Asymptotics of Spherical Superfunctions

on Rank One Riemannian Symmetric

Superspaces

Alexander Alldridge, Wolfgang Palzer

Received: June 21, 2014 Revised: August 8. 2014

Communicated by Patrick Delorme

Abstract. We compute the Harish-Chandra c-function for a generic class of rank-one purely non-compact Riemannian symmetric superspaces X = G/K in terms of Euler Γ functions, proving that it is meromorphic. Compared to the even case, the poles of the c-function are shifted into the right half-space. We derive the full asymptotic Harish-Chandra series expansion of the spherical superfunctions on X. In the case where the multiplicity of the simple root is an even negative number, they have a closed expression as Jacobi polynomials for an unusual choice of parameters.

2010 Mathematics Subject Classification: Primary 22E45, 17B15; Secondary 58A50 Keywords and Phrases: Harish-Chandra c-function, Lie supergroup, Riemannian symmetric superspace, spherical superfunction

1

1This research was supported by Deutsche Forschungsgemeinschaft, grant no. DFG ZI 513/2-1 (AA), SFB/TR 12 “Symmetries and University in Mesoscopic Systems” (AA, WP), and by the Institutional Strategy of the University of Cologne within the German Excellence Initiative (AA).

1. Introduction

Let X = G/K be a Riemannian symmetric space. The mainstay of harmonic analysis on X is the study of the asymptotic behaviour of the spherical functions ϕ_{λ} [20, 23, 24]. These are K-invariant joint eigenfunctions of the G-invariant differential operators D on X, viz.

Dϕ_{λ} = Γ(D)(λ)ϕ_{λ},

where Γ(D)(λ) is the value of the Harish-Chandra homomorphism. A basic observation is that ϕ_{λ}(a) admits an asympotic series expansion whose leading contribution is c(λ)e^{(λ−ϱ)(a)}, with ϱ the half sum of the positive roots. Here, c(λ) is Harish-Chandra’s famous c-function. Moreover, c(λ) admits an extension, as a meromorphic function, to the entire dual 𝔞^{∗} of the complex Cartan subspace, due to the Gindikin–Karpelevič formula

c(λ) = c_{0} ∏_{α} \frac{2^{λ_{α}−1}Γ(λ_{α})}{Γ(\frac{1}{2}(λ_{α} + \frac{m_{α}}{2} + 1))Γ(\frac{1}{2}(λ_{α} + \frac{m_{α}}{2} + m_{2α}))},

where the product is over all positive indivisible roots and λ_{α} := \frac{⟨λ,α⟩}{⟨α,α⟩}. This equation is basic in the proof of the fundamental theorems of harmonic analysis on Riemannian symmetric spaces: the support theorem for wave packets, the inversion, Paley–Wiener, Plancherel and Schwartz isomorphism theorems for the spherical Fourier transform. Although the situation is more complicated for the Helgason–Fourier transform (for τ-spherical functions) and even more so for reductive non-Riemannian symmetric spaces, the basic philosophy of studying the asymptotics of spherical functions (or their replacements by more general Eisenstein integrals) remains valid, cf. Ref. [34]. In the present paper, we study the asymptotics of spherical functions defined on a supersymmetric generalisation of Riemannian symmetric spaces. These play a basic role in the so-called Efetov supersymmetry method of condensed matter physics. Indeed, as shown by Zirnbauer [39, 40], harmonic analysis on such superspaces can be used to give precise analytic expressions for the mean conductance of quasi-one dimensional disordered fermionic systems. Remarkably, contrary to the classical case, there are discrete contributions in the Plancherel decomposition, and this is visible and relevant in the physics of the corresponding systems. Since the Plancherel measure is governed by the c-function, the failure of absolute continuity can already be observed from the location of the c-function zeroes. Indeed, let (G, K) be one of the symmetric pairs listed in Table 4.1 below. Fix an indivisible restricted root α and h_{0} with α(h_{0}) = 1. Let ϱ = \frac{1}{2}(m_{α}α + 2m_{2α}α) where m_{α} and m_{2α} are the multiplicities of α and 2α, respectively, and identify λ ≡ λ(h_{0}). Our first main result is the following theorem.

Theorem A. For every ℜλ > 0, the limit

c(λ) = lim_{t⟶∞} ϕ_{λ}(e^{th_{0}})e^{−t(λ−ϱ)}

exists. For some choice of c_{0} ≡ c_{0}(ϱ) ≠ 0, it is given by

(1.1)  c(λ) = c_{0} \frac{2^{−λ}Γ(λ)}{Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + 1)) Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + m_{2α}))}

if α is anisotropic, and if α is isotropic, then it is given by

(1.2)  c(λ) = c_{0}λ.

Formally, this result takes the same form as in the classical situation for the case of an anisotropic root α. However, for an isotropic root α (of multiplicity m_{α} = −2), one would expect c(λ) ≃ (λ − 1) from Equation (1.1) and the duplication formula. This differs from the true result in Equation (1.2) by the absence of a ‘ϱ-shift’. This situation is similar for the Harish-Chandra homomorphism, see Ref. [1]. Moreover, in the present supersymmetric setting, m_{α} and hence ϱ may be arbitrarily large negative numbers, drastically changing the asymptotic behaviour of ϕ_{λ}. Moreover, due to the shift in the denominator in Equation (1.1), c(λ) picks up zeroes in the right half plane. These lead to discrete contributions in the Plancherel formula, as we will show in a forthcoming paper. The class of symmetric pairs considered in Theorem A is a choice of rank one pairs that is generic in the sense that every rank one subpair of a reductive symmetric pair (of even type) associated with a choice of indivisible root (even, odd, ot both) is generated by copies of the pairs listed in Table 4.1 below: An anisotropic root α such that 2α is a root corresponds to the 𝔤𝔩 case; an anisotropic root such that 2α is not a root corresponds to the 𝔬𝔰𝔭 case, even it is purely odd (here the parameter p = 0); finally, an isotropic root α corresponds to the 𝔤𝔩(1|1) case. Choosing real forms of these symmetric pairs in such a way that the underlying symmetric spaces become Riemannian introduces extra conditions if one insists on taking real forms also of the odd part of the Lie superalgebra. However, these conditions are artificial from a physical point of view and moreover unnecessary for the analysis to go through. Therefore, we adopt the setting of ‘cs manifolds’ invented by Joseph Bernstein, that is, of real manifolds with complex sheaves of superfunctions. In many respects, this theory is parallel to that of real supermanifolds; however, there are some caveats, and we carefully lay the foundations to help the reader navigate these impasses in Section 2. As the statement of Theorem A suggests, the general form of c-function for a supersymmetric symmetric pair is not given by a simple-minded generalisation of the Gindikin–Karpelevič formula, since the contributions from isotropic roots are of a different form. Compare Ref. [6] for details. The proof of Theorem A is somewhat more difficult than in the classical case. The reason lies again in the changed growth behaviour of the exponential e^{t(λ−ϱ)}: Upon parametrising the geodesic sphere at infinity of X = G/K in stereographical coordinates, the integrals in question exhibit a singular behaviour. One therefore has to make a careful choice of cutoffs in a stereographical atlas parametrised by the Weyl group and keep track of this in the limit of t ⟶ ∞. In order for this to work, one needs to establish the Weyl group symmetry of the symmetric superfunctions ϕ_{λ} (Corollary 3.20). Although this appears to be quite innocent, it requires the extension of the usual integral formulæ for the Iwasawa and Bruhat decompositions to the ‘parameter-dependent’ setting of supermanifolds over a general base supermanifold. For this reason, we are obliged to develop some foundational material in Subsections 2.6 and 3.2. Once one has an explicit formula for the c-function, Harish-Chandra’s series expansion of the spherical function carries over. (This also relies on Corollary 3.20.) One has the following statement.

Theorem B. Let (G, K) be such that the root α is even and ℜλ > 0, λ ∉ \frac{1}{2}ℤ. Then we have

ϕ_{λ}|_{A^{+}} = ∑_{w∈W_{0}} Φ_{wλ}, Φ_{±λ}(e^{th_{0}}) = e^{t(±λ−ϱ)} ∑_{ℓ=0}^{∞} γ_{ℓ}(λ)e^{−2ℓt},

where A^{+} is the positive Weyl chamber, W_{0} is the Weyl group, γ_{0}(±λ) = c(±λ), and γ_{ℓ}(λ) follow a two-term recursion.

In most cases, W_{0} = {±1}, but for a suitable choice of parameters in case G is an orthosymplectic supergroup, it can be trivial. A similar situation occurs when α is an odd anisotropic root (Proposition 4.16). In this case, the complexification of G is GL(1|1, ℂ). Remarkably, when ϱ is a negative integer (as can happen in the 𝔬𝔰𝔭 case), the above series expansion is finite, and ϕ_{λ} admits a simple closed expression in terms of Jacobi polynomials (Corollary 5.6) P_{n}^{(a,b)}. Here, the parameters are chosen such that n = −ϱ, so that in this case, the spherical superfunctions are exponential polynomials of a fixed degree independent of λ.

We end this introduction by a synopsis of the paper’s contents. Section 2 is devoted to a brief collection of basic facts on cs manifolds. We highlight some points that to our knowledge are missing in the literature: Proposition 2.3 is a generalisation of Leites’s theorem for morphisms from cs manifolds to complex manifolds. Subsection 2.4 contains an account of the exponential morphism for cs Lie supergroups. Subsection 2.5 explains cs forms of complex supergroups. Subsection 2.6 is a brief account of the basic theory of Berezin fibre integrals. In Subsection 2.7 a localization formula for Berezin integrals over superspheres is derived. Section 3 introduces the main players of this article, the spherical superfunctions. It begins by setting the stage for symmetric superpairs in Subsection 3.1. In Subsection 3.2, we collect some integral formulæ necessary for the manipulation of the Harish-Chandra integral for ϕ_{λ}. Notably, we generalise the classical formula for the Haar measure on a Lie group in exponential coordinates in Proposition 3.12, the nilpotent case being of particular importance to us. The definition of the spherical functions is stated in Subsection 3.3; here, we also show a basic symmetry property of the spherical functions (Corollary 3.20) that is important in the proof both of Theorem A (Theorem 4.2) and of Theorem B (Theorem 5.5). Section 4 contains the statement and proof of Theorem A (Theorem 4.2). It is also the place where the main analytic problems have to be tackled and where the theory departs most from the classical cases (see the comments above). The theorem is stated in Subsection 4.1 and proved in Subsection 4.2. The proof proceeds case-by-case: the unitary case is contained in Subsubsection 4.2.1, the orthosymplectic one in Subsubsection 4.2.2, and the case of an anisotropic purely odd root (the ‘GL(1|1, ℂ) case’) in Subsection 4.2.3. The latter also contains the analytic expression for the spherical superfunctions ϕ_{λ} in this case. The final Section 5 is devoted to the asymptotic series expansion of ϕ_{λ}, that is, the proof of Theorem B (Theorem 5.5). At this point, the main analytic difficulties have been overcome, so we can follow the ususal procedure of making a pertubation ansatz for the solutions of the eigenvalue equation for the radial part of the Laplacian on X, deducing a two-term recursion for the coefficients, and proving convergence via Gangolli estimates. The estimates miraculously also go through in cases of negative multiplicity; if the half sum ϱ of positive roots is a negative integer, the coefficients can even be computed explicitly, and the series terminates. In this case, we derive an expression for ϕ_{λ} in terms of Jacobi polynomials (Corollary 5.6) whose degree is fixed independent of λ.

Acknowledgements. This article is based in large parts on the second named author’s doctoral thesis under the first named author’s guidance. We wish to thank Martin Zirnbauer for constructive comments on early versions of our results and the anonymous referee for the careful reading of our paper and for giving detailed suggestions that helped to improve it.

2. Basic facts on cs manifolds

In this section, we collect some basic facts on cs manifolds and cs Lie supergroups. As remarked above, this setting is necessary to cover the all the symmetric pairs we are interested in a Riemannian incarnation. We shall be suitably brief, including proofs only for those facts which so far have remained undocumented in the literature.

2.1. Basic definitions. We will work with cs manifolds and complex supermanifolds. The latter are covered by a wide literature, e.g. by Refs. [13,31,37]. The former, introduced by Joseph Bernstein, are covered to some degree in Ref. [15]. In many respects, they are similar to real supermanifolds, so one may follow the standard texts on that subject [15,30,37]. The differences that exist are quite subtle, and we will comment on these to the extent required in our applications. Specifically, we shall follow the conventions of Ref. [4].

In particular, we will work in the framework of ℂ-superspaces, that is, of locally superringed spaces over Spec ℂ. Our notation will always be X = (X_{0}, O_{X}) for ℂ-superspaces, and φ = (φ_{0}, φ^{♯}) : X ⟶ Y for morphisms, where we may consider φ^{♯} as a sheaf map O_{Y} ⟶ φ_{0∗}O_{X} or φ_{0}^{−1}O_{Y} → O_{X} by the fundamental adjunction of the direct and inverse image functors, see e.g. Refs. [11, I.4, Equation (5)] or [26, II.4, Theorem 4.8]. Somewhat abusing notation, we will also write X_{0} for the reduced superspace associated with X, see [4, Construction 3.9]. There is a canonical closed embedding

j_{X_{0}} : X_{0} ⟶ X

and for any f ∈ O_{X}(U), where U ⊆ X_{0} is open, and any x ∈ U, we denote by f(x) := j_{X_{0}}^{♯}(f)(x) ∈ κ(x) := O_{X,x}/𝔪_{X,x} the value of f at x. In the cases of interest to us, we will always have κ(x) = ℂ. We define the cs and complex affine superspaces as

𝔸^{p|q} := (ℝ^{p}, C_{ℝ^{p}}^{∞} ⊗ ⋀(ℂ^{q})^{∗}), 𝔸_{hol}^{p|q} := (ℂ^{p}, H_{ℂ^{p}} ⊗ ⋀(ℂ^{q})^{∗}),

where C_{ℝ^{p}}^{∞} is the sheaf of smooth complex-valued functions on ℝ^{p} and H_{ℂ^{p}} is the sheaf of holomorphic functions on ℂ^{p}, and we consider the Euclidean topology. Then, by definition, a cs manifold is a ℂ-superspace locally isomorphic to 𝔸^{p|q}, whereas a complex supermanifold is one locally isomorphic to 𝔸_{hol}^{p|q}. We are tempted speak of cs manifolds simply as “supermanifolds”, but for the sake of convention, we will stick in this paper to the original appellation, which was introduced by Joseph Bernstein, cf. Ref. [15]. More generally, given two cs manifolds X and S, a cs manifold over S is a morphism X → S that is locally in X isomorphic to the projection of a direct product S × Y → S. In this case dim_{S} X := dim Y is called the fibre dimension. Usually, we will just write X/S without mentioning the structural morphism explicitly, denoting it by p_{X} where necessary. In particular, 𝔸_{S}^{p|q} : = S × 𝔸^{p|q} is a cs manifold over S. A morphism f : X/S → Y/S over S is a morphism f : X → Y respecting the structural morphisms. Given an open embedding X/S → 𝔸_{S}^{p|q}/S over S, the pullback of the standard coordinate functions on 𝔸^{p|q} is called a fibre coordinate system (over S). We recover the usual cs manifolds upon setting S = ∗, the terminal object of the category of ℂ-superspaces (i.e. the singleton space, together with the constant sheaf ℂ), but the relative point of will be important in Subsections 2.3, 2.6, 2.4, and 3.2. In Ref. [30], the corresponding notion for real supermanifolds is discussed under the name of families. In Ref. [4], which we follow, it is defined over base superspaces S more general than cs manifolds. A coordinate-free view on affine superspaces will be useful. To that end, we define a cs vector space to be a real super-vector space V = V¯0 ⊕ V¯1 with a distinguished complex structure on V¯1. We set

𝔸(V) := (V¯0, C_{V¯0}^{∞} ⊗ ⋀ V ∗ ; ¯1)

for any finite-dimensional cs vector space V. In particular, 𝔸^{p|q} = 𝔸(ℝ^{p}⊕Πℂ^{q}). Similarly, for any finite-dimensional complex super-vector space V, we define

𝔸_{hol}(V) := (V¯0, H_{V¯0} ⊗ ⋀ V ∗ ; ¯1),

so that 𝔸_{hol}^{p|q} = 𝔸_{hol}(ℂ^{p|q}), where we write ℂ^{p|q} := ℂ^{p} ⊕ Πℂ^{q}. We take note of the following trivial observation

V¯0 = 0 ⟹ 𝔸(V) = 𝔸_{hol}(V).

