = dim C H(W s ). In such a situation the identity principle of complex analysis implies that there exists at most one extension (holomorphic or anti-holomorphic) of an involution from M to H(W s ). Therefore, it is enough to prove the existence of extensions.
Since h ∈ H(W s ) is uniquely representable as h = gm with g ∈ Mp and m ∈ M, the involution h → h † is extended by gm → (gm) † = m † g † . Similarly, h → shs extends by gm → (sgs)(sms), and h → sh † s does so by the composition of the other two.
The equivariance property of ψ follows from the fact that g → sg † s on Mp coincides with the operation of taking the inverse, g → g −1 .

Oscillator semigroup representation. -
Here we construct the fundamental representation of the semigroup H(W s ) on the Hilbert space A V , which in the present context we call Fock space. Our approach is parallel to that of Howe [8]: the Fock space we use is related to the L 2 -space of Howe's work by the Bargmann transform [6]. (Using the language of physics, one would say that Howe works with the position wave function while our treatment relies on the phase space wave function.) In particular, following Howe we take advantage of a realization of H(W s ) as the complement of a certain determinantal variety in the Siegel upper half plane.

Cayley transformation.
-Let us begin with some background information on the Cayley transformation, which is defined to be the meromorphic mapping If g ∈ Sp , then from A(gw, gw ′ ) = A(w, w ′ ) we have for all w, w ′ ∈ W . By assuming that (Id W − g) is invertible and then replacing w and The inverse of the Cayley transformation is given by Reversing the above argument, one shows that if (X + Id W ) is invertible and X ∈ sp , then C −1 (X ) ∈ Sp . Moreover, by the relation X + Id W = 2(Id W − g) −1 for C(g) = X , if Id W − g is regular, then so is X + Id W , and vice versa. Thus if we introduce the sets the following is immediate.

Proposition 3.7. -The Cayley transformation defines a bi-holomorphic map
Now we consider the restriction of C to the semigroup H(W s ). Letting † be the Hermitian conjugation of the previous section, denote by Re(X ) = 1 2 (X + X † ) the real part of an operator X ∈ End(W ) and define the associated Siegel upper half space S to be the subset of elements X ∈ End(W ) which are symmetric with respect to the canonical symmetric bilinear form S on W = V ⊕ V * with Re(X ) > 0 . Notice the relations S(w, w ′ ) = A(w, sw ′ ) and A(sw, sw ′ ) = −A(w, w ′ ), from which it is seen that X is symmetric if and only if sX ∈ sp . Define D S := {X ∈ S | Det(sX + Id W ) = 0}, let ζ + s := S \ D S , and define a slightly modified Cayley transformation by Translating Proposition 3.7, it follows that a defines a bi-holomorphic map from Sp \ D Sp onto the set of S-symmetric operators with D S removed. This result is an immediate consequence of the following identity.
In particular, one has the following equivalence: . Using a(g) = X one directly computes that 1 2 (Id W − g) = (sX + Id W ) −1 and 1 2 (Id W + g) = (sX + Id W ) −1 sX . The desired identity follows by inserting these relations in the previous equation. Since ζ + s is obviously stable under Hermitian conjugation, this is another proof of the stability of H(W s ) under the involution ψ ; cf. Corollary 3.2.

Construction of the semigroup representation.
-Let us now turn to the main goal of this section. Recall that we have a Lie algebra representation of sp on a(V ) = S(V * ) which is defined by its canonical embedding in w 2 (W ). We shall now construct the corresponding representation of the semigroup H(W s ) on the Fock space A V .
It will be seen later that the character of this representation on the lifted toral semigroup T + is Det − 1 2 (s − sh). This extends to M = Int(Mp)T + by the invariance of the character with respect to the conjugation action of Mp . Since H(W s ) is connected and M is totally real of maximal dimension in H(W s ), the identity principle then implies that if a semigroup representation of H(W s ) can be constructed with a holomorphic character, this character must be given by the square root function h → Det − 1 2 (s − sh). There is no difficulty discussing the square root on the simply connected submanifold M. However, in order to make sense of the square root function on the full semigroup, we must lift all considerations to H(W s ). For convenience of notation, given h ∈ H(W s ) we let a h := a(h), and for x ∈ H(W s ) we simply write a x ≡ a(τ H (x)) where namely H(W s ) → H(W s ), it follows that we may define φ on H(W s ) as desired. For the construction of the oscillator representation it is useful to observe that φ can be extended to a slightly larger space. This extension is constructed as follows.
Regard the complex symplectic group Sp as the total space of an Sp R -principal bundle π : Sp → π(Sp), g → gσ (g −1 ). Recall that the restricted map π : M → M is a diffeomorphism, and that M contains the neutral element Id ∈ Sp in its boundary. We choose a small ball B centered at Id in the base π(Sp), and using the fact that M can be identified with a cone in isp R we observe that A : The latter is not the case, as φ is not purely imaginary on the nonempty set Fix( ψ).
The semigroup representation R : H(W s ) → End(A V ) will be given by a certain averaging process involving the standard representation of the Heisenberg group. The latter representation is defined as follows. For elements w = v + cv of the real vector space W R , the operator δ (v) + µ(cv) is self-adjoint and its exponential T v+cv := e iδ (v)+iµ (cv) converges and is unitary (see, e.g., [9]). These operators satisfy the relation where ω := iA| W R is the induced real symplectic structure. If T → T † denotes the adjoint operation in End(A V ), it follows from δ (v) † = µ(cv) that Thus if H := W R × U 1 is equipped with the Heisenberg group multiplication law, It is well known that up to equivariant isomorphisms there is only one such representation. The oscillator representation x → R(x) of H(W s ) is now defined by Here dvol is the Euclidean volume element on W R which we normalize so that It should be stressed that we often parameterize W R ≃ V by the map v → v + cv = w . Notice that by the positivity of Re(a x ) the Gaussian function w → γ x (w) decreases rapidly, so that all integrals involved in the discussion above and below are easily seen to converge. In particular, since the unitary operator T w (for w ∈ W R ) has L 2 -norm T w = 1, it follows for any x ∈ H(W s ) that where the bound C(x) by direct computation of the integral is a finite number: Thus R(x) is a bounded linear operator on A V . In Proposition 3.17 we will establish the uniform bound R(x) ≤ C(x) < 1 for all x ∈ M. It is also clear that the operator R(x) depends continuously on x ∈ H(W s ).

