Haar expectations of ratios of random characteristic polynomials
 A. Huckleberry^{1}Email author,
 A. Püttmann^{2} and
 M. R. Zirnbauer^{3}
https://doi.org/10.1186/s4062701500053
© Huckleberry et al. 2016
Received: 20 August 2015
Accepted: 3 December 2015
Published: 13 January 2016
Abstract
We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups \(K = \mathrm {O}_N\,\), \(\mathrm {SO}_N\,\), and \(\mathrm {USp}_N\,\). To that end, we start from the Clifford–Weyl algebra in its canonical realization on the complex \(\fancyscript {A}_V\) of holomorphic differential forms for a \(\mathbb {C} \)vector space \(V_0\,\). From it we construct the Fock representation of an orthosymplectic Lie superalgebra \(\mathfrak {osp}\) associated to \(V_0\,\). Particular attention is paid to defining Howe’s oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of \(\mathfrak {sp} \subset \mathfrak {osp}\). 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 \(V_0 = \mathbb {C}^n \otimes \mathbb {C}^N\) where \(\mathbb {C}^N\) is equipped with the standard Krepresentation, and focus on the subspace \(\fancyscript {A}_V^K\) of Kequivariant forms. By Howe duality, this is a highestweight irreducible representation of the centralizer \(\mathfrak {g}\) of \(\mathrm {Lie}(K)\) in \(\mathfrak {osp}\). We identify the KHaar expectation of n ratios with the character of this \(\mathfrak {g} \)representation, which we show to be uniquely determined by analyticity, Weylgroup invariance, certain weight constraints, and a system of differential equations coming from the LaplaceCasimir invariants of \(\mathfrak {g}\,\). 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 Weyltype character formula for all integers \(N \ge 1\). This completes earlier work of Conrey, Farmer, and Zirnbauer for the case of \(\mathrm {U}_N\,\).
1 Background
The Haar average I(t) can be regarded as the (numerical part of the) character of an irreducible representation of a Lie supergroup \((\mathfrak {g}\, ,G)\) restricted to a suitable subset of a maximal torus of G. The Lie superalgebra \(\mathfrak {g}\) is the Howe dual partner of the compact group K in an orthosymplectic Lie superalgebra \(\mathfrak {osp} \,\). It is naturally represented on a certain infinitedimensional spinor–oscillator module \(\mathfrak {a} (V)\) —more concretely, the complex of holomorphic differential forms on the vector space \(\mathbb {C}^n \otimes \mathbb {C}^N\)—and the irreducible representation is that on the subspace \(\mathfrak {a}(V)^K\) of Kequivariant forms.
To even define the character, we must exponentiate the representation of the Lie algebra part of \(\mathfrak {osp}\) on \(\mathfrak {a}(V)\). This requires going to a completion \(\fancyscript {A}_V\) of \(\mathfrak {a}(V)\), and can only be done partially. Nevertheless, the represented semigroup contains enough structure to derive LaplaceCasimir differential equations for its character.
1.1 Comparison with results of other approaches
The very same formulas for \(\mathrm {SO}_N\) and \(\mathrm {O}_N\) were derived in the recent literature [3, 8] but, again, only in the much narrower range \(q \le \mathrm {Int}[N/2]\). There exist a number of other interesting recent works which emphasize the Lie superalgebraic and combinatorial side of the picture (see, e.g., [4–6]).
1.2 Howe duality and weight expansion
Averaging the product of ratios Z(t, k) with respect to the compact group K corresponds to the projection from \(\mathfrak {a}(V)\) onto the vector space \(\mathfrak {a}(V)^K\) of Kinvariants (Corollary 4.1). Now, Howe duality (Proposition 2.2) implies that \(\mathfrak {a}(V)^K\) is the representation space for an irreducible highestweight representation \(\rho _*\) of the Howe dual partner \(\mathfrak {g}\) of K in the orthosymplectic Lie superalgebra \(\mathfrak {osp}(W)\). This representation \(\rho _*\) is constructed by realizing \(\mathfrak {g} \subset \mathfrak {osp}(W)\) as a subalgebra in the space of degreetwo elements of the Clifford–Weyl algebra \(\mathfrak {q}(W)\). Precise definitions of these objects, their relationships, and the Howe duality statement can be found in Sect. 2.
Using the decomposition \(\mathfrak {a}(V)^K = \oplus _{\gamma \in \Gamma }\, V_\gamma \) into weight spaces, Howe duality leads to the weight expansion \(I(\mathrm {e}^H) = \mathrm {STr}_{\mathfrak {a}(V)^K} \, \mathrm {e}^{\rho _*(H)} = \sum _{\gamma \in \Gamma } B_\gamma \, \mathrm {e}^{\gamma (H)}\) for \(t = \mathrm {e}^H \in T_1 \times T_+\,\). Here \(\mathrm {STr}\) denotes the supertrace. There are strong restrictions on the set of weights \(\Gamma \). Namely, if \(\gamma \in \Gamma \), then \(\gamma = \sum _{j=1}^n (\mathrm {i} m_j \psi _j  n_j \phi _j)\) and \( \frac{N}{2}\le m_j \le \frac{N}{2} \le n_j\) for all \(j\,\). The coefficients \(B_\gamma = \mathrm {STr}_{V_\gamma } (\mathrm {Id})\) are the dimensions of the weight spaces (multiplied by parity). Note that the set of weights of the representation \(\rho _*\) of \(\mathfrak {g}\) on \(\mathfrak {a}(V)^K\) is infinite.
1.3 Group representation and differential equations
Before outlining the strategy for computing our character in the infinitedimensional setting of representations of Lie superalgebras and groups, we recall the classical situation where \(\rho _*\) is an irreducible finitedimensional representation of a reductive Lie algebra \(\mathfrak {g}\) and \(\rho \) is the corresponding Lie group representation of the complex reductive group G. In that case the character \(\chi \) of \(\rho \), which automatically exists, is the trace \(\mathrm {Tr}\, \rho \), which is a radial eigenfunction of every LaplaceCasimir operator. These differential equations can be completely understood by their behavior on a maximal torus of G.
In our case we must consider the infinitedimensional irreducible representation \(\rho _*\) of the Lie superalgebra \(\mathfrak {g} = \mathfrak {osp}\) on \(\mathfrak {a}(V)^K\). Casimir elements, LaplaceCasimir operators of \(\mathfrak {osp}\), and their radial parts have been described by Berezin [1]. In the situation \(U_0 \simeq U_1\) at hand we have the additional feature that every \(\mathfrak {osp}\)Casimir element I can be expressed as a bracket \(I = [ \partial , F ]\) where \(\partial \) is the holomorphic exterior derivative when we view \(\mathfrak {a} (V)^K\) as the complex of Kequivariant holomorphic differential forms on \(V_0\,\).
Thinking classically we consider the even part of the Lie superalgebra \(\mathfrak {osp}(W_0 \oplus W_1)\), which is the Lie algebra \(\mathfrak {o}(W_1) \oplus \mathfrak {sp}(W_0)\). The real structures at the Lie supergroup level come from a real form \(W_\mathbb {R}\) of W. The associated real forms of \(\mathfrak {o} (W_1)\) and \(\mathfrak {sp}(W_0)\) are the real orthogonal Lie algebra \(\mathfrak {o}(W_{1,\mathbb {R}})\) and the real symplectic Lie algebra \(\mathfrak {sp}(W_{0,\mathbb {R}})\). These are defined in such a way that the elements in \(\mathfrak {o}(W_{1,\mathbb {R}}) \oplus \mathfrak {sp} (W_{0,\mathbb {R}})\) and \(\mathrm {i} W_\mathbb {R}\) are mapped as elements of the Clifford–Weyl algebra via the spinor–oscillator representation to antiHermitian operators on \(\mathfrak {a}(V)\) with respect to a compatibly defined unitary structure. In this context we frequently use the unitary representation of the real Heisenberg group \(\exp ( \mathrm {i} W_{0, \mathbb {R}}) \times \mathrm {U}_1\) on the completion \(\fancyscript {A}_V\) of the module \(\mathfrak {a}(V)\).
Since \(\wedge (V^*_1)\) has finite dimension, exponentiating the spinor representation of \(\mathfrak {o}(W_{1,\mathbb {R}})\) causes no difficulties. This results in the spinor representation of \(\mathrm {Spin}(W_{1,\mathbb {R}})\), a 2:1 covering of the compact group \(\mathrm {SO}(W_{1,\mathbb {R}})\). So in this case one easily constructs a representation \(R_1 : \, \mathrm {Spin} (W_{1, \mathbb {R}}) \rightarrow \mathrm {U} (\mathfrak {a}(V))\) which is compatible with \(\rho _*_{\mathfrak {o} (W_{1,\mathbb {R}})}\).
Exponentiating the oscillator representation of \(\mathfrak {sp} (W_{0,\mathbb {R}})\) on the infinitedimensional vector space \(\mathrm {S}(V_0^*)\) requires more effort. In Sect. 3.4, following Howe [11], we construct the Shale–Weil–Segal representation \(R' : \, \mathrm {Mp}(W_{0, \mathbb {R}}) \rightarrow \mathrm {U} (\fancyscript {A}_V)\) of the metaplectic group \(\mathrm {Mp}(W_{0, \mathbb {R}})\) which is the 2:1 covering group of the real symplectic group \(\mathrm {Sp} (W_{ 0,\mathbb {R}})\). This is compatible with \(\rho _*\vert _{\mathfrak {sp} (W_{0, \mathbb {R}})}\). Altogether we see that the even part of the Lie superalgebra representation integrates to \(G_\mathbb {R} = \mathrm {Spin} (W_{1 ,\mathbb {R}}) \times _{\mathbb {Z}_2} \mathrm {Mp}(W_{0,\mathbb {R}})\,\).
The construction of \(R^\prime \) uses a limiting process coming from the oscillator semigroup \(\widetilde{\mathrm {H}}(W_0^s)\), which is the double covering of the contraction semigroup \(\mathrm {H}(W_0^s) \subset \mathrm {Sp}(W_0)\) and has \(\mathrm {Mp}(W_{0,\mathbb {R}})\) in its boundary. Furthermore, we have \(\widetilde{ \mathrm {H}} (W_0^s) = \mathrm {Mp}(W_{0,\mathbb {R}}) \times M\) where M is an analytic totally real submanifold of maximal dimension which contains a real form of the torus \(T_+\) (see Sect. 3.2). The representation \(R_0 : \,\widetilde{\mathrm {H}}(W_0^s)\rightarrow \mathrm {End}(\fancyscript {A}_V)\) constructed in Sect. 3.3 facilitates the definition of the representation \(R^\prime \) and of the character \(\chi \) in Sect. 4.2 and Sect. 5.1. It should be underlined that Proposition 3.24 ensures convergence of the superfunction \(\chi (h)\), which is defined as a supertrace and exists for all \(h \in \widetilde{\mathrm {H}}(W_0^s)\).
On that basis, the key idea of our approach is to exploit the fact that every Casimir invariant \(I \in \mathsf {U}(\mathfrak {g})\) is exact in the sense that \(I = [ \partial , F]\). 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 \(\chi \) the differential equations \(D(I) \chi = 0\) where D(I) is the LaplaceCasimir 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 \(\chi \).
In the case of \(K = \mathrm {O}_N\) the Lie group associated to the even part of the real form of the Howe partner \(\mathfrak {g}\) is embedded in a simple way in the full group \(G_\mathbb {R}\) described above. It is itself just a lowerdimensional group of the same form. In the case of \(K = \mathrm {USp}_N\) a sort of reversing procedure takes places and the analogous real form is \(\mathrm {USp}_{2n} \times \mathrm {SO}_{2n}^*\). Nevertheless, the precise data which are used as input into the series developments, the uniqueness theorem, and the final calculations of \(\chi \) are essentially the same in the two cases. Therefore there is no difficulty handling them simultaneously.
2 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, \(\mathfrak {osp}\,\), 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 \(\mathfrak {osp}\) and of all Howe dual pairs inside of \(\mathfrak {osp} \,\). Roots and weights of the relevant representations are described in detail.
2.1 Notion of Lie superalgebra
A \(\mathbb {Z}_2\) grading of a vector space V over \(\mathbb {K} = \mathbb {R}\) or \(\mathbb {C}\) is a decomposition \(V = V_0\oplus V_1\) of V into the direct sum of two \(\mathbb {K}\)vector spaces \(V_0\) and \(V_1\,\). The elements in \((V_0 \cup V_1) {\setminus} \{ 0\}\) are called homogeneous. The parity function \(\, :(V_0 \cup V_1) {\setminus} \{ 0\} \rightarrow \mathbb {Z}_2\,\), \(v\in V_s \mapsto v = s\,\), assigns to a homogeneous element its parity. We write \(V \simeq \mathbb {K}^{p q}\) if \(\dim _\mathbb {K} V_0 = p\) and \(\dim _\mathbb {K} V_1 = q\,\).
 1.
\([\mathfrak {g}_s , \mathfrak {g}_{s^\prime } ] \subset \mathfrak {g}_{s + s^\prime }\,\), i.e., \([X,Y]= X + Y\) (mod 2) for homogeneous elements X, Y.
 2.
Skew symmetry: \([X,Y] = (1)^{XY}[Y,X]\) for homogeneous X, Y.
 3.Jacobi identity, which means that \(\mathrm {ad}(X) = [X, \; ] : \, \mathfrak {g} \rightarrow \mathfrak {g}\) is a (super)derivation:$$\begin{aligned} \mathrm {ad}(X)\, [Y,Z] = [\mathrm {ad}(X)Y , Z] + (1)^{XY}[Y,\mathrm {ad}(X)Z]. \end{aligned}$$
Example 2.1
In fact, for every \(\mathbb {Z}_2\)graded associative algebra \(\fancyscript {A}\) the bracket \([\, , \,] : \, \fancyscript {A} \times \fancyscript {A} \rightarrow \fancyscript {A}\) defined by \([X,Y] = XY  (1)^{X Y} YX\) satisfies the Jacobi identity.
Example 2.2
Example 2.3
2.1.1 Supertrace
 1.
\(\mathfrak {g}_0\) and \(\mathfrak {g}_1\) are Borthogonal to each other;
 2.
B is symmetric on \(\mathfrak {g}_0\) and skew on \(\mathfrak {g}_1\,\);
 3.
\(B([X,Y],Z) = B(X,[Y,Z])\) for all \(X, Y, Z \in \mathfrak {g}\,\).
Lemma 2.1
If \(\mathfrak {g}\) is a Lie superalgebra in \(\mathrm {End}(V)\) , the trace form \(B(X,Y) = \mathrm {STr}\, (XY)\) is an \(\mathrm {ad} \) invariant bilinear form. One has \(\mathrm {STr} \, [X,Y] = 0\,\).
Recalling the setting of Example 2.2, note that the supertrace for \(W = V \oplus V^*\) is odd under the \(\mathfrak {gl}\)automorphism \(\tau \) fixing \(\mathfrak {osp}(W)\), i.e., \(\mathrm {STr}_W \circ \tau =  \mathrm {STr}_W\,\). It follows that \(\mathrm {STr}_W X = 0\) for any \(X \in \mathfrak {osp}(W)\). Moreover, \(\mathrm {STr}_W (X_1 X_2 \cdots X_{2n+1}) = 0\) for any product of an odd number of \(\mathfrak {osp}\)elements.
2.1.2 Universal enveloping algebra
Lemma 2.2
If \(\mathfrak {g}\) is a Lie superalgebra, the supercommutator \(\{\,,\,\}\) gives \(\mathsf {U}(\mathfrak {g})\) the structure of another Lie superalgebra in which \(\{ \mathsf {U}_n(\mathfrak {g}) , \mathsf {U}_{n^\prime }(\mathfrak {g})\} \subset \mathsf {U}_{n + n^\prime  1} (\mathfrak {g})\).
Proof
Compatibility with the \(\mathbb {Z}_2\)grading, skew symmetry, and Jacobi identity are properties of \(\{ \, , \, \}\) that are immediate at the level of the tensor algebra \(\mathsf {T}(\mathfrak {g})\). They descend to the corresponding properties at the level of \(\mathsf {U} (\mathfrak {g})\) by the definition of the twosided ideal \(\mathsf {J} (\mathfrak {g})\). Thus \(\mathsf {U}(\mathfrak {g})\) with the bracket \(\{ \, , \, \}\) is a Lie superalgebra.
By definition, the supercommutator of \(\mathsf {U}(\mathfrak {g})\) and the bracket of \(\mathfrak {g}\) agree at the linear level: \(\{X,Y\} \equiv [X,Y]\) for \(X, Y \in \mathfrak {g}\,\). It is therefore reasonable to drop the distinction in notation and simply write \([ \, , \,]\) for both of these product operations. This we now do.
2.2 Structure of \(\mathfrak {osp}(W)\)
Lemma 2.3
Proof
We now review how \(\mathfrak {sp}(W_0)\) and \(\mathfrak {o}(W_1)\) decompose for our case \(W_s = V_s^{} \oplus V_s^{*} \,\). For that purpose, if U is a vector space with dual vector space \(U^*\), let \(\mathrm {Sym}(U,U^*)\) and \(\mathrm {Alt}(U, U^*)\) denote the symmetric resp. alternating linear maps from U to \(U^*\).