2.2. Morphisms to affine superspaces. One essential respect in which cs manifolds differ from real or complex supermanifolds is that morphisms to affine superspace are characterised in a subtly different fashion. Namely, for any ℂ-superspace X, denote by O_{X,ℝ} the sheaf whose local sections are the superfunctions f (i.e. local sections of O_{X}) whose values f(x) are real, for any x ∈ X_{0} at which f is defined. Then we have the following generalisation of Leites’s theorem on morphisms.

Proposition 2.1 ([4, Lemma 3.13, Corollary 4.15]). Let T be a cs manifold and denote the standard coordinate system on 𝔸^{p|q} by (t_{a}). Then the natural map

𝔸^{p|q}(T) := Hom(T, 𝔸^{p|q}) ⟶ Γ(O_{X,ℝ,¯0}^{p} × O_{X,¯1}^{q}) : f ⟼ (f^{♯}(t_{a}))

is a bijection. Here and in what follows, Hom denotes morphisms of ℂsuperspaces and Γ denotes the global sections functor for sheaves.

Expanding on the above conventions, we let, for any cs manifolds X and T, X(T) := Hom(T, X) denote the set of morphisms T ⟶ X. We call any such morphism x : T ⟶ X a T -valued point and write x ∈_{T} X. For a morphism f : X → Y and x ∈_{T} X, we denote by f(x) := f ∘ x ∈_{T} Y. Given a T -valued point x ∈_{T} X where T/S is a cs manifold over S, we write x ∈_{T/S} X/S (or x ∈_{T/S} X for brevity) if x is over S as a morphism T → X. We will use this suggestive notation constantly. Together with the Yoneda lemma, it provides a convenient way of defining and handling morphisms. We will be constantly using this point of view, so Proposition 2.1 is basic to our study. A coordinate-free version of Proposition 2.1 can be formulated as follows: Following Ref. [4], we define

(V ⊗ W)¯0 := V¯0 ⊗_{ℝ} W¯0 ⊕ V¯1 ⊗_{ℂ} W¯1.

With this notation, the above proposition admits the following reformulation.

Corollary 2.2 ([4, Corollary 3.24]). Let V be a finite-dimensional cs vector space and T a cs manifold. Then there is a natural bijection

𝔸(V)(T) = Hom(T, 𝔸(V)) ⟶ Γ((O_{T,ℝ} ⊗ V)¯0).

On the other hand, as the holomorphic version of Leites’s theorem states, morphisms from a complex supermanifold T to 𝔸_{hol}^{p|q} constitute the same data as tuples of superfunctions on T without any reality condition. Remarkably, this carries over to the situation where T is a cs manifold, by the results of Ref. [4].

Proposition 2.3. Let V be a finite-dimensional complex super-vector space and T a cs manifold. Then there is a natural bijection

𝔸_{hol}(V)(T) = Hom(T, 𝔸_{hol}(V)) ⟶ Γ((O_{T} ⊗_{ℂ} V)¯0).

Proof. Broadly following Ref. [4], a ℂ-superspace T is called holomorphically regular if the above natural map is bijective for any V and any open subspace of T. Thus, our claim is that any cs manifold is holomorphically regular. We will use the terminology of Ref. [4]. Since its use will be localized to this proof, we refer to that article for any undefined notions. Firstly, we may assume that V = ℂ^{n}, by Lemma 4.15 (op. cit.). Then 𝔸_{hol}(V) = 𝔸_{hol}^{n} and 𝔸_{hol}^{n}(T) = 𝔸_{hol}^{n}(T ¯0), where T ¯0 := (T_{0}, O_{T,¯0}) is the even part of T, so it is sufficient to prove that T ¯0 is holomorphically regular. But by Proposition 5.24 (op. cit.), T ¯0 (and hence, any open subspace thereof) admits a tidy embedding into some 𝔸^{N}. In view of Proposition 5.11 and Lemma 5.2 (op. cit.), it is therefore sufficient to prove that 𝔸^{N} is holomorphically regular. By Lemma 5.2 (op. cit.) again, the natural map is injective for any open subspace of 𝔸^{N}. Let us prove that it is also surjective. So, let U ⊆ ℝ^{N} be open and f_{1}, …, f_{n} ∈ O_{𝔸^{N}} (U) = C^{∞}(U, ℂ). We may define f_{0} := (f_{1}, …, f_{n}) : U → ℂ^{n}, and this map is smooth. Setting

f_{V}^{♯} (h) := h ∘ (f_{0}|_{f_{0}^{−1}(W)}), h ∈ O_{𝔸_{hol}^{n}}(W) = H(W),

for any open W ⊆ ℂ^{n}, defines a morphism f = (f_{0}, f^{♯}) : 𝔸^{N}|_{U} → 𝔸_{hol}^{n} such that f^{♯}(z_{a}) = f_{a}, where (z_{a}) are the standard coordinates on 𝔸_{hol}^{n}. □

For any finite-dimensional complex super-vector space V, we define a set-valued cofunctor 𝔸^{ℂ}(V) on the category of cs manifolds by setting

𝔸^{ℂ}(V)(T) := Γ((O_{T} ⊗_{ℂ} V)¯0)

on objects T, and by the obvious definition on morphisms. In these terms, Proposition 2.3 states that 𝔸^{ℂ}(V) is the restriction to the category of cs manifolds of the point functor of the complex supermanifold 𝔸_{hol}(V). (Of course, the restriction of the point functor of the algebraic affine superspace Spec S(V^{∗}) to cs manifolds is also the same, but this is somewhat less remarkable.) Proposition 2.3 has the following consequence, which will be important in the applications to supergroups below.

Corollary 2.4. Let X and Y be complex supermanifolds and T a cs manifold. Then the natural map

(X × Y)(T) ⟶ X(T) × Y (T)

is a bijection.

2.3. The tangent functor. Preparing for our discusson of the exponential morphism of cs Lie supergroups below, we introduce the ‘total space’ of the ‘tangent bundle’. A subtlety for cs manifolds X is that such a ‘total space’ representing the (complex) tangent sheaf does not literally exist as a cs manifold. In this section, we introduce a replacement on the level of functors. We then single out a representable subfunctor, the ‘real tangent bundle’. We will work with cs manifolds over an arbitrary base S. This will be used to the define the exponential morphism. It will also be important for the machinery of Berezinian fibre integration that we introduce in Subsection 2.6, on which our proof of the Weyl symmetry of the spherical superfunctions in Corollary 3.20 hinges.

Definition 2.5 (The tangent functor). Let X/S be a cs manifold over S. We let T_{X/S} be the sheaf of p_{X,0}^{−1}O_{S}-linear superderivations of O_{X}, and call this the tangent sheaf over S. Locally, it is spanned by fibre coordinate derivations and hence locally free. We define the tangent functor T_{S}X of X over S as the set-valued cofunctor on cs manifolds given on objects T/S by

(T_{S}X)(T/S) := {(x, v) | x ∈_{T/S} X, v ∈ Γ((x^{∗}T_{X/S})¯0)}

and by the obvious definition on morphisms. There are canonical morphisms of functors

0_{X} : X ⟶ T_{S}X : x ⟼ (x, 0), π_{X/S} : T_{S}X ⟶ X : (x, v) ⟼ x,

called the zero section and the projection, respectively. If S = ∗, we omit the corresponding subscripts. The construction of T_{S}X is functorial: For any morphism φ : X/S → Y/S of cs manifolds over S, the tangent morphism T_{S}φ : T_{S}X → T_{S}Y is defined by

(T_{S}φ)_{T} (x, v) := (φ(x), T_{x/S}φ(v)), (T_{x/S}φ)(v)(f) := v(φ^{♯}(f))

for all f ∈ (φ_{0}^{−1}O_{Y})(U), U ⊆ Y_{0} open. Here, we recall that x^{∗}T_{X/S} is the sheaf of vector fields over S and along x. Clearly, the functor T_{S} : X ↦ T_{S}X preserves fibre products over S, so T_{S}S = S and T ∗ = ∗, the point functor of the singleton space.

This definition of the tangent functor is compatible with the definition of the tangent spaces. If x ∈ X_{0} is a point, considered as a morphism ∗ ⟶ X, then the fibre product of functors (T_{S}X)_{x} = ∗ ×_{X} T_{S}X is given by

(T_{S}X)_{x}(T) = (∗ ×_{X} T_{S}X)(T) = Γ((x_{T}^{∗} T_{X/S})¯0)
= {φ ∈ Hom_{S}(T × Spec 𝔻¯0, X) | φ|_{ε=0} = x_{T} }
= 𝔸^{ℂ}(T_{x/S}X)(T).

(2.1)

Here, x_{T} is the specialization of x, i.e. the composite T ⟶ ∗ ⟶ X, and T_{x/S}X is the tangent space of X over S at x—that is, the super-vector space over ℂ whose homogeneous elements are the O_{S,p_{0}(x)}-linear maps v : O_{X,x} → ℂ such that

v(fg) = v(f)g + (−1)^{|f||v|}fv(g), ∀f, g.

In other words, the fibre of T_{S}X ⟶ X over x is the ℂ-affine superspace of T_{x/S}X. Moreover, Spec 𝔻¯0 = (∗, ℂ[ε]/(ε^{2})), and (·)|_{ε=0} denotes the canonical morphism T ⟶ T × Spec 𝔻¯0 that pulls back functions in O_{T} [ε]/(ε^{2}) by dropping the linear term in ε. The last of the equalities in Equation (2.1) amounts to the routine check that the map φ^{♯} = x_{T}^{♯} + εv is an algebra morphism if and only if v is a vector field over S along x_{T}. Similarly, one checks that for any morphism φ : X → Y, the action of T φ on the fibre (T_{S}X)_{x} is simply given by the application of the functor 𝔸^{ℂ}(·) to the ordinary tangent map T_{x/S}φ : T_{x}X → T_{φ_{0}(y)}Y.

Besides the (complex) tangent space T_{x/S}X over S, we consider the real tangent space, defined as the cs vector space

T_{x/S}^{ℝ}X := {v ∈ T_{x/S}X | v(O_{X,ℝ,¯0}) ⊆ ℝ}

of real tangent vectors. More generally, we introduce the following.

Definition 2.6 (The real tangent bundle). The real tangent sheaf T_{X/S}^{ℝ} of X over S as the subsheaf of T_{X/S} whose local sections over an open set U ⊆ X_{0} are those v ∈ T_{X/S}(U) such that

v(O_{X,ℝ,¯0}(V)) ⊆ O_{X,ℝ}(V)

for all open subsets V ⊆ U. If (x_{a}) = (u, ξ) are local fibre coordinates of X over S defined on U, then

T_{X/S}^{ℝ}|_{U} = ⨁_{i} O_{X,ℝ} \frac{∂}{∂u_{i}} ⊕ ⨁_{j} O_{X} \frac{∂}{∂ξ_{j}} ,

where \frac{∂}{∂u_{i}} and \frac{∂}{∂ξ_{j}} are the fibre coordinate derivations introduced by (u, ξ). In this sense, T_{X/S}^{ℝ} is a ‘locally free graded module over the ring extension (O_{X}, O_{X,ℝ})’. We define the real tangent functor T_{S}^{ℝ}X as the subfunctor of T_{S}X given on objects by

(T_{S}^{ℝ}X)(T/S) := {(x, v) ∈ T_{S}X(T/S) | v ∈ Γ((x^{∗}T_{X/S}^{ℝ})¯0)}.

The condition on v in the last equation amounts to

v((x_{0}^{−1}O_{X,ℝ,¯0})(U)) ⊆ O_{T,ℝ}(U)

for all open subsets U ⊆ T_{0}. Due to the local freeness of T_{X/S}^{ℝ}, the functor T_{S}^{ℝ}X is representable by a cs manifold, which is the total space of a vector bundle over X. This vector bundle, also denoted by T_{S}^{ℝ}X, is called the real tangent bundle of X. By Proposition 2.1 or Corollary 2.2, its fibre at x ∈ X_{0} is computed to be

(T_{S}^{ℝ}X)_{x} = 𝔸(T_{x/S}^{ℝ}X),

the cs affine superspace associated with the cs vector space T_{x/S}^{ℝ}X.

2.4. The exponential morphism of a cs Lie supergroup. It is a wellknown fact that there is an equivalence between the categories of Lie supergroups and supergroup pairs, cf. Ref. [13, 15, 28, 29]. This is true in the real and complex, smooth and analytical cases, and the arguments valid in these cases carry over unchanged to the setting of cs Lie supergroups. However, for applications, in particular those to Harmonic Superanalysis we are concerned with in this paper, it useful to have a ‘geometric’ view on this equivalence. This uses the exponential morphism and the Campbell–Hausdorff series, neither of which has as yet been given a treatment in the literature in this setting. The exponential morphism will also be used extensively in Section 4, for the derivation of the Iwawasa H-projection and the integration in exponential coordinates on the nilpotent Iwawasa N supergroup (on the basis of Proposition 3.12), both of which are primordial for the c-function asymptotics. Moreover, the cs Lie supergroups we will consider (see below for the definition) will be ‘forms’ of complex Lie supergroups. We will widely use their functors of points, and the avalability of a manageable expression thereof will be crucial. As we show below, in Proposition 2.14, such an expression can be derived by means of the exponential morphism. We begin with some basic definitions.

Definition 2.7. By definition, a cs Lie supergroup is a group object G in the category of cs manifolds.

Let G be a cs Lie supergroup. We set

𝔤 := T_{1}G, 𝔤_{ℝ} := T_{1}^{ℝ}G,

where we write T_{1}^{ℝ}G = T_{1/∗}^{ℝ}G and otherwise use the notation from Subsection 2.3. We have

𝔤 = {x | g = 1 + εx¯0 + τx¯1 ∈ G(Spec 𝔻)}, 𝔻 := ℂ[ε|τ]/(ε^{2}, ετ),

where we set Spec 𝔻 := (∗, 𝔻). Hence, for x, y ∈ 𝔤, we may define [x, y] by

a = 1 + ε[x, y]¯0 + τ[x, y]¯1, a := (ghg^{−1}h^{−1})|_{ε=ε_{1}ε_{2}=τ_{1}τ_{2},τ=τ_{1}ε_{2}=ε_{1}τ_{2}},

where

g := 1 + ε_{1}x¯0 + τ_{1}x¯1, h := 1 + ε_{2}y¯0 + τ_{2}y¯1.

With this bracket, 𝔤 acquires the structure of a complex Lie superalgebra. Moreover, we have 𝔤_{ℝ,¯0} = 𝔤_{0} and 𝔤¯0 = 𝔤_{0} ⊗_{ℝ} ℂ, where 𝔤_{0} is the Lie algebra of the real Lie group G_{0}. Note that by Corollary 2.2, we have

𝔸(𝔤_{ℝ})(𝔸^{0|q}) = T_{1}^{ℝ}((ΠT^{q})G)_{0},

the Lie algebra of the real Lie group ((ΠT)^{q}G)_{0} = G(𝔸^{0|q}). When φ : G → H is a morphism of cs Lie supergroups, then φ(1) = 1, so that we have a map dφ := T_{1}φ : 𝔤 = T_{1}G → T_{1}H = 𝔥. Using the definition of the bracket, it is immediate that dφ is a morphism of complex Lie superalgebras. Moreover, we have dφ(𝔤_{ℝ}) ⊆ 𝔥_{ℝ}.

We may now transfer the definition of the exponential morphism of a real Lie supergroup [21] to the setting of cs Lie supergroups. We shall be terse on the parts that are similar, highlighting only the use of the tangent functor.

Proposition 2.8. Let G be a cs Lie supergroup. There is a unique morphism

exp_{G} : 𝔸(𝔤_{ℝ}) → G

of cs manifolds whose action on 𝔸^{0|q}-valued points is precisely the exponential map of the real Lie group ((ΠT)^{q}G)_{0} = G(𝔸^{0|q}). The morphism exp_{G} is a local isomorphism in a neighbourhood of zero.

Proof. The uniqueness is obvious, since the 𝔸^{0|q} form a set of generators for the category of cs manifolds. For the existence, we construct a certain even vector field on G × 𝔸(𝔤_{ℝ}). If m is the multiplication of G, we consider

0_{G} × ι : G × 𝔸^{ℂ}(𝔤) → T G × T G, T m : T G × T G = T (G × G) → T G,

where ι is the canonical morphism 𝔸^{ℂ}(𝔤) = (T G)_{1} → T G. Let L denote their composite. Explicitly, on points, it is given by

L(g, v) = T m(g, 1, 0, v) = (g, v_{g}), v_{g}(f) := (id ⊗ v)(m(g, ·)^{♯}(f)).