The main point now is to prove the semigroup multiplication rule R(xy) = R(x)R(y). For this we apply the Heisenberg multiplication formula (3.2) to the inner integral of
to see that R(xy) = R(x)R(y) is equivalent to the multiplication rule γ xy = γ x ♯γ y where the right-hand side means the twisted convolution For the proof of the formula γ x ♯γ y = γ xy , we will need to know that φ transforms as . (3.5) and similarly for the other terms. We then holomorphically extend the right-hand side to w ′ in W , and by making a shift of integration variables we bring the convolution integral into the form where a x • a y = −s + (a y + s)(a x + a y ) −1 (a x + s) = a x − (a x − s)(a x + a y ) −1 (a x + s) is the semigroup multiplication on ζ + s . Using A(w, sw ′ ) = w, w ′ for w ∈ W R and the defining relation a x • a y = a xy , we see that the first factor on the right-hand side is e − 1 4 w, a xy w . By the transformation rule (3.5) the integral is evaluated as and multiplying factors it follows that which is the desired semigroup property.
is a representation of the semigroup H(W s ).
We conclude this section by deriving a formula for the adjoint.

Proposition 3.13. -The adjoint of R(x) is computed as R(x)
Proof. -We recall the relations φ = φ • ψ from Proposition 3.10 and (a h ) † = a ψ(h) from Remark 3. 3. Since w, a h w = w, (a h ) † w , it follows that The desired formula, R(x) † = R( ψ(x)), now results from this equation and the identities T † w = T −w and γ x (−w) = γ x (w). With this in hand, the second statement R(x)R(x) † = R(x ψ(x)) is a consequence of the semigroup property.

Basic conjugation formula.
-Here we compute the effect of conjugating (in the semigroup sense) operators of the form q(w), w ∈ W , with operators R(x) coming from the semigroup. This is an immediate consequence of an analogous result for the operators T w . For this we first allow T w to be defined for w = v + ϕ ∈ W by T w := e iq(w) = e iδ (v)+iµ(ϕ) .
These operators are no longer defined on Fock space, but are defined on O(V ). They satisfy Note that for x ∈ H(W s ) and w ∈ W the operators R(x)T w and T τ H (x)w R(x) are bounded on A V . Thus we interpret the following as a statement about operators on that space.

Proposition 3.14. -For w ∈ W and x ∈ H(W s ) one has the relation
Proof. -For convenience of notation we write Thus for all w 1 , w 2 ∈ W , one simplifies the exponent to obtain

Reading (3.8) backwards one sees that this expression equals T hw R(x).
The basic conjugation rule now follows immediately. Proof. -Apply Proposition 3.14 for w replaced by tw and differentiate both sides of the resulting formula at t = 0.

Spectral decomposition and operator bounds.
-Numerous properties of R are derived from a precise description of the spectral decomposition of R(x) for x ∈ M .
Since every orbit of Sp R acting by conjugation on M has nonempty intersection with T + , it is important to understand this decomposition when x ∈ T + . For this we begin with the case where V is one-dimensional.

Proposition 3.16. -Suppose that V is one-dimensional and that the T + -action on
Thus the Gaussian function γ x (w) in the present case is To apply the operator T v+cv to f m we use the description Decomposing Our goal is to compute where dvol(v) corresponds to dvol(w) by the isomorphism V ≃ W R . Expanding the exponential e iµ(cv) and using µ(cv The only terms which survive are those with k = l . Thus , and φ (x) = 2λ −1/2 (1 − λ −1 ) −1 , since we are to take the positive square root at points we employ the standard multi-index notation f m := f m 1 1 · · · f m d d and λ m := λ m 1 1 · · · λ m d d . In this case the multi-dimensional integrals split up into products of one-dimensional integrals. Thus, the following is an immediate consequence of the above.
One would expect the same result for the spectrum to hold for every conjugate gT + g −1 , and this expectation is indeed borne out. However, in the approach we are going to take here, we first need the existence and basic properties of the oscillator representation of the metaplectic group. The following is a first step in this direction.

3). That function C(x) clearly is invariant under conjugation
Evaluating it for the case of an element x ≡ t ∈ T + with eigenvalues λ i one obtains The inequality C(t) < 1 now follows from the fact that λ i > 1 for all i .

Representation of the metaplectic group. -
Recall that we have realized the metaplectic group Mp in the boundary of the oscillator semigroup H(W s ) and that H(W s ) contains the lifted manifold T + in such a way that the neutral element Id ∈ Mp is in its boundary. Here we show that for x ∈ T + and g ∈ Mp the limit lim x→Id R(gx) is a well-defined unitary operator R ′ (g) on Fock space and R ′ : Mp → U(A V ) is a unitary representation. The basic properties of this oscillator representation are then used to derive important facts about the semigroup representation R . Convergence will eventually be discussed in the so-called bounded strong* topology (see [8], p. 71). For the moment, however, we shall work with the slightly weaker notion of bounded strong topology where one only requires uniform boundedness and pointwise convergence of the operators themselves (with no mention made of their adjoints). Note that since R(gx) < 1 by Corollary 3.7, we need only prove the convergence of R(gx) f on a dense set of functions f ∈ A V . Let us begin with g = Id .

Lemma 3.4.
-If a sequence x n ∈ T + converges to Id ∈ Mp , then the sequence R(x n ) converges in the bounded strong topology to the identity operator on Fock space. Proof. -If f is any T + -eigenfunction, the sequence R(x n ) f converges to f by the explicit description of the spectrum given in Corollary 3. 6. The statement then follows because the subspace generated by these functions is dense.
Using this lemma along with the semigroup property, we now show that the limiting operators exist and are well-defined.