Lemma 2.4
Proof
The situation for \(\mathfrak {sp}(W_0)\) is identical but for a sign change: the symmetric form S is replaced by the alternating form \(A\,\), and this causes the parity of \(\mathsf {B}, \mathsf {C}\) to be reversed. \(\square \)
By adding up dimensions, Lemmas 2.3 and 2.4 entail the following consequence.
Corollary 2.1
As a \(\mathbb {Z}_2\) graded vector space, \(\mathfrak {osp}(V \oplus V^*)\) is isomorphic to \(\mathbb {K}^{pq}\) where \(p = d_0 (2d_0 + 1) + d_1 (2d_1  1)\), \(q = 4 d_0 d_1\,\) , and \(d_s = \dim V_s \,\).
Lemma 2.5
Lemma 2.6
The embedding \(\mathrm {End}(V)\rightarrow \mathrm {End}(V) \oplus \mathrm {End} (V^*)\) by \(\mathsf {A} \mapsto \mathsf {A} \oplus (\mathsf {A}^ \mathrm {st})\) projected to \(\mathfrak {osp}(W)\) is an isomorphism of Lie superalgebras \(\mathfrak {gl}(V) \rightarrow \mathfrak {g}^{(0)}\).
Proof
In the following subsections we will often write \(\mathfrak {osp}(W) \equiv \mathfrak {osp}\) for short.
2.2.1 Roots and root spaces
A Cartan subalgebra of a Lie algebra \(\mathfrak {g}_0\) is a maximal commutative subalgebra \(\mathfrak {h} \subset \mathfrak {g}_0\) such that \(\mathfrak {g}_0\) (or its complexification if \(\mathfrak {g}_0\) is a real Lie algebra) has a basis consisting of eigenvectors of \(\mathrm {ad}(H)\) for all \(H \in \mathfrak {h} \,\). Recall that \([X,Y]=X+Y\) for homogeneous elements X, Y of a Lie superalgebra \(\mathfrak {g}\,\). From the vantage point of decomposing \(\mathfrak {g}\) by eigenvectors or root spaces, it is therefore reasonable to call a Cartan subalgebra of \(\mathfrak {g}_0\) a Cartan subalgebra of \(\mathfrak {g}\,\). We will see that \(X \in \mathfrak {osp}_1\) and \([X,H] = 0\) for all \(H \in \mathfrak {h} \subset \mathfrak {osp}_0\) imply \(X = 0\,\), i.e., there exists no commutative subalgebra of \(\mathfrak {osp}\) that properly contains a Cartan subalgebra. Lie superalgebras with this property are called of type I in [1].
Lemma 2.7
2.2.2 Casimir elements
Now recall that \(\mathsf {U}(\mathfrak {g})\) comes with a canonical bracket operation, the supercommutator \([\, , \,] : \, \mathsf {U} (\mathfrak {g}) \times \mathsf {U}(\mathfrak {g}) \rightarrow \mathsf {U} (\mathfrak {g})\). An element \(X \in \mathsf {U}(\mathfrak {g})\) is said to lie in the center of \(\mathsf {U} (\mathfrak {g})\), and is called a Casimir element, iff \([X,Y]= 0\) for all \(Y \in \mathsf {U} (\mathfrak {g})\). By the formula (2.2), a necessary and sufficient condition for that is \([X,Y] = 0\) for all \(Y \in \mathfrak {g}\,\).
In the case of \(\mathfrak {g} = \mathfrak {osp}\,\), for every \(\ell \in \mathbb {N}\) there is a Casimir element \(I_\ell \) of degree \(2\ell \), which is constructed as follows. Consider the bilinear form \(B : \, \mathfrak {osp} \times \mathfrak {osp} \rightarrow \mathbb {K}\) given by the supertrace (in some representation), \(B(X,Y) : = \mathrm {STr}\, (XY)\). Recall that this form is \(\mathrm {ad}\)invariant, which is to say that \(B([X,Y],Z) = B(X,[Y,Z])\) for all \(X, Y, Z \in \mathfrak {g}\,\).
Lemma 2.8
For all \(\ell \in \mathbb N\) the element \(I_\ell \) is Casimir, and \(I_\ell  = 0\).
Proof
The other statement, \(I_\ell  = 0\,\), follows from \(\widetilde{e}_i  = e_i\), the additivity of the \(\mathbb {Z}_2\)degree and the fact that \(\mathrm {STr}\,(a) = 0\) for any odd element \(a \in \mathsf {U}(\mathfrak {g})\). \(\square \)
Lemma 2.9
Let \(\mathfrak {osp}(V \oplus V^*)\) be the orthosymplectic Lie superalgebra for a \(\mathbb {Z}_2\) graded vector space V with isomorphic components \(V_0 \simeq V_1\,\) . Then for all \(\ell \in \mathbb {N}\) the Casimir element \(I_\ell \) is expressible as a bracket: \(I_\ell = [\partial , F_\ell ]\,\).
Proof
As we shall see in Sect. 5.1.3, Lemma 2.9 has the drastic consequence that all \(\mathfrak {osp}\)Casimir elements \(I_\ell \) are zero in a certain class of representations of \(\mathfrak {osp}(V \oplus V^*)\) for \(V_0 \simeq V_1\,\).
2.3 Howe pairs in \(\mathfrak {osp}(W)\)
In the present context, a pair \((\mathfrak {h},\mathfrak {h}')\) of subalgebras \(\mathfrak {h}, \mathfrak {h}' \subset \mathfrak {g}\) of a Lie superalgebra \(\mathfrak {g}\) is called a dual pair whenever \(\mathfrak {h}'\) is the centralizer of \(\mathfrak {h}\) in \(\mathfrak {g}\) and vice versa. In this subsection, let \(\mathbb {K} = \mathbb {C}\,\).
Given a \(\mathbb {Z}_2\)graded complex vector space \(U = U_0 \oplus U_1\) we let \(V := U \otimes \mathbb {C}^N\), where \(\mathbb {C}^N\) is equipped with the standard representation of \(\mathrm {GL} (\mathbb {C}^N)\), \(\mathrm {O}(\mathbb {C}^N)\,\), or \(\mathrm {Sp} (\mathbb {C}^N)\,\), as the case may be. As a result, the Lie algebra \(\mathfrak {k}\) of whichever group is represented on \(\mathbb {C}^N\) is embedded in \(\mathfrak {osp}(V \oplus V^*)\). We will now describe the dual pairs \((\mathfrak {h},\mathfrak {k})\) in \(\mathfrak {osp} (W)\) for \(W = V \oplus V^*\). These are known as dual pairs in the sense of R. Howe.
Let us begin by recalling that for any representation \(\rho :\, K \rightarrow \mathrm {GL}(E)\) of a group K on a vector space \(E\,\), the dual representation \(\rho ^*: \, K \rightarrow \mathrm {GL}(E^*)\) on the linear forms on E is given by \((\rho (k)\varphi )(x) = \varphi (\rho (k)^{1}x)\). By this token, every representation \(\rho :\, K \rightarrow \mathrm {GL}(\mathbb {C}^N)\) induces a representation \(\rho _W = (\mathrm {Id} \otimes \rho ) \times (\mathrm {Id} \otimes \rho ^*)\) of K on \(W = V \oplus V^*\).
Lemma 2.10
Let \(\rho : \, K \rightarrow \mathrm {GL}(\mathbb {C}^N)\) be any representation of a Lie group \(K\,\) . If \(V = U \otimes \mathbb {C}^N\) for a \(\mathbb {Z}_2\) graded complex vector space \(U = U_0 \oplus U_1\,\) , the induced representation \({\rho _W}_*(\mathfrak {k})\) of the Lie algebra \(\mathfrak {k}\) of K on \(W = V \oplus V^*\) is a subalgebra of \(\mathfrak {osp}(W)_0\,\).
Proof
Lemma 2.11
\(\rho _W(k) = \Psi \circ (\mathrm {Id} \otimes k) \circ \Psi ^{1}\) for all \(k \in K\,\).
Proof
In the following we often write \(\mathrm {O}(\mathbb {C}^N) \equiv \mathrm {O}_N\) and \(\mathrm {Sp}(\mathbb {C}^N) \equiv \mathrm {Sp}_N\) for short.
Corollary 2.2
For \(K = \mathrm {O}_N\) and \(K = \mathrm {Sp}_N\,\) , the map \(X \mapsto \Psi \circ (X \otimes \mathrm {Id}) \circ \Psi ^{1}\) defines a Lie superalgebra embedding into \(\mathfrak {osp}(W)\) of \(\, \mathfrak {osp}(U \oplus U^*)\) resp. \(\mathfrak {osp} (\widetilde{U} \oplus \widetilde{U}^*)\).
Proof
For \(K = \mathrm {O}_N\) the bilinear form B of \(\mathbb {C}^N\) is symmetric and the bilinear form Q of W pulls back—see (2.6)—to the standard orthosymplectic form of \(U \oplus U^*\).
For \(K = \mathrm {Sp}_N\,\), the form B is alternating. Its pullback, the orthosymplectic form of \(U \oplus U^*\) twisted by the sign factor \((1)^\delta \), is restored to standard form by switching to the \(\mathbb {Z}_2\)graded vector space \(\widetilde{U} \oplus \widetilde{U}^*\) with the twisted \(\mathbb {Z}_2\)grading. \(\square \)
To go further, we need a statement concerning \(\mathrm {Hom}_G(V_1 ,V_2)\), the space of Gequivariant homomorphisms between two modules \(V_1\) and \(V_2\) for a group \(G\,\).
Lemma 2.12
Proof
\(\mathrm {Hom}(X_1 \otimes Y_1\, , X_2 \otimes Y_2)\) is canonically isomorphic to \(X_1^*\otimes Y_1^*\otimes X_2 \otimes Y_2\) as a Grepresentation space, with Gequivariant maps corresponding to Ginvariant tensors. Since the Gaction on \(X_1^*\otimes X_2^{}\) is trivial, one sees that \(\mathrm {Hom}_G(X_1 \otimes Y_1\, , X_2 \otimes Y_2)\) is isomorphic to the tensor product of \(X_1^*\otimes X_2^{} \simeq \mathrm {Hom}(X_1, X_2)\) with the space of Ginvariants in \(Y_1^*\otimes Y_2^{} \,\). The latter in turn is isomorphic to \(\mathrm {Hom}_G(Y_1, Y_2)\). \(\square \)
Proposition 2.1
Writing \(\mathfrak {g}_N \equiv \mathfrak {g}(\mathbb {C}^N)\) for \(\mathfrak {g} = \mathfrak {gl}\,\), \(\mathfrak {o}\,\), \(\mathfrak {sp}\,\), the following pairs are dual pairs in \(\mathfrak {osp}(W):\) \((\mathfrak {gl}(U),\mathfrak {gl}_N)\), \((\mathfrak {osp}(U\oplus U^*), \mathfrak {o}_N)\), \((\mathfrak {osp}(\widetilde{U}\oplus \widetilde{U}^*), \mathfrak {sp}_N)\).
Proof
Here we calculate the centralizer of \(\mathfrak {k}\) in \(\mathfrak {osp}(W)\) for each of the three cases \(\mathfrak {k} = \mathfrak {gl}_N\,\), \(\mathfrak {o}_N\,\), \(\mathfrak {sp}_N\) and refer the reader to [10] for the remaining details.
In the case of \(K = \mathrm {O}_N\,\), the discussion is shortened by recalling Lemma 2.11 and the Kequivariant isomorphism \(\Psi : \, (U \oplus U^*) \otimes \mathbb {C}^N \rightarrow W\). By Schur’s lemma, these imply \(\mathrm {End}_K(W) \simeq \mathrm {End}(U \oplus U^*)\). From Corollary 2.2 it then follows that the intersection \(\mathfrak {osp}(W) \cap \mathrm {End}_K(W)\) is isomorphic as a Lie superalgebra to \(\mathfrak {osp}(U \oplus U^*)\). Passing to the Lie algebra level for K, we get the second dual pair, \((\mathfrak {osp}(U \oplus U^*), \mathfrak {o}_N)\).
Finally, if \(K = \mathrm {Sp}_N\,\), the situation is identical except that Corollary 2.2 compels us to switch to the \(\mathbb {Z}_2\)twisted structure of orthosymplectic Lie superalgebra in \(\mathrm {End}_K(W) \simeq \mathrm {End}(U \oplus U^*)\). This gives us the third dual pair, \((\mathfrak {osp}(\widetilde{U} \oplus \widetilde{U}^*), \mathfrak {sp}_N)\). \(\square \)
2.4 Clifford–Weyl algebra \(\mathfrak {q}(W)\)
Let \(\mathbb {K} = \mathbb {C}\) or \(\mathbb {K} = \mathbb {R}\) (in this subsection the choice of number field again is immaterial) and recall from Example 2.3 the definition of the Jordan–Heisenberg Lie superalgebra \(\widetilde{W} = W \oplus \mathbb {K}\,\), where \(W = W_0 \oplus W_1\) is a \(\mathbb {Z}_2\)graded vector space with components \(W_1^{} = V_1^{} \oplus V_1^*\) and \(W_0^{} = V_0^{} \oplus V_0^*\). The universal enveloping algebra of the Jordan–Heisenberg Lie superalgebra is called the Clifford–Weyl algebra (or quantum algebra). We denote it by \(\mathfrak {q}(W) \equiv \mathsf {U}(\widetilde{W})\).
Lemma 2.13
\([\mathfrak {q}_n(W) , \mathfrak {q}_{n^\prime }(W)] \subset \mathfrak {q}_{n + n^\prime  2}(W)\).
Proof
Lemma 2.2 asserts the commutation relation \([\mathsf {U}_n (\mathfrak {g}), \mathsf {U}_{n^\prime } (\mathfrak {g}) ] \subset \mathsf {U}_{n + n^\prime 1} (\mathfrak {g})\) for the general case of a Lie superalgebra \(\mathfrak {g}\) with bracket \([ \mathfrak {g}, \mathfrak {g}] \subset \mathfrak {g}\,\). For the specific case at hand, where the fundamental bracket \([W , W] \subset \mathbb {K}\) has zero component in W, the degree \(n + n^\prime  1\) is lowered to \(n + n^\prime 2\) by the very argument proving that lemma. \(\square \)
Lemma 2.14
\([\mathfrak {s} , \mathfrak {s}] \subset \mathfrak {s}\,\).
Proof
From the definition of \(\mathfrak {s}\) and \([W,W] \subset \mathbb {K}\) it is clear that \([\mathfrak {s}, \mathfrak {s}] \subset \mathbb {K} \oplus \mathfrak {s}\,\). The statement to be proved, then, is that [a, b] for \(a,\, b \in \mathfrak {s}\) has zero scalar part.
2.5 \(\mathfrak {osp}(W)\) inside \(\mathfrak {q}(W)\)
Lemma 2.15
The map \(\tau :\, \mathfrak {s} \rightarrow \mathfrak {osp}(W)\) is an isomorphism of Lie superalgebras.
Proof
Remark 2.1
By the isomorphism \(\tau \) every representation \(\rho \) of \(\mathfrak {s} \subset \mathfrak {q}(W)\) induces a representation \(\rho \circ \tau ^{1}\) of \(\mathfrak {osp}(W)\).
2.6 Spinor–oscillator representation
As before, starting from a \(\mathbb {Z}_2\)graded \(\mathbb {K}\)vector space \(V = V_0 \oplus V_1\,\), let the direct sum \(W = V\oplus V^*\) be equipped with the orthosymplectic form Q and denote by \(\mathfrak {q}(W)\) the Clifford–Weyl algebra of W.
The algebra \(\mathfrak {a}(V)\) now is to become a representation space for \(\mathfrak {q}(W)\). Four operations are needed for this: the operator \(\varepsilon (\varphi _1): \, \wedge ^k(V_1^*) \rightarrow \wedge ^{k+1} (V_1^*)\) of exterior multiplication by a linear form \(\varphi _1 \in V_1^*\,\); the operator \(\iota (v_1): \, \wedge ^k( V_1^*) \rightarrow \wedge ^{k1}(V_1^*)\) of alternating contraction with a vector \(v_1\in V_1\,\); the operator \(\mu (\varphi _0):\, \mathrm {S}^l (V_0^*) \rightarrow \mathrm {S}^{l+1}(V_0^*)\) of multiplication with a linear function \(\varphi _0 \in V_0^*\,\); and the operator \(\delta (v_0): \, \mathrm {S}^l (V_0^*) \rightarrow \mathrm {S}^{l1} (V_0^*)\) of taking the directional derivative by a vector \(v_0 \in V_0\,\).
Moreover, being a representation of \(\widetilde{W}\), the map q yields a representation of the universal enveloping algebra \(\mathsf {U}( \widetilde{W}) \equiv \mathfrak {q}(W)\). This representation is referred to as the spinor–oscillator representation of \(\mathfrak {q}(W)\). In the sequel we will be interested in the \(\mathfrak {osp} (W)\)representation induced from it by the isomorphism \(\tau ^{1}\).