The inverse morphism L^{−1} : T G → G × 𝔸^{ℂ}(𝔤) is given on points by

L^{−1}(g, v) = (g, v_{g^{−1}}),

so that L is an isomorphism of functors. Now, we define the morphism Θ : G × 𝔸(𝔤_{ℝ}) ⟶ T G as the composite

G × 𝔸(𝔤_{ℝ})  G × 𝔸(𝔤)  T G._{L}

By the Yoneda Lemma, Θ corresponds to a unique element (x, v) of

T (G)(G × 𝔸(𝔤_{ℝ})).

It is clear that π_{G} ∘ Θ = p_{1} : G × 𝔸(𝔤_{ℝ}) → G, so that v ∈ Γ((p_{1}^{∗}T_{G})¯0). We may promote v to an even vector field on G × 𝔸(𝔤_{ℝ}) over 𝔸(𝔤_{ℝ}). Arguing as in [21, Lemma 4.1], one sees that v is real and complete. Let

γ_{v} : 𝔸^{1} × G × 𝔸(𝔤_{ℝ}) → G × 𝔸(𝔤_{ℝ})

be its global flow. We define exp_{G} as the composite

𝔸(𝔤_{ℝ})  𝔸^{1} × G × 𝔸(𝔤_{ℝ})  G × 𝔸(𝔤_{ℝ})  G. (1,1)×id γ_{v} p_{1}

Using the naturality of the morphism L, we see that v corresponds on G(𝔸^{0|q}) to a similarly defined vector field that is known to generate the exponential flow [27]. This implies the claim. □

Corollary 2.9. Let φ : G → H be a morphism of cs Lie supergroups. Then

exp_{H} ∘ 𝔸(dφ) = φ ∘ exp_{G} .

Proof. It is sufficient to check the equality on 𝔸^{0|n}-valued points, where it is immediate from Proposition 2.8 and the classical theory [25]. □

The following is a form of the equivalence of supergroups and supergroup pairs, stating that any supergroup G is G_{0}-equivariantly split. (In fact, it gives the essential bijection on objects; the equivalence on morphisms then follows from Corollary 2.9.)

Corollary 2.10. Let G be a cs Lie supergroup. Then the morphism

G_{0} × 𝔸(𝔤¯1) ⟶ G : (g, x) ⟼ g exp_{G} x

is a G_{0}-equivariant isomorphism of cs manifolds.

Proof. Since d exp_{G} = id_{𝔤} by Proposition 2.8 and the morphism is G_{0}- equivariant, it has invertible tangent map at every point. The underlying map is id_{G_{0}}, so the claim follows from the inverse function theorem [13, 15, 30]. □

We end this subsection by a discussion of the adjoint action. This is the action Ad_{G} := Ad of G on 𝔸^{ℂ}(𝔤) = (T G)_{1}, given as follows: For g ∈_{S} G and x ∈ (T G)_{1}(S), regard x as an element of G(S × Spec 𝔻¯0) such that x|_{ε=0} = 1_{S}. (See the comments after Equation (2.1) for explanation.) Then

Ad(g)(x) := gxg^{−1} ∈ G(S × Spec 𝔻¯0),

where g is specialized to S × Spec 𝔻¯0 via the first projection. It follows that

Ad(g)(x)|_{ε=0} = gg^{−1} = 1_{S},

so Ad(g)(x) ∈_{S} (T G)_{1} = 𝔸^{ℂ}(𝔤). Using the definitions, it is easy to check that for any morphism φ : G → H of cs Lie supergroups, we have

Ad_{H}(φ(g))(𝔸^{ℂ}(dφ)(x))) = 𝔸^{ℂ}(dφ)(Ad_{G}(g)(x))

for any g ∈_{S} G, x ∈_{S} 𝔸^{ℂ}(𝔤). If we let G_{0} × G_{0} act on G_{0} × 𝔸(𝔤¯1) by (g_{1}, g_{2})(g, x) = (g_{1}gg_{2}^{−1}, Ad(g_{2})(x)) for (g_{1}, g_{2}) ∈_{T} G_{0} × G_{0}, then we see that the isomorphism stated in Corollary 2.10 is even (G_{0} × G_{0})-equivariant. Remarkably, the adjoint action of G passes to 𝔸(𝔤_{ℝ}). Namely, let g ∈_{S} G and x ∈_{S} 𝔸(𝔤_{ℝ}). We need to check that for any f ∈ Γ(O_{G,ℝ,¯0}), we have y(f) ∈ Γ(O_{S,ℝ}) for y := Ad(g)(x). By the definition, we have

y(f) = (id ⊗ x)(f(g(·)g^{−1})),

where id ⊗ x denotes the promotion of x to a vector field over S (which exists because of the local freeness of the tangent sheaf over S, cf. [30, Lemma 2.2.3]). Since f(g(·)g^{−1}) = (m(g, m(·, g^{−1})))^{♯}(f) is an even and real-valued superfunction on S × G, the assertion is immediate from the assumption on x.

Corollary 2.11. Let G be a cs Lie supergroup. If g ∈_{S} G and x ∈_{S} 𝔸(𝔤_{ℝ}), then

exp_{G} Ad(g)(x) = g(exp_{G} x)g^{−1}.

Proof. By the above, both sides of the equation are well-defined. Hence, it is sufficient to check the equality on 𝔸^{0|q}-valued points, for every q. But on that level, it follows from Proposition 2.8 and the classical theory [25]. Alternatively, one may use a parameter version of Corollary 2.9. □

Remark 2.12. Although 𝔤_{ℝ} is in general not a Lie algebra, the bracket is well-defined on 𝔸(𝔤_{ℝ}). Indeed, 𝔤 has a homogeneous basis (e^{a}) contained in 𝔤_{ℝ}. Then x, y ∈ Γ(O_{T,¯1} ⊗ 𝔤¯1) admit representations x = ∑_{a} x_{a}e^{a} and y = ∑_{a} y_{a}e^{a}, and

[x, y] = − ∑_{abc} x_{a}y_{b}C_{c}^{ab}e^{c}

where C_{c}^{ab} ∈ ℂ are the structure constants of 𝔤. But x_{a}y_{b}C_{c}^{ab} has the value zero, so that it is a section of O_{T,ℝ,¯0}.

2.5. Complex supergroups and cs forms. Our interest in cs supergroups comes from the fact that their are many ‘cs forms’ of complex Lie supergroups, whereas there are comparatively few real forms. (See the examples we will be considering in Section 4 and beyond, and compare Ref. [15] for a more extensive list of reasons. In any case, any real Lie supergroup defines a cs Lie supergroup by the complexification of its structure sheaf.) Since we will encounter many such cs forms, it will be useful to have a uniform description of their functors of points. We will derive such a description by the use of the exponential morphism.

Definition 2.13 (Forms of complex supergroups in cs manifolds). Let G_{ℂ} be a complex Lie supergroup with associated complex supergroup pair (𝔤, G_{ℂ,0}). Let G_{0} be a real form of G_{ℂ,0}, i.e. a closed subgroup whose Lie algebra is a real form of 𝔤¯0. The cs Lie supergroup G associated with (𝔤, G_{0}) is called a cs form of G_{ℂ}.

Any cs form G of a complex Lie supergroup G_{ℂ} comes with a canonical morphism G ⟶ G_{ℂ} of ℂ-superspaces. It can be given an expression in terms of the exponential morphism, as follows. The real Lie supergroup (G_{ℂ})_{ℝ} associated with G_{ℂ} has an exponential morphism [21]. Since its differential is complex linear for the complex structure induced by 𝔤, it is holomorphic, and therefore induced by a unique morphism denoted by exp_{G_{ℂ}} : 𝔸_{hol}(𝔤) → G_{ℂ}. By the above and Corollary 2.4, the canonical morphism G → G_{ℂ} is given on T -valued points as follows:

G ⟶ G_{ℂ} : g exp_{G}(x) ⟼ g exp_{G_{ℂ}}(x), g ∈_{T} G_{0}, x ∈_{T} 𝔸(𝔤¯1) = 𝔸_{hol}(𝔤¯1)

Thus, the T -valued points of G can be characterised within G_{ℂ} as follows.

Proposition 2.14. Let G_{ℂ} be a complex Lie supergroup and G a cs form of G_{ℂ}. For any cs manifold T, the T -valued points of G are given by

G(T) = {g ∈_{T} G_{ℂ} | g_{0} ∈_{T_{0}} G_{0}}.

Proof. Express a T -valued point of G_{ℂ} as g exp_{G_{ℂ}}(x) where g ∈_{T} G_{ℂ,0}, x ∈_{T} 𝔸_{hol}(𝔤¯1). By the above considerations, g exp_{G_{ℂ}}(x) ∈ G(T) if and only if g ∈_{T} G_{0}. Thus, we need to see that

g ∈_{T} G_{0} ⟺ g_{0} ∈_{T_{0}} G_{0}.

Since (exp_{G_{ℂ}}(x))_{0} = 1, this will already be sufficient to prove the claim. Certainly, the left-hand statement implies that on the right-hand side. On the other hand, assume that g_{0} ∈_{T_{0}} G_{0}. This pins down the map underlying g. Local coordinates on G_{0} can be chosen in such a way that they are the restriction of holomorphic local coordinates (z_{a}) on G_{ℂ,0}. Then g ∈_{T} G_{0} translates to the requirement that g^{♯}(z_{a}) be real-valued. Since (g^{♯}(z_{a}))_{0} = z_{a} ∘ g_{0}, this is immediate by the assumption, thereby proving the claim. □

2.6. Integration on relative cs manifolds. Below, in Corollary 3.20, we prove the Weyl symmetry of the spherical superfunctions ϕ_{λ} by the use of integral geometry. Without this fact, we would not be able to prove convergence in the delicate c-function expansion, nor could we derive the Harish-Chandra series, so the result is absolutely essential. In the proof of loc. cit., we will need to handle integrals with parameters. A suitable formalism is that of fibre integrals over relative cs manifolds. We very briefly collect the basic definitions and facts to it set up.

Definition 2.15 (Relative Berezinian densities). Assume given a cs manifold X/Y over Y. For any fibre coordinate neighbourhood U ⊆ X_{0}, we let

Ber_{X/Y} |_{U} := Ber(T_{X/Y}^{∗} (U)).

(Compare Ref. [31] for the definition of the Berezinian module of a free module.) This defines a locally free O_{X}-module Ber_{X/Y} with local basis of sections

Dx = D(u, ξ) = du_{1} ⋯ du_{p} \frac{∂}{∂ξ_{1}} ⋯ \frac{∂}{∂ξ_{q}}

of parity ≡ q (2), for any local fibre coordinate system x = (x_{a}) = (u, ξ). Twisting by the relative orientation sheaf, we obtain |Ber|_{X/Y} := or_{X_{0}/Y_{0}} ⊗_{ℤ} Ber_{X/Y}, with corresponding local basis of sections |Dx| = |D(u, ξ)|. The local sections of the latter sheaf are called relative Berezinian densities.

Definition 2.16 (Relative Berezin integral). Let X/Y be a cs manifold over Y. A retraction r : X → X_{0} (i.e. a left inverse of j_{X_{0}} : X_{0} → X) is called a retraction over Y if there is a retraction r_{Y} : p_{X}(X) → p_{X}(X)_{0} such that r_{Y} ∘ p_{X} = p_{X_{0}} ∘ r_{X}. Here, p_{X}(X) ⊆ Y is the open subspace of Y over the open set p_{X,0}(X_{0}) ⊆ Y_{0}. Fix a retraction r of X over Y. A local system (u, ξ) of fibre coordinates is called r-adapted if u = r^{♯}(u_{0}) for some fibre coordinate system u_{0} of X_{0}/Y_{0}. Let ω ∈ Γ(|Ber|_{X/Y}). Then we may define

Y ⨏_{X}^{r} ω ∈ O_{Y} (p_{X,0}(X_{0})),

the relative Berezin integral or fibre integral of ω, as follows. Let (U_{i}) be a collection of open subspaces of X, such that supp ω ⊆ ⋃_{i} U_{i,0}, there are r-adapted fibre coordinate systems (u^{i}, ξ^{i}) over Y defined on U_{i}, and there are r_{Y} -adapted coordinate systems (v^{i}, η^{i}) on V_{i} := p_{X}(U_{i}), such that (u^{i}, p_{X}^{♯}(v^{i}), ξ^{i}, p_{X}^{♯}(η^{i})) form coordinate systems of U_{i}. Let (χ^{i}) be a (not necessarily compactly supported) partition of unity on ⋃_{i} U_{i,0}, subordinate to the cover. We may expand

ω|_{U_{i,0}} = |D(u^{i}, ξ^{i})| f^{i}, f^{i} = ∑_{I,J} ξ^{I}p_{X}^{♯}(η^{J})r^{♯}(g_{IJ}^{i}), g_{IJ}^{i} ∈ O_{X_{0}}(U_{i,0}),

with I ranging through the subsets of {1, …, q}, p|q := dim_{Y} X, and J ranging though the subsets of {1, …, n}, m|n := dim Y. With these data, we define

Y ⨏_{X}^{r} ω := ∑_{i,J} η^{J}p_{Y}^{♯} (_{Y_{0}}_{⨏X_{0}}|du_{0}^{i}| χ_{i}g_{qJ}^{i}),

whenever the integrals and the series converge absolutely. Here, q = {1, …, q} and _{Y_{0}}_{⨏X_{0}} ϖ denotes the function y ⟼ ∫_{p_{X,0}^{−1}(y)} ϖ|_{p_{X,0}^{−1}(y)}, defined on p_{X,0}(X_{0}).

Remark 2.17. In Definition 2.16, the retraction r_{Y} is uniquely determined by r. In particular, not every retraction of X is over Y. For example, the retraction r on 𝔸^{2|2}, given by r(s) := (s_{1} + s_{2}s_{3}s_{4}, s_{2}) for an T -valued point s ∈_{T} 𝔸^{2|2}, has no counterpart under the submersion ψ: 𝔸^{2|2} ⟶ 𝔸^{1|2}, (s_{1}, s_{2}, s_{3}, s_{4}) ⟼ (s_{1}, s_{3}, s_{4}).

Up to some computations in coordinates, the following fact is no harder to prove than the absolute situation where Y = ∗, cf. Ref. [3].

Theorem 2.18. The relative Berezin integral is well-defined independent of all choices and depends only on the choice of a retraction. In case the integrand ω ∈ Γ(|Ber|_{X/Y}) is compactly supported in the fibres over Y, i.e. p_{X,0} : supp ω → Y_{0} is a proper map, then the integral is independent of the retraction.

Recalling the definition of the direct image with proper supports from Ref. [11], the condition in the above theorem be succintly rephrased as follows:

ω ∈ Γ(p_{X,0!}|Ber|_{X/Y}).

We will use this notation in the sequel.

Corollary 2.19. Let φ : X′/Y → X/Y be a morphism of cs manifolds. Let r and r′ be retractions of X and X′ over Y, respectively. Assume that r ∘ φ = φ_{0} ∘ r′. Then for any ω ∈ Γ(|Ber|_{X/Y}), we have

Y ⨏_{X′}^{r′} φ^{♯}(ω) = _{Y} ⨏_{X}^{r} ω

i.e. both integrals exist if only one of them does, in which case they coincide.

2.7. Integral localization in polar coordinates. It is known that under suitable symmetry assumptions, there are remarkable integral localization theorems for supermanifolds [16, 35]. Here, we show that for the special case of the supersphere, a precise form thereof can obtained by the use of polar coordinates. We shall make extensive use of the formalism of T -valued points. Compare the remarks in Subsection 2.2; the reader may also consult Ref. [1, Appendix B]. Denote by y = (y_{a}) = (v, η) the standard coordinates on 𝔸^{p|2q} and the Berezin– Lebesgue density by |Dλ| := (−2π)^{−q}|Dy|. The retraction associated with y is

r(x) := x¯0 := (x_{1}, …, x_{p}), x ∈ 𝔸^{p|2q}(T) = Γ(O_{T,¯0,ℝ}^{p} × O_{T,¯1}^{q}).