Letting t = t(m, n) → Id it follows from Corollary 3.7 that
Thus the Cauchy property of R(x n ) f is passed on to R(gx n ) f and therefore the sequence R(gx n ) f converges in the Hilbert space A V . Let lim n→∞ R(gx n ) f =: R ′ (g) f .
To show that the limit is well-defined, pick from T + another sequence y n → Id , let lim n→∞ R(gy n ) f =: R ′′ (g) f , and notice that R ′ (g) f − R ′′ (g) f is no bigger than Using the same reasoning as above, the middle term is estimated as In the limit n → ∞ this yields the desired result R ′ (g) = R ′′ (g). Remark 3. 6. -Since R(gx n ) < 1 the sequence R(gx n ) converges to R ′ (g) in the bounded strong topology. Such convergence preserves the product of operators, which is to say that if A n → A and B n → B, then A n B n → AB . Indeed, and convergence follows from A n < 1 and A n → A, B n → B. Note in particular that if R(gx n ) → R ′ (g) and The bounded strong * topology also requires pointwise convergence of the sequence of adjoint operators. Therefore we must also consider sequences of the form R(gx n ) † . For this (see the proof of Theorem 3.1 below) we will use the following fact.
and use the uniform boundedness of A n to show that A n f converges.
Applying this with A n = R(gx n g −1 ), B n = R(x n ) and C n = A n B n = R(gx n )R(g −1 x n ), we have the following statement about convergence along the conjugate gT + g −1 .
Proposition 3. 19. -For g ∈ Mp and {x n } any sequence in T + with x n → Id ∈ Mp , it follows that R(gx n g −1 ) converges pointwise to R ′ (g)R ′ (g −1 ).
Next, if we take three sequences in T + and write then it follows that the sequence R(x n g −1 ) converges to an operator In particular, the operator R ′ (g) is injective for all g ∈ Mp . Finally, we define y n by y 2 n = x n and write R(gx n ) = R(gy n g −1 )R(gy n ). Taking the limit of both sides of this equation entails that and since R ′ (g) is injective, this now allows us to reach the main goal of this section.
is surjective, and thus R ′ (g) ∈ GL(A V ) by exchanging g ↔ g −1 . For the homomorphism property we write R(g 1 x n )R(g 2 y n ) = R(g 1 x n g 2 y n ) = R(g 1 x n g −1 1 )R(g 1 g 2 y n ) and take limits to obtain R ′ (g 1 )R ′ (g 2 ) = R ′ (g 1 g 2 ). Convergence in the bounded strong * topology also requires convergence of the adjoint. This property follows from R(gx n ) † = R( ψ(gx n )) = R(x n g −1 ) and the discussion after (3.10), since R ′ (g) is now known to be an isomorphism. Unitarity of the representation is then immediate from R(gx n ) † → R ′ (g) † and R( Finally, we must show that R ′ : Mp → U(A V ) is continuous. This amounts to showing that if {g k } is a sequence in Mp which converges to g , then R ′ (g k ) f → R ′ (g) f for any f ∈ A V . Hence, we let {x n } be a sequence in T + with x n → Id and choose t = t(m, n) as in the proof of Proposition 3.18 so that and then let t → Id . Using the uniform boundedness of R(g k t) as t → Id , this shows that the convergence R(g k x n ) → R ′ (g k ) is uniform in g k . Since we have g k x n → gx n for every fixed n , the continuity of x → R(x) f then implies that R ′ (g k ) f → R ′ (g) f .
Let us underline two important consequences. Proposition 3.20. -For g 1 , g 2 ∈ Mp and x ∈ H(W s ) it follows that Proof. -If y m and z n are sequences in T + which converge to Id , then, since x → R(x) f is continuous for all f in Fock space, R(g 1 y m x g 2 z n ) converges pointwise to R(g 1 x g 2 ).
On the other hand, we have R(g 1 y m x g 2 z n ) = R(g 1 y m )R(x)R(g 2 z n ) by the semigroup property, and the right-hand side converges pointwise to R ′ (g 1 )R(x)R ′ (g 2 ).
We refer to R ′ : Mp → U(A V ) as the oscillator representation of the metaplectic group. It has the following fundamental conjugation property.
Proof. -Since the inverse operator R ′ (g) −1 is now available, this follows from the conjugation property at the semigroup level (see Proposition 3.15).
Note that analogously we have the classical conjugation formula for the representation of the Heisenberg group on the Fock space A V , i.e., for all g ∈ Mp and w ∈ W R (see Proposition 3.14).

The trace-class property. -The concrete formula for the eigenvalues of R(x)
which is given in Proposition 3.6 shows that if x ∈ T + , then R(x) is of trace class. Using the conjugation property proved above, we now show that this holds for all x ∈ H(W s ).

Proposition 3.22. -For every x ∈ H(W s ) the operator R(x) is of trace class.
Proof. -We must show that the operator R(x)R(x) † has finite trace. To verify this property observe that R(x)R(x) † = R(y) with y := x ψ(x) ∈ M . Since y = gt 2 g −1 for some t ∈ T + and R(gt 2 g −1 ) = R ′ (g)R(t)R ′ (g) −1 , the desired result follows from the explicit formula in Proposition 3.6 for the eigenvalues of t .

Proposition 3.23. -For every x ∈ H(W s ) one has
Proof. -Since the representation of the Heisenberg group H on A V is irreducible, the space of bounded linear operators End(A V ) H that commute with the H-action is just CId A V . Now the dual of the space of trace-class operators is the space of bounded linear operators itself, where a bounded linear operator A is realized as a functional on the trace-class operators by F A (B) := Tr (AB) (see [9]). Invariance of F A by the Hrepresentation is equivalent to the invariance of A as an operator. Thus, an H-invariant bounded linear functional on the space of trace-class operators is just a multiple of Tr .
We construct such a functional, Φ , by letting 1 be the vacuum in A V and averaging the expectation in T w 1 ∈ A V over all w ∈ W R : (3.12) Indeed, the representation of H is unitary and by using T † w = T −1 w along with the basic property T w T w ′ = T w+w ′ e i 2 ω(w,w ′ ) , one shows that Φ is invariant, i.e., Φ(T w LT −1 w ) = Φ(L) for all w ∈ W R . If L := 1 1, · A V is the projector on the vacuum, then it is a simple matter to show that Thus, this formula holds for any bounded linear operator L of trace class. Now Φ(R(x)) is computed as Due to the fact that γ x (w ′ ) is rapidly decreasing, we may take the expectation inside the inner integral: Again using the basic properties of T w , we see that Thus the inner integral over w ′ is a Gaussian integral, and evaluating it we obtain Integrating this quantity with respect to dvol(w) we obtain and the desired result for the trace follows from (3.13) and dimW R = 2 dim C V .

Proposition 3.24.
-For every P 1 , P 2 in the Weyl algebra and every x ∈ H(W s ) the operator q(P 1 )R(x)q(P 2 ) is of trace class on the Fock space A V . Furthermore, the function H(W s ) → C , x → Tr q(P 1 )R(x)q(P 2 ), is holomorphic.
Proof. -(Sketch) It is enough to show that q(P)R(x) is of trace class for the case that P is a monomial operator, i.e., q(P) = µ(cv) k δ (v ′ ) ℓ . We must then compute the trace of the square root of C(y) = q(P)R(y)q(P) † for y = xψ(x) ∈ M . For this, recall that the †-operation interchanges multiplication and differentiation. Direct computation of | C(y) f m | shows that this number is of the order of magnitude of λ −m m k+ℓ where λ is the torus element corresponding to √ y . Thus the conjugated operator has effectively the same trace as R( √ y) itself and in particular Tr C(y) < ∞ .
Turning to holomorphicity, if we insert the equation L(x) = q(P 1 )R(x)q(P 2 ) in the formula (3.12) for the trace and use the definitions to compute the various integrals, then the function F(x) = Φ(L(x)) is of the form where g(x , ·) is a rapidly decreasing function on W R and g(·, w) is holomorphic on the semigroup H(W s ). Since the∂ x -operator can be exchanged with the integral, it is immediate that F is holomorphic and therefore so is Tr q(P 1 )R(x)q(P 2 ).