2.6.1 Weight constraints
Since the Lie algebra \(\mathfrak {k}\) is defined on \(\mathbb {C}^N\), the \(\mathfrak {k}\)action on \(\mathfrak {a}(V)\) preserves the degree. This action exponentiates to an action of the complex Lie group K on \(\mathfrak {a}(V)\).
Proposition 2.2
Proof
This is a restatement of Theorems 8 and 9 of [10]. \(\square \)
Remark 2.2
In the case of \((\mathfrak {g}, \mathfrak {k}) = (\mathfrak {osp}(U \oplus U^*), \mathfrak {o}_N )\) it matters that \(K = \mathrm {O}_N\,\), as the connected Lie group \(K = \mathrm {SO}_N\) has invariants in \(\mathfrak {a}(V)\) not contained in \(\langle \mathfrak {g}^{(2)}.1 \rangle _\mathbb {C} \,\).
Proposition 2.2 has immediate consequences for the weights of the \(\mathfrak {g}\)representation on \(\mathfrak {a}(V)^K\). Using the notation of Sect. 2.2.1, let \(\{ H_{sj} \}\) be a standard basis of \(\mathfrak {h}\) and \(\{ \vartheta _{sj} \}\) the corresponding dual basis. We now write \(\vartheta _{0j} =: \phi _j\,\) and \(\vartheta _{1j} =: \mathrm {i}\psi _j\) (\(j = 1, \ldots , n\)).
Corollary 2.3
The representations of \(\mathfrak {osp}(U\oplus U^*)\) on \(\mathfrak {a}(V)^{\mathrm {O}_N}\) and \(\mathfrak {osp}(\widetilde{U} \oplus \widetilde{U}^*)\) on \(\mathfrak {a}(V)^{ \mathrm {Sp}_N}\) each have highest weight \(\lambda _N = \frac{N}{2}\sum _{j=1}^n (\mathrm {i}\psi _j  \phi _j)\) . Every weight of these representations is of the form \(\sum _{j=1}^n (\mathrm {i} m_j \psi _j  n_j \phi _j)\) with \(\frac{N}{2}\le m_j \le \frac{N}{2} \le n_j\,\).
Proof
The restriction \(m_j \ge \frac{N}{2}  N\) results from \(\wedge ( V_1^*) = \wedge (U_1^*\otimes (\mathbb {C}^N)^*)\) being isomorphic to \(\otimes _{j=1}^n \wedge (\mathbb {C}^N)^*\) and the vanishing of \(\wedge ^k (\mathbb {C}^N)^*= 0\) for \(k > N\,\). \(\square \)
Corollary 2.4
For each of our two cases \(\mathfrak {g} = \mathfrak {osp}(U \oplus U^*)\) and \(\mathfrak {g} = \mathfrak {osp}(\widetilde{U}\oplus \widetilde{U}^*)\) the element \({C} =  \sum _{s,j} H_{s j} \subset \mathfrak {g}\) is represented on \(\mathfrak {a}(V)^K\) by the degree operator.
Proof
2.6.2 Positive and simple roots
2.6.3 Unitary structure
We now equip the spinor–oscillator module \(\mathfrak {a}(V)\) for \(V = V_0 \oplus V_1\) with a unitary structure. The idea is to think of the algebra \(\mathfrak {a}(V)\) as a subset of \(\mathcal {O}(V_0\, , \, \wedge V_1^*)\), the holomorphic functions \(V_0 \rightarrow \wedge (V_1^*)\). 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 \(\mathbb {R}\), i.e., \(V = V_\mathbb {R} \otimes \mathbb {C}\), and that V be reinterpreted as a real vector space \(V^\prime := V_\mathbb {R} \oplus J\, V_\mathbb {R}\) with complex structure \(J \simeq \mathrm {i}\,\). Needless to say, this is done in a manner consistent with the \(\mathbb {Z}_2\)grading, so that \(V^\prime = V_0^\prime \oplus V_1^\prime \) and \(V_s^\prime = V_{s,\mathbb {R}} \oplus J\, V_{s,\mathbb {R}} \simeq V_{s,\mathbb {R}} \otimes \mathbb {C} = V_s\,\).
The subspace \(V_\mathbb {R} \subset V^\prime \) has played no role so far, but now we use it to decompose the complexification \(V^\prime \otimes \mathbb {C}\) into holomorphic and antiholomorphic parts: \(V^\prime \otimes \mathbb {C} = V \oplus \overline{V}\) and determine an operation of complex conjugation \(V^*\rightarrow \overline{V^*}\). We also fix on \(V = V_0 \oplus V_1\) a Hermitian scalar product (a.k.a. unitary structure) \(\langle \, , \, \rangle \) so that \(V_0 \perp V_1\,\). This scalar product determines a paritypreserving complex antilinear bijection \(c : \, V \rightarrow V^*\) by \(v \mapsto cv = \langle v , \, \rangle \). Composing c with complex conjugation \(V^*\rightarrow \overline{V^*}\) we get a \(\mathbb {C}\)linear isomorphism \(V \rightarrow \overline{V^*}\), \(v \mapsto \overline{cv}\,\).
Lemma 2.16
Proof
Note that although \(\delta (v)\) and \(\mu (\varphi )\) do not exist as operators on the Hilbert space \(\fancyscript {A}_V\,\), they do extend to linear operators on \(\mathcal {O}(V_0 \, , \wedge V_1^*)\) for all \(v \in V_0\) and \(\varphi \in V_0^*\,\).
2.7 Real structures
In this subsection we define a real structure for the complex vector space \(W = V\oplus V^*\) and describe, in particular, the resulting real forms of the (\(\mathbb {Z}_2\)even components of the) Howe dual partners introduced above.
Proposition 2.3
The elements of \(\mathfrak {o}(W_{1,\mathbb {R}}) \oplus \mathfrak {sp}(W_{0,\mathbb {R}}) \subset \mathfrak {osp}(W)\) are mapped via \(\tau ^{1}\) and the spinor–oscillator representation to antiHermitian operators in \(\mathrm {End}(\mathfrak {a}(V))\).
Proof
Given the real structure \(W_\mathbb {R}\) of W, we now ask how \(\mathrm {End}( W_\mathbb {R})\) intersects with the Howe pairs \((\mathfrak {osp}(U \oplus U^*), \mathfrak {o}_N)\) and \((\mathfrak {osp}(\widetilde{U} \oplus \widetilde{U}^*), \mathfrak {sp}_N)\) embedded in \(\mathfrak {osp}(W)\). By the observation that Q restricted to \(W_\mathbb {R}\) is not realvalued, \(\mathfrak {osp}(U \oplus U^*) \cap \mathrm {End}( W_\mathbb {R})\) fails to be a real form of the complex Lie superalgebra \(\mathfrak {osp}(U \oplus U^*)\), and the same goes for \(\mathfrak { osp} (\widetilde{U} \oplus \widetilde{U}^*)\). Nevertheless, it is still true that the even components of these intersections are real forms of the complex Lie algebras \(\mathfrak {osp}(U\oplus U^*)_0\) and \(\mathfrak {osp} (\widetilde{U}\oplus \widetilde{U}^*)_0\,\).
Now, to get an understanding of the intersections \(\mathfrak {o}_N \cap \mathrm {End}(W_\mathbb {R})\) and \(\mathfrak {sp}_N \cap \mathrm {End}(W_\mathbb {R})\), recall the relation \(\mathsf {D} =  \mathsf {A}^\mathrm {t}\) for \(X \in \mathfrak {osp}(W)_0\) and the fact that the action of the complex Lie algebras \(\mathfrak {o}_N = \mathfrak {o}(\mathbb {C}^N)\) and \(\mathfrak {sp}_N = \mathfrak {sp} (\mathbb {C}^N)\) on W stabilizes the decomposition \(W = V \oplus V^*\), with the implication that \(\mathsf {B} = \mathsf {C} = 0\) in both cases. Combining \(\mathsf {D} =  \mathsf {A}^ \mathrm {t}\) with \(\mathsf {D} = \overline{\mathsf {A}}\) one gets the antiHermitian property \(\mathsf {A} =  \overline{\mathsf {A}}^ \mathrm {t}\), which means that \(\mathfrak {o}_N \cap \mathrm {End}(W_\mathbb {R})\) and \(\mathfrak {sp}_N \cap \mathrm {End}(W_\mathbb {R})\) are compact real forms of \(\mathfrak {o}_N\) and \(\mathfrak {sp}_N\,\).
Let a real structure \((U \oplus U^*)_\mathbb {R}\) of \(U \oplus U^*\) be defined in the same way as the real structure \(W_\mathbb {R} = (V \oplus V^*)_\mathbb {R}\) of \(W = V \oplus V^*\).
Proposition 2.4
\(\mathfrak {osp}(U \oplus U^*)_0 \cap \mathrm {End}(W_\mathbb {R}) \simeq \mathfrak {o}( (U_1^{} \oplus U_1^*)_\mathbb {R} ) \oplus \mathfrak {sp} ((U_0^{} \oplus U_0^*)_\mathbb {R})\).
Proof
The intersection is computed by transferring the conditions \(\mathsf {D} = \overline{\mathsf {A}}\) and \(\mathsf {C} = \overline{ \mathsf {B}}\) to the level of \(\mathfrak {osp}(U \oplus U^*)_0\,\). Of course \(\mathsf {D} = \overline{\mathsf {A}}\) just reduces to the corresponding condition \(\mathsf {d} = \overline{\mathsf {a}}\,\). Because the isometry \(\psi : \, \mathbb {C}^N \rightarrow (\mathbb {C}^N)^*\) in the present case is symmetric one has \(\overline{\psi ^{1}} = \psi ^\mathrm {t} = + \psi \), so the condition \(\mathsf {C} = \overline{\mathsf {B}}\) transfers to \(\mathsf {c} = \overline{\mathsf {b}}\,\). For the same reason, the parity of the maps \(\mathsf {b}, \mathsf {c}\) is identical to that of \(\mathsf {B}, \mathsf {C}\), i.e., \(\mathsf {b} \vert _{U_0^*\rightarrow U_0^{}}\) is symmetric, \(\mathsf {b} \vert _{U_1^*\rightarrow U_1^{}}\) is skew, and similar for \(\mathsf {c}\,\). Hence, computing the intersection \(\mathfrak {osp}(U \oplus U^*)_0 \cap \mathrm {End}(W_\mathbb {R})\) amounts to the same as computing \(\mathfrak {osp}(V \oplus V^*)_0 \cap \mathrm {End}(W_\mathbb {R})\), and the statement follows from our previous discussion of the latter case. \(\square \)
Let us summarize this result.
Proposition 2.5
\(\mathfrak {osp}(\widetilde{U} \oplus \widetilde{U}^*)_0 \cap \mathrm {End}(W_\mathbb {R}) \simeq \mathfrak {usp} (U_1^{} \oplus U_1^*) \oplus \mathfrak {so}^*(U_0^{} \oplus U_0^*)\).
3 Semigroup representation
Viewing the other summand \(\mathfrak {sp}(W_0)\) of \(\mathfrak {g}_0\) as being in the infinitedimensional Weyl algebra \(\mathfrak {w}(W_0)\), it is definitely not possibly to exponentiate it in such a naive way. This is in particular due to the fact that for most \(x \in \mathfrak {sp}(W_0)\) the formal series \(\mathrm {e}^x\) is not contained in any space \(\mathfrak {w}_n (W_0)\) of the filtration of \(\mathfrak {w} (W_0)\).
As a first step toward remedying this situation, we consider \(\mathfrak {q}(W)\) as a space of densely defined operators on the completion \(\fancyscript {A}_V\) (cf. Sect. 2.6.3) of the spinor–oscillator module \(\mathfrak {a}(V)\). 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 \(\mathfrak {w}(W)\). Recall that this representation on \(\mathfrak {a}(V)\) is defined by multiplication \(\mu ( \varphi )\) for \(\varphi \in V^*\) and the directional derivative \(\delta (v)\) for \(v \in V\).
For \(x \in \mathfrak {w}(W)\) there is at least no formal obstruction to the exponential series of x existing in \(\mathrm {End}(\mathcal A_V)\). 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.
3.1 The oscillator semigroup
Here we introduce the basic semigroup in the complex symplectic group. Various structures are lifted to its canonical 2:1 covering. Actions of the real symplectic and metaplectic groups are discussed along with the role played by the cone of elliptic elements.
3.1.1 Contraction semigroup: definitions, basic properties
Letting \(\langle \, , \, \rangle \) be the unitary structure on V which was fixed in the previous chapter, we recall the complex antilinear bijection \(c : \, V \rightarrow V^*\), \(v \mapsto \langle v \, , \, \rangle \,\). There is an induced map \(C : \, W \rightarrow W\) on \(W = V \oplus V^*\) by \(C(v + \varphi ) = c^{1}\varphi + cv\,\). As before, we put \(W_\mathbb {R} := \mathrm {Fix}(C) \subset W\).
Since we have restricted our attention to the symplectic side, the vector spaces W and \(W_\mathbb {R}\) are now equipped with the standard complex symplectic structure A and real symplectic form \(\omega = \mathrm {i}A\) , respectively. From here on in this chapter we abbreviate the notation by writing \(\mathrm {Sp} := \mathrm {Sp}(W)\) and \(\mathfrak {sp} := \mathfrak {sp} (W)\). Let an antiunitary involution \(\sigma : \, \mathrm {Sp} \rightarrow \mathrm {Sp}\) be defined by \(g \mapsto C g \, C^{1}\). Its fixed point group \(\mathrm {Fix} (\sigma )\) is the real form \(\mathrm {Sp} (W_\mathbb {R}\)) of main interest. We here denote it by \(\mathrm {Sp}_\mathbb {R}\) and let \(\mathfrak {sp}_\mathbb {R}\) stand for its Lie algebra.
It is immediate that \(\mathrm {H}(W^s)\) is an open semigroup in \(\mathrm {Sp}\) with \(\mathrm {Sp}_\mathbb {R}\) on its boundary. Furthermore, \(\mathrm {H}(W^s)\) is stabilized by the action of \(\mathrm {Sp}_\mathbb {R} \times \mathrm {Sp}_\mathbb {R}\) by \(h\mapsto g_1 h \, g_2^{1}\).
The map \(\pi :\, \mathrm {Sp} \rightarrow \mathrm {Sp}\,\), \(h \mapsto h \sigma (h^{1})\), will play an important role in our considerations. It is invariant under the \(\mathrm {Sp}_\mathbb {R}\)action by right multiplication, \(\pi (hg^{1}) = \pi (h)\), and is equivariant with respect to the action defined by left multiplication on its domain of definition and conjugation on its image space, \(\pi (gh) = g \pi (h) g^{1}\). Direct calculation shows that in fact the \(\pi \)fibers are exactly the orbits of the \(\mathrm {Sp}_\mathbb {R}\)action by right multiplication. Observe that if \(h = \exp (\mathrm {i}X)\) for \(X \in \mathfrak {sp}_\mathbb {R}\,\), then \(\sigma (h) = h^{1}\) and \(\pi (h) = h^2\). In particular, if \(\mathfrak {t}\) is a Cartan subalgebra of \(\mathfrak {sp}\) which is defined over \(\mathbb {R}\,\), then \(\pi \vert _{\exp (\mathrm {i} \mathfrak {t}_\mathbb {R})}\) is just the squaring map \(t \mapsto t^2\).
3.1.2 Actions of \(\mathrm {Sp}_{\mathbb {R}}\,\)
We now fix a Cartan subalgebra \(\mathfrak {t}\) having the property that \(T_\mathbb {R} = \exp (\mathfrak {t}_\mathbb {R})\) is contained in the unitary maximal compact subgroup defined by \(\langle \, ,\, \rangle \) of \(\mathrm {Sp} (W_\mathbb {R})\). This means that T acts diagonally on the decomposition \(W = V \oplus V^*\) and there is a (unique up to order) orthogonal decomposition \(V = E_1 \oplus \ldots \oplus E_d\) into onedimensional subspaces so that if \(F_j := c(E_j)\), then T acts via characters \(\chi _j\) on the vector spaces \(P_j := E_j \oplus F_j\) by \(t(e_j\, ,f_j) = (\chi _j(t) e_j \, , \chi _j(t)^{1} f_j)\).
We now wish to analyze \(\mathrm {H}(W^s)\) via the map \(\pi :\, h \mapsto h \sigma (h^{1})\). However, for a technical reason related to the proof of Proposition 3.1 below, we must begin with the opposite map, \(\pi ^\prime : \, h \mapsto \sigma (h^{1}) h\,\). Thus let \(M := \pi ^\prime (\mathrm {H}(W^s))\) and write \(\pi ^\prime :\, \mathrm {H} (W^s)\rightarrow M\,\). The toral semigroup \(T_+ := \exp (\mathrm {i} \mathfrak {t}_\mathbb {R})\,\cap \, \mathrm {H}(W^s)\) consists of those elements \(t \in \exp ( \mathrm {i} \mathfrak {t}_\mathbb {R})\) that act as contractions on \(V^*\), i.e., \(0 < \chi _j(t)^{1} < 1\) for all \(j\,\). The restriction \(\pi ^\prime \vert _{T_+} = \pi \vert _{T_+}\) is, as indicated above, the squaring map \(t \mapsto t^2\); in particular we have \(T_+ \subset M\) and the set \(\{ g\, t g^{1} \mid t \in T_+ \, , \, g \in \mathrm {Sp}_\mathbb {R} \}\) is likewise contained in \(M\,\).