One should avoid to confuse x¯0 ∈ 𝔸^{p}(T) with the underlying morphism x_{0} ∈ 𝔸^{p}(T_{0}). Furthermore, ∥ · ∥^{2} : 𝔸^{p|2q} → 𝔸^{1} shall be given by

(2.2)  ∥x∥^{2} := ∑_{i=1}^{p} x_{i}^{2} + 2 ∑_{j=1}^{q} x_{p+2j−1}x_{p+2j}

for x ∈_{T} 𝔸^{p|2q}. Using the positive square root, this yields

∥ · ∥ := √ ∘ ∥ · ∥^{2} : 𝔸_{≠0}^{p|2q} → 𝔸_{>0}^{1},

where 𝔸_{≠0}^{p|2q} := 𝔸^{p|2q}|_{ℝ^{p}\{0}} and similarly for the subscript “> 0”.

Definition 2.20 (Rotationally invariant superfunctions). Let f ∈ Γ(O_{S×𝔸^{p|2q}}), where S is any cs manifold. In case p > 0, f will be called rotationally invariant over S if for some ε > 0, there exists an f^{∘} ∈ O_{S×𝔸^{1}}(S_{0} × (−ε, ∞)) such that

f(s, x) = f^{∘}(s, ∥x∥), (s, x) ∈_{T} S × 𝔸_{≠0}^{p|2q}.

In case p = 0, ∥ · ∥ is not defined, so the definition of rotational invariance has to be modified as follows: f is called rotationally invariant over S if there exists g ∈ Γ(O_{S×𝔸^{1}}), with

f(s, x) = g(s, ∥x∥^{2}), (s, x) ∈_{T} 𝔸^{p|2q}.

In this case, we define f^{∘}(s, t) := g(s, t^{2}).

Remark 2.21. In the case p > 0, the super function f^{∘}|_{S_{0}×(0,∞)} extends to a superfunction on S × 𝔸^{1} such that f(s, t) = f(s, −t). Such an extension is given by f^{∘}(s, t) := f(s, te_{1}), where t ∈_{T} 𝔸^{1} and e_{1} is the first standard basis vector of ℝ^{p}. Since f^{∘} is even in the second component, there is an extension g of f^{∘} ∘ (id_{S} × √ ) to S × 𝔸^{1}, and f(s, x) = g(s, ∥x∥^{2}) for all (s, x) ∈_{T} S × 𝔸^{p|2q}.

Here and in what follows, when handling Berezin (fibre) integrals, we will use Theorem 2.18 and Corollary 2.19 implicitly.

Proposition 2.22. Let f ∈ Γ(O_{S×𝔸^{p|2q}}) be rotationally invariant over S. Then

S⨏_{S×𝔸^{p|2q}}^{r}|Dλ(x)| f(s, x) = \begin{cases}\frac{π^{\frac{p−2q}{2}} (−1)^{q}}{Γ(\frac{p}{2})} ∫_{0}^{∞} dr r^{\frac{p}{2} −1}∂_{r}^{q}f^{∘}(s, \sqrt{r}), & p > 0, \\ (−π)^{−q}∂_{r=0}^{q}f^{∘}(s, \sqrt{r}), & p = 0, \end{cases}

in the sense that the integral exists if and only if the right-hand side exists, and in this case, they are equal.

Proof. Consider the superfunction g from above. Applying Taylor expansion yields

g(s, t + t′) ≡ ∑_{k=1}^{q} \frac{1}{k!}t^{′k}∂_{t′=0}^{k}g(s, t + t′) ≡ ∑_{k=1}^{q} \frac{1}{k!}t^{′k}∂_{t}^{k}g(s, t) mod (t^{′q+1})

with t = t′ = id_{𝔸^{1}}. Hence, we have

f(s, y) = g(s, ∥v∥^{2} + ∥η∥^{2}) = ∑_{k=1}^{q} \frac{1}{k!}∂_{2}^{k}g(s, ∥v∥^{2})∥η∥^{2k},

where y = id_{𝔸^{p|q}}, v = id_{𝔸^{p}}, η = id_{𝔸^{0|q}}, and (∂_{2}g)(s, t) := ∂_{t}g(s, t). The expression ∥η∥^{2k} contains η_{1} ⋯ η_{2q} if and only if k = q. In this case, it equals 2^{q}q!η_{1} ⋯ η_{2q}. For p = 0, this means that the integral is (−π)^{−q}∂_{t=0}^{q}g(s, t), as claimed. Similarly, if p > 0, then it takes the form

S⨏_{S×𝔸^{p|2q}}^{r}|Dλ(x)| f(s, x) = (−π)^{−q} ∫_{ℝ^{p}}|dv_{0}| ∂_{2}^{q}g(s, ∥v_{0}∥^{2}).

Applying polar coordinates for p ⩾ 2, we obtain for C = 2π^{\frac{p−2q}{2}} (−1)^{q}Γ(\frac{p}{2})^{−1}

= C ∫_{0}^{∞} dr r^{p−1}∂_{2}^{q}g(s, r^{2}) = \frac{C}{2} ∫_{0}^{∞} dr r^{\frac{p}{2} −1}∂_{r}^{q}g(s, r).

In case p = 1, one obtains the same result by symmetry. □

We obtain the following localization formula.

Corollary 2.23. Let k ⩽ min(\frac{p}{2}, q) and f ∈ Γ(O_{S×𝔸^{p|2q}}) be rotationally invariant and compactly supported in the fibres over S. Then

S⨏_{S×𝔸^{p|2q}}^{r}|Dλ(x)| f(s, x) = _{S}⨏_{S×𝔸^{p−2k|2q−2k}}^{r}|Dλ(x)| f^{∘}(s, x).

Proof. Use Proposition 2.22 and integration by parts for k < \frac{p}{2}. The fundamental theorem of calculus needs to be applied for k = \frac{p}{2}. □

Combining our results, the integral of rotationally invariant superfunctions takes the following form, which depends only on p − 2q and not on p|2q.

Corollary 2.24. Let f ∈ Γ(O_{S×𝔸^{p|2q}}) be rotationally invariant and compactly supported along the fibres over S. Then

S⨏_{S×𝔸^{p|2q}}^{r}|Dλ(x)| f(s, x) = \frac{2π^{\frac{p−2q}{2}}}{Γ(\frac{p−2q}{2} )} ∫_{0}^{∞} dr r^{p−2q−1}f^{∘}(s, r), p − 2q > 0.

If p − 2q ⩽ 0, then the following two cases occur:

\begin{cases}(−π)^{\frac{p−2q}{2}} ∂r=0^{\frac{2q−p}{2}} f^{∘}(s, \sqrt{r}) = (−π)^{\frac{p−2q}{2}} \frac{(\frac{2q−p}{2})!}{(2q − p)!}∂_{r=0}^{2q−p}f^{∘}(s, r), & 2 | p − 2q ⩽ 0, \\ (−π)^{\frac{p−1−2q}{2}} ∫_{0}^{∞} dr r^{− \frac{1}{2}} ∂r^{\frac{2q+1−p}{2}} f^{∘}(s, \sqrt{r}), & 2 ∤ p − 2q < 0. \end{cases}

Proof. In view of Proposition 2.22 and Corollary 2.23, the only case that needs some consideration is that of p = 0. Here, Faà di Bruno’s formula gives

∂_{r=0}^{2q}f^{∘}(s, r) = ∂_{r=0}^{2q}f^{∘}(s, \sqrt{r^{2}}) = ∑_{k_{1}+2k_{2}=2q} \frac{(2q)!}{k_{1}!k_{2}!}∂_{t=0}^{k_{1}+k_{2}}f^{∘}(s, \sqrt{t}) 0^{k_{1}}1^{k_{2}}.

All summands except for k_{2} = q vanish. □

3. Symmetric superspaces and spherical superfunctions

In this section, we introduce our main objects, the spherical superfunctions. Before proceeding to the definition of the spherical superfunctions, we collect some ancillary facts concerning symmetric superpairs and integration formulæ.

3.1. Symmetric superpairs. We review some facts on symmetric superpairs, referring to Refs. [1, 6] for omitted details.

Definition 3.1 (Symmetric superpairs and Cartan decomposition). A symmetric superpair is a pair (𝔤, θ), where 𝔤 be a complex Lie superalgebra and θ an involutive automorphism of 𝔤. The eigenspace decomposition

𝔤 = 𝔨 ⊕ 𝔭, 𝔨 := ker(1 − θ), 𝔭 := ker(1 + θ)

is called Cartan decomposition of (𝔤, θ). Let (G, K, θ) be given, where G is a cs Lie supergroup with Lie superalgebra 𝔤, θ is an involutive automorphism of G, and K is a closed subsupergroup, θ|_{K} = id_{K}, and its Lie superalgebra is 𝔨 = ker(1 − θ) (denoting the derivative of θ by the same letter). Then (G, K, θ) is called a symmetric supertriple, and the symmetric superpair (𝔤, θ) is called the infinitesimal superpair associated with (G, K, θ). Then (G, K, θ) is said to admit a global Cartan decomposition if the morphism

K × 𝔸(𝔭_{ℝ}) ⟶ G : (k, x) ⟼ ke^{x} (3.1)

is an isomorphism of cs manifolds. Here, we write e^{x} := exp_{G}(x).

Proposition 3.2. A pair (G, K, θ) admits a global Cartan decomposition if and only if this is true for (G_{0}, K_{0}, θ_{0}).

Proof. The morphism in (3.1) is a local isomorphism at (1, z), for any z ∈ 𝔸(𝔭_{ℝ})_{0}, since its derivative is given by 𝔨 × 𝔭 ⟶ 𝔤 : (y, x) ⟼ y + x. Since the morphism is K-equivariant, it is everywhere a local isomorphism. By the inverse function theorem [30], it is an isomorphism if and only if the underlying map is a bijection. □

Definition 3.3 (Reductive and even type conditions). The notions of a reductive, strongly reductive, or even type symmetric superpair (𝔤, θ) are defined in Ref. [1]. If (𝔤, θ) is the infinitesimal superpair of a symmetric supertriple (G, K, θ), then we accordingly apply these adjectives to (G, K, θ).

Definition 3.4. Let (𝔤, θ) be a reductive symmetric superpair of even type with even Cartan subspace 𝔞. Then

(3.2)  𝔤 = 𝔪 ⊕ 𝔞 ⊕ ⨁_{α∈Σ} 𝔤^{α}, 𝔪 := 𝔷_{𝔨}(𝔞), 𝔤^{α} := ⋂_{h∈𝔞} ker(ad h − α(h)),

where Σ ⊆ 𝔞^{∗} \ 0, called the set of restricted roots, is the finite set defined by this equation. A root α ∈ Σ is called even and odd, if, respectively, 𝔤α ; ¯0 ≠ 0 and 𝔤α ; ¯1 ≠ 0. Note that roots may simultaneously be even and odd. Given h ∈ 𝔞_{ℝ} such that α(h) ∈ ℝ \ 0 for all α ∈ Σ, the subset

Σ^{+} := {α ∈ Σ | α(h) > 0}

is called a positive system. Roots α ∈ Σ^{+} for which \frac{α}{2} ∉ Σ are called indivisible. We set ϱ := \frac{1}{2} ∑_{α∈Σ^{+}} m_{α}α, where m_{α} := sdim 𝔤^{α} = dim 𝔤α ; ¯0 − dim 𝔤α ; ¯1. Fixing a positive system Σ^{+}, Equation (3.2) takes on the form

𝔤 = 𝔫̄ ⊕ 𝔪 ⊕ 𝔞 ⊕ 𝔫, 𝔫 := ⨁_{α∈Σ^{+}} 𝔤^{α}, 𝔫̄ := θ(𝔫) = ⨁_{α∈Σ^{+}} 𝔤^{−α},

the Bruhat decomposition of 𝔤. Since 𝔤^{α} ⊕ 𝔤^{−α} is θ-invariant, we have

𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫,

the Iwasawa decomposition of 𝔤.

For the remainder of this subsection, let (G, K, θ) be a reductive symmetric supertriple of even type. Fix an even Cartan subspace and a positive system of roots. Denote by N, N̄, and A the analytic subsupergroups of G corresponding to 𝔫, 𝔫̄, and 𝔞, respectively. Here, the analytic subsupergroup of G corresponding to a subalgebra 𝔥 ⊆ 𝔤 such that 𝔥_{¯0,ℝ} := 𝔥 ∩ 𝔤_{¯0,ℝ} is a real form of 𝔥¯0 is defined to be the cs Lie supergroup associated with the supergroup pair (𝔥, H_{0}), where H_{0} is the analytic (i.e. , connected) subgroup of G_{0} with the Lie algebra 𝔥_{¯0,ℝ}. Moreover, let M be the closed subsupergroup associated with the supergroup pair (𝔪, M_{0}), where M_{0} := Z_{K_{0}}(𝔞). The following proposition follows in the same way as Proposition 3.2.

Proposition 3.5 (Global Bruhat decomposition). The morphism

N̄ × M × A × N ⟼ G : (n̄, m, a, n) ⟼ n̄man

is an open embedding if and only if this is true for the underlying map. In this case, we say that (G, K, θ) admits a global Bruhat decomposition.

For the following proposition, consult Refs. [1, 6].

Proposition 3.6 (Global Iwasawa decomposition). Both of the morphisms

K × A × N ⟶ G : (k, a, n) ⟼ kan,
N × A × K ⟶ G : (n, a, k) ⟼ nak

are isomorphisms if and only if this is already true for one of the underlying maps. In this case, we say that (G, K, θ) admits a global Iwasawa decomposition.

Remark 3.7. Observe that M centralises 𝔞, since

Ad(me^{x})(h) = Ad(m)(e^{ad(x)}(h)) = h

for m ∈_{S} M_{0}, x ∈_{S} 𝔸(𝔪¯1), h ∈_{S} 𝔸(𝔞_{ℝ}). Similarly, M normalises N and N̄.

Let (G, K, θ) admit a global Iwasawa decomposition. We define morphisms

k, u: G → K, A, H: G → 𝔸(𝔞_{ℝ}), n, n_{1} : G → N

by requiring, for g ∈_{S} G, that

(3.3)  g = k(g)e^{H(g)}n(g) = n_{1}(g)e^{A(g)}u(g).

Then

(3.4)  n_{1}(g) = n(g^{−1})^{−1}, A(g) = −H(g^{−1}), u(g) = k(g^{−1})^{−1}.

In view of Remark 3.7, H and k are right M-invariant, i.e.

(3.5)  H(gm) = H(g), k(gm) = k(g), g ∈_{S} G, m ∈_{S} M.

Moreover, we note for later reference that

(3.6)  H(gh) = H(gk(h)) + H(h), k(gh) = k(gk(h)),

for all g, h ∈_{S} G. This follows from a straightforward computation on points. The importance of the above decompositions is that they give rise to natural coordinate systems on certain homogeneous superspaces. Here and in the sequel, we will use quotients of cs Lie supergroups. They are defined in the same way as for real Lie supergroups, see Refs. [2,13] for the latter. We will need the fact that for a closed cs Lie subsupergroup H of G, the sheaf of superfunctions on G/H is the direct image under the canonical projection G_{0} ⟶ G_{0}/H_{0} of the sheaf of H-invariant superfunctions on G.

Proposition 3.8. Let (G, K, θ) admit a global Iwasawa decomposition. The morphism

G ⟶ N × A : g ↦ (n_{1}(g), e^{A(g)})

induces an isomorphism G/K ⟶ N × A.

Let Q = MAN be the closed subsupergroup of G generated by M, A, and N.

Proposition 3.9. Let (G, K, θ) admit global Bruhat and Iwasawa decompositions. The composite

G  K  K/M_{k}

induces an isomorphism G/Q → K/M. The composite

N̄  G  G/Q  K/M_{k̇}

is an open embedding, which is also denoted by k.

3.2. Integral formulæ for supergroups and symmetric superspaces. In this subsection, we derive integral formulæ for the decompositions given above. These are crucial for the proof of the central Corollary 3.20 and therefore for the proof of the c-function asymptotics and the Harish-Chandra series. Moreover, the proof of loc. cit. hinges on the parameter-dependent version of these formulæ, which are therefore technically not a simple-minded copy of their classical relatives. We begin with some generalities, cf. Ref. [2].

Definition 3.10 (Invariant Berezinian densities). Let G be a cs Lie supergroup and a: G × X → X an action. Consider the cs manifold X_{G} := G × X over G. A Berezinian density ω ∈ Γ(|Ber|_{X}) is said to be G-invariant if

(a, p_{X})^{♯}(p_{X}^{♯}(ω)) = p_{X}^{♯}(ω).