Compatibility with Lie algebra representation. -We now show that the semigroup representation
Proof. -Recall that the operator R(x) is the result of integrating the Heisenberg translations T w against the Gaussian density γ x (w) dvol(w). Thus (3.14) For w 1 , w 2 ∈ W the linear transformation Y : w → w 1 A(w 2 , w) + w 2 A(w 1 , w) is in sp , and sp is spanned by such transformations. It is therefore sufficient to prove the statement of the lemma for Y of this form. Hence let Y := w 1 A(w 2 , ·) + w 2 A(w 1 , ·) and observe that the corresponding element in the Weyl algebra is τ −1 (Y ) = 1 2 (w 1 w 2 + w 2 w 1 ) . Now, defining T w for w ∈ W by T w = e iq(w) as before, we have Therefore, forw := t 1 w 1 + t 2 w 2 consider the expression Using TwT w = e − 1 2 A(w,w) Tw +w and shifting integration variables w → w −w we obtain Comparing Eqs. (3.15, 3.16) with (3.14) we see that the formula of the lemma is true if But checking this equation is just a simple matter of computing derivatives. Recall that γ x (w) = φ (x) e − 1 4 A(w, sa x w) and φ (x) = Det 1/2 (a x + s). Writing h := τ H (x) and using TrY = 0 one computes the left-hand side to be On substituting Y = w 1 A(w 2 , ·) + w 2 A(w 1 , ·), this expression immediately agrees with the result of taking the two derivatives on the right-hand side.

The extended character
Having prepared the algebraic foundations ( §2) and the necessary representationtheoretic tools ( §3), we now turn to the main part of our work, which is to prove the formula (1.1) for the K-Haar expectation value I(t) of a product of ratios of characteristic polynomials. We will achieve this goal by exploiting the fact that I(t) is the same as the character χ(t) of the irreducible g-representation on A K V . The key property determining χ is a system of differential equations coming from the Casimir invariants of the Lie superalgebra g : acting as differential operators on a certain supermanifold F of Lie supergroup type, these invariants annihilate χ as a section of F . End

Generalities on
The Grassmann envelope End Ω (V ) is given the structure of an associative algebra by and it also has the structure of a Lie algebra by the usual commutator: More generally, if g = g 0 ⊕ g 1 ⊂ End(V ) is a complex Lie (sub)superalgebra, the Grassmann envelopeg(Ω) of g by Ω is still defined in the same way: and it still carries the same structure of associative algebra and Lie algebra. The supertrace STr :g(Ω) → Ω 0 is defined by ω ⊗ X → ω STr X . Note that this satisfies STr [ξ , η] = 0 for all ξ , η ∈g(Ω).
The Grassmann envelopeg(Ω) is Z 2 -graded byg(Ω) =g 0 ⊕g 1 whereg s = Ω s ⊗ g s . There also exists a Z-grading, which is induced by that of Ω; this isg(Ω) = kg is a nilpotent ideal. The subspaceg 0 is a Lie subalgebra and we have g 0 =g (0) ⊂g 0 . The following is an elementary statement found, e.g., in [1]. -Each elementg ∈ G(Ω) has a unique factorization of the formg = gθ 1 = θ 2 g with g ∈ G(Ω) 0 and θ 1 , θ 2 ∈ G(Ω) 1 . Each element g ∈ G(Ω) 0 can be uniquely represented in the form g = g 0 n 1 = n 2 g 0 with g 0 ∈ G and n 1 , n 2 ∈ N(Ω) 0 . Lie supergroup is a pair (g , G) consisting of a complex Lie superalgebra g = g 0 ⊕ g 1 and a complex Lie group G that exponentiates g 0 , i.e., Lie(G) = g 0 . Given a Lie superalgebra g there exist, in general, several choices of G. If g → End(V ) is a faithful finite-dimensional representation of g , i.e., g ⊂ End(V ), then one choice is G :

Lie supergroups. -A complex
A superfunction on a complex Lie supergroup (g , G) is a section in the sheaf F ≡ F (G, g 1 ) of germs of holomorphic functions on G with values in ∧(g * 1 ). This sheaf F is a locally free sheaf of O G -modules where O G is the sheaf of germs of holomorphic functions on G. The gradings on ∧(g * 1 ) give a Z-grading F = k F (k) and a Z 2 - is the decomposition of a superfunction f with respect to the former, then num(

Representation of g by superderivations.
-In the theory of ordinary Lie groups one has a realization of the Lie algebra by left-or right-invariant vector fields acting as derivations of the algebra of differentiable functions on the Lie group. In the same spirit, one wants to construct a representation of the Lie superalgebra g by superderivations on the sheaf of algebras F . The basic tool for this construction is a correspondence between superfunctions f ∈ F and functions on G(Ω) with values in Ω . There exists no canonical correspondence of such kind; one possible choice is as follows.
Then if X ∈ g 0 andX R denotes the corresponding left-invariant vector field on functions ϕ on G, where α i ∈ Ω 0 and β j ∈ Ω 1 .
In that case, the inverse mapping Φ f → f is given by restriction to elementsg = g e ∑ ξ j ⊗F j with g ∈ G .
The advantage of passing to the right-hand side of the correspondence f ↔ Φ f is that functions on G(Ω) can be shifted by elements of G(Ω) by multiplication from the left or right. This possibility now puts us in a position to construct the desired representation of the Lie superalgebra g by superderivations on the sheaf F (G, g 1 ).
On basic grounds [1] one then has the following statement. The character χ associated with the representation (ρ * , ρ;V ) of a Lie supergroup (g, G) is the superfunction , provided that the supertrace exists. Note that the numerical part num(χ) of the character is just the supertrace of the Lie group representation ρ : Proof. -Let χ be the character associated with the representation (ρ * , ρ;V ) of a Lie supergroup (g, G). Then by the definition of the correspondence χ ↔ Φ χ one has where the notation above (g 0 ∈ G, α i ∈ Ω 0 , β j ∈ Ω 1 ) is being employed. This satisfies Φ χ (h) = Φ χ (g −1 hg), since STr (XY ) = STr (Y X ) for X ,Y ∈ End Ω (V ) and the representations ρ * and ρ are compatible. For X ∈ g 1 the infinitesimal (or linearized) version of the same argument gives (X L +X R )Φ χ = 0 and hence (X L +X R )χ = 0 .