In the sequel, we will often encounter the action of \(\mathrm {Sp}_ \mathbb {R}\) on \(T_+\) and M by conjugation. We therefore denote this action by a special name, \(\mathrm {Int}(g)\, t := g\, t g^{1}\).
Proposition 3.1
\(M = \mathrm {Int}(\mathrm {Sp}_\mathbb {R}) T_+ \,\).
Proof
For \(g \in \mathrm {Sp}\) one has \(\sigma (g^{1}) = C g^{1} C^{1} = s g^\dagger s\,\). Hence if \(M \ni m = \sigma (h^{1})h\) with \(h \in \mathrm {H}(W^s)\), then \(m = s h^\dagger s h\,\). Consequently, \(\langle w, m w \rangle _s = \langle h w , h w \rangle _s < \langle w , w \rangle _s\) for all \(w \in W {\setminus} \{ 0 \}\). In particular, \(\langle w , m w \rangle _s\in \mathbb {R}\) and if \(w \not = 0\) is an meigenvector with eigenvalue \(\lambda \), it follows that \(\lambda \in \mathbb {R}\) and \(\lambda \langle w , w \rangle _s < \langle w , w \rangle _s \not = 0\,\).
Now we have \(C h C^{1} = \sigma (h)\) and hence \(C m C^{1} = m^{1}\). As a result, if \(w \not = 0\) is an meigenvector with eigenvalue \(\lambda \), then so is C w with eigenvalue \(\lambda ^{1}\). Since \(C s C^{1} =  s\), the product of \(\langle w , w \rangle _s\) with \(\langle Cw , Cw \rangle _s\) is always negative. If \(\langle w , w \rangle _s > 0\) it follows that \(\lambda < 1\) and \(\lambda ^{1} > 1\); if \(\langle Cw , Cw \rangle _s > 0\) then \(\lambda ^{1} < 1\) and \(\lambda > 1\). In both cases \(0 < \lambda \not = 1\).
Since m does indeed have at least one eigenvector, we have constructed a complex 2plane \(Q_1\) as the span of the linearly independent vectors w and Cw. The plane \(Q_1\) is defined over \(\mathbb {R}\) and, because \(0 \not = \langle w , w \rangle _s = A(Cw,w)\), it is Anondegenerate. Its Aorthogonal complement \(Q_1^\perp \) is therefore also nondegenerate and defined over \(\mathbb {R}\).
The transformation \(m \in \mathrm {Sp}\) stabilizes the decomposition \(W = Q_1 \oplus Q_1^\perp \,\). Hence, proceeding by induction we obtain an Aorthogonal decomposition \(W = Q_1 \oplus \ldots \oplus Q_d\,\). Since the \(Q_j\) are minvariant symplectic planes defined over \(\mathbb {R}\), there exists \(g \in \mathrm {Sp}_\mathbb {R}\) so that \(t := g m g^{1}\) stabilizes the above Tinvariant decomposition \(W = P_1 \oplus \ldots \oplus P_d\,\). Exchanging w with C w if necessary, we may assume that t acts diagonally on \(P_j = E_j \oplus F_j\) by \((e_j\, ,f_j) \mapsto (\lambda _j \, e_j \, , \lambda _j^{1} f_j)\) with \(\lambda _j > 1\). In other words, \(t \in T_+\,\). \(\square \)
Corollary 3.1
The semigroup \(\mathrm {H}(W^s)\) decomposes as \(\mathrm {H}(W^s) = \mathrm {Sp}_\mathbb {R} T_+ \mathrm {Sp}_\mathbb {R}\,\) . In particular, \(\mathrm {H}(W^s)\) is connected.
Proof
Because \(\mathrm {Sp}_\mathbb {R}\) and \(T_+\) are connected, so is \(\mathrm {H}(W^s) = \mathrm {Sp}_\mathbb {R} T_+ \mathrm {Sp}_\mathbb {R} \,\). \(\square \)
It is clear that \(M = \mathrm {Int}( \mathrm {Sp}_\mathbb {R} ) T_+ \subset \mathrm {H}(W^s)\). Furthermore, since both \(T_+\) and \(\mathrm {Sp}_\mathbb {R}\) are invariant under the operation of Hermitian conjugation \(h\mapsto h^\dagger \) and under the involution \(h \mapsto s h s\), we have the following consequences.
Corollary 3.2
\(\mathrm {H}(W^s)\) is invariant under \(h \mapsto h^\dagger \) and also under \(h \mapsto s h s\) . In particular, \(\mathrm {H}(W^s)\) is stabilized by the map \(h \mapsto \sigma (h^{1}) = s h^\dagger s\).
Remark 3.1
Letting \(h^\prime := \sigma (h)^{1}\) one has \(\pi ^\prime (h) = \sigma (h)^{1} h = h^\prime \sigma (h^\prime )^{1} = \pi (h^\prime )\) and hence \(M = \pi ^\prime (\mathrm {H}(W^s)) = \pi (\mathrm {H}(W^s))\). The stability of \(\mathrm {H}(W^s)\) under \(h \mapsto \sigma (h)^{1}\) was not immediate from our definition of \(\mathrm {H}(W^s)\), which is why we have been working from the viewpoint of \(\mathrm {H}(W^s) = {\pi ^\prime }^{1}(M)\) so far. Now that we have it, we may regard \(\mathrm {H}(W^s)\) as the total space of an \(\mathrm {Sp}_\mathbb {R}\)principal bundle \(\pi :\, \mathrm {H}(W^s) \rightarrow M\). We are going to see in Corollary 3.3 that this principal bundle is trivial.
Next observe that, since \(\sigma (m) = m^{1}\) for \(m = \sigma (h)^{1} h = h^\prime \sigma (h^\prime )^{1} \in M\,\), the maps \(\pi : \, M \rightarrow M\) and \(\pi ^\prime : \, M \rightarrow M\) coincide and are just the square \(m \mapsto m^2\). Thus the claim that the elements of M have a unique square root in M can be formulated as follows.
Proposition 3.2
The squaring map \(\pi = \pi ^\prime : \, M \rightarrow M\) is bijective.
Proof
Recall from Proposition 3.1 that every \(m \in M\) is diagonalizable in the sense that \(M = \mathrm {Int}( \mathrm {Sp}_\mathbb {R}) T_+\,\). Since \(\pi : \, T_+\rightarrow T_+\) is surjective, the surjectivity of \(\pi : \, M \rightarrow M\) is immediate. For the injectivity of \(\pi \) we note that the meigenspace with eigenvalue \(\lambda \) is contained in the \(m^2\)eigenspace with eigenvalue \(\lambda ^2\). The result then follows from the fact that all eigenvalues of m are positive real numbers. \(\square \)
Corollary 3.3
Each of the two \(\mathrm {Sp}_\mathbb {R}\) equivariant maps \(\mathrm {Sp}_\mathbb {R} \times M \rightarrow \mathrm {H} (W^s)\) defined by \((g,m) \mapsto gm\) and by \((g,m) \mapsto m g^{1}\) , is a bijection.
Proof
Consider the map \((g,m) \mapsto gm\,\). Surjectivity is evident from Corollary 3.1 and \(M = \mathrm {Int}(\mathrm {Sp}_\mathbb {R}) T_+\,\). For the injectivity it suffices to prove that if \(m_1 , m_2 \in M\) and \(g \in \mathrm {Sp}_\mathbb {R}\) with \(g m_1 = m_2\,\), then \(m_1 = m_2\,\). But this follows directly from \(\pi ^\prime (m_1) = \pi ^\prime (g m_1) = \pi ^\prime (m_2)\) and the fact that \(\pi ^\prime \vert _M\) is the bijective squaring map.
The proof for the map \((g,m) \mapsto m g^{1}\) is similar, with \(\pi \) replacing \(\pi ^\prime \). \(\square \)
3.1.3 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 \(m = \sigma (m^{1})\). We regard \(\psi :\, \mathrm {H}(W^s) \rightarrow \mathrm {H}(W^s) \,\), \(h \mapsto \sigma (h^{1})\), as an antiholomorphic involution and reformulate this condition as \(M \subset \mathrm {Fix} (\psi )\). In the present section we are going to show that M is a closed, connected, realanalytic submanifold of \(\mathrm {H}(W^s)\) which locally agrees with \(\mathrm {Fix} (\psi )\). This implies in particular that M is totally real in \(\mathrm {H}(W^s)\) with \(\dim _\mathbb {R} M = \dim _\mathbb {C} \mathrm {H}(W^s)\). We will also show that the exponential map identifies M with a precisely defined open cone in \(\mathrm {i} \mathfrak {sp}_\mathbb {R}\,\). We begin with the following statement.
Lemma 3.1
The image M of \(\pi \) is closed as a subset of \(\mathrm {H}(W^s)\).
Proof
Let \(h \in \mathrm {cl}(M) \subset \mathrm {H} (W^s)\). By the definition of M, we still have \(h \sigma (h)^{1} =: m \in M\,\). If \(h_n\) is any sequence from M with \(h_n \rightarrow h\,\), then \(h_n \sigma (h_n)^{1} \rightarrow m\,\). But m has a unique square root \(\sqrt{m} \in M\) and \(h_n = \sigma (h_n)^{1} \rightarrow \sqrt{m} \,\). Hence \(h = \sqrt{m} \in M\,\). \(\square \)
Remark 3.2
M of course fails to be closed as a subset of \(\mathrm {Sp}\,\). For example, \(g = \mathrm {Id}\) is in the closure of \(M \subset \mathrm {Sp}\) but is not in \(M\,\).
Lemma 3.2
The exponential map \(\exp : \, \mathfrak {sp} \rightarrow \mathrm {Sp}\) has maximal rank along \(\mathfrak {t}_+\,\).
Proof
In fact, much stronger regularity holds. For the statement of this result we recall the antiholomorphic involution \(\psi :\, \mathrm {H} (W^s) \rightarrow \mathrm {H}(W^s)\) defined by \(h \mapsto \sigma (h^{1})\) and let \(\mathrm {Fix}(\psi )^0\) denote the connected component of \(\mathrm {Fix} (\psi )\) that contains M.
Proposition 3.3
The image \(M \subset \mathrm {H}(W^s)\) of \(\pi : \, h \mapsto h \sigma (h^{1})\) is the closed, connected, totally real submanifold \(\mathrm {Fix}(\psi )^0\), which is halfdimensional in the sense that \(\dim _\mathbb {R} M = \dim _\mathbb {C} \mathrm {H} (W^s)\). The set \(\mathcal {C} = \mathrm {Ad}(\mathrm {Sp}_\mathbb {R}) \mathfrak {t}_+\,\), which is an open positive cone in \(\mathrm {i} \mathfrak {sp}_\mathbb {R}\,\), is in bijection with M by the realanalytic diffeomorphism \(\exp : \, \mathcal {C} \rightarrow M\).
Proof
Now \(\psi \) is an antiholomorphic involution. Therefore, \(\mathrm {Fix}(\psi )^0\) is a totally real, halfdimensional closed submanifold of \(\mathrm {H}(W^s)\), and since \(\mathcal {C}\) is open in \(\mathrm {i} \mathfrak {sp}_\mathbb {R}\,\), we also know that \(\dim _\mathbb {C} \mathcal {C} = \dim _\mathbb {R} \mathrm {Fix}(\psi )^0\). The maximal rank property of \(\exp \) then implies that \(M = \mathrm {im}(\exp : \mathcal {C} \rightarrow \mathrm {Fix} (\psi )^0)\) is open in \(\mathrm {Fix} (\psi )^0\). In Lemma 3.1 it was shown that M is closed in \(\mathrm {H} (W^s)\). Thus it is open and closed in the connected manifold \(\mathrm {Fix}(\psi )^0\), and consequently \(\exp : \, \mathcal {C} \rightarrow M = \mathrm {Fix}(\psi )^0\) is a local diffeomorphism of manifolds. Since we already know that \(\exp : \, \mathcal {C} \rightarrow M\) is bijective, the desired result follows. \(\square \)
Corollary 3.4
The two identifications \(\mathrm {Sp}_\mathbb {R} \times M = \mathrm {H}(W^s)\) defined by \((g,m) \mapsto gm\) and \((g,m) \mapsto m g^{1}\) are realanalytic diffeomorphisms. The fundamental group of \(\mathrm {H}(W^s)\) is isomorphic to \(\pi _1(\mathrm {Sp}_\mathbb {R}) \simeq \mathbb {Z}\,\).
Proof
3.2 Oscillator semigroup and metaplectic group
Recall that we are concerned with the Lie algebra representation of \(\mathfrak {sp}_\mathbb {R} \subset \mathfrak {sp}\) which is defined by the identification of \(\mathfrak {sp}\) with the set of symmetrized elements of degree two in the Weyl algebra \(\mathfrak {w}(W)\) and the representation of \(\mathfrak {w}(W)\) on \(\mathfrak {a}(V)\). In Sect. 3.4 we construct the oscillator representation of the metaplectic group \(\mathrm {Mp}\,\), a 2:1 cover of \(\mathrm {Sp}_\mathbb {R}\,\), which integrates this Lie algebra representation. Observe that since \(\pi _1(\mathrm {Sp}_\mathbb {R}) \simeq \mathbb {Z}\) and \(\mathbb {Z}\) has only one subgroup of index two, there is a unique such covering \(\tau : \, \mathrm {Mp} \rightarrow \mathrm {Sp}_\mathbb {R}\,\). The method of construction [11] we use first yields a representation of the 2:1 covering space \(\widetilde{\mathrm {H}} (W^s)\) of \(\mathrm {H}(W^s)\) and then realizes the oscillator representation of \(\mathrm {Mp}\) by taking limits that correspond to going to \(\mathrm {Sp}_\mathbb {R}\) in the boundary of \(\mathrm {H}(W^s)\). This representation of the oscillator semigroup \(\widetilde{\mathrm {H}}(W^s)\) is for our purposes at least as important as the representation of \(\mathrm {Mp}\,\).
The goal of the present section is to lift all essential structures on \(\mathrm {H}(W^s)\) to \(\widetilde{\mathrm {H}}(W^s)\).
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 \(u \in U\) in the fiber over \(g \in G\) is a homotopy class \(u \equiv [\alpha _g]\) of paths \(\alpha _g : \, [0,1] \rightarrow G\) connecting g with the neutral element \(e \in G\,\); and an associative product \(U \times U \rightarrow U\), \((u_1,u_2) \mapsto u_1 u_2\,\), 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 \(\pi _1(G) \equiv \pi _1(G,e)\) acts on U by monodromy, i.e., if \([\alpha _g] = u \in U\) and \([c] = \gamma \in \pi _1(G)\), then one sets \(\gamma (u) := [ \alpha _g *c] \in U\) where \(\alpha _g *c\) is the path from g to e which is obtained by composing the path \(\alpha _g\) with the loop c based at \(e\,\). This \(\pi _1(G)\)action satisfies the compatibility condition \(\gamma _1 (u_1) \gamma _2(u_2) = (\gamma _1 \gamma _2)(u_1 u_2)\) and in that sense is central.
The situation for our semigroup \(\mathrm {H} (W^s)\) is analogous except for the minor complication that the distinguished point \(e = \mathrm {Id}\) does not lie in \(\mathrm {H}(W^s)\) but lies in the closure of \(\mathrm {H}(W^s)\). Hence, by the same principles, the universal cover U of \(\mathrm {H}(W^s)\) comes with a product operation and there is a central action of \(\pi _1(\mathrm {H} (W^s))\) on U. Moreover, the product \(U \times U \rightarrow U\) still is associative. To see this, first notice that the subsemigroup \(T_+ \subset \mathrm {H}(W^s)\) is simply connected and as such is canonically embedded in U. Then for \(u_1, u_2, u_3 \in U\) observe that \(u_1 (u_2 u_3) = \gamma ((u_1 u_2) u_3)\) where \(\gamma \in \pi _1 (\mathrm {H}(W^s))\) could theoretically depend on the \(u_j\,\). However, any such dependence has to be continuous and the fundamental group is discrete, so in fact \(\gamma \) is independent of the \(u_j\) and, since \(\gamma \) is the identity when the \(u_j\) are in \(T_+\) (lifted to U), the associativity follows.