Here, p_{X} : X_{G} → X is the projection.

The following is straightforward.

Lemma 3.11. Let ω ∈ Γ(|Ber|_{X}) be a Berezinian density. Then ω is Ginvariant if and only if, the following is true:

∫_{X} ω(x)f(gx) = ∫_{X} ω(x)f(x)

for any g ∈_{S} G, any cs manifold S, and any f ∈ Γ_{c}(p_{X,0!}O_{X_{G}}).

The G-superspace X is called analytically unimodular if there is a non-zero G-invariant Berezinian density. If the action is transitive and X is analytically unimodular, then |Ber|_{X} has a global module basis, given by the choice of such a Berezinian density; moreover, it is unique up to a multiplicative constant [2]. Sufficient conditions for the analytic unimodularity of homogeneous Gsuperspaces are stated in [1, Proposition A.2].

Proposition 3.12. Let G be analytically unimodular and |Dg| a non-zero Ginvariant Berezinian density. Let U = −U ⊆ 𝔸(𝔤_{ℝ})_{0} be an open neighbourhood of 0 such that exp_{G} : 𝔸(𝔤_{ℝ})|_{U} → G is an open embedding. Then

exp_{G}^{♯}(|Dg|) = |Dλ|(x) |Ber|(\frac{1 − e^{− ad x}}{ad x} )

on 𝔸(𝔤_{ℝ})|_{U}. Here, |Dλ| is an adequately normalized Berezin–Lebesgue density.

When G is nilpotent, then str ad x = 0 for the generic point x of 𝔸(𝔤_{ℝ}). Hence, by [1, Proposition A.2] and Proposition 3.12, the following is immediate.

Corollary 3.13. Let G be a nilpotent cs Lie supergroup. Then G is analytically unimodular and

exp_{G}^{♯}(|Dg|) = |Dλ|

for an appropriate normalization of the invariant density |Dg| and the Berezin– Lebesgue density |Dλ|.

Proof of Proposition 3.12. The claim follows along the lines of the classical result [23, Chapter I, § 2, Theorem 1.14]. If V is a sufficiently small neighbourhood of 0 ∈ 𝔤_{0}, then the Campbell– Hausdorff morphism

C : 𝔸(𝔤_{ℝ})|_{U} × 𝔸(𝔤_{ℝ})|_{V} → 𝔸(𝔤_{ℝ}), exp_{G} C(x, y) = (exp_{G} x)(exp_{G} y)

is well-defined. We let

X := 𝔸(𝔤_{ℝ})|_{U} × 𝔸(𝔤_{ℝ})|_{V} , S := 𝔸(𝔤_{ℝ}),

and consider X as a cs manifold over S via p_{1}. Set φ := (p_{1}, C) and let exp^{♯}(|Dg|) = |Dλ|ρ for some ρ ∈ O_{𝔸(𝔤_{ℝ})}(U). For any f ∈ Γ_{c}(O_{𝔸(𝔤_{ℝ})}|_{U}), we have

∫_{𝔸(𝔤_{ℝ})}|Dλ| ρf = ∫_{G}|Dg| f(log g) = ∫_{G}|Dg| f(log(exp_{G}(−x)g))
= ∫_{𝔸(𝔤_{ℝ})}|Dλ(y)| ρ(y)f(C(−x, y))
= ∫_{𝔸(𝔤_{ℝ})}|Dλ(y)| |Ber|(T_{φ/S})ρ(C(x, y))f(y).

Here, x is the generic point of T := 𝔸(𝔤_{ℝ})|_{U} and the identity y = C(x, C(−x, y)) was applied. Since f was chosen arbitrarily, this implies

ρ(y) = |Ber|(T_{φ/S})(x, y)ρ(C(x, y)), y ∈_{T} 𝔤|_{V} .

Setting y = 0 shows

ρ(x) = |Ber|(T_{φ/S})(x, 0)^{−1}ρ(0).

Hence, the claim will follow after suitable normalization and computing T_{φ/S}|_{y=0}. For the latter, it suffices to prove

T_{(x,0)/S}(φ) = \frac{ad x}{1 − e^{− ad x}}

where we consider ad x as an O_{T} -linear endomorphism of O_{T} ⊗ 𝔤. Interpreting vector fields as morphisms via dual numbers 𝔻¯0 (see Subsection 2.3) this reduces to an equation between morphisms, which can be checked on 𝔸^{0|q}-valued points, and hence follows from the classical situation [25]. □

For the remainder of this subsection, let (G, K, θ) be a reductive symmetric supertriple of even type, such that a global Iwasawa decomposition exists. Further, assume that K_{0} is compact. Then the G × G-superspace G, the Gsuperspace G/K, and the K-superspace K/M are all analytically unimodular. We choose corresponding invariant Berezinian densities |Dg|, |Dġ|, and |Dk̇|, as well as invariant densities |Dn| and |Dn̄| = θ^{♯}(|Dn|) on N and N̄, respectively, and a Haar density da on A. The normalization of these relative to each other is fixed by the Propositions 3.14, 3.15, and 3.17, which are simple generalizations of integral formulæ from Refs. [1, 6].

We will use without further mention the fact that for compactly supported integrands, the Berezin (fibre) integral is independent of the choice of retraction (Theorem 2.18) and thus admits coordinate transformations (Corollary 2.19).

Proposition 3.14 ([1, Proposition 2.2]). The pullback of the invariant Berezin density |Dġ| via the Iwasawa isomorphism is |Dk| ⊗ da ⊗ |Dn|. In particular,

∫_{G}|Dg| f(s, g) = ∫_{K}|Dk| ∫_{A} da ∫_{N}|Dn| f(s, kan)e^{2ϱ(log a)}

for f ∈ Γ_{c}(p_{1!}O_{S×G}).

Proposition 3.15 ([6, Lemma 4.2]). Let f, h ∈ Γ_{c}(p_{1!}O_{S×K/M}). Then

∫_{K}|Dk| f(s, k(g^{−1}k))h(s, k) = ∫_{K}|Dk| f(s, k)h(s, k(gk))e^{−2ϱ(H(gk))}

for any s ∈_{T} S and all cs manifolds T.

Corollary 3.16. Let f ∈ Γ(O_{S×K/M}). Then for any s ∈_{T} S, we have

∫_{K/M}|Dk̇| f(s, k(g^{−1}k)) = ∫_{K/M}|Dk̇| f(s, k)e^{−2ϱ(H(gk))}.

Proof. In the classical case, this equation follows directly from Proposition 3.15. In the super setting, the volume of K may vanish, so one has to argue with greater care. Let χ ∈ Γ_{c}(O_{K}), such that ∫_{K}|Dk′| χ(k′) = 1. Writing B = K/M,

∫_{B}|Dk̇| f(s, k(g^{−1}k)) = ∫_{K}|Dk′| χ(k′) ∫_{B}|Dk̇| f(s, k(g^{−1}k′k))
= ∫_{B}|Dk̇| ∫_{K}|Dk′| χ(k′k^{−1})f(s, k(g^{−1}k′)),

by the left-invariance of |Dk̇| and the right-invariance of |Dk′|. By Proposition 3.15, this equals

= ∫_{B}|Dk̇| ∫_{K}|Dk′| χ(k(gk′)k^{−1})f(s, k′)e^{−2ϱ(H(gk′))},

so on applying the right-invariance of |Dk′| and the left-invariance |Dk̇| again, we obtain the equalities

= ∫_{K}|Dk′| ∫_{B}|Dk̇| χ(k(gk′k)k^{−1})f(s, k′k)e^{−2ϱ(H(gk′k))}

= ∫_{K}|Dk′| ∫_{B}|Dk̇| χ(k(gk)k^{−1}k′)f(s, k)e^{−2ϱ(H(gk))}.

Finally, the left-invariance of |Dk′| gives

= ∫_{B}|Dk̇|(∫_{K}|Dk′| χ(k′))f(s, k)e^{−2ϱ(H(gk))}
= ∫_{B}|Dk̇| f(s, k)e^{−2ϱ(H(gk))},

which is the desired statement. □

We have the following fact from Ref. [6].

Proposition 3.17 ([6, Proposition 4.4]). The pullback of the invariant Berezin density |Dk̇| on K/M via the open embedding k from Proposition 3.9 is |Dn̄|e^{−2ϱ(H(n̄))}.

Remark 3.18. Note that the standard retraction on N̄ is in general not compatible via k with any globally defined retraction on K/M. Hence, Theorem 2.18 does not apply, and the equality

∫_{K/M}|Dk̇| f(s, k) = ∫ r ; N̄|Dn̄| f(s, k(n̄))e^{−2ϱ(H(n̄))}

will in general not hold for superfunctions f ∈ Γ(O_{S×K/M}), unless the intersection supp f with the image of S × N̄ is not compact along the fibres of p_{1}. Below, where we derive the leading asymptotics of the spherical superfunctions on G/K, we will resolve these issues by introducing an atlas of similar charts and cutting off the integrand in these charts by the choice of a partition of unity. A similar problem occurs if we use polar coordinates on G/K, in fact, it is more severe, leading to singularities at the boundary of the Weyl chamber. In a forthcoming paper, where we treat the inversion formula for the spherical Fourier transform in rank one, we will discuss these singularities and the ‘boundary terms’ that they introduce at length.

3.3. Definition of the spherical superfunctions. Keeping the above assumptions, we can now define, for λ ∈ 𝔞^{∗}, the spherical superfunction ϕ_{λ} ∈ Γ(O_{X}), where X := G/K, as follows. For g ∈_{T} G, where T is any cs manifold, we let

(3.7)  ϕ_{λ}(g) := ∫_{K/M}|Dk̇| e^{(λ−ϱ)(H(gk))}.

By Lemma 3.11, this defines a superfunction on X. We will derive some alternative integral expressions for ϕ_{λ} in this subsection— in particular, we will obtain the symmetry property stated in Corollary 3.20, which will be essential for the Harish-Chandra series expansion of ϕ_{λ}. The statements we give here are essentially identical to the classical case, and their derivations are parallel to those given in Refs. [23,24]. However, they are based on Corollary 3.16, whose proof was a little more subtle than classically, so we briefly give the details.

Proposition 3.19. For any λ ∈ 𝔞^{∗} and g, h ∈_{T} G, we have

ϕ_{λ}(hg^{−1}) = ∫_{K/M}|Dk̇| e^{(λ−ϱ)(H(hk))}e^{(−λ−ϱ)(H(gk))}

Proof. On applying Equation (3.6) twice, we obtain

H(hg^{−1}k) = H(hk(g^{−1}k)) + H(g^{−1}k) = H(hk(g^{−1}k)) − H(gk(g^{−1}k)),

so that

ϕ_{λ}(hg^{−1}) = ∫_{K/M}|Dk̇| e^{(λ−ϱ)(H(hk(g^{−1}k)))}e^{−(λ−ϱ)(H(gk(g^{−1}k)))}.

Setting S := G × G and f(g, h, k) := e^{(λ−ϱ)(H(hk))}e^{−(λ−ϱ)(H(gk))} in Corollary 3.16, we arrive at our claim. □

Setting h = 1 in Proposition 3.19, the following fact is immediate. It will be instrumental in deriving a series expansion for ϕ_{λ}.

Corollary 3.20. For λ ∈ 𝔞^{∗} and g ∈_{T} G, we have ϕ_{λ}(g^{−1}) = ϕ_{−λ}(g).

This allows us to derive an alternative expression for ϕ_{λ}, as follows: Consider the isomorphism ĩ_{K} : M\K → K/M induced by the inversion morphism i_{K} of K. The Berezinian density

|Dk̇_{r}| := ĩ_{K}^{♯}(|Dk̇|),

where |Dk̇| is the K-invariant Berezinian density on K/M, is right K-invariant. We have the following formula, which will be used in a subsequent paper.

Corollary 3.21. For any λ ∈ 𝔞^{∗} and g ∈_{T} G, we have

ϕ_{λ}(g) = ∫_{M\K}|Dk̇_{r}| e^{(λ+ϱ)(A(kg))}.

where A(·) is defined in Equation (3.3).

Proof. We have by the definition of |Dk̇_{r}| and Equation (3.4)

ϕ_{λ}(g) = ∫_{K/M}|Dk̇|e^{(λ−ϱ)(H(gk))}
= ∫_{K/M}|Dk̇|e^{(−λ+ϱ)(A(k^{−1}g^{−1}))} = ∫_{M\K}|Dk̇_{r}|e^{(−λ+ϱ)(A(kg^{−1}))},

so the assertion follows from Corollary 3.20. □

4. The Harish-Chandra c-function

In this section, unless something else is stated, let (G, K, θ) be a reductive symmetric supertriple of even type admitting global Iwasawa and Bruhat decompositions. A fixed even Cartan subspace 𝔞 and positive system Σ^{+} are understood, and in all the cases we will consider, the rank dim 𝔞−dim 𝔞∩dim 𝔷(𝔤) will be one. The notation introduced in Subsection 3.1 will be used freely.

We will derive the leading asymptotics of the spherical superfunctions ϕ_{λ}. These are governed by the Harish-Chandra c-function. We compute these explicitly in terms of Euler Γ functions. While the general structure of these functions is similar to the even setting, the occurrence of negative root multiplicities leads to a shift in the location of poles and zeroes. By a rank reduction procedure, this result has, in Ref. [6], been extended to arbitrary rank, on the basis of our results. In general, isotropic restricted roots occur, leading to terms in the general c-function formula that are not simply shifted versions of the ‘purely even’ c-function. Moreover, contrary to the ‘classical’ setting of Riemannian symmetric spaces of the non-compact type, where the proof of convergence of the leading asymptotic term of ϕ_{λ} is a simple argument based on the dominated convergence theorem, in the super setting, this becomes a quite subtle issue. We need to cover the ‘maximal boundary’ B = K/M by a whole atlas of Weyl-group related ‘stereographical charts’. In the situations we consider (whose common feature is that they are of rank one), B is a ‘geodesic supersphere at infinity’, and this makes for two chart domains. These are essential, since the integrals on the individual domains are both divergent unless the integrand is cut off appropriately. For the divergences in this expansion to cancel, the Weyl symmetry of the spherical functions (Corollary 3.20), derived above without any restriction on the rank, by the use of relative cs manifolds and fibre integrals, will be of central importance. Based on the derivation of the leading asymptotics, we prove the existence of a full Harish-Chandra series expansion, similar to the even case. We will show the existence of the c-function in three cases of rank one and compute it explicitly. Contrary to the classical situation, the proof of existence is by far the more difficult of these steps. Both (the proof of existence and the explicit determination) will be performed on a case-by-case basis.

4.1. Definition and statement of the main result.

Definition 4.1. Let h_{0} ∈ 𝔞 such that α(h_{0}) > 0 for all α ∈ Σ^{+}. The Harish-Chandra c-function is defined as follows:

(4.1)  c(λ) := lim_{t→∞} e^{−(λ−ϱ)(th_{0})}ϕ_{λ}(e^{th_{0}}),

for λ ∈ 𝔞^{∗}, ℜλ(h_{0}) > 0, provided that the limit exists. Unless the rank of (G, K, θ), defined to be dim 𝔞 − dim 𝔞 ∩ 𝔷(𝔤), is one, it is not obvious per se that this definition is independent of the choice of h_{0}.

We will consider the cases stated in Table 4.1 below, where we abbreviate 𝔤𝔩(p|q, ℂ) by 𝔤𝔩_{p|q}, etc., and allow arbitrary integers p, q ⩾ 0. We refer the reader to the following subsubsections for the precise definitions, in particular, regarding the cs supergoups G and K. For any of these symmetric pairs, we have the following statement.

𝔤  𝔨  G  K
𝔤𝔩_{2+p|q}  𝔤𝔩_{1} × 𝔤𝔩_{1+p|q}  U_{cs}(1, 1 + p|q)  U(1) × U_{cs}(1 + p|q)
𝔬𝔰𝔭_{2+p|2q}  𝔬𝔰𝔭_{1+p|2q} SOSp_{cs}^{+}(1, 1 + p|2q) SOSp_{cs}(1 + p|2q)
𝔤𝔩_{1|1} × 𝔤𝔩_{1|1}  𝔤𝔩_{1|1}  (GL × GL)_{ℂ|ℝ}^{+}(1|1)  UGL_{ℝ}^{+}(1|1)

Table 1. Riemannian symmetric superspaces under consideration

Theorem 4.2. Let be (G, K) be one of the symmetric pairs listed in Table 4.1 and let α be a choice of positive indivisible restricted root. The c-function c(λ) exists for ℜλ > 0. For some c_{0} ≡ c_{0}(ϱ) ≠ 0, it is given by

c(λ) = c_{0} \frac{2^{−λ}Γ(λ)}{Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + 1)) Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + m_{2α}))}

if α is anisotropic, and if α is isotropic, then it is given by

c(λ) = c_{0}λ.