Spinor-oscillator character as a radial superfunction.
-Given the representations R 1 and R 0 , we form the semigroup representation We already know the representation R to be compatible with ρ * : osp → gl(A V ). Now we define a superfunction γ on Spin(W 1 ) where {F 1 , . . . , F d 1 } is a basis of osp 1 and {ξ 1 , . . . , ξ d 1 } is its dual basis. We refer to γ as the spinor-oscillator character. By the circumstance that R and ρ * are representations on the infinite-dimensional vector space A V we are obliged to discuss the domain of definition of γ . For this, notice that Spin(W 1 ) acts non-trivially only on the finitedimensional (or spinor) part of A V . Expanding e ∑ ξ j ρ * (F j ) we obtain a finite sum e ∑ ξ j ρ * (F j ) = ∑ J ρ * (P J )ξ j 1 · · · ξ j k with P J ∈ q(W ). (Here we recall that osp 1 can be viewed as part of the Clifford-Weyl algebra q(W ) by the isomorphism τ −1 : osp → s ⊂ q(W ).) By Proposition 3.24 the operator R(g)ρ * (P J ) is of trace class. Thus the character γ is defined on the full domain Spin(W 1 ) × Z 2 H(W s 0 ). Moreover, γ(g) depends analytically on g . Does there exist any good sense in which to think of the spinor-oscillator character γ as a radial superfunction? There is no question that for every X ∈ osp we do have superderivationsX L andX R on F (Spin × H), and the function γ by its definition as a character does satisfy (X L +X R )χ = 0 . However, our semigroup elements g ∈ Spin(W 1 ) × Z 2 H(W s 0 ) do not possess an inverse in the spinor-oscillator representation and we therefore shouldn't expect such a relation as Φ χ (g −1 hg) = Φ χ (h).
A substitute is this. Let Spin R ⊂ Spin(W 1 ) be the compact real subgroup which is obtained by exponentiating the skew-symmetric degree-two elements of the Clifford algebra generated by a Euclidean vector space W 1,R ⊂ W 1 Then on the real submanifold Spin R × M Sp (with M Sp formerly denoted by M) the superfunction γ is G R -radial, since the representations R ′ of Mp and R 0 on M Sp ⊂ H(W s 0 ) are compatible. As a result, we will later be able to apply Berezin's theory of radial parts of Laplace-Casimir operators.

Numerical part of the character.
-Here we show that the restriction of the numerical part of γ to a toral set T 1 × Z 2 T + in Spin(W 1 ) × Z 2 H(W s 0 ) gives the autocorrelation function described in §1. That is, we show that the ratios of characteristic polynomials Det(Id N − e −φ k) and their averages with respect to a compact group K (with k ∈ K ⊂ U N and φ being a parameter), can be expressed as supertraces of operators on the spinor-oscillator module a(V ) and the submodule of K-invariants a(V ) K .
We begin with a summary of the relevant facts. From Proposition 3.15 we recall the basic conjugation rule for the oscillator representation: . On the side of the spinor representation, the corresponding conjugation formula is This defines the 2:1 covering homomorphism Spin(W 1 ) → SO(W 1 ), y → τ S (y), which exponentiates the isomorphism of Lie algebras τ : s ∩ c 2 (W 1 ) → o(W 1 ).
Recall also (from Proposition 3.23) the formula for the oscillator character: An analogous result is known for the case of the spinor representation; see, e.g., the textbook [2]. Defining the spinor character as the supertrace with respect to the canonical Z 2 -grading of the spinor module, one has STr R 1 (y) = ψ(y) , ψ(y) 2 = Det(Id W 1 − τ S (y)) .
Thus the spinor character, just like the oscillator character, is a square root. By taking the supertrace over the total Fock representation space, we obtain the formula For W 0 = W 1 , the case of our interest, Spin(W 0 ) intersects with H(W s 0 ) and the square root ψ(y) is defined in such a way that ψ( . This action serves to realize the group G := (GL(U 1 )×GL(U 0 ))× C × GL(C N ) as a subgroup of GL(V 1 ) × GL(V 0 ) ⊂ SO(W 1 ) × Sp(W 0 ). For the purpose of letting G act in the spinor-oscillator module a(V ), let this representation be lifted to that of a double cover-ingG ֒→ Spin(W 1 ) × Z 2 H(W s 0 ). The following statement gives the value of the spinoroscillator character on (t 1 ,t 0 ; g) ∈ G where g ∈ GL(C N ) and t s = diag(t s,1 , . . . ,t s, n ) are diagonal matrices in GL(U s ) (for s = 0, 1).

Lemma 4.5.
-If dimU 0 = dimU 1 = n and |t 0, j | > 1 for all j = 1, . . . , n , then Proof. -Since t 1 and t 0 are assumed to be of diagonal form, the statement holds true for the case of general n if it does so for the special case n = 1. Hence let n = 1.
In that case t 1 and t 0 are single numbers and t s g acts on which turns into the stated formula on pulling out a factor of Det(−t 1 g)/Det(−t 0 g) from under the square root. (Of course, the double covering of GL(U 1 ) × GL(U 0 ) is to be used in order to define this square root globally.) In the formula of Lemma 4.5 we now set t 1, j = e iψ j and t 0, j = e φ j . We then put g −1 ≡ k ∈ K and integrate against Haar measure dk of unit mass on K. This integral and the summation that defines the supertrace can be interchanged, as STr a(V ) R(t 1 ,t 0 ; k −1 ) is a finite sum of power series and the conditions Re φ j > 0 ensure uniform and absolute convergence. The representation of K on a(V ) is induced by the representation of K on V . Therefore, averaging over K with respect to Haar measure has the effect of projecting from a(V ) to the K-trivial isotypic component a(V ) K , and we arrive at In the case of an even dimension N, the domain of definition of this formula is a complex torus T := T 1 × T + where T 1 = (C × ) n and T + ⊂ (C × ) n is the open subset determined by the conditions |t 0, j | = e Re φ j > 1 for all j . For odd N we must continue to work with a double cover (also denoted by T ) to take the square root e (N/2) ∑ j (iψ j −φ j ) . Let g be the Howe dual partner of Lie(K) in osp(W ). We know from Proposition 2.1 that g = osp(U ⊕ U * ) for K = O N and g = osp( U ⊕ U * ) for K = USp N . Recall also from §2. 6.1 that the g-representation on a(V ) K is irreducible and of highest weight λ N = (N/2) ∑ j (iψ j − φ j ). Denote by Γ λ the set of weights of this representation. Let B γ = (−1) |γ| dim a(V ) K γ be the dimension of the weight space a(V ) K γ multiplied with the correct sign to form the supertrace.
-On the right-hand side we recognize the correlation function (see §1) which is the object of our study and, as we have explained, is related to the character of the irreducible g-representation on a(V ) K . The left-hand side gives this character (restricted to the toral set T ) in the form of a weight expansion, some information about which has already been provided by Corollary 2.3 of §2. 6.1.

Formula for the character. -
Being an eigenfunction of (the radial parts of) the Laplace-Casimir operators that represent the center of the universal enveloping algebra U(g), our character χ = ∑ γ B γ e γ satisfies a certain system of differential equations. Here we first describe the origin and explicit form of these differential equations.
We then prove that χ is determined uniquely by these in combination with the weight constraints for γ ∈ Γ λ . Finally, we provide the explicit function with these properties.