Let now \(\Gamma \simeq 2\mathbb {Z}\) denote the subgroup of index two in \(\pi _1 (\mathrm {H}(W^s)) \simeq \mathbb {Z}\) and consider \(\widetilde{ \mathrm {H}} (W^s) := U/\Gamma \), which is our object of interest. Since the \(\Gamma \)action on U is central, i.e., \(\gamma _1 (u_1) \gamma _2 (u_2) = (\gamma _1 \gamma _2) (u_1 u_2)\) for all \(\gamma _1 , \gamma _2 \in \Gamma \) and \(u_1 , u_2 \in U\), the product \(U \times U \rightarrow U\) descends to a product \(U/\Gamma \times U/\Gamma \rightarrow U/\Gamma \,\). Thus \(U/ \Gamma = \widetilde{\mathrm {H}} (W^s)\) is a semigroup, and the situation at hand is summarized by the following statement.
Proposition 3.4
The 2:1 covering \(\tau _H : \, U / \Gamma = \widetilde{\mathrm {H}} (W^s) \rightarrow \mathrm {H}(W^s)\), \([\alpha _h]\, \Gamma \mapsto h\,\), is a homomorphism of semigroups.
3.2.2 Actions of the metaplectic group
Recall that we have two 2:1 coverings: a homomorphism of groups \(\tau :\, \mathrm {Mp} \rightarrow \mathrm {Sp}_\mathbb {R}\,\), along with a homomorphism of semigroups \(\tau _H : \, \widetilde{\mathrm {H}}(W^s) \rightarrow \mathrm {H}(W^s)\). Now, a pair of elements \((g^\prime ,g) \in \mathrm {Sp}_\mathbb {R} \times \mathrm {Sp}_\mathbb {R}\) determines a transformation \(h \mapsto g^\prime h g^{1}\) of \(\mathrm {H} (W^s)\), and by the homotopy lifting property of covering maps a corresponding action of \(\mathrm {Mp} \times \mathrm {Mp}\) on \(\widetilde{ \mathrm {H}}(W^s)\) is obtained as follows.
Now, we have another realanalytic diffeomorphism \(\mathrm {Sp}_\mathbb {R} \times M \rightarrow \mathrm {H}(W^s)\) by \((g,m) \mapsto m g^{1}\), which transfers left translation in \(\mathrm {Sp}_\mathbb {R}\) to right multiplication on \(\mathrm {H}(W^s)\), and by using it we can repeat the above construction. The result is another identification \(\widetilde{ \mathrm {H}}(W^s) \simeq \mathrm {Mp} \times M\) and another \(\mathrm {Mp}\)action on \(\widetilde{\mathrm {H}}(W^s)\). Altogether we then have two actions of \(\mathrm {Mp}\) on \(\widetilde{\mathrm {H}} (W^s)\). The essence of the next statement is that they commute.
Proposition 3.5
Proof
Notice that since the submanifold \(M \subset \mathrm {H} (W^s)\) is simply connected, there exists a canonical lifting of M (which we still denote by M) to the cover \(\widetilde{\mathrm {H}}(W^s)\); this is the unique lifting by which \(T_+ \subset M\) is embedded as a subsemigroup in \(\widetilde{\mathrm {H}}(W^s)\). Proposition 3.5 then allows us to write \(\widetilde{ \mathrm {H}}(W^s) = \mathrm {Mp}.M.\mathrm {Mp}\).
3.2.3 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 \(\varphi : \, \mathrm {Sp}_\mathbb {R} \rightarrow \mathrm {Sp}_\mathbb {R}\) uniquely lifts to a Lie group automorphism \(\widetilde{\varphi }\) of the universal covering group \(\widetilde{\mathrm {Sp}}_\mathbb {R}\,\), and the latter induces an automorphism of the fundamental group \(\pi _1(\mathrm {Sp}_\mathbb {R}) \simeq \mathbb {Z}\) viewed as a subgroup of the center of \(\widetilde{\mathrm {Sp}}_\mathbb {R}\,\). Now \(\mathrm {Aut}( \pi _1( \mathrm {Sp}_\mathbb {R} )) \simeq \mathrm {Aut} (\mathbb {Z}) \simeq \mathbb {Z}_2\) and both elements of this automorphism group stabilize the subgroup \(\Gamma \simeq 2\mathbb {Z}\) in \(\pi _1(\mathrm {Sp}_\mathbb {R})\). Therefore \(\widetilde{ \varphi }\) induces an automorphism of \(\mathrm {Mp} = \widetilde{\mathrm {Sp} }_\mathbb {R} / \Gamma \,\).
Since the operation \(h\mapsto h^{1}\) canonically lifts from \(\mathrm {Sp}_\mathbb {R}\) to \(\mathrm {Mp}\) and \(h \mapsto (h^{1} )^\dagger \) is a Lie group automorphism of \(\mathrm {Sp}_\mathbb {R}\,\), it follows that Hermitian conjugation \(h\mapsto h^\dagger \) has a natural lift to \(\mathrm {Mp}\,\). The same goes for the Lie group automorphism \(h \mapsto s h s\) of \(\mathrm {Sp}_\mathbb {R}\,\).
Proposition 3.6
Hermitian conjugation \(h \mapsto h^\dagger \) and the involution \(h \mapsto s h s\) lift to unique maps with the property that they stabilize the lifted manifold M. In particular, the basic antiholomorphic map \(\psi :\, \mathrm {H}(W^s) \rightarrow \mathrm {H}(W^s)\), \(h \mapsto \sigma (h^{1})= s h^\dagger s\) lifts to an antiholomorphic map \(\widetilde{\psi }: \, \widetilde{\mathrm {H}} (W^s)\rightarrow \widetilde{\mathrm {H}}(W^s)\) which is the identity on M and \(\mathrm {Mp} \times \mathrm {Mp}\) equivariant in that \(\widetilde{\psi } (g_1 x\, g_2^{1}) = g_2 \widetilde{\psi }(x) g_1^{1}\) for all \(g_1, g_2\in \mathrm {Mp}\) and \(x \in \widetilde{\mathrm {H}}(W^s)\).
Proof
Recall that the simply connected space \(M \subset \mathrm {H} (W^s)\) has a canonical lifting (still denoted by M) to \(\widetilde{ \mathrm {H}}(W^s)\). Since all of our involutions stabilize M as a submanifold of \(\mathrm {H}(W^s)\), they are canonically defined on the lifted manifold \(M\,\). In particular, the involution \(\psi \) on M is the identity map, and therefore so is the lifted involution \(\widetilde{\psi }\).
Note furthermore that the involution defined by \(h \mapsto s h s\) is holomorphic on \(\mathrm {H}(W^s)\) and that the other two are antiholomorphic. Now \(\widetilde{\mathrm {H}}(W^s)\) is connected and the lifted version of M is a totally real submanifold of \(\widetilde{\mathrm {H}}(W^s)\) with \(\dim _\mathbb {R} M = \dim _\mathbb {C} \widetilde{ \mathrm {H}}(W^s)\). In such a situation the identity principle of complex analysis implies that there exists at most one extension (holomorphic or antiholomorphic) of an involution from M to \(\widetilde{ \mathrm {H}}(W^s)\). Therefore, it is enough to prove the existence of extensions.
Since \(h \in \widetilde{\mathrm {H}}(W^s)\) is uniquely representable as \(h = g m\) with \(g \in \mathrm {Mp}\) and \(m \in M\), the involution \(h \mapsto h^\dagger \) is extended by \(gm \mapsto (gm)^\dagger = m^\dagger g^\dagger \). Similarly, \(h \mapsto s h s\) extends by \(gm \mapsto (sgs)(sms)\), and \(h \mapsto s h^\dagger s\) does so by the composition of the other two.
The equivariance property of \(\widetilde{\psi }\) follows from the fact that \(g \mapsto s g^\dagger s\) on \(\mathrm {Mp}\) coincides with the operation of taking the inverse, \(g \mapsto g^{1}\). \(\square \)
3.3 Oscillator semigroup representation
Here we construct the fundamental representation of the semigroup \(\widetilde{\mathrm {H}}(W^s)\) on the Hilbert space \(\fancyscript {A}_V\), which in the present context we call Fock space. Our approach is parallel to that of Howe [11]: the Fock space we use is related to the \(L^2\)space of Howe’s work by the Bargmann transform [9]. (Using the language of physics one would say that Howe works with the position space wave function while our treatment relies on the phase space wave function.) In particular, following Howe we take advantage of a realization of \(\mathrm {H} (W^s)\) as the complement of a certain determinantal variety in the Siegel upper half plane.
3.3.1 Cayley transformation
Proposition 3.7
Proposition 3.8
This result is an immediate consequence of the following identity.
Lemma 3.3
Proof
Remark 3.3
3.3.2 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 \(\mathfrak {sp}\) on \(\mathfrak {a}(V) = \mathrm {S}(V^*)\) which is defined by its canonical embedding in \(\mathfrak {w}_2(W)\). We shall now construct the corresponding representation of the semigroup \(\widetilde{\mathrm {H}}(W^s)\) on the Fock space \(\fancyscript {A}_V\).
It will be seen later that the character of this representation on the lifted toral semigroup \(T_+\) is \(\mathrm {Det}^{\frac{1}{2}}(s  sh)\). This extends to \(M = \mathrm {Int}(\mathrm {Mp}) T_+\) by the invariance of the character with respect to the conjugation action of \(\mathrm {Mp}\,\). Since \(\widetilde{\mathrm {H}}(W^s)\) is connected and M is totally real of maximal dimension in \(\widetilde{\mathrm {H}} (W^s)\), the identity principle then implies that if a semigroup representation of \(\widetilde{\mathrm {H}}(W^s)\) can be constructed with a holomorphic character, this character must be given by the square root function \(h \mapsto \mathrm {Det}^{\frac{1}{2}}(s  sh)\).
Regard the complex symplectic group \(\mathrm {Sp}\) as the total space of an \(\mathrm {Sp}_\mathbb {R}\)principal bundle \(\pi : \, \mathrm {Sp} \rightarrow \pi (\mathrm {Sp})\), \(g \mapsto g \sigma (g^{1})\). Recall that the restricted map \(\pi : \, M \rightarrow M\) is a diffeomorphism, and that M contains the neutral element \(\mathrm {Id} \in \mathrm {Sp}\) in its boundary. We choose a small ball B centered at \(\mathrm {Id}\) in the base \(\pi (\mathrm {Sp})\), and using the fact that M can be identified with a cone in \(\mathrm {i}\mathfrak {sp}_\mathbb {R}\) we observe that \(A := B \, \cup \, M \subset \pi (\mathrm {Sp})\) is contractible. Now \(U := \pi ^{1}(A)\) is diffeomorphic to a product \(\mathrm {Sp}_\mathbb {R} \times A\) and thus comes with a 2:1 covering \(\widetilde{U} \rightarrow U\) defined by \(\tau : \, \mathrm {Mp} \rightarrow \mathrm {Sp}_\mathbb {R}\,\). The covering space \(\widetilde{U}\) contains \(\widetilde{ \mathrm {H}}(W^s)\), and is invariant under the \(\mathrm {Mp}\)action by right multiplication. By construction it also contains the metaplectic group \(\mathrm {Mp}\,\), which covers the group \(\mathrm {Sp}_\mathbb {R}\) in \(\mathrm {Sp}\,\).
Recall the definition of the determinant variety \(D_{\mathrm {Sp}} = \{g\in \mathrm {Sp} \mid \mathrm {Det}(\mathrm {Id}_W  g) = 0 \}\). Let \(\widetilde{D}_{\mathrm {Sp}}\) denote the set of points in \(\widetilde{U}\) which lie over \(D_{\mathrm {Sp}} \cap U\) by the covering \(\widetilde{U} \rightarrow U\).
Proposition 3.9
There is a unique continuous extension of \(\phi \) from \(\widetilde{\mathrm {H}}(W^s)\) to its closure in \(\widetilde{U}\) so that \(\phi ^2\) agrees with the lift of f from U. The intersection of \(\widetilde{D}_{\mathrm {Sp}}\) with any \(\mathrm {Mp}\) orbit in \(\widetilde{U}\) is nowhere dense in that orbit and the restriction of the extended function \(\phi \) to the complement of that intersection is realanalytic.
Remark 3.4
Before beginning the proof, it should be clarified that at the points of the lifted determinant variety, i.e., where the lifted square root \(\phi \) of the function \(f(g) = \mathrm {Det}(2s\,(\mathrm {Id}_W  g)^{1})\) has a pole, continuity of the extension means that the reciprocal of \(\phi \) extends to a continuous function which vanishes on that set.
Proof
The intersection of \(D_{\mathrm {Sp}}\) with any \(\mathrm {Sp}_\mathbb {R}\)orbit in U is nowhere dense in that orbit; therefore the same holds for the intersection of \(\widetilde{D}_{\mathrm {Sp}}\) with any \(\mathrm {Mp}\)orbit in \(\widetilde{U}\).
Let \(x \in \widetilde{U} {\setminus} \widetilde{D}_{\mathrm {Sp}}\) be a point of the boundary of \(\widetilde{\mathrm {H}}(W^s)\). Choose a local contractible section \(\Sigma \subset \widetilde{U}\) of \(\widetilde{U} \rightarrow A\) with \(x \in \Sigma \) and a neighborhood \(\Delta \) of the identity in \(\mathrm {Mp}\) so that the map \(\Delta \times \Sigma \rightarrow \widetilde{U}\,\), \((g,s) \mapsto s g^{1}\), realizes \(\Delta \times \Sigma \) as a neighborhood \(\widetilde{V}\) of x which has empty intersection with \(\widetilde{D}_{\mathrm {Sp}}\,\). By construction \(\widetilde{V} \cap \widetilde{\mathrm {H}}(W^s)\) is connected and is itself simply connected. Thus the desired unique extension of \(\phi \) exists on \(\widetilde{V}\). At x this extension is simply defined by taking limits of \(\phi \) along arbitrary sequences \(\{ x_n \}\) from \(\widetilde{\mathrm {H}}(W^s)\). Thus the extended function (still called \(\phi \)) is welldefined on the closure of \(\widetilde{\mathrm {H}}(W^s)\) and is realanalytic on the complement of \(\widetilde{D} _{\mathrm {Sp}}\) in every \(\mathrm {Mp}\)orbit in that closure. It extends as a continuous function on the full closure of \(\widetilde{ \mathrm {H}} (W^s)\) by defining it to be identically \(\infty \) on \(\widetilde{D}_{ \mathrm {Sp}}\,\), i.e., its reciprocal vanishes identically at these points. \(\square \)
Now let us proceed with our main objective of defining the semigroup representation on \(\widetilde{\mathrm {H}}(W^s)\). Recall the involution \(\psi : \, \mathrm {H}(W^s) \rightarrow \mathrm {H}(W^s)\), \(h \mapsto \sigma (h)^{1}\), and its lift \(\widetilde{\psi }\) to \(\widetilde{\mathrm {H}}(W^s)\). The following will be of use at several points in the sequel.
Proposition 3.10
\(\phi \circ \widetilde{\psi } = \overline{\phi }\,\).
Proof
By direct calculation, \(f \circ \psi = \overline{f}\,\). Thus, since \(f = \phi ^2\), we have either \(\phi \circ \widetilde{\psi } = \overline{\phi }\) or \(\phi \circ \widetilde{\psi } =  \overline{ \phi }\,\). The latter is not the case, as \(\phi \) is not purely imaginary on the nonempty set \(\mathrm {Fix} (\widetilde{\psi })\). \(\square \)
Proposition 3.11
Now we come to the main point.
Proposition 3.12
The twisted convolution for \(x , y \in \widetilde{\mathrm {H}}(W^s)\) satisfies \(\gamma _x \sharp \gamma _y = \gamma _{xy}\).
Proof
Corollary 3.5
We conclude this section by deriving a formula for the adjoint.
Proposition 3.13
The adjoint of R(x) is computed as \(R(x)^\dagger = R(\widetilde{ \psi }(x))\). In particular, \(R(x)R(x)^\dagger = R(x\, \widetilde{ \psi }(x))\).
Proof
3.3.3 Basic conjugation formula
Proposition 3.14
Proof
The basic conjugation rule now follows immediately.
Proposition 3.15
Proof
Apply Proposition 3.14 for w replaced by tw and differentiate both sides of the resulting formula at \(t=0\). \(\square \)
3.3.4 Spectral decomposition and operator bounds
Numerous properties of R are derived from a precise description of the spectral decomposition of R(x) for \(x \in M\,\). Since every orbit of \(\mathrm {Sp}_\mathbb {R}\) acting by conjugation on M has nonempty intersection with \(T_+\,\), it is important to understand this decomposition when \(x \in T_+\,\). For this we begin with the case where V is onedimensional.
Proposition 3.16
Proof
Remark 3.5
Note that as \(x \in T_+\) goes to the unit element (or, equivalently, \(\lambda \rightarrow 1\)), the expression \(R(x) f^m\) converges to \(f^m\) in the strong sense for all \(m \in \mathbb {N} \cup \{0\}\,\).