Here, we identify λ ≡ \frac{⟨λ,α⟩}{⟨α,α⟩} if α is anisotropic and λ ≡ ⟨λ, α⟩ otherwise.

In particular, up to the constant c_{0}, the value of the c-function c(λ) is independent of the choice of h_{0} with α(h_{0}) > 0. After suitable normalisation of h_{0}, the constants c_{0} depend on p and q only via ϱ, see below for details.

4.2. Proof of Theorem 4.2. We now begin with the examination of the c-function for the Riemannian symmetric superspaces X = G/K from Table 4.1, thereby proving Theorem 4.2 case-by-case. For the three individual cases, the formula is proved below in Theorem 4.8, Theorem 4.14, and Corollary 4.17, respectively. We will constantly be using the functor of points, employing Proposition 2.1, Corollary 2.2 and Proposition 2.3 to compute it explicitly.

4.2.1. The unitary case. In the following, let 𝔤 := 𝔤𝔩(2 + p|q, ℂ). Moreover, we let G := U_{cs}(1, 1 + p|q) denote the cs form of GL(2 + p|q, ℂ) corresponding to

G_{0} := U(1, 1 + p) × U(q),

which is a the real form of the complex Lie group G_{ℂ,0} := GL(2 + p, ℂ) × GL(q, ℂ). An involution θ of G resp. of 𝔤 is given by θ(x) := σxσ where

σ := ( \begin{matrix}−𝟙_{1} & 0 & 0 \\ 0 & 𝟙_{1+p} & 0 \\ 0 & 0 & 𝟙_{q} \end{matrix} ) . (4.2)

Under this involution, 𝔤 decomposes as 𝔤 = 𝔨 ⊕ 𝔭 with

𝔨 = ( \begin{matrix}∗ & 0 & 0 \\ 0 & ∗ & ∗ \\ 0 & ∗ & ∗ \end{matrix} ) , 𝔭 = ( \begin{matrix}0 & ∗ & ∗ \\ ∗ & 0 & 0 \\ ∗ & 0 & 0 \end{matrix} ) .

Documenta Mathematica 19 (2014) 1317–1366

Here, dashed lines in the matrices indicate the action of the involution, whereas full lines signify the grading. A non-degenerate invariant supersymmetric even bilinear form b on 𝔤 is given by b(x, y) := str(xy). We let 𝔞_{ℝ} ⊆ 𝔭_{0} be the subspace generated by the matrix

(4.3)  h_{0} := ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) .

Here and in what follows, the dimension of the rows and columns in all matrices will be 1, 1, p, and q, respectively. As ungraded vector spaces, 𝔤 = 𝔤𝔩(2 + p + q, ℂ), so since h_{0} is even, the root decomposition is the same as in the classical case. Defining α ∈ 𝔞^{∗} by α(h_{0}) = 1, there are exactly four roots: ±α, ±2α. The general elements of 𝔤^{α} and 𝔤^{−α} are respectively

(4.4)  ( \begin{matrix}0 & 0 & B¯0 & B¯1 \\ 0 & 0 & B¯0 & B¯1 \\ −C¯0 & C¯0 & 0 & 0 \\ −C¯1 & C¯1 & 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & −B¯0 & −B¯1 \\ 0 & 0 & B¯0 & B¯1 \\ C¯0 & C¯0 & 0 & 0 \\ C¯1 & C¯1 & 0 & 0 \end{matrix} ) .

The general elements of 𝔤^{2α} and 𝔤^{−2α} are respectively

( \begin{matrix}−A & A & 0 & 0 \\ −A & A & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , ( \begin{matrix}−A & −A & 0 & 0 \\ A & A & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) .

Moreover, 𝔪 consists of the following matrices:

( \begin{matrix}D & 0 & 0 & 0 \\ 0 & D & 0 & 0 \\ 0 & 0 & E & F \\ 0 & 0 & G & H \end{matrix} ) .

We let Σ^{+} := {α, 2α}, so that 𝔫 = 𝔤^{α} ⊕ 𝔤^{2α} and 𝔫̄ = 𝔤^{−α} ⊕ 𝔤^{−2α}. From the above, we see that

m_{±α} = 2(p − q), m_{±2α} = 1, ϱ = (1 + p − q)α.

The analytic subsupergroup K of G with Lie superalgebra 𝔨 is U(1) × U_{cs}(1 + p|q). The underlying Lie group is K_{0} = U(1) × U(1 + p) × U(q). Since the Riemannian symmetric space G_{0}/K_{0} = U(1, 1 + p)/(U(1) × U(1 + p)) is of noncompact type, G_{0} admits global Iwasawa and Bruhat decompositions [22]. In view of Proposition 3.6 and Proposition 3.5, the same is true of G. We parametrise 𝔫 and 𝔫̄ by setting τ := ( \begin{matrix}1 & 1 & 0 \\ 0 & 0 & 𝟙_{p+q} \end{matrix} )^{t} and

(4.5)  X(D, E, F) := τ (\begin{matrix}D & E \\ F & 0 \end{matrix}) τ^{t}σ, X̄(D, E, F) := στ (\begin{matrix}D & E \\ F & 0 \end{matrix}) τ^{t},

so that

𝔫 = {X(D, E, F) | D ∈ ℂ, E ∈ ℂ^{1×p|q}, F ∈ ℂ^{p|q×1}},
𝔫̄ = {X̄(D, E, F) | D ∈ ℂ, E ∈ ℂ^{1×p|q}, F ∈ ℂ^{p|q×1}}.

Let 𝔤_{ℝ} := 𝔤_{¯0,ℝ} ⊕ 𝔤¯1, 𝔤_{¯0,ℝ} being the Lie algebra of G_{0}, and A_{ℝ} := A ∩ 𝔤_{ℝ} for any A ⊆ 𝔤. Then by Proposition 2.14, we have an isomorphism φ, given by

φ: 𝔸^{1} × 𝔸^{p|q} × 𝔸^{p|q} ⟶ 𝔸(𝔫̄_{ℝ}),
(a, b, c) ⟼ X(\frac{1}{2i}a, E(b, c), F (b, c)),

on T -valued points b = b¯0 + b¯1, c = c¯0 + c¯1 ∈_{T} 𝔸^{p|q}, where we set

E(b, c) := (−(b¯0 − ic¯0)^{t} −(b_{¯1,1} − c_{¯1,1}) ⋯ −(b_{¯1,q} − c_{¯1,q})) ,
F (b, c) := ((b¯0 + ic¯0)^{t} b_{¯1,1} + c_{¯1,1} ⋯ b_{¯1,q} + c_{¯1,q})^{t} .

A similar statement holds for 𝔫. In what follows, we will again make extensive use of the formalism of T -valued points, see above for explanations. For any cs manifold T, the T -valued points of N are of the form

n_{DEF} := e^{X(D,E,F)} = 1 + X(D, E, F) + \frac{1}{2}X(D, E, F)^{2}
= 𝟙_{2+p+q} + X(D + \frac{1}{2}EF, E, F )
n̄_{DEF} := e^{X̄(D,E,F)} = 𝟙_{2+p+q} + X̄(D + \frac{1}{2}EF, E, F ),

with D, E, F constrained appropriately, since τ^{t}στ = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ). Obviously, we have the equality θ(n_{DEF}) = n̄_{DEF}.

Lemma 4.3. The restriction of H : G ⟶ 𝔸(𝔞_{ℝ}) to N̄ is given by

H(n̄) = \frac{1}{2} log((1 − BC)^{2} − 4A^{2})h_{0}, n̄ = n̄_{DEF} ∈_{T} N̄.

Proof. Formally, the calculations are the same as in the classical case, the only difference being that one has to work with generalised instead of ordinary points. Write n̄ = n̄_{DEF} = ke^{th_{0}}n_{D′E′F′}, so that H(n̄) = th_{0}. Then

(4.6)  n_{(−A)(−B)(−C)}n̄_{DEF} = θ(n̄)^{−1}n̄ = n̄_{(−D′)(−E′)(−F′)}e^{2th_{0}}n_{D′E′F′}.

Let v := (1, 1, 0). The simple identities

vστ = 0, vτ = (2 0)

give

(4.7)  vn_{DEF} = (1 − 2D − EF 1 + 2D + EF 2E) , vn̄_{DEF} = v,
(4.8) n_{DEF} v^{t} = v^{t},  n̄_{DEF} v^{t} = (−2D − EF 1 + 2D + EF 2F )^{t} .

Applying these on the right-hand side of Equation (4.6) gives

vθ(n̄)^{−1}n̄v^{t} = cosh 2t + sinh 2t = 2e^{2t}.

On the other hand, applying these on the left-hand side, we obtain

vθ(n̄)^{−1}n̄v^{t} = 2((1 − EF)^{2} − 4D^{2}).

This proves the lemma. □

The classical approach to derive c(λ) is to write

ϕ_{λ}(g) = ∫_{K/M}|Dk̇|e^{(λ−ϱ)(H(gk))} = ∫N̄|Dn̄|e^{(λ−ϱ)(H(gk(n̄)))}.

However, if q ≠ 0, this cannot be done, since the standard retraction on N̄ does not extend to K/M. Since ϱ can be an arbitarily large negative multiple of α, the integral on the right-hand side does not even have to exist. The next guess might be to interchange the limit in Equation (4.1) with the integral over K/M. However, this is not permitted, since exp(λ−ϱ)(H(e^{th_{0}}ke^{−th_{0}})) does admit a smooth limit, even in the classical case. The solution we propose is to apply Proposition 3.17 after cutting off the integrand inside k_{0}(N̄_{0}). Thus, fix χ ∈ C∞ c(0, ∞) such that χ = 1 on a neighbourhood of 1, and define Ξ ∈ Γ(O_{K/M}) by

Ξ(k(n̄)) := χ((1 − EF)^{2} − 4D^{2}), n̄ = n̄_{DEF} ∈_{T} N̄

on the open subspace k(N̄) ⊆ K/M, and by zero otherwise. Clearly, this superfunction is well-defined and has compact support in the image k(N̄) of k: N̄ → K/M. Then for any w ∈ M′_{0} := N_{K_{0}}(𝔞), we have

ϕ_{λ}(g) = ∫_{K/M}|Dk̇| e^{(λ−ϱ)(H(gk))}Ξ(k) + ∫_{K/M}|Dk̇| e^{(λ−ϱ)H(gwk)}(1 − Ξ)(wk) (4.9)

by K-invariance of |Dk|. Denote the summands by I_{0}^{λ}(g) and I_{∞}^{λ}(g), respectively. Proposition 3.17 applies to the first of these two integrals. To do the same for the second, a good choice for w has to be made. The Weyl group W_{0} = M′_{0}/M_{0} = {±1}; let w_{0} ∈ M′_{0} be a representative of the non-trivial element. By the Bruhat decomposition [22, Theorem 1.3], we have

(K_{0}/M_{0}) \ k_{0}(N̄_{0}) = w_{0}o, o := k(1) = 1M_{0}.

Since 1 − Ξ vanishes in a neighbourhood of o, this shows that the second integrand in Equation (4.9) has compact support inside (K_{0}/M_{0}) \ {w_{0}o} = k_{0}(N̄_{0}). On applying Theorem 2.18, Corollary 2.19, Equation (3.6), and Proposition 3.17, we obtain

\begin{aligned}I_{0}^{λ}(g) & = ∫N̄|Dn̄|e^{(λ−ϱ)(H(gn̄))}e^{−(λ+ϱ)(H(n̄))}Ξ(k(n̄)) \\ I_{∞}^{λ}(g) & = ∫N̄|Dn̄|e^{(λ−ϱ)(H(gw_{0}n̄))}e^{−(λ+ϱ)(H(n̄))}(1 − Ξ)(w_{0}k(n̄)). \end{aligned} (4.10)

For the following considerations, let g ∈_{T} A. Moreover, we will identify 𝔞 with ℂ via 1 ⟼ h_{0} and 𝔞^{∗} with ℂ via λ ⟼ λ(h_{0}). It is obvious from Corollary 2.11 that

e^{th_{0}} n̄_{DEF} e^{−th_{0}} = exp_{G} e^{t ad h_{0}}(X̄(D, E, F)) = n̄_{e^{−2t}D,e^{−t}E,e^{−t}F} ,

and thus, we have for n̄ = n̄_{DEF}:

(4.11)  H(e^{th_{0}} n̄e^{−th_{0}}) = \frac{1}{2} log((1 − e^{−2t}EF)^{2} − 4e^{−4t}D^{2}),
(4.12)  H(e^{th_{0}}w_{0}n̄e^{−th_{0}}) \begin{matrix}= H(e^{−th_{0}} n̄e^{th_{0}}) − 2t \\ = \frac{1}{2} log((e^{−2t} − EF)^{2} − 4D^{2}), \end{matrix}

(4.13)  H(e^{th_{0}}k(n̄)e^{−th_{0}}) = \frac{1}{2} log(\frac{(1 − e^{−2t}EF)^{2} − 4e^{−4t}D^{2}}{(1 − EF)^{2} − 4D^{2}} ),
(4.14)  H(e^{th_{0}}w_{0}k(n̄)e^{−th_{0}}) = \frac{1}{2} log(\frac{(e^{−2t} − EF)^{2} − 4D^{2}}{(1 − EF)^{2} − 4D^{2}} ),

where in the third and fourth line, the identity H(ak(g)a^{−1}) = H(aga^{−1}) − H(g), which follows from Equation (3.6), was applied. Using Equation (4.13) and Corollary 2.11, one sees that

χ(e^{−2H(a_{t}k(n̄)a_{t}^{−1})}) = χ(\frac{(1 − EF)^{2} − 4D^{2}}{(1 − e^{−2t}EF)^{2} − 4e^{−4t}D^{2}} ) ⟶ Ξ(k(n̄)) (t → ∞)

for a_{t} = e^{th_{0}} and n̄ = n̄_{DEF}. The convergence is uniform with derivatives on compact sets in n̄. In particular, it holds with k(n̄) replaced by the generic point k of K. On applying Equation (4.12), we see that

Ξ(w_{0}k(n̄)) = lim_{t→∞} χ(e^{−2H(a_{t}w_{0}k(n̄)a_{t}^{−1})})

= lim_{t→∞} χ(\frac{(1 − EF)^{2} − 4D^{2}}{(e^{−2t} − EF)^{2} − 4D^{2}} ) = χ(\frac{(1 − EF)^{2} − 4D^{2}}{(EF)^{2} − 4D^{2}} )

We now apply the coordinates introduced by φ. We write (s, y) for the coordinates and for the T -valued points of 𝔸^{1} × 𝔸^{2p|2q}, where y = (b c). In terms of the norm squared function ∥·∥^{2} defined in Equation (2.2), we have

−4D^{2} = s^{2}, −EF = ∥y∥^{2}, X̄(D, E, F) = φ(s, y).

By Corollary 3.13 and since the isomorphism φ is linear, the pullback of |Dk̇| via exp ∘ φ is a Berezin–Lebesgue density, which, by taking |Dk̇| to be normalised adequately, may be assumed to be that from Subsection 2.7. We will denote the latter by |Dμ| instead of |Dλ| to avoid confusion with the parameter λ.

Therefore, it is immediate that Equation (4.10) takes the form

\begin{aligned}I_{0}^{λ}(e^{th_{0}})e^{−t(λ−ϱ)} & = ∫_{𝔸^{1+2p|2q}}|Dμ| \frac{ψ(1, e^{−t}y, e^{−2t}s)^{\frac{1}{2} (λ−ϱ)}}{ψ(1, y, s)^{\frac{1}{2} (λ+ϱ)}} χ(ψ(1, y, s)) \\ I_{∞}^{λ}(e^{th_{0}})e^{−t(λ−ϱ)} & = ∫_{𝔸^{1+2p|2q}}|Dμ| \frac{ψ(e^{−2t}, y, s)^{\frac{1}{2} (λ−ϱ)}}{ψ(1, y, s)^{\frac{1}{2} (λ+ϱ)}} (1 − χ) (\frac{ψ(1, y, s)}{ψ(0, y, s)}) \end{aligned} (4.15)

for ψ defined by

ψ(c, y, s) := (c + ∥y∥^{2})^{2} + s^{2}.