Extended character.
-Formula (4.2) and Corollary 4.1 express the character χ as a function on the toral set T . Our next step is to describe a supermanifold with a g-action where χ exists as a G R -radial superfunction and Laplace-Casimir operators can be applied. Here we give only a rough sketch, leaving the details to the reader. First of all, the symmetry group G R for χ has to be identified. Recall from the end of §4.2.1 that the good real group acting in the spinor-oscillator representation A V is Spin(W 1,R ) × Z 2 Mp(W 0,R ) =: G ′ , which contains K = O N and K = USp N as subgroups. Since we are studying the character χ of the g-representation on the subspace A K V of K-invariants, we now seek the subgroup G R ⊂ G ′ which centralizes K; this means that we are asking the exponentiated version of a question which was answered at the Lie algebra level in §2. 7. Here, restricting the group G ′ to the centralizer of K we find We observe that G R for the case of K = O N is just the lower-dimensional copy of G ′ which corresponds to U s taking the role of V s . We also see immediately that the Lie algebras Lie(G R ) coincide with the real forms described in Propositions 2.4 and 2. 5.
The second object to construct is a real domain The choice we make for T R is the one singled out by the parametrization t 1, j = e iψ j and t 0, j = e φ j with real-valued coordinates ψ j and φ j . We also want the elements of M R to commute with those of K as endomorphisms of W . By these requirements, the good real domain M R to consider in the case of K = O N is the lower-dimensional copy of Spin R × M Sp which, again, corresponds to U s replacing V s . By the detailed analysis of §3 (where M Sp was simply denoted by M) and the fact that the diagonal elements constitute a maximal torus in Spin R , we infer the desired property M R = G R .T R .
In the case of K = USp N the same requirements lead to the choice , where the construction of M SO is fully parallel to that of M Sp : if SO denotes the complex orthogonal group of the vector space U 0 ⊕ U * 0 , we introduce the semigroup H ⊂ SO which is defined by the inequality g † sg < g for s = −Id U 0 ⊕ Id U * 0 and then take M SO ⊂ H to be the totally real submanifold M SO ⊂ H of pseudo-Hermitian elements m = sm † s . For this choice we easily check that Having constructed a manifold M R with an action of G R by conjugation, we now consider the supermanifold F (M R , g 1 ) which is the sheaf of algebras of analytic functions on M R with values in ∧(g * 1 ) where, once again, we remind the reader that g = osp(U ⊕U * ) for K = O N and g = osp( U ⊕ U * ) for K = USp N . By construction, for every point x ∈ M R we have T x M R ⊗ C ≃ g 0 , the even part of the Lie superalgebra g . Hence by the basic principles reviewed at the beginning of this Chapter, the supermanifold F (M R , g 1 ) carries two representations of g by superderivations (X →X L and X →X R ). The benefit from all this is that for every Casimir invariant I ∈ U(g) we now get a Laplace-Casimir operator D(I) on F by replacing each X ∈ g in the polynomial expression for I by the corresponding differential operatorX L (orX R ).
Utilizing the present setting the character χ, which was given in (4.2) as a function on T , will now be extended to a section of F (M R , g 1 ). By construction, the direct product M R × K is contained as a subspace in Spin(W 1 ) × Z 2 H(W s 0 ), and by restriction of the representation R we get a mapping R : M R × K → End(A V ) whose image still lies in the subspace of trace-class operators. A good definition of χ ∈ F therefore is where {F j } now is a basis of g 1 and {ξ j } is its dual basis. The symbol ρ * here denotes the g-representation on A K V . It is clear that by restricting the numerical part of χ(m) to the toral set T R ⊂ M R we recover the function described in (4.2).