Now let V be of arbitrary dimension and assume that \(x \in T_+\) is diagonalized on \(W = V \oplus V^*\) in a basis \(\{ e_1 , \ldots , e_d , c e_1,\ldots ,c e_d \}\) with eigenvalues \(\lambda _1, \ldots \lambda _d , \lambda _1^{1}, \ldots ,\lambda _d^{1}\) respectively. Since \(x \in T_+\), we have \(\lambda _i > 1\) for all i. For \(f_i := c e_i\) and \(m := (m_1,\ldots , m_d)\) we employ the standard multiindex notation \(f^m := f_1^{m_1} \cdots f_d^{m_d}\) and \(\lambda ^m := \lambda _1^{m_1} \cdots \lambda _d^{m_d}\). In this case the multidimensional integrals split up into products of onedimensional integrals. Thus, the following is an immediate consequence of the above.
Corollary 3.6
Let \(x \in T_+\) be diagonal in a basis \(\{ e_i \}\) of V with eigenvalues \(\lambda _i\) ( \(i = 1, \ldots , d\) ). If \(f^m\) is a monomial \(f^m \equiv \prod _i (c e_i)^{m_i}\) , then \(R(x)f^m = \lambda ^{m1/2} f^m\).
One would expect the same result for the spectrum to hold for every conjugate \(g T_+ g^{1}\), 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.
Proposition 3.17
The operator norm function \(\mathrm {Mp} \times T_+ \rightarrow \mathbb {R}^{>0}\), \((g,t) \mapsto \Vert R(g\,t g^{1}) \Vert \) is bounded by a continuous \(\mathrm {Mp}\) independent function \(C(t) < 1\,\).
Proof
Since \(R(x)^\dagger = R(\widetilde{\psi }(x))\) and \(\Vert R(t g) \Vert ^2 = \Vert R(t g)^\dagger R(t g) \Vert = \Vert R(g^{1} t^2 g) \Vert \), we infer the following estimates.
Corollary 3.7
For all \(t \in T_+\) and \(g \in \mathrm {Mp}\) one has \(\Vert R(t g) \Vert < 1\) and \(\Vert R(gt) \Vert < 1\,\).
3.4 Representation of the metaplectic group
Recall that we have realized the metaplectic group \(\mathrm {Mp}\) in the boundary of the oscillator semigroup \(\widetilde{\mathrm {H}} (W^s)\) and that \(\widetilde{\mathrm {H}}(W^s)\) contains the lifted manifold \(T_+\) in such a way that the neutral element \(\mathrm {Id} \in \mathrm {Mp}\) is in its boundary. Here we show that for \(x \in T_+\) and \(g \in \mathrm {Mp}\) the limit \(\lim _{x \rightarrow \mathrm {Id}} R(gx)\) is a welldefined unitary operator \(R'(g)\) on Fock space and \(R' :\, \mathrm {Mp} \rightarrow \mathrm {U} (\fancyscript {A}_V)\) 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 socalled bounded strong* topology (see [11], 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 \(\Vert R(gx) \Vert < 1\) by Corollary 3.7, we need only prove the convergence of R(gx) f on a dense set of functions \(f \in \fancyscript {A}_V\). Let us begin with \(g = \mathrm {Id}\,\).
Lemma 3.4
If a sequence \(x_n \in T_+\) converges to \(\mathrm {Id} \in \mathrm {Mp}\,\), 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. \(\square \)
Using this lemma along with the semigroup property, we now show that the limiting operators exist and are welldefined.
Proposition 3.18
If \(x_n \in T_+\) converges to \(\mathrm {Id} \in \mathrm {Mp}\,\), then for every \(g \in \mathrm {Mp}\) the sequence of operators \(R(g x_n)\) converges pointwise, i.e., \(R(g x_n) f \rightarrow R^\prime (g) f\) for all f in \({\fancyscript{A}}_V\). The limiting operator \(R'(g)\) is independent of the sequence \(\{ x_n \}\,\).
Proof
Remark 3.6
The bounded strong\(^*\) topology also requires pointwise convergence of the sequence of adjoint operators. Therefore we must also consider sequences of the form \(R(g x_n)^\dagger \). For this (see the proof of Theorem 3.1 below) we will use the following fact.
Lemma 3.5
Let \(\{ A_n \}\) and \(\{ B_n \}\) be sequences of bounded operators and let \(C_n := A_n B_n\,\). If \(C_n\) and \(B_n\) converge pointwise with \(B_n \rightarrow B\) and the sequence \(\{ A_n \}\) is uniformly bounded, then \(A_n\) converges pointwise on the image of \(B\,\).
Proof
Applying this with \(A_n = R(g x_n g^{1})\), \(B_n = R(x_n)\) , and \(C_n = A_n B_n = R(g x_n)R(g^{1} x_n)\), we have the following statement about convergence along the conjugate \(g T_+ g^{1}\).
Proposition 3.19
For \(g \in \mathrm {Mp}\) and \(\{ x_n \}\) any sequence in \(T_+\) with \(x_n \rightarrow \mathrm {Id} \in \mathrm {Mp}\,\), it follows that \(R(g x_n g^{1})\) converges pointwise to \(R'(g) R'(g^{1})\).
Theorem 3.1
For every \(g \in \mathrm {Mp}\) and every sequence \(\{ x_n \} \subset T_+\) with \(x_n \rightarrow \mathrm {Id}\) the sequence \(\{R( g x_n) \}\) converges in the bounded strong \(^*\) topology. The limit \(R'(g)\) is independent of the sequence and defines a unitary representation \(R':\,\mathrm {Mp} \rightarrow \mathrm {U} ({\fancyscript{A}}_V)\).
Proof
From (3.11) we have \(R'(g)(\mathrm {Id}_{ \fancyscript {A}_V}  R'(g^{1}) R'(g)) = 0\) and, since \(R'(g)\) is injective, \(R'(g^{1}) R'(g) = \mathrm {Id}_{\fancyscript {A}_V}\,\). Hence \(R'(g^{1})\) is surjective, and thus \(R'(g) \in \mathrm {GL}(\fancyscript {A}_V)\) by exchanging \(g \leftrightarrow g^{1}\). 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 \(R(g x_n)^\dagger = R(\widetilde{\psi }(g x_n)) = R(x_n g^{1})\) and the discussion after (3.10), since \(R'(g)\) is now known to be an isomorphism. Unitarity of the representation is then immediate from \(R(g x_n)^\dagger \rightarrow R'(g)^\dagger \) and \(R(x_n g^{1}) \rightarrow B(g^{1}) = R'(g)^{1}\).
Let us underline two important consequences.
Proposition 3.20
Proof
If \(y_m\) and \(z_n\) are sequences in \(T_+\) which converge to \(\mathrm {Id}\,\), then, since \(x \mapsto R(x) f\) 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 righthand side converges pointwise to \(R'(g_1) R(x) R'(g_2)\). \(\square \)
We refer to \(R' :\, \mathrm {Mp} \rightarrow \mathrm {U}(\fancyscript {A}_V)\) as the oscillator representation of the metaplectic group. It has the following fundamental conjugation property.
Proposition 3.21
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). \(\square \)
3.4.1 The traceclass property
Proposition 3.22
For every \(x \in \widetilde{\mathrm {H}}(W^s)\) the operator R(x) is of trace class.
Proof
Recall that \(R(x) R(x)^\dagger = R(y)\), where \(y = x \widetilde{\psi }(x)\in M\). Since \(y = g t^2 g^{1}\) for some \(t\in T_+\) and \(\sqrt{R(g t^2 g^{1})} = R'(g)R(t)R'(g)^{1}\), the desired result follows from the explicit formula in Corollary 3.6 for the eigenvalues of t.
Proposition 3.23
Proof
Proposition 3.24
For every \(P_1,P_2\) in the Weyl algebra and every \(x \in \widetilde{\mathrm {H}}(W^s)\) the operator \(q(P_1) R(x) q(P_2)\) is of trace class on the Fock space \(\fancyscript {A}_V\). Furthermore, the function \(\widetilde{\mathrm {H}} (W^s) \rightarrow \mathbb {C}\,\), \(x \mapsto \mathrm {Tr}\; q(P_1) R(x) q(P_2)\), is holomorphic.
Proof
3.5 Compatibility with Lie algebra representation
Lemma 3.6
Proof
4 Spinor–oscillator character
The purpose of this chapter is to introduce the character of the spinor–oscillator representation of a certain supersemigroup \((\widetilde{H},\mathcal {F})\) in the orthosymplectic Lie supergroup of \(W = V \oplus V^*= W_0 \oplus W_1\,\). A summary of this short chapter is as follows.
Referring to [1] and [12] for details, we begin by briefly recalling the basic notions of Lie supergroups (in this case semigroups) and their representations. Next, we recall from Sect. 2.6 the infinitedimensional representation of the complex Lie superalgebra \(\mathfrak {osp}(W)\) on the spinor–oscillator module \(\fancyscript {A}_V\) (a.k.a. Fock space). The complex Lie group \(G = \mathrm {SO}(W_1) \times \mathrm {Sp}(W_0)\) associated to the even part of \(\mathfrak {osp}(W)\) is the base manifold of the associated Lie supergroup \(\mathrm {OSp}\,\). As a 2 : 1 covering space of the domain \(\mathrm {SO} (W_1) \times \mathrm {H} (W_0^s)\) in G, the semigroup \(\widetilde{H} := \mathrm {Spin}(W_1) \times _{\mathbb {Z}_2} \widetilde{\mathrm {H}}(W_0^s)\) inherits complex supermanifold structure. The \(\mathfrak {osp}\)representation is integrated to \(\widetilde{H}\) as a supersemigroup representation on \(\fancyscript {A}_V\). Using the character of a supergroup representation as a model, we introduce a superfunction \(\chi \) on \(\widetilde{H}\) which we regard as the character of this semigroup representation. We refer to it as the spinor–oscillator character for short.
In the last subsection of the chapter we let \(V = U \otimes \mathbb {C}^N\) and recall the setting of a Howe dual pair \((\mathfrak {g},\mathfrak {k}) = (\mathfrak {osp}(U \oplus U^*),\mathfrak {o}_N)\) or \((\mathfrak {osp} (\widetilde{U} \oplus \widetilde{U}^*),\mathfrak {sp}_N)\). In this setting, we evaluate the spinor–oscillator character \(\chi \) in two respects: (1) we Haar average it over K, which amounts to projecting from \(\fancyscript {A}_V\) to the submodule \(\fancyscript {A}_V^K\) of Kinvariants, and (2) we restrict it to a toral set \(T^+ \subset \widetilde{H}\) in a supersemigroup over \(\mathfrak {g}\,\). We then show that the restricted character \(\chi _{T^+} (t)\) coincides with the integral function I(t) of Eq. (1.1).
4.1 Background on Lie supergroups and their representations
Given a (finitedimensional) complex Lie superalgebra \(\mathfrak {g}\,\), an associated complex Lie supergroup is a ringed space \((G,\mathcal {F})\) where G is a complex Lie group associated to \(\mathfrak {g}_0\) which in addition integrates the representation of \(\mathfrak {g}_0\) on \({\mathfrak g}_1\). The group operations on G lift to sheaf morphisms that satisfy the natural conditions imposed by associativity, inverse, and fixing the identity. Uniqueness theorems allow us to choose \(\mathcal {F}\) as the sheaf of germs of holomorphic functions with values in the Grassmann algebra \(\Lambda := \wedge \mathfrak {g}_1^*\). Here we follow Berezin’s construction of the group structure, in particular his construction of the derivations associated to \(\mathfrak {g}\) which are defined by left and right ‘multiplication’.
Grassmann analytic continuation (GAC) is a process that extends functions in the structure sheaf \(\mathcal {F}\) of G to holomorphic functions with values in \(\Lambda \) on the complex Lie group \(\tilde{G}\) ([1], p. 250–257; see also [12], Sect. §\(1\)). The Lie supergroup structure morphisms of \(\mathcal {F}\) are defined by the standard complex Lie group structure of \(\tilde{G}\). Indeed, the left and right representations of \(\mathfrak {g}\) as derivations on \(\mathcal {F}\) are defined via the standard invariant vector fields defined by \(\tilde{\mathfrak {g}}\) on \(\tilde{G}\,\); and representations of the Lie supergroup \((G,\mathcal {F})\) are defined by representations (with coefficients in \(\Lambda \)) of the complex Lie group \(\tilde{G}\). We will sketch some aspects of this below, referring to [1] and [12] for details.
Our goal here is to introduce the spinor–oscillator representation and define its character. While this is a supersemigroup representation on an infinitedimensional space, we begin by recalling the basics of Lie supergroup representations on finitedimensional spaces. In abstract terms, a representation of a Lie supergroup \((G, \mathcal {F}_G)\) is a morphism \((\rho , \rho ^*)\) of Lie supergroups to \((\mathrm {GL}(V), \mathcal {F}_{\mathrm {GL}(V)})\), where \(V = V_0 \oplus V_1\) is some graded vector space. Here we are interested in holomorphic representations, so that \(\mathrm {GL}(V)\) is the complex Lie group \(\mathrm {GL}(V_0)\times \mathrm {GL}(V_1)\) and \(\mathcal {F}_{\mathrm {GL} (V)}\) is its standard matrix structure sheaf with values in the Grassmann algebra \(\Lambda \,\). The map \(\rho : \; G \rightarrow \mathrm {GL}(V_0) \times \mathrm {GL}(V_1)\) is a holomorphic homomorphism of complex Lie groups.
Let us assume that we are given a representation \(\rho _*:\;\mathfrak {g}\rightarrow \mathfrak {gl}(V)\), which we extend to \(\rho _*: \; \tilde{\mathfrak {g}} \rightarrow \Lambda _0 \otimes \mathfrak {gl}(V)_0 + \Lambda _1 \otimes \mathfrak {gl} (V)_1\) by \(\rho _*(\alpha \otimes X) = \alpha \otimes \rho _*(X)\). With \(\rho \) as above we then explicitly construct the morphism \(\rho ^*\), and hence the character \(\rho ^*(\mathrm {STr})\), as follows. First, writing elements \(\tilde{g} \in \tilde{G}\) as \(\tilde{g} = g \exp (\Xi )\) with \(g\in G\) and \(\Xi \in \mathfrak {n}\,\), we consider the Lie group representation \({\tilde{\rho}}:{\tilde{G}}\rightarrow \Lambda\otimes \text{End}(V)\) given by \(g{\text{e}}^{\Xi}\mapsto\rho(g){\text{e}}^{\rho_{\ast}(\xi)}.\) By the \(\mathbb{Z}_2\)grading \(V=V_0\otimes V_1\) the image matrices are of the form \(\left(\begin{array}{cc}A&B\\ C&D \end{array}\right),\) where the coefficients in A and D are elements of \(\Lambda_0\) and those in B and C are elements of \(\Lambda_1.\)Now if f is a superfunction in \(\mathcal {F}_{\mathrm {GL}(V)}\) with Grassmann analytic continuation \(\tilde{f}\), then \(\tilde{f} \circ \tilde{\rho }\) is a \(\Lambda \)valued holomorphic function on \(\tilde{G}\) which is the GAC of a function \(\rho ^*(f)\) in \(\mathcal {F}_G\,\). To determine the latter, one restricts \(\tilde{f} \circ \tilde{\rho }\) to a characteristic subset \(\Gamma (\xi _1, \ldots ,\xi _m)\) which is defined by a basis \(\{\xi _1, \ldots ,\xi _m\}\) of \(\mathfrak {g}_1^*\). If \(\{F_1, \ldots ,F_m\}\) denotes the dual basis, \(\Gamma (\xi _1,\ldots ,\xi _m)\) is the image of the map \(G\rightarrow \tilde{G}\) given by \(g \mapsto g\,\mathrm {e}^{\,\sum \xi _j \, F_j}\), so that \(\rho ^*(f) := \tilde{f}(\rho (g)\,\mathrm {e}^{\,\sum \xi _j\, \rho _* (F_j)})\). The character of the representation \((\rho , \rho ^*)\) then is the superfunction in \(\mathcal {F}_G\) which is defined in the expected way: \(\chi (g) := \mathrm {STr}\,(\rho (g)\, \mathrm {e}^{\,\sum \xi _j\, \rho _*(F_j)})\).
4.2 Character of the spinor–oscillator representation
In Sect. 3 we constructed a semigroup representation \(R \equiv R_0 :\; \widetilde{\mathrm {H}}(W_0^s)\rightarrow \mathrm {End}(\fancyscript {A}_{V_0})\) and also a group representation \(R^\prime :\; \mathrm {Mp}\rightarrow \mathrm {U} (\fancyscript {A}_{V_0})\), which exponentiate the oscillator representation \(\mathfrak {sp}(W_0) \rightarrow \mathrm {End}(\fancyscript {A}_{V_0})\). These are compatible in that \(\mathrm {Mp}\) acts on \(\widetilde{\mathrm {H}} (W_0^s)\) by translation on the left and right, and \(R_0(g_1 x \, g_2) = R'(g_1) R_0(x) R'(g_2)\) for all \(g_1 , g_2 \in \mathrm {Mp}\) and \(x \in \widetilde{\mathrm {H}} (W_0^s)\).