Since the integrands above are rotationally invariant over 𝔸^{1}, Corollary 2.24 applies. (Here, y is the variable in which we rotate, and s is the base parameter.) In particular, ϕ_{λ} only depends on m_{α} and m_{2α}. Therefore, if m_{α} > 0, we may assume that q = 0 and use the well-known formula for the c-function in the classical case [23, Chapter IV, Theorem 6.4] to arrive at the following conclusion.

Proposition 4.4. Let m_{α} > 0 and ℜλ > 0. Then c(λ) exists and

c(λ) = c_{0} \frac{2^{−λ}Γ(λ)}{Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + 1)) Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + m_{2α}))}

for some constant c_{0}.

In case m_{α} ⩽ 0, we apply Proposition 2.22. Again, the rotation variable is y and the parameter is s. If f is one of the integrands in Equation (4.15), then f^{∘} is determined by replacing ψ with ψ^{∘}, defined by

(4.16)  ψ^{∘}(c, r, s) := (c + r)^{2} + s^{2}.

We obtain the following intermediate result.

Lemma 4.5. Let m_{α} ⩽ 0 and ℜλ > 0. Then

\begin{aligned}c(λ) & = C_{0} ∫_{0}^{∞} ds ∂_{r=0}^{− \frac{mα}{2}} ψ^{∘}(1, r, s)^{−(λ+ϱ)/2}χ(ψ^{∘}(1, r, s)) \\ & + C_{0} ∫_{0}^{∞} ds ∂_{r=0}^{− \frac{mα}{2}} \frac{ψ^{∘}(0, r, s)^{(λ−ϱ)/2}}{ψ^{∘}(1, r, s)^{(λ+ϱ)/2}} (1 − χ) (\frac{ψ^{∘}(1, r, s)}{ψ^{∘}(0, r, s)}) \end{aligned} (4.17)

and c(λ) is meromorphic.

Proof. The integrands have compact support, so we may interchange lim_{t→∞} with the integral. The same can be done for ∂_{λ}. □

In order to finally derive c(λ), the function χ needs to be removed. The result is the following integral expression:

Proposition 4.6. If m_{α} ⩽ 0, and ℜλ > 0, then c(λ) exists, and we have

c(λ) = C_{0} ∫_{0}^{∞} ds ∂_{r=0}^{1−ϱ}((1 + r)^{2} + s^{2})^{−(λ+ϱ)/2}. (4.18)

In the proof, the following estimate will be used repeatedly.

Lemma 4.7. For all r, s ⩾ 0, z ∈ ℂ, and k ∈ ℕ, we have

|∂_{r}^{k}((1 + r)^{2} + s^{2})^{z/2}| < c_{k}((1 + r)^{2} + s^{2})^{(ℜz−k)/2},

where c_{k} is some z-dependent constant independent of r, s.

Proof. As one can see by induction,

∂_{r}^{k}((1 + r)^{2} + s^{2})^{z/2} = p_{k}(\frac{1 + r}{s} )s^{k}((1 + r)^{2} + s^{2})^{(z−2k)/2},

where p_{k} is a polynomial of order at most k. Since lim_{t→∞}(t^{2} + 1)^{− \frac{k}{2}} p_{k}(t) exists, there is a constant c_{k} such that |(t^{2} + 1)^{− \frac{k}{2}} p_{k}(t)| < c_{k} for all t ⩾ 0. Taking t = s^{−1}(1 + r), this implies

|∂_{r}^{k}((1 + r)^{2} + s^{2})^{z/2}| < c_{k}((1 + r)^{2} + s^{2})^{(ℜz−k)/2}.  □

Proof of Proposition 4.6. Firstly, note that −\frac{m_{α}}{2} = 1 − ϱ. Denote the second integral in Equation (4.17) by c_{II}(λ). Since ψ^{∘}(c, ut, st) = t^{2}ψ^{∘}(ct^{−1}, u, s), the substitution r = su in the derivative yields

\begin{aligned}C_{0}^{−1}c_{II}(λ) & = ∫_{0}^{∞} ds ∂_{u=0}^{1−ϱ}s^{ϱ−1} \frac{ψ^{∘}(0, su, s)^{(λ−ϱ)/2}}{ψ^{∘}(1, su, s)^{(λ+ϱ)/2}} (1 − χ)(\frac{ψ^{∘}(1, su, su)}{ψ^{∘}(0, su, s)}) \\ & = ∫_{0}^{∞} ds ∂_{u=0}^{1−ϱ}s^{λ−1} \frac{ψ^{∘}(0, u, 1)^{(λ−ϱ)/2}}{ψ^{∘}(1, su, s)^{(λ+ϱ)/2}} (1 − χ)(\frac{ψ^{∘}(s^{−1}, u, 1)}{ψ^{∘}(0, u, 1)}) \end{aligned} (4.19)

The next step is to exchange ∫ ds and ∂_{u}^{q}. Since only the limit u → 0 is of interest, u may be assumed to be small. Let ε < 1 be small enough for (1 − χ) = 0 on the interval [1, 1 + 4ε^{2}]. Since

\frac{ψ^{∘}(s^{−1}, u, 1)}{ψ^{∘}(0, u, 1)} = \frac{(s^{−1} + u)^{2} + 1}{u^{2} + 1} ⩽ 1 + 4ε^{2}, ∀s : sε > 1,

the function

[0, ε] × (0, ∞) ⟶ [0, 1] : (u, s) ⟼ (1 − χ) (\frac{(s^{−1} + u)^{2} + 1}{u^{2} + 1})

is of compact support. Therefore, ∂_{u=0}^{1−ϱ} and ∫_{δ}^{∞} ds can be exchanged for any δ > 0. Now choose 0 < δ < 1 sufficiently small, such that

(1 − χ)(\frac{(s^{−1} + u)^{2} + 1}{u^{2} + 1}) = 1, ∀s ∈ (0, δ).

By Lemma 4.7, we have, for u ⩽ ε, s ⩽ δ, and k ⩽ q, that

|s^{λ−1}∂_{u}^{k}ψ^{∘}(1, su, s)^{−(λ+ϱ)/2}| < c_{k}s^{ℜλ−1+k}ψ^{∘}(1, su, s)^{−(ℜλ+ϱ+k)/2}
⩽ c_{k}5^{−ϱ/2}s^{ℜλ−1}.

Since s^{ℜλ−1} is integrable on [0, δ], we may also exchange ∫_{0}^{δ} ds and ∂_{u=0}^{1−ϱ}.

Therefore, the right-hand side of Equation (4.19) equals

∂_{u=0}^{1−ϱ} ∫_{0}^{∞} ds s^{λ−1} \frac{ψ^{∘}(0, u, 1)^{(λ−ϱ)/2}}{ψ^{∘}(1, su, s)^{(λ+ϱ)/2}} (1 − χ)(\frac{ψ^{∘}(s^{−1}, u, 1)}{ψ^{∘}(0, u, 1)})

= ∂_{u=0}^{1−ϱ} ∫_{0}^{∞} \frac{ds}{s} \frac{ψ^{∘}(0, \sqrt{s}u, \sqrt{s})^{(λ−ϱ)/2}}{ψ^{∘}(1/\sqrt{s}, \sqrt{s}u, \sqrt{s})^{(λ+ϱ)/2}} (1 − χ)(\frac{ψ^{∘}(1/\sqrt{s}, \sqrt{s}u, \sqrt{s})}{ψ^{∘}(0, \sqrt{s}u, \sqrt{s})}).

We substitute t^{−1} = ψ^{∘}(0, \sqrt{s}u, \sqrt{s}) = s(u^{2}+1) in the integral. Since s^{−1}ds and ψ^{∘}(1/\sqrt{s}, \sqrt{s}u, \sqrt{s}) = s^{−1} +2u+s(u^{2} +1) are invariant under this substitution, this leads to

∂_{u=0}^{1−ϱ} ∫_{0}^{∞} \frac{dt}{t} \frac{t^{−(λ−ϱ)/2}}{ψ^{∘}(1/\sqrt{t}, \sqrt{t}u, \sqrt{t})^{(λ+ϱ)/2}} (1 − χ)(tψ^{∘}(1/\sqrt{t}, \sqrt{t}u, \sqrt{t})),

= ∂_{u=0}^{1−ϱ} ∫_{0}^{∞} dt t^{ϱ−1}ψ^{∘}(1, tu, t)^{−(λ+ϱ)/2}(1 − χ)(ψ^{∘}(1, tu, t)).

Again, the derivatives and the integral have to be exchanged. Possibly after shrinking ε, one may assume that

(1 − χ)((1 + tu)^{2} + t^{2}) = 0, ∀t, u : 0 < t, u ⩽ ε.

Therefore, it suffices to consider the integral ∫_{ε}^{∞} dt. Clearly, ∫_{ε}^{R} dt and ∂_{u=0}^{1−ϱ} may be exchanged, where R is large enough to arrange for

(1 − χ)((1 + tu)^{2} + t^{2}) = 0, ∀t : t ⩾ R.

On applying Lemma 4.7 once again, we see that

|∂_{u}^{k}t^{ϱ−1}ψ^{∘}(1, tu, t)^{−(λ+ϱ)/2}| < c_{k}t^{ϱ−1+k}ψ^{∘}(1, tu, t)^{−(ℜλ+ϱ+k)/2}
⩽ c_{k}5^{−ϱ/2}t^{−ℜλ−1},

which is integrable over (R, ∞) for ℜλ > 0. Thus, ∫_{R}^{∞} dt and ∂_{u=0}^{1−ϱ} can also be exchanged. Therefore, we find

c_{II}(λ) = C_{0} ∫_{0}^{∞} dt ∂_{u=0}^{1−ϱ}t^{ϱ−1}ψ^{∘}(1, tu, t)^{−(λ+ϱ)/2}(1 − χ)(ψ^{∘}(1, tu, t))
= C_{0} ∫_{0}^{∞} dt ∂_{r=0}^{1−ϱ}ψ^{∘}(1, r, t)^{−(λ+ϱ)/2}(1 − χ)(ψ^{∘}(1, r, t))

upon substituting r = tu in the derivative. Up to a replacement of χ by 1 − χ, this is equal to the first integral in Equation (4.17), and so, the assertion follows. □

We now arrive at our conclusion in the unitary case.

Theorem 4.8. The c-function c(λ) for the symmetric pair of G = U_{cs}(1, 1 + p|2q) and K = U(1) × U_{cs}(1 + p|q) exists for ℜλ > 0. Explicitly, it is given by

c(λ) = c_{0} \frac{2^{−λ}Γ(λ)}{Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + 1)) Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + m_{2α}))}, c_{0} ≡ c_{0}(ϱ) ≠ 0.

Documenta Mathematica 19 (2014) 1317–1366

Proof. The case of m_{α} > 0 is the content of Proposition 4.4, so let m_{α} ⩽ 0. Since c(λ) is meromorphic, we may assume that ℜλ > −ϱ. Proposition 4.6 gives

C_{0}^{−1}c(λ) = ∫_{0}^{∞} ds ∂_{r=0}^{1−ϱ}((1 + r)^{2} + s^{2})^{− \frac{λ+ϱ}{2}} = ∂_{r=0}^{1−ϱ} ∫_{0}^{∞} ds ((1 + r)^{2} + s^{2})^{− \frac{λ+ϱ}{2}} .

Here, integral and derivative may be exchanged due to Lemma 4.7, since

|∂_{r}^{k}((1 + r)^{2} + s^{2})^{−(λ+ϱ)/2}| < c_{k}((1 + r)^{2} + s^{2})^{−(ℜλ+ϱ+k)/2}
⩽ c_{k}(1 + s^{2})^{−(ℜλ+ϱ)/2}.

for k ⩽ 1 − ϱ. This is integrable by assumption. Substitution with s = (1 + r)\sqrt{t} yields

c(λ) = \frac{C_{0}}{2} ∂_{r=0}^{1−ϱ}(1 + r)^{−λ−ϱ+1} ∫_{0}^{∞} dt t^{− \frac{1}{2}} (1 + t)^{−(λ+ϱ)/2}

= \frac{C_{0}(1 − ϱ)!(−1)^{1−ϱ}}{2} \binom{λ − 1}{1 − ϱ}\frac{Γ(\frac{1}{2})Γ(\frac{1}{2}(λ + ϱ − 1))}{Γ(\frac{1}{2}(λ + ϱ))}

= \frac{C_{0}\sqrt{π}(−1)^{1−ϱ}}{2} \frac{Γ(λ)Γ(\frac{1}{2}(λ + ϱ − 1))}{Γ(λ + ϱ − 1)Γ(\frac{1}{2}(λ + ϱ))} = C_{0}(−2)^{1−ϱ}π \frac{2^{−λ}Γ(λ)}{Γ(\frac{1}{2}(λ + ϱ))^{2}} .

Here, we have applied the integral formula for the Euler beta function, the identity \binom{−x}{k} = (−1)^{k}\binom{x+k−1}{k}, and the duplication formula

Γ(z) = \frac{1}{\sqrt{π}} 2^{z−1}Γ(\frac{1}{2}z)Γ(\frac{1}{2}(z + 1)).

The claim now follows from m_{2α} = 1 and m_{α} = 2(ϱ − 1). □

4.2.2. The ortho-symplectic case. Let 𝔤 := 𝔬𝔰𝔭(1, 1 + p|2q, ℂ) be the complex Lie subsuperalgebra of 𝔤𝔩(2 + p|2q, ℂ), given by

𝔬𝔰𝔭(1, 1 + p|2q, ℂ) = {x ∈ 𝔤𝔩(2 + p|2q, ℂ) | x^{st^{3}}J + Jx = 0},

where

(\begin{matrix}R & S \\ T & V \end{matrix})^{st^{3}} := (\begin{matrix}R^{t} & T^{t} \\ −S^{t} & V^{t} \end{matrix})

and

J = ( \begin{matrix}−𝟙_{1} & 0 & 0 \\ 0 & 𝟙_{1+p} & 0 \\ 0 & 0 & J_{q} \end{matrix} ) . (4.20)

Here J_{q} denotes the 2q × 2q matrix with q copies of the 2 × 2 matrix ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) on the diagonal. Therefore, 𝔤 consists of the matrices of the form

( \begin{matrix}0 & X_{12} & X_{13} \\ X_{12}^{t} & X_{22} & X_{23} \\ J_{q}X_{13}^{t} & −J_{q}X_{23}^{t} & X_{33} \end{matrix} ) , X_{22} ∈ 𝔰𝔬(1 + p, ℂ), X_{33} ∈ 𝔰𝔭(2q, ℂ), (4.21)

Documenta Mathematica 19 (2014) 1317–1366

with no further restrictions on the other entries. We let

𝔤_{ℝ} := 𝔤_{¯0,ℝ} ⊕ 𝔤¯1, 𝔤_{¯0,ℝ} := 𝔰𝔬(1 + p, ℝ) × 𝔲𝔰𝔭(2q),

where 𝔲𝔰𝔭(2q) is the compact form of 𝔰𝔭(2q, ℂ), given by 𝔲𝔰𝔭(2q) := 𝔲(2q) ∩ 𝔰𝔭(2q, ℂ). Let G = SOSp_{cs}^{+}(1, 1 + p|2q) be the cs form of OSp(2 + p|2q, ℂ) given by the underlying Lie group SO(1, 1 + p, ℝ) × USp(2q). Here, USp(2q) := U(2q) ∩ Sp(2q, ℂ). The involution θ is the restriction of the one considered in the previous subsubsection. Let 𝔤 = 𝔨 ⊕ 𝔭 be the corresponding eigenspace decomposition. The analytic subsupergroup K corresponding to 𝔨 is K = SOSp_{cs}(1 + p|2q), with underlying Lie group SO(1 + p, ℝ) × USp(2q) and Lie cs algebra 𝔨. For p = 0, K is the semi-direct product USp(2q) ⋉ 𝔸^{0|2q}, where 𝔸^{0|2q} is given the additive supergroup structure of the super-vector space ℂ^{0|2q}. Let 𝔞 ⊆ 𝔭¯0 be the even Cartan subalgebra generated by the element h_{0} defined in Equation (4.3). Since σ and J commute, the restricted root space decomposition for 𝔤 is obtained by restricting that for 𝔤𝔩 from Equation (4.4) to 𝔬𝔰𝔭(1, 1 + p|2q, ℂ). Only two restricted roots remain: α and −α, where α(h_{0}) = 1. The general elements of 𝔤^{α} and 𝔤^{−α} are respectively

(4.22) ( \begin{matrix}0 & 0 & B¯0 & B¯1 \\ 0 & 0 & B¯0 & B¯1 \\ Bt ; ¯0 & −Bt ; ¯0 & 0 & 0 \\ J_{q}Bt ; ¯1 & −J_{q}Bt ; ¯1 & 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & B¯0 & B¯1 \\ 0 & 0 & −B¯0 & −B¯1 \\ Bt ; ¯0 & Bt ; ¯0 & 0 & 0 \\ J_{q}Bt ; ¯1 & J_{q}Bt ; ¯1 & 0 & 0 \end{matrix} ) .