Weyl group.
-Consider now the numerical part of the section χ ∈ F . Being a G R -radial function, num(χ) is invariant under the action of the Weyl group W which normalizes T R with respect to the G R -action on M R by conjugation. Since the latter group action decomposes as a direct product of two factors, so does W .
For both cases (K = O N , USp N ) the second factor of the Weyl group W is just the permutation group S n . As a matter of fact, conjugation of a diagonal element t 0 ∈ M Sp or t 0 ∈ M SO by g ∈ Mp((U 0 ⊕U * 0 ) R ) or g ∈ SO * (U 0 ⊕U * 0 ) can return another diagonal element only by permutation of the eigenvalues e φ 1 , . . ., e φ n of t 0 . (No inversion e φ j → e −φ j is possible, as this would mean transgressing the oscillator semigroup.) This factor S n of W will play no important role in the following, as the expressions we will encounter are automatically invariant under such permutations.
The first factors of W are of greater significance. For the two cases of K = O N and K = USp N these are the Weyl groups W SO 2n and W Sp 2n respectively. An explicit description of these groups is as follows. Let {e 1 , . . ., e n } be an orthonormal basis of U and decompose U ⊕U * into a direct sum of 2-planes, where P j is spanned by the vector e j and the linear form ce j = e j , · ( j = 1, . . . , n). In both cases at hand, i.e., for the symmetric form S as well the alternating form A, this is an orthogonal decomposition. The real torus under consideration is parameterized by (e iψ 1 , . . . , e iψ n ) ∈ (U 1 ) n acting by e iψ j .(e j ) = e iψ j e j and e iψ j .(ce j ) = e −iψ j ce j .
The Weyl group W Sp is generated by the permutations of these planes and the involutions which are defined by conjugation by the mapping that sends e j → ce j and ce j → −e j . The Weyl group W SO is generated by the permutations together with the involutions which are induced by the mappings that simply exchange e j with ce j . Since we are in the special orthogonal group and the determinant for a single exchange e j ↔ ce j is −1, the number of involutions in any word in W SO has to be even.
In summary, the W -action on our standard bases of linear functions, {iψ j } and {φ j }, is given by the respective permutations together with the action of the involutions defined by sign change, iψ j → −iψ j . In the sequel, the Weyl group action will be understood to be either this standard action or alternatively, depending on the context, the corresponding action on the exponentiated functions {e iψ j } and {e φ j }.
As a final remark on the subject, let us note that the Weyl group symmetries of the function χ(t) can also be read off directly from the explicit expression (4.2). In particular, the absence of reflections φ j → −φ j is clear from the conditions Re φ j > 0 . 4.3.3. Laplace-Casimir eigenvalues. -We are now ready to start deriving a system of differential equations for χ. Recall that every Casimir element I ∈ U(g) determines an invariant differential operator D(I), called a Laplace-Casimir operator, by replacing each element X ∈ g in the polynomial expression of I byX L orX R . Lemma 4. 6. -Let χ be the character of an irreducible representation (ρ * , ρ) of a Lie supergroup (g , G) on a complex vector space. Then for any Casimir element I ∈ U(g) the character χ is an eigenfunction of the differential operator D(I).
Recall now from §2.2.2 that for every ℓ ∈ N we have a Casimir element I ℓ ∈ U(osp) of degree 2ℓ . Recall also that under the assumption Let us insert here the following comment. While ∂ , ∂ , Λ were abstractly defined as osp-generators, they acquire a transparent meaning when represented as operators on the spinor-oscillator module a(V ). In fact, using formula (2.8) one finds Here, for notational brevity, we make no distinction between osp-elements ∂ , ∂ , etc. and the operators representing them on a(V ). It is now natural to identify the spinoroscillator module a(V ) (or rather, a suitable completion thereof) with the complex of holomorphic differential forms on V 0 . Writing ε( f 1, j ) ≡ dz j and δ (e 0, j ) ≡ ∂ /∂ z j we then see that ∂ is the holomorphic exterior derivative: ∂ = ι(v) becomes the operator of contraction with the vector field v = ∑ z j ∂ /∂ z j generating scale transformations z j → e t z j , while Λ = [∂ , ∂ ] = ∂ • ι(v) + ι(v) • ∂ is interpreted as the Lie derivative with respect to that vector field. Consider now any irreducible osp-representation on a Z 2 -graded vector space V with the property that the V -supertrace of e −tΛ (t > 0) exists. Let λ (I ℓ ) be the scalar value of the Casimir invariant I ℓ in the representation V . Then a short computation using I ℓ = [∂ , F ℓ ] and [∂ , Λ] = 0 shows that λ (I ℓ ) multiplied by STr V e −tΛ vanishes: since the supertrace of any bracket is zero. Thus we are facing a dichotomy: either we have STr V e −tΛ = 0 , or else λ (I ℓ ) = 0 for all ℓ ∈ N . Now our representation a(V ) K realizes the latter alternative, which leads to the following consequence.  Proof. -For any real parameter t > 0 the supertrace of the operator e −tΛ on a(V ) K certainly exists and is non-zero. In fact, using formula (4.2) one computes the value as STr a(V ) K e −tΛ = STr R(e t Id n , e t Id n ; Id N ) = K Det n (Id N − e −t k) Det n (Id N − e −t k) dk = 1 = 0 .
The dichotomy of λ (I ℓ ) STr a(V ) K e −tΛ = 0 therefore gives D(I ℓ )χ = λ (I ℓ )χ = 0 . 4. 3.4. Differential equations for the character. -Since the Casimir elements commute with all elements of the Lie superalgebra, the Laplace-Casimir operators leave the set of radial superfunctions invariant. We denote byḊ(I ℓ ) the radial parts of the Laplace-Casimir operators D(I ℓ ) for ℓ ∈ N . These operators, which arise by restricting the Laplace-Casimir operators to the space of radial functions, are given by differential operators on the torus T R . They have been described by Berezin [1], and his results will now be stated in a form well adapted to our purposes. Recall from §2.2.1 that osp-roots are of the form ±ϑ s j ± ϑ tk . Recall also the relations ϑ 0 j = φ j and ϑ 1 j = iψ j ( j = 1, . . ., n). In the following we regard the variables φ j and ψ j as real (local) coordinates for the real torus T R .
For ℓ ∈ N let D ℓ be the degree-2ℓ differential operator Let where ∆ + = ∆ + 0 ∪ ∆ + 1 is a system of even and odd positive roots (cf. §2. 6.2).  The methods of this section can be used to derive differential equations for the character of a certain class of irreducible representations of gl(U ) ≃ g (0) . Define Here, {i(ψ j − ψ k ), φ j − φ k | j < k} and {φ j − iψ k } are the sets of even and odd positive roots of g (0) . The following statement is Corollary 4.12 of [4] adapted to the present context and notation. The idea of the proof is the same as that of Lemma 4.7.  4. 3. 5. Uniqueness theorem. -Recall that the main goal of this paper can be stated as that of explicitly computing the character χ of a local representation defined on the space of invariants a(V ) K in the spinor-oscillator module. We have restricted ourselves to the cases where K is either O N or USp N . The representation on a(V ) K is defined at the infinitesimal level on the full complex Lie superalgebra g which is the Howe partner of K in the canonical realization of osp in the Clifford-Weyl algebra of V ⊕V * . It has been shown that χ : T R → C satisfies the differential equations D ℓ (Jχ) = 0. By analytic continuation it satisfies the same equations on the complex torus T .
Recall that Γ λ denotes the set of weights of the g-representation on a(V ) K . Recall also from Corollary 2.3 that the weights γ = ∑ n j=1 (im j ψ j − n j φ j ) ∈ Γ λ satisfy the weight constraints − N 2 ≤ m j ≤ N 2 ≤ n j . The highest weight is λ = N 2 ∑(iψ j − φ j ). By the definition of the torus T the weights γ ∈ Γ λ are analytically integrable and we now view e γ as a function on T . -The character χ : T → C is annihilated by the differential operators D ℓ • J for all ℓ ∈ N , and it has a convergent expansion χ = ∑ B γ e γ where the sum runs over weights γ = ∑ n j=1 (im j ψ j − n j φ j ) satisfying the constraints − N 2 ≤ m j ≤ N 2 ≤ n j . For the case of K = USp N it is the unique function on T with these two properties and B λ = 1. For K = O N it is the unique W -invariant function on T with these two properties and B λ = 1 , B λ −iNψ n = 0 . Remark 4. 3. -To verify the property B λ −iNψ n = 0 which holds for the case of K = O N , look at the right-hand side of the formula of Corollary 4.1: in order to generate a term e γ = e λ −iNψ n in the weight expansion, you must pick the term e −iNψ n in the expansion of the determinant for j = n in the numerator; but the latter term depends on k as Det(−k) which vanishes upon taking the Haar average for K = O N . By Winvariance the property B λ −iNψ n = 0 is equivalent to B λ −iNψ j = 0 for all j .
In view of this Remark and Corollaries 2.3 and 4.2, it is only the uniqueness statement of Theorem 4.1 that remains to be proved here. This requires a bit of preparation, in particular to appropriately formulate the condition D ℓ (Jχ) = 0 . For that we develop Jχ in a series Jχ = ∑ τ a τ f τ where the f τ are D ℓ -eigenfunctions for every ℓ ∈ N .
The first step is to determine an appropriate expansion for J. Recall that .
Given a factor in the denominator of this representation, we wish to factor out, e.g., e − β 2 to obtain a term (1 − e −β ) −1 which we will attempt to develop in a geometric series. In order for this to converge uniformly on compact subsets of T it is necessary and sufficient for Re β to be positive on t . This of course depends on the root β .
Fortunately, the sets of odd positive roots for our two cases of K = O N and K = USp N are the same (see §2. 6.2): So indeed, if we factor out e − β 2 from each term in the denominator and do the same in the numerator, we obtain the expression , and it is possible to expand each term of the denominator in a geometric series. Here which converges uniformly on compact subsets of T . In this expression b and σ denote the vectors b = (b 1 , . . ., b r ) and σ = (σ 1 , . . . , σ r ), respectively, and bσ := ∑ b i σ i . Following the usual multi-index notation, b ≥ 0 means b i ≥ 0 for all i . Note A 0 = 1. Now we know that the character has a convergent series representation Thus we may write For convenience of the discussion we letγ := γ + bσ and reorganize the sums as where the inner sum is a finite sum which runs over all b ≥ 0 such thatγ − bσ ∈ Γ λ .
We are now in a position to explain the recursion procedure which shows that χ is unique. Start by applying D ℓ to Jχ as represented in the expression (4.3). Since δ +γ is of the form ∑(im k ψ k − n k φ k ), we immediately see that it is an eigenfunction with eigenvalue E(ℓ,γ) := (−1) ℓ ∑(m 2ℓ k − n 2ℓ k ). The functions e δ +γ in the sum are independent eigenfunctions. Hence it follows that 0 = E(ℓ,γ) ∑ A b Bγ −bσ (4.4) for allγ fixed and then for all ℓ ∈ N .
From now on we consider the equations (4.4) only in those cases whereγ is itself a weight of our representation. (We have license to do so as only the uniqueness part of Theorem 4.1 remains to be proved.) In this case we have the following key fact.
Proof. -Our first job is to compute δ . For the list of even and odd positive roots we refer the reader to §2. 6