In the same vein, the complex spinor representation \(\mathfrak {o}(W_1) \rightarrow \mathfrak {gl}(\wedge V_1^*)\) exponentiates to a holomorphic Lie group representation \(R_1:\, \mathrm {Spin}(W_1) \rightarrow \mathrm {GL}(\wedge V_1^*)\). Consider now the tensor product \(\fancyscript {A}_V := \wedge V_1^*\otimes \fancyscript {A}_{V_0}\,\). By trivial extension, our representations \(R_0, R_1\) give rise to representations \(R_0 :\; \widetilde{\mathrm {H}} (W_0^s)\rightarrow \mathrm {End} (\fancyscript {A}_V)\) and \(R_1 :\, \mathrm {Spin} (W_1) \rightarrow \mathrm {GL} (\fancyscript {A}_V)\). Note that \(R_1\) and \(R_0\) commute, as they act on different factors of the tensor product \(\fancyscript {A}_V\). Note also that the group \(\mathbb {Z}_2\) acts on \(\widetilde{\mathrm {H}}(W_0^s)\) and \(\mathrm {Spin} (W_1)\) by deck transformations of the 2 : 1 coverings \(\widetilde{\mathrm {H}} (W_0^s) \rightarrow \mathrm {H}(W_0^s)\) and \(\mathrm {Spin}(W_1) \rightarrow \mathrm {SO} (W_1)\). The nontrivial element of \(\mathbb {Z}_2\) is represented by a sign change, \(R_0 \rightarrow  R_0\) and \(R_1 \rightarrow  R_1\,\).
4.3 Identification of the restricted character with I(t)
Here we show that restriction of \(\chi \) to a certain toral set in \(\widetilde{H}\) yields the integrand Z(t, k) of the autocorrelation function I(t) described in Sect. 1. In other words, we show that Z(t, k) can be expressed as the supertrace of an operator on the spinor–oscillator module \({\fancyscript {A}}_V\). Then, by taking the Haar average over the compact group K, we identify I(t) with the supertrace of an operator on the submodule \(\fancyscript {A}_V^K\) of Kinvariants.
Lemma 4.1
Proof
Since \(t_1\) and \(t_0\) are assumed to be of diagonal form, the statement holds true for a general value of n if it does so for the special case of \(n = 1\). Hence let \(n = 1\).
Let now \(\mathfrak {g}\) be the Howe dual partner of \(\mathrm {Lie}(K)\) in \(\mathfrak {osp}(W)\). We know from Proposition 2.1 that \(\mathfrak {g} = \mathfrak {osp}(U \oplus U^*)\) for \(K = \mathrm {O}_N\) and \(\mathfrak {g} = \mathfrak {osp}(\widetilde{U} \oplus \widetilde{U}^*)\) for \(K = \mathrm {USp}_N\,\). Recall also from Sect. 2.6.1 that the \(\mathfrak {g}\)representation on \(\mathfrak {a}(V)^K\) is irreducible and of highest weight \(\lambda _N = (N/2) \sum _j (\mathrm {i}\psi _j  \phi _j)\). Denote by \(\Gamma _\lambda \) the set of weights of this representation. Let \(B_\gamma = (1)^{ \gamma } \dim {\mathfrak {a}(V)^K}_\gamma \) be the dimension of the weight space \({\mathfrak {a}(V)^K}_\gamma \) multiplied with the correct sign to form the supertrace.
Corollary 4.1
Remark 4.1
On the righthand side we recognize the correlation function (see Sect. 1) which is the object of our study and, as we have explained, is related to the character of the irreducible \(\mathfrak {g}\)representation on \(\mathfrak {a}(V)^K\). The lefthand 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 Sect. 2.6.1.
5 Proof of the character formula
Throughout we will be concerned with the irreducible \(\mathfrak {g} \)representation on the subspace \({\fancyscript {A}}_V^K\) of Kinvariants in Fock space, where \(\mathfrak {g} \subset \mathfrak {osp}(V \oplus V^*)\) denotes the Howe partner defined by the Kaction on \(V \oplus V^*\), \(V = U \otimes \mathbb {C}^N\). Here our dealings with the ‘big’ Lie superalgebra \(\mathfrak {osp}(V \oplus V^*)\) in Sect. 4.2 are repeated at the level of the ‘small’ Lie superalgebra \(\mathfrak {g}\,\). In particular, we associate with \(\mathfrak {g}\) a complex supersemigroup \((\widetilde{H}^\prime , \mathcal {F})\) which serves to partially integrate the \(\mathfrak {g}\)representation on \(\fancyscript {A}_V^K\).
We then focus our attention on the spinor–oscillator character \(\chi \) pulled back to \(\widetilde{H}^\prime \times K\) and Haar averaged over K. Let \(\chi ^\prime \) be the resulting superfunction on \(\widetilde{H}^\prime \). Its restriction \(\chi _{T^+}^\prime \) to a toral set \(T^+ \subset \widetilde{H}^\prime \) is the numerical function that we were led to consider in Sect. 4.3; by Eq. (4.2) it is the function I(t) which is to be computed.
To prove the formula (5.1) for the character I(t), we study the full superfunction \(\chi ^\prime : \; \widetilde{H}^\prime \rightarrow \wedge \mathfrak {g}_1^*\). General methods show that \(\chi ^\prime \) has two distinctive properties: (i) it is radial with respect to the vector fields given by \(\mathfrak {g}\,\), and (ii) it is an eigenfunction for every LaplaceCasimir operator, i.e., every differential operator D(I) associated to a Casimir invariant I, on \((\widetilde{H} ^\prime , \mathcal {F})\). Hence we look closely at the differential equations \(D(I) \chi ^\prime = \lambda \chi ^\prime \). For \(I_\ell = \sum \big ( \phi _j^{2\ell }  (1)^\ell \psi _j^{2\ell } \big )\), \(\ell \in \mathbb {N}\), regarded as an element of the center of the universal enveloping algebra of \(\mathfrak {g}\), we show that \(D(I_\ell ) \chi ^\prime = 0\). It follows that the radial part of \(D(I_\ell )\), which is the differential operator corresponding to \(D(I_\ell )\) on \(T^+\), annihilates the restricted character \(I(t) = \chi _{T^+}^\prime (t)\).
5.1 Properties of the character \(\chi ^\prime \)

a complex Lie superalgebra \(\mathfrak {g} = \mathfrak {osp}(U \oplus U^*)\) or \(\mathfrak {osp}(\widetilde{U} \oplus \widetilde{U}^*)\);

a complex Lie supersemigroup \((\widetilde{H}^\prime , \mathcal {F})\) over \(\mathfrak {g}\,\);

the character \(\chi ^\prime \) of a representation \((\rho ,\rho _*)\) of \((\widetilde{H}^\prime , \mathcal {F}, \mathfrak {g})\) on \(\fancyscript {A}_V^K\).
First of all, to construct \(\widetilde{H}^\prime \) we take \(G \subset \mathrm {SO}(W_1) \times \mathrm {Sp}(W_0)\) to be the complex Lie group associated to the even part \(\mathfrak {g}_0 \subset \mathfrak {g}\) and let \(H^\prime \subset G\) be the semigroup which is defined by intersecting G with \(\mathrm {SO}(W_1) \times \mathrm {H} (W_0^s)\). We then define \(\widetilde{H}^\prime \) to be the preimage of \(H^\prime \) in the 2:1 covering space \(\widetilde{H} = \mathrm {Spin}(W_1) \times _{\mathbb {Z}_2} \widetilde{\mathrm {H}}(W_0^s)\) of \(\mathrm {SO}(W_1) \times \mathrm {H} (W_0^s)\).
In the current subsection we show that the character \(\chi ^\prime \) is, as would be expected, a radial superfunction. We also show that it is an eigenfunction of every LaplaceCasimir operator D(I) and if \(\mathrm {dim}\, U_0 = \mathrm {dim}\, U_1 = \mathbb {C}^n\), i.e., if we are dealing with \(\mathfrak {g} = \mathfrak {osp}_{2n2n}\,\), then the LaplaceCasimir operators annihilate \(\chi ^\prime \).
To simplify our notation, we now drop the primes and write \(\widetilde{H}, \chi \) instead of \(\widetilde{H}^\prime , \chi ^\prime \).
5.1.1 Radiality of \(\chi \)
A holomorphic superfunction f on \(\widetilde{H}\) is radial if and only if for every \(X \in \mathfrak {g}\) the sum \(L_X + R_X\) of the derivations defined by the left and right representations of X annihilate it. For a homogeneous element X of \(\mathfrak {g}\) the action of these derivations on f is defined as follows (see [1], p. 258, and [12], Sect. \(1\)). First, one considers the Grassmann analytic continuation (GAC) \(\tilde{f}\) of f. If X is even, then one differentiates \(\tilde{f}\) with respect to the local action of the 1parameter group \(\mathrm {e}^{tX}\). If X is odd, then one chooses an arbitrary element \(\alpha \in \Lambda _1\) and differentiates \(\tilde{f}\) with respect to the local action of \(\mathrm {e}^{tY}\) where \(Y = \alpha X\). One shows in this latter case that the result is of the form \(\alpha L_X(\tilde{f})\) where \(L_X\) is an odd derivation which does not depend on \(\alpha \). Of course \(\alpha L_X\) could be identically zero; so it might be necessary to extend the Grassmann algebra in order to prevent this from happening unless \(L_X\) vanishes identically. Thus in both the odd and even cases we have an operator \(L_X\) on the sheaf of \(\Lambda \)valued holomorphic functions on the complex Lie group \(\tilde{G}\). One checks that these operators stabilize the subspace of functions on \(\tilde{G}\) which arise through GAC from \((G,\mathcal {F}_G)\) and that the resulting map \(\mathfrak {g} \rightarrow \mathrm {Der}(\mathcal {F}_G)\), \(X\mapsto L_X\) is a Lie superalgebra morphism. Carrying this out in the analogous way by multiplying the 1parameter groups \(\mathrm {e}^{tX}\) and \(\mathrm {e}^{ tY}\) on the right, one obtains the morphism defined by \(X\mapsto R_X\,\).
The key for showing that the operators \(L_X+R_X\) annihilate the character \(\chi \) is the fact that the GAC \(\tilde{\chi }\) of \(\chi \) is \(\mathrm {STr}\; \tilde{\rho }\), where \(\tilde{\rho }\) is the associated complex Lie semigroup representation of \(\widetilde{H} N\). One defines this by \(\tilde{\rho }(x\, \mathrm {e}^{\Xi }) := \rho (x)\, \mathrm {e}^{\rho _*(\Xi )}\), as before.
Proposition 5.1
The character \(\chi \) is a radial holomorphic superfunction on \(\widetilde{H}\).
Proof
Let \(X \in \mathfrak {g}_1\) and \(Y = \alpha X\) be as above. By using the multiplicative semigroup property, the fact that \(\mathrm {STr}\; [ \tilde{\rho }(g) , \rho _*(Y) ] = 0\), and using the \(\Lambda \)linearity of \(\mathrm {STr}\) to factor out \(\alpha \), we observe that \(\alpha (L_X + R_X)\) annihilates \(\mathrm {STr}\, \tilde{\rho }\). As we mentioned above, in order to conclude that \(L_X + R_X\) annihilates this, it may be necessary to extend the Grassmann coefficients. For \(X \in \mathfrak {g}_0\) the argument is even simpler, as it isn’t necessary to multiply by \(\alpha \). \(\square \)
5.1.2 The character \(\chi \) is a LaplaceCasimir eigenfunction
Proposition 5.2
\(\chi \) is an eigenfunction of every LaplaceCasimir operator D(I).
Proof
Since I lies in the center of \(\mathsf {U}(\mathfrak {g})\), the operator \(\rho _*(I)\) commutes with all operators defined by \(\mathsf {U} (\mathfrak {g})\) on \(\mathfrak {a}(V)^K\). Now according to Proposition 2.2 the subalgebra \(\mathfrak {g}^{(2)} \oplus \mathfrak {g}^{(0)} \subset \mathfrak {g}\) of degreenonincreasing operators stabilizes the vacuum space \(\langle 1 \rangle _\mathbb {C} \subset \mathfrak {a}(V)^K\). By the irreducibility of the \(\mathfrak {g}\)representation on \(\mathfrak {a}(V)^K\) this subalgebra stabilizes no other proper subspace of \(\mathfrak {a}(V)^K\). Therefore, the linear operator \(\rho _*(I)\) stabilizes \(\langle 1 \rangle _\mathbb {C}\) with some eigenvalue \(\lambda (I)\). Furthermore, \(1 \in \mathbb {C} \subset \mathfrak {a}(V)^K\) is a cyclic vector for the action of \(\mathsf {U}(\mathfrak {g})\) on \(\mathfrak {a}(V)^K\). Thus \(\rho _*(I) \equiv \lambda (I)\, \mathrm {Id}_{\mathfrak {a}(V)^K}\) and the desired result follows. \(\square \)
5.1.3 Vanishing of the \(D(I_\ell )\)eigenvalues
Recall now from Sect. 2.2.2 that for every \(\ell \in \mathbb {N}\) we have a Casimir element \(I_\ell \in \mathsf {U} (\mathfrak {osp})\) of degree \(2\ell \). Recall also that under the assumption of equal dimensions \(V_0 \simeq V_1\) we introduced \(\partial , \widetilde{\partial } \in \mathfrak {osp}_1\, \), \(C = [\partial , \widetilde{\partial } ] \in \mathfrak {osp}_0 \,\), and \(F_\ell \in \mathsf {U} (\mathfrak {osp})\) such that \(I_\ell = [\partial ,F_\ell ]\) and \([\partial ,C] = 0\,\). For the proof of Proposition 5.3 below, we will make use of these objects at the level of \(U_0 \simeq U_1\,\).
Proposition 5.3
Let \(U = U_0 \oplus U_1\) be a \(\mathbb {Z}_2\) graded vector space with \(U_0 \simeq U_1\,\) and \(\chi \) be the character of the supersemigroup representation of \(\widetilde{H}\) which is the integrated form of the irreducible \(\mathfrak {g}\)representation on \(\mathfrak {a}(V)^K\) for \(V = U \otimes \mathbb {C}^N\). Then \(D(I_\ell )\chi = 0\) for all \(\ell \in \mathbb {N}\,\).
Remark 5.1
The condition \(U_0 \simeq U_1\) is needed in order for the formula \(I_\ell = [ \partial , F_\ell ]\) of Lemma 2.9 to be available.
Proof
5.2 Derivation of the differential equations
Here we outline a foundational result which leads to a proof that the differential operators \(D_\ell \circ J\), where J is the square root of a certain (super)Jacobian and \(\ell \in \mathbb N\), annihilate I(t). Due primarily to Berezin [1], this result has been adapted to our context in [12].
5.2.1 Radial operators
At this stage, another object enters: a space \(T^+\) which plays the role of maximal complex torus in \(\widetilde{H}\). To introduce it, we recall from the beginning of Sect. 5.1 that we are given a complex semigroup \(H^\prime \) inside the complex Lie group G with Lie algebra \(\mathfrak {g}_0\,\). In terms of this structure, the space \(T^+\) is defined as the preimage in \(\widetilde{H}\) of the intersection of the standard Cartan torus \(T \subset G\) with \(H^\prime \).
From Sect. 2.2.2 we again recall that for every \(\ell \in \mathbb {N}\) we have a Casimir element \(I_\ell \in \mathsf {U} (\mathfrak {osp})\) of degree \(2\ell \). We also recall the expression (5.2) for the differential operators \(D_\ell \) in terms of the local coordinates \(\phi _1, \ldots , \phi _n, \psi _1, \ldots , \psi _n\) we have been using all along.
Theorem 5.1
Remark 5.2
While some choice of domain B is necessary to ensure that both J(t) and \(J(t)^{1}\) exist for all \(t \in B \cap T^+\), the expression for \(\dot{D}(I_\ell )\) does not depend on B.
5.2.2 The differential equations
In view of the formula for \(\dot{D}(I_\ell )\), and knowing that \(D(I_\ell ) \chi = 0\) for all \(\ell \), the following is a key technical step.
Lemma 5.1
For all \(\ell \in \mathbb {N}\) we have \(D_\ell \, J = 0\).
Proof
The reasoning for the case of \(\mathfrak {g} = \mathfrak {osp}(\widetilde{U} \oplus \widetilde{U}^*)\) is no different. \(\square \)
Corollary 5.1
The restriction \(\chi _{T^+}\) of the character \(\chi \) from \(\widetilde{H}\) to \(T^+\) satisfies the system of differential equations \(D_\ell \, J \, \chi _{T^+} = 0\) for all \(\ell \in \mathbb {N}\,\).
Proof
Since \(D(I_\ell ) \chi = 0\) and hence by restriction \(\dot{D}(I_\ell ) \chi _{T^+} = 0\), it follows from Theorem 5.1 that \(J^{1} (D_\ell + Q_{\ell 1}) J \chi _{T^+} = 0\) for every \(\ell \in \mathbb {N}\,\). Now by applying the operator \(\dot{D}(I_\ell )\) to the constant function 1 and using Lemma 5.1, we obtain \(0 = J \dot{D}(I_\ell ) 1 = D_\ell \, J + Q_{\ell 1}\, J = c_{\ell 1}\,J\), where \(c_{\ell 1}\) is the constant term of the differential operator \(Q_{\ell 1}\,\), and from this we conclude that \(c_{\ell  1} = 0\) for all \(\ell \in \mathbb {N}\,\). It then follows by induction on \(\ell \) that \(D_\ell \, J \, \chi = 0\) for all \(\ell \in \mathbb {N}\,\). \(\square \)
Corollary 5.2
Let \(\gamma \) be the (restricted) character of an irreducible representation of the Lie supergroup \((\mathfrak {gl}(U),\mathrm {GL}(U_0) \times \mathrm {GL}(U_1))\) on a finitedimensional \(\mathbb {Z}_2\) graded vector space \(V = V_0 \oplus V_1\,\). If \(U_0 \simeq U_1\) but \(\mathrm {dim}(V_0) \not = \mathrm {dim}(V_1)\), then \(D_\ell J_0\,\gamma = 0\) for all \(\ell \in \mathbb {N}\,\).
5.3 Global \(G_\mathbb R\)invariance and the Weyl group
Recall that \(\chi \) is only invariant by the local action of the supergroup \((G,\mathfrak {g})\) on \(\widetilde{H}\). However, there exists a real form \(G_\mathbb {R}\) which acts globally on \(\widetilde{H}\) by conjugation and therefore \(\chi \) is invariant by this action.
Since \(\chi \) is invariant under the \(G_\mathbb {R}\)action by conjugation, its restriction to a real toral semigroup in \(T^+\) is invariant under the action of the Weyl group W defined by \(G_\mathbb {R}\,\). Since \(\chi \) is holomorphic, its restriction to the complexification \(T^+\) is likewise Winvariant. Now \(G_\mathbb {R}\) decomposes as a direct product of two factors and so W also decomposes in this way. For both cases (\(K = \mathrm {O}_N\), \(\mathrm {USp}_N\)) the second factor of the Weyl group W is just the permutation group \(\mathrm {S}_n\,\). As a matter of fact, conjugation of a diagonal element \(t_0 \in M_\mathrm {Sp}\) or \(t_0 \in M_\mathrm {SO}\) by \(g \in \mathrm {Mp}((U_0^{} \oplus U_0^*)_\mathbb {R})\) or \(g \in \mathrm {SO}^*( U_0^{ } \oplus U_0^*)\) can return another diagonal element only by permutation of the eigenvalues \(\mathrm {e}^{\phi _1} , \ldots , \mathrm {e}^{\phi _n}\) of \(t_0\,\). (No inversion \(\mathrm {e}^{\phi _j} \rightarrow \mathrm {e}^{ \phi _j}\) is possible, as this would mean transgressing the oscillator semigroup.) This factor \(\mathrm {S}_n\) of W will play no important role in the following, as the expressions we will encounter are automatically invariant under such permutations.
The Weyl group \(W_{\mathrm {Sp}}\) is generated by the permutations of these planes and the involutions which are defined by conjugation by the mapping that sends \(e_j \mapsto c e_j\) and \(c e_j \mapsto  e_j\,\). The Weyl group \(W_\mathrm {SO}\) is generated by the permutations together with the involutions which are induced by the mappings that simply exchange \(e_j\) with \(c e_j\,\). Since we are in the special orthogonal group and the determinant for a single exchange \(e_j \leftrightarrow c e_j\) is \(1\), the number of involutions in any word in \(W_\mathrm {SO}\) has to be even.
In summary, the Waction on our standard bases of linear functions, \(\{ \mathrm {i}\psi _j\}\) and \(\{\phi _j \}\), is given by the respective permutations together with the action of the involutions defined by sign change, \(\mathrm {i} \psi _j \mapsto  \mathrm {i} \psi _j\,\). 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 \(\{ \mathrm {e}^{ \mathrm {i} \psi _j} \}\) and \(\{ \mathrm {e}^{\phi _j} \}\). As a final remark, let us note that the Weylgroup symmetries of the function \(\chi _{T^+}\) can also be read off directly from the explicit expression (4.2). In particular, the absence of reflections \(\phi _j \rightarrow \phi _j\) is clear from the conditions \(\mathfrak {Re}\, \phi _j > 0\,\).
5.4 Formula for \(\chi _{T^+}\)
Recall that the main goal of this paper is to compute the restriction \(\chi _{T^+}\) to \(T^+\) of the character \(\chi \) which is defined on \(\widetilde{H}\) and plays the role of a character of the \(\widetilde{H}\)representation on the space of invariants \({\fancyscript{A}}_V^K\) in the spinor–oscillator module. Here \(\widetilde{H}\) is the 2 : 1 cover of an open semigroup in the complex Lie group G of the Howe partner supergroup of K. From now on we will only deal with the restriction of this numerical part and therefore we simplify notation by denoting it by \(\chi _{T^+} \equiv \chi \).
We have restricted ourselves to the cases where K is either \(\mathrm {O}_N\) or \(\mathrm {USp}_N\,\). The representation on \(\mathfrak {a}(V)^K\) is defined at the infinitesimal level on the full complex Lie superalgebra \(\mathfrak {g}\) which is the Howe partner of K in the canonical realization of \(\mathfrak {osp}\) in the Clifford–Weyl algebra of \(V \oplus V^*\). It has been shown that \(\chi : \, T^+ \rightarrow \mathbb {C}\) satisfies the differential equations \(D_\ell \, J\, \chi = 0\,\). We now recall that the nonzero weights of the Fourier expansion of \(\chi \) are constrained to a certain region and show that \(\chi \) is the unique holomorphic function satisfying both the weight constraints and the differential equations.
5.4.1 Uniqueness
Recall that \(\Gamma _\lambda \) denotes the set of weights of the \(\mathfrak {g}\)representation on \(\mathfrak {a}(V)^K\). From Corollary 2.3 we know that the weights \(\gamma = \sum _{j = 1}^n (\mathrm {i} m_j \psi _j  n_j \phi _j) \in \Gamma _\lambda \) satisfy the weight constraints \( \frac{N}{2} \le m_j \le \frac{N}{2} \le n_j\,\). The highest weight is \(\lambda = \frac{N}{2} \sum (\mathrm {i} \psi _j  \phi _j)\). By the definition of the torus \(T^+\) the weights \(\gamma \in \Gamma _\lambda \) are analytically integrable and we now view \(\mathrm {e}^\gamma \) as a function on \(T^+\).
Theorem 5.2
The character \(\chi : \, T^+ \rightarrow \mathbb {C}\) is annihilated by all differential operators \(D_\ell \circ J\) for \(\ell \in \mathbb {N}\,\), and it has a convergent expansion \(\chi = \sum B_\gamma \, \mathrm {e}^{\gamma }\) where the sum runs over weights \(\gamma = \sum _{j = 1}^n (\mathrm {i} m_j \psi _j  n_j \phi _j)\) satisfying the constraints \( \frac{N}{2} \le m_j \le \frac{N}{2} \le n_j\,\). For the case of \(K = \mathrm {USp}_N\) it is the unique Winvariant function on \(T^+\) with these two properties and \(B_\lambda = 1\). For \(K = \mathrm {O}_N\) it is the unique Winvariant function on \(T^+\) with these two properties and \(B_\lambda = 1\,\), \(B_{\lambda  \mathrm {i}N\psi _n} = 0\,\).
Remark 5.4
To verify the property \(B_{\lambda  \mathrm {i}N\psi _n} = 0\) which holds for the case of \(K = \mathrm {O}_N\,\), look at the righthand side of the formula of Corollary 4.1: in order to generate a term \(\mathrm {e}^\gamma = \mathrm {e}^{\lambda  \mathrm {i}N\psi _n}\) in the weight expansion, you must pick the term \(\mathrm {e}^{ \mathrm {i} N\psi _n}\) in the expansion of the determinant for \(j = n\) in the numerator; but the latter term depends on k as \(\mathrm {Det}(k)\) which vanishes upon taking the Haar average for \(K = \mathrm {O}_N \,\). By Winvariance the property \(B_{\lambda  \mathrm {i}N\psi _n} = 0\) is equivalent to \(B_{\lambda  \mathrm {i} N\psi _j} = 0\) for all \(j\,\).
In view of this Remark and Corollaries 2.3 and 5.1, it is only the uniqueness statement of Theorem 5.2 that remains to be proved here. This requires a bit of preparation, in particular to appropriately formulate the condition \(D_\ell \, J \chi = 0\,\). For that we develop \(J \chi \) in a series \(J \chi = \sum _\tau a_\tau f_\tau \) where the \(f_\tau \) are \(D_\ell \)eigenfunctions for every \(\ell \in \mathbb {N}\,\).
From now on we consider the Eq. (5.4) only in those cases where \(\tilde{\gamma }\) is itself a weight of our representation. (We have license to do so as only the uniqueness part of Theorem 5.2 is at stake here). In this case we have the following key fact.
Lemma 5.2
If \(\gamma \in \Gamma _\lambda \) and the eigenvalue \(E(\ell , \gamma )\) vanishes for all \(\ell \in \mathbb {N}\,\), then \(\gamma \) is the highest weight \(\lambda \).
Proof
We are now able to give the proof of the uniqueness statement of Theorem 5.2.
Proof
In the case of \(K = \mathrm {O}_N\) we are confronted with the fact that the weight \(\gamma = \lambda  \mathrm {i} N \psi _n\) satisfies the weight constraints and yet gives \(E(\ell ,\gamma ) = 0\) for all \(\ell \). However, in this exceptional situation the conditions of Theorem 5.2 provide that \(B_{\lambda  \mathrm {i}N\psi _n} = 0\). Thus the expansion coefficients \(B_\gamma \) are still uniquely determined by our recursion procedure. \(\square \)
5.4.2 Explicit solution of the differential equations
Lemma 5.3
The function \(S_W (\mathrm {e}^{\lambda } Z)\) is holomorphic on \(\cap _{j=1}^n \{ \mathfrak {Re}\, \phi _j > 0 \}\).
Proof
Lemma 5.4
For all \(\ell \, , N \in \mathbb {N}\) the function \(\varphi :\, T^+ \rightarrow \mathbb {C}\) defined by \(\varphi = S_W(\mathrm {e}^{\lambda _N} Z)\) is a solution of the differential equation \(D_\ell \, J \varphi = 0\,\).
Proof
The factor \(\mathrm {e}^{\lambda _N + \delta ^\prime }\) is the character of the representation \((\frac{N \mp 1}{2} \mathrm {STr}\, , \mathrm {SDet}^{\frac{N \mp 1}{2}})\) of (a double cover of) the Lie supergroup \((\mathfrak {g}^{(0)} , \mathrm {GL}(U_0) \times \mathrm {GL}(U_1))\). This representation is onedimensional, and from Corollary 5.2 we have \(D_\ell (J_0\, \mathrm {e}^{ \lambda _N + \delta ^\prime }) = 0\) for all \(\ell , N \in \mathbb {N}\,\).
The statement of the lemma now follows by applying the Winvariant differential operator \(D_\ell \) to the formula for \(\mathrm {ord}(W_\lambda ) J \varphi \) above. \(\square \)
5.4.3 Weight constraints
Here we carry out the final step in proving the explicit formula for the character \(\chi \) of our representation. Since the formula in the case of \(K = \mathrm {SO}_N\) follows directly from that for \(K = \mathrm {O}_N\) (see Sect. 1) and the case of \(K = \mathrm {U}_N\) has been handled in [7], we need only discuss the cases of \(K = \mathrm {O}_N\) and \(K = \mathrm {USp}_N\,\).
We have shown above that \(\varphi = S_W(\mathrm {e}^\lambda Z)\) is holomorphic on the product of the full complex torus of the variables \(\mathrm {e}^{\mathrm {i} \psi _k}\) with the domain defined by \(\mathfrak {Re}\, \phi _k > \,0\,\). Although the individual terms \(\varphi _{ [w]}\) in the representation of \(\varphi \) have poles (which cancel in the Weylgroup averaging process) we may still develop each term of \(\varphi \) in a series expansion; this will in fact yield the desired weight constraints.
Lemma 5.5
If \(\gamma = \sum (\mathrm {i} m_k \psi _k  n_k \phi _k)\) and \(B_\gamma \not = 0\,\), then \(m_1 \le \frac{N}{2}\, \).
Proof
If \(w(\mathrm {i} \psi _1) = \mathrm {i} \psi _1\,\), then by the same argument as in the case of \([w] = [\mathrm {Id}]\) we see that \(\mathrm {e}^{ \mathrm {i} \psi _1}\) occurs in the series development of \(\varphi _{[w]}\) with a power \(m_1\) of at most \(\frac{N}{2}\,\).
Using Weylgroup invariance, this estimate for \(m_1\) will now yield the desired result.
Lemma 5.6
Suppose that \(K = \mathrm {O}_N\) and let \(\varphi = \sum B_\gamma \, \mathrm {e}^\gamma \) be the globally convergent series expansion of the proposed character \(\varphi = S_W(\mathrm {e}^\lambda Z)\). Then for every weight \(\gamma = \sum (\mathrm {i} m_k \psi _k  n_k \phi _k)\) with \(B_\gamma \not = 0\) it follows that \( \frac{N}{2} \le m_k \le \frac{N}{2} \le n_k\,\).
Proof
The inequality \(n_k \ge \frac{N}{2}\) was proved above as an immediate consequence of the fact that the Weyl group W effectively acts only on the \(\psi _j\,\).
Above we showed that on the region R the proposed character \(\varphi \) has a series development where in every \(\gamma \) the coefficient \(m_1\) of \(\mathrm {i} \psi _1\) is at most \(\frac{N}{2}\,\). Recalling the fact that the function \(\varphi \) is holomorphic on \(T^+\), we infer that \(m_1 \le \frac{N}{2}\) also holds true for the globally convergent series development \(\sum B_\gamma \, \mathrm {e}^\gamma \).
To get the same statement for \(\mathrm {i} \psi _k\) with \(k \not = 1\) we just change the definition of R to R(k) defined by the inequalities \(\mathfrak {Re} (\mathrm {i} \psi _k) > \mathfrak {Re} (\mathrm {i} \psi _1) > \ldots > 0 \, \). Arguing for general k as we did for \(k = 1\) in the above lemma, we show that the coefficient \(m_k\) of \(\mathrm {i} \psi _k\) in every \(\gamma \) in the series expansion of every \(\varphi _{[w]}\) on R(k) is at most \(\frac{N}{2}\,\). By the holomorphic property, the same is true for the global series expansion of the proposed character \(\varphi \).
Hence, to complete the proof we need only show the inequality \(m_k \ge \frac{N}{2}\,\). But for this it suffices to note that for every k there is an element w of the Weyl group with \(w(\mathrm {i} \psi _k) =  \mathrm {i}\psi _k\,\). Indeed, using the Weyl invariance of \(\varphi \), if there was some \(\gamma \) where \(m_k <  \frac{N}{2}\,\), then the coefficient of \(\mathrm {i} \psi _k\) in \(w(\gamma )\) would be larger than \(\frac{N}{2}\,\). \(\square \)
To complete our work, we must prove Lemma 5.6 for the case \(K = \mathrm {USp}_N\,\). For this we use the same notation as above for the basic linear functions, namely \(\mathrm {i} \psi _k\) and \(\phi _k\,\). Here, compared to the \(\mathrm {O}_N\) case, there are only slight differences in the \(\lambda \)positive roots and the Weyl group. The only difference in the roots is in \(\Delta ^+_{\lambda ,0}\) where \(\mathrm {i} \psi _j + \mathrm {i} \psi _k\) occurs in the larger range \(j \le k\) and \(\phi _j + \phi _k\) in the smaller range \(j < k\,\). The Weyl group acts by permutation of indices on both the \(\mathrm {i} \psi _j\) and \(\phi _j\) and by sign reversal on the \(\mathrm {i} \psi _j\, \). 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 5.6 in this case, we need only go through the argument in the \(\mathrm {O}_N\) case and make minor adjustments. In fact, the main step is to prove Lemma 5.5 and, there, the only change is that the range of j for the factor \(1  \mathrm {e}^{\mathrm {i}(\psi _1 + \psi _j)}\) is larger. This is only relevant in the case \(w(\mathrm {i} \psi _1) =  \mathrm {i} \psi _1\,\), where we rewrite the additional denominator term \((1  \mathrm {e}^{w(2\mathrm {i} \psi _1)})^{1}\) as \( \mathrm {e}^{2\mathrm {i} \psi _1}(1  \mathrm {e}^{ 2\mathrm {i} \psi _1})^{1}\). Hence the factor in front of the ratio of products on the r.h.s. of equation (5.7) gets an additional factor of \(\mathrm {e}^{2\mathrm {i}\psi _1}\) and now is \(\mathrm {e}^{\mathrm {i} (\frac{N}{2} + 1)\psi _1}\). Thus \(m_1 \le \frac{N}{2} 1\) which certainly implies \(m_1 \le \frac{N}{2}\,\).
Let us summarize this discussion.
Theorem 5.3
Declarations
Acknowledgements
The work for this project was carried out with the support of SFB/Tr 12, Symmetries and Universality in Mesoscopic Systems, of the Deutsche Forschungsgemeinschaft.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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