.2. Direct computation shows that if
The same computation for the case of K = USp N shows that δ = n ∑ k=1 k (iψ k − φ n−k+1 ) . Now we write γ = ∑ k (im k ψ k − n k φ k ) with the weight constraints − N 2 ≤ m k ≤ N 2 ≤ n k . The assumption that E(ℓ, γ) vanishes for all ℓ means that ∑ k (m n−k+1 + k − 1) 2ℓ = ∑ k (n k + k − 1) 2ℓ for all ℓ in the case of K = O N . In the case of K = USp N it means that ∑ k (m n−k+1 + k) 2ℓ = ∑ k (n k + k) 2ℓ for all ℓ .
In the second case the only solution for m k and n k satisfying the weight constraints is the highest weight λ itself. In the first case there is one other solution, namely that which is obtained from the highest weight by replacing m n = N 2 by m n = − N 2 . However, one directly checks that in the O N case, where 2iψ n is not a root, it is not possible to obtain such a γ by adding some combination of roots from g (2) to λ .
We are now able to give the proof of the uniqueness statement of Theorem 4.1.
Proof. -We will determine B γ recursively, starting from B λ = 1. Let γ = λ be a weight that satisfies the weight constraints. Then if K = USp N we know that E(ℓ, γ) is non-zero for some ℓ . It therefore follows from equation (4.4) and A 0 = 1 that By definition, the function S W (e λ Z) is holomorphic on ∩ n j=1 {Re φ j > 0} \ ∪ α∈∆ + 0 (ψ) Σ α . Now it is a theorem of complex analysis that if a function is holomorphic outside an analytic set of complex codimension at least two, then this function is everywhere holomorphic. Therefore, since the intersection of two or more of the submanifolds Σ α is of codimension at least two in T , it suffices to show that for any α ∈ ∆ + 0 (ψ) our function S W (e λ Z) extends holomorphically to Hence let α be some fixed root in ∆ + 0 (ψ). There exists a Weyl group element w ∈ W and a w-invariant neighborhood U of D α such that w : U → U is a reflection fixing the points of D α . Let z α : U → C be a complex coordinate which is transverse to D α in the sense that w(z α ) = −z α . Because the root α occurs at most once in the product Z, the function S W (e λ Z) has at most a simple pole in z α . We may choose U in such a way that S W (e λ Z) is holomorphic on U \ D α . Doing so we have a unique decomposition where A and B are holomorphic in U . Since S W (e λ Z) is W -invariant, we conclude that w(A) = −A and hence A = 0 along D α . The factor e λ N +δ ′ is the character of the representation ( N∓1 2 STr , SDet 2 ) of (a double cover of) the Lie supergroup (g (0) , GL(U 0 ) × GL(U 1 )). This representation is one-dimensional, and from Corollary 4.3 we have D ℓ (J 0 e λ N +δ ′ ) = 0 for all ℓ, N ∈ N .
The statement of the lemma now follows by applying the W -invariant differential operator D ℓ to the formula for ord(W λ )Jϕ above. 4. 3.7. Weight constraints. -Here we carry out the final step in proving the explicit formula for the character χ of our representation. Since the formula in the case of K = SO N follows directly from that for K = O N (see §1) and the case of K = U N has been handled in [4], we need only discuss the cases of K = O N and K = USp N .
In the situation where w(iψ 1 ) = −iψ 1 we rewrite the factors in the denominator as (1 − e −w(α) ) −1 = −e w(α) (1 − e w(α) ) −1 and expand, and convergence in R is again guaranteed. Adding these series we obtain a series representation ϕ = ∑ [w] ϕ [w] = ∑ γ in the larger range j ≤ k and φ j + φ k in the smaller range j < k . The Weyl group acts by permutation of indices on both the iψ j and φ j and by sign reversal on the iψ j . In this case, as opposed to the case above where only an even number of sign reversals were allowed, every sign reversal transformation is in the Weyl group.
In order to prove Lemma 4.12 in this case, we need only go through the argument in the O N case and make minor adjustments. In fact, the main step is to prove Lemma 4.11 and, there, the only change is that the range of j for the factor 1 − e −i(ψ 1 +ψ j ) is larger. This is only relevant in the case w(iψ 1 ) = −iψ 1 , where we rewrite the additional denominator term (1 − e −w(2iψ 1 ) ) −1 as −e −2iψ 1 (1 − e −2iψ 1 ) −1 . Hence the factor in front of the ratio of products on the r.h.s. of equation (4.7) gets an additional factor of e −2iψ 1 and now is e −i( N 2 +1)ψ 1 . Thus m 1 ≤ − N 2 − 1 which certainly implies m 1 ≤ N 2 . Let us summarize this discussion. Moreover, using the fact that the Weyl group transformations for K = O N always involve an even number of sign changes, one sees that B λ −iNψ n = 0 in that case. As a consequence of the uniqueness theorem (Theorem 4.1) we therefore have (1 − e −w(α) ) in both the O N and USp N cases. Since the SO N case has been handled as a consequence of the result for O N , our work is now complete.