The general element of 𝔪 = 𝔷_{𝔨}(𝔞) is

( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & E & F \\ 0 & 0 & −J_{q}G^{t} & H \end{matrix} ) .

We take Σ^{+} := {α} as our positive system, hence 𝔫 = 𝔤^{α} and 𝔫̄ = 𝔤^{−α}. From Equation (4.22), we have m_{α} = p − 2q, so ϱ = \frac{1}{2}(p − 2q)α. Using the notation from Equation (4.5), we set

X(F) := X(0, E, F), X̄(F) := X̄(0, E, F), E := (−F t ; ¯0 −F t ; ¯1J_{q}) .

Since the Iwasawa decomposition of 𝔤 is the restriction of that for 𝔤𝔩 considered above, we see that

𝔫 = {X(F) | F ∈ ℂ^{p|2q×1}}, 𝔫̄ = {X̄(E) | E ∈ ℂ^{p|2q×1}}.

Then by Proposition 2.14, we have an isomorphism φ, given by

φ : 𝔸^{p|2q} ⟶ 𝔸(𝔫̄_{ℝ}) : y ⟼ X̄(y),

and similarly for 𝔫. The T -valued points of N and N̄ are

n_{y} := exp_{G} X(y) = 𝟙_{2+p+2q} + X(y), n̄_{b} := 𝟙_{2+p+2q} + X̄(y),

where y ∈_{T} 𝔸^{p|2q} and T is any cs manifold. As in the previous subsubsection, we will identify 𝔞 and 𝔞^{∗} with ℂ via the bases h_{0} and α. The Iwasawa decompositions for 𝔤𝔩 and 𝔬𝔰𝔭 are compatible, so that we have the following lemma.

Lemma 4.9. The restriction to N̄ of the morphism H : N̄ → 𝔸(𝔞_{ℝ}) is given by

H(n̄) = log(1 + ∥y∥^{2}), n̄ = n̄_{y}, y ∈_{T} 𝔸^{p|2q},

for any cs manifold T, where ∥·∥^{2} is defined in Equation (2.2).

Proof. Lemma 4.3 applies, since n̄_{y} = n̄_{0yb} where b = −(yt ; ¯0 yt ; ¯1J_{q}). We obtain

H(n̄) = \frac{1}{2} log((1 − by)^{2}) = log(1 − by) = log(1 + ∥y∥^{2}),

by the definition of the norm squared function. □

Equations (4.13) and (4.14) specialise to

H(e^{th_{0}}k(n̄_{y})e^{−th_{0}}) = log(\frac{1 + e^{−2t}∥y∥^{2}}{1 + ∥y∥^{2}} ),
H(e^{th_{0}}w_{0}k(n̄_{y})e^{−th_{0}}) = log(\frac{e^{−2t} + ∥y∥^{2}}{1 + ∥y∥^{2}} ),

where w_{0} ∈ M′_{0} again denotes a representative of the non-trivial Weyl group element. Note that this only makes sense if p > 0. Otherwise, the root α is not even and the Weyl group would be trivial in this case. Therefore, we assume p > 0 for the time being, postponing the case of p = 0 to the end of this subsubsection. Let χ be as before and define Ξ ∈ Γ(O_{K/M}) by

Ξ(k(n̄_{y})) := χ(1 + ∥y∥^{2})

for n̄_{y} ∈_{T} N̄. As functions on K/M, both Ξ and k ↦ (1 − Ξ)(w_{0}k) have compact support inside k_{0}(N̄_{0}), and

(1 − Ξ)(w_{0}k(n̄_{y})) = (1 − χ) (\frac{1 + ∥y∥^{2}}{∥y∥^{2}} ) .

In analogy with Equations (4.9), (4.10), and (4.15), we have by Theorem 2.18 and Corollary 2.19 that

\begin{aligned}ϕ_{λ}(g) & = I_{0}^{λ}(g) + I_{∞}(g), \\ I_{0}^{λ}(g) & := ∫_{K/M}|Dk̇| e^{(λ−ϱ)(H(gk))}Ξ(k), \\ I_{∞}^{λ}(g) & := ∫_{K/M}|Dk̇| e^{(λ−ϱ)H(gwk)}(1 − Ξ)(w_{0}k), \\ I_{0}^{λ}(e^{th_{0}})e^{−t(λ−ϱ)} & = ∫_{𝔸^{p|2q}}|Dμ(y)|\frac{(1 + e^{−2t}∥y∥^{2})^{λ−ϱ}}{(1 + ∥y∥^{2})^{λ+ϱ}} χ(1 + ∥y∥^{2}), \\ I_{∞}^{λ}(e^{th_{0}})e^{−t(λ−ϱ)} & = ∫_{𝔸^{p|2q}}|Dμ(y)|\frac{(e^{−2t} + ∥y∥^{2})^{λ−ϱ}}{(1 + ∥y∥^{2})^{λ+ϱ}} (1 − χ) (\frac{1 + ∥y∥^{2}}{∥y∥^{2}} ) . \end{aligned} (4.23)

Here, again, |Dμ| denotes the Berezin–Lebesgue density on 𝔸^{p|2q}. By Corollary 2.23, ϕ_{λ} and therefore c(λ) only depends on m_{α} and not on p and q separately in this case, too. Therefore, for m_{α} > 0, [23, Chapter IV, Theorem 6.4] implies the following.

Proposition 4.10. Let m_{α} > 0. Then c(λ) exists for ℜλ > 0, and we have

c(λ) = c_{0} \frac{2^{−λ}Γ(λ)}{Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + 1)) Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + m_{2α}))}, c_{0} ≡ c_{0}(ϱ) ≠ 0.

For 2ϱ = m_{α} ⩽ 0, we get a similar expression.

Proposition 4.11. Let m_{α} ⩽ 0 and p > 0. Then c(λ) exists for ℜλ > 0, and

c(λ) = c′_{0} \frac{Γ(λ)}{Γ(λ + ϱ)}, c′_{0} ≡ c′_{0}(ϱ).

Proof. Following Corollary 2.24, we distinguish between the even and odd cases. Firstly, assume that m_{α} be even. In this case, the integrals in Equation (4.23) reduce to a pointwise derivative, which may be exchanged with limits to arrive at

c(λ) = C∂_{r=0}^{−ϱ} ((1 + r)^{−(λ+ϱ)}χ(1 + r) + r^{λ−ϱ}(1 + r)^{−(λ+ϱ)}(1 − χ) (\frac{1 + r}{r} )) .

for some constant C. As functions of r, χ(1 + r) and (1 − χ)(r^{−1}(1 + r)) are constant near zero. Furthermore, ∂_{r=0}^{k}r^{λ−ϱ} = 0 for k ⩽ −ϱ, so that

c(λ) = Cq!\binom{−(λ + ϱ)}{−ϱ} = C(−1)^{ϱ}q!\binom{λ − 1}{−ϱ} = C(−1)^{ϱ} \frac{Γ(λ)}{Γ(λ + ϱ)}.

Now, assume that m_{α} < 0 is odd. By Lemma 4.7, we may exchange limits and integrals, to obtain

\begin{aligned}c(λ) = C & ∫_{0}^{∞} dr r^{− \frac{1}{2}} ∂_{r}^{\frac{1}{2} −ϱ}((1 + r)^{−(λ+ϱ)}χ(1 + r)) \\ & + C ∫_{0}^{∞} dr r^{− \frac{1}{2}} ∂_{r}^{\frac{1}{2} −ϱ}(r^{λ−ϱ}(1 + r)^{−(λ+ϱ)}(1 − χ)(r^{−1}(1 + r))). \end{aligned} (4.24)

Denote the two integrals by c_{I}(λ) and c_{II}(λ), respectively. As in the proof of Proposition 4.6, we will rewrite these in order to show that the respective contributions of the cutoff function χ cancel. This will be done by integrating by parts. Recall that (1 − χ)(r^{−1}(1 + r)) = 1 for small r. Therefore, for ℜλ > 0 and k ⩽ \frac{1}{2} − ϱ, we have

lim_{r→0}(∂_{r}^{k−1}r^{− \frac{1}{2}} )∂_{r}^{−ϱ− \frac{1}{2} −k}(r^{λ−ϱ}(1 + r)^{−(λ+ϱ)}(1 − χ)(r^{−1}(1 + r)))

= lim_{r→0} r^{\frac{1}{2} −k} ∑_{ℓ=0}^{\frac{1}{2} −ϱ−k} c_{ℓ}r^{λ−ϱ−( \frac{1}{2} −ϱ−k−ℓ)}(1 + r)^{−(λ−ϱ)−ℓ}

= lim_{r→0} ∑_{ℓ=0}^{\frac{1}{2} −ϱ−k} c_{ℓ}r^{λ+ℓ}(1 + r)^{−(λ−ϱ)−ℓ} = 0

Therefore, no boundary terms occur upon integrating by parts, and

c_{II}(λ) = (\frac{1}{2} − ϱ)!\binom{−ϱ}{\frac{1}{2} − ϱ} ∫_{0}^{∞} \frac{dr}{r} r^{λ}(1 + r)^{−(λ+ϱ)}(1 − χ)(r^{−1}(1 + r))

= (\frac{1}{2} − ϱ)!\binom{−ϱ}{\frac{1}{2} − ϱ} ∫_{0}^{∞} \frac{ds}{s} s^{−λ}(s^{−1}(1 + s))^{−(λ+ϱ)}(1 − χ)(1 + s)

= (\frac{1}{2} − ϱ)!\binom{−ϱ}{\frac{1}{2} − ϱ} ∫_{0}^{∞} \frac{ds}{s} s^{ϱ}(1 + s)^{−(λ+ϱ)}(1 − χ)(1 + s).

where r^{ϱ−1}r^{λ−ϱ} = r^{λ−1} and the substitution s = r^{−1} were applied. Since similarly, for k ⩽ \frac{1}{2} − ϱ, we have as r → ∞

∂_{r}^{k−1}r^{− \frac{1}{2}} ∂_{r}^{\frac{1}{2} −ϱ−k}((1 + r)^{−(λ+ϱ)}(1 − χ)(1 + r)) ≃ r^{\frac{1}{2} −k}(1 + r)^{−λ− \frac{1}{2} +k} ⟶ 0,

we may integrate by parts to find

∫_{0}^{∞} dr r^{− \frac{1}{2}} ∂_{r}^{\frac{1}{2} −ϱ}((1 + r)^{−(λ+ϱ)}(1 − χ)(1 + r))

= (\frac{1}{2} − ϱ)!\binom{−ϱ}{\frac{1}{2} − ϱ} ∫_{0}^{∞} \frac{ds}{s} s^{ϱ}(1 + s)^{−(λ+ϱ)}(1 − χ)(1 + s),

since (1 − χ)(1 + r) vanishes for r small and equals 1 for r large. Combining c_{I}(λ) and c_{II}(λ) leads to

c(λ) = C ∫_{0}^{∞} dr r^{− \frac{1}{2}} ∂_{r}^{\frac{1}{2} −ϱ}(1 + r)^{−(λ+ϱ)} = C(−1)^{−ϱ− \frac{1}{2}} \sqrt{π} \frac{Γ(λ)}{Γ(λ + ϱ)}

for ℜλ > −ϱ, as in the proof of Theorem 4.8. Moreover, Equation (4.24) shows that c(λ) is holomorphic in λ for ℜλ > 0, since the integrands are sufficiently bounded. Therefore, the case of 0 < ℜλ ⩽ −ϱ follows by analytic continuation. □

The case where p = 0 is still open. This situation is interesting, in that there are only purely odd restricted roots (which, however, are anisotropic). Fortunately, one can evaluate ϕ_{λ} explicitly in this case.

Proposition 4.12. If p = 0, then ϕ_{λ} is for some constant c_{1} given by

ϕ_{λ}(e^{th_{0}}) = c_{1}e^{λt} ∑_{k=0}^{−ϱ} \binom{λ − ϱ}{k}\binom{−λ − ϱ}{−ϱ − k}e^{(−ϱ−2k)t}

Proof. The cs manifolds K/M and N̄ admit only one retraction, since the underlying spaces are trivial. Hence, we may by Theorem 2.18 and Corollary 2.19 pull back the defining integral of ϕ_{λ} and apply Equation (3.6), to obtain

ϕ_{λ}(e^{th_{0}})e^{−t(λ−ϱ)} = ∫N̄|Dn̄| e^{(λ−ϱ)(H(e^{th_{0}} n̄e^{−th_{0}} ))}e^{−(λ+ϱ)(H(n̄))}

= ∫_{𝔸^{0|2q}}|Dμ(y)| (1 + e^{−2t}∥y∥^{2})^{λ−ϱ}(1 + ∥y∥^{2})^{−(λ+ϱ)}

≃ ∂_{r=0}^{q}(1 + e^{−2t}r)^{λ−ϱ}(1 + r)^{−(λ+ϱ)}

≃ q! ∑_{k=0}^{q} \binom{λ − ϱ}{k}\binom{−λ − ϱ}{q − k}e^{−2kt}.

This proves the claim. □

Corollary 4.13. If p = 0, the c(λ) exists for ℜλ > 0 and

c(λ) = c′_{0} \frac{Γ(λ)}{Γ(λ + ϱ)}, c′_{0} ≡ c′_{0}(ϱ).

Using the duplication formula, Proposition 4.10, Proposition 4.11, and Corollary 4.13 combine to the following result.

Theorem 4.14. The c-function c(λ) for the symmetric pair of the cs Lie supergroups G = SOSp_{cs}^{+}(1, 1+p|2q) and K = SOSp_{cs}(1+p|2q) exists for ℜλ > 0. Explicitly,

c(λ) = c_{0} \frac{2^{−λ}Γ(λ)}{Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + 1)) Γ (\frac{1}{2}(λ + \frac{m_{α}}{2} + m_{2α}))}, c_{0} ≡ c_{0}(ϱ) ≠ 0.

4.2.3. The case of GL(1|1). Let 𝔤 := 𝔤𝔩(1|1, ℂ)×𝔤𝔩(1|1, ℂ). Then 𝔤¯0 is Abelian. We will write the elements of 𝔤 as double matrices of the form

( \begin{matrix}A & B & E & F \\ C & D & G & H \end{matrix} ) .

Let 𝔤_{ℝ} := 𝔤_{¯0,ℝ} ⊕ 𝔤¯1 be given by requiring 𝔤_{¯0,ℝ} to be the Lie algebra of the Lie group G_{0} whose general elements are

( \begin{matrix}z & 0 & z̄^{−1} & 0 \\ 0 & r & 0 & s \end{matrix} ) , z ∈ ℂ^{×}, r, s > 0.

The cs form G of G_{ℂ} = GL(1|1, ℂ) × GL(1|1, ℂ) defined by G_{0} will be denoted by (GL × GL)_{ℂ|ℝ}^{+}(1|1). An involution θ on 𝔤 can be defined by

θ ( \begin{matrix}A & B & E & F \\ C & D & G & H \end{matrix} ) = ( \begin{matrix}E & F & A & B \\ G & H & C & D \end{matrix} ) .

The general elements of the eigenspaces 𝔨 and 𝔭 are respectively

( \begin{matrix}A & B & A & B \\ C & D & C & D \end{matrix} ) , ( \begin{matrix}A & B & −A & −B \\ C & D & −C & −D \end{matrix} ) .

Let K be the analytic cs Lie subgroup of G with Lie cs algebra 𝔨, denoted by K = UGL_{ℝ}^{+}(1|1). The general element of K_{0} is

( \begin{matrix}z & 0 & z & 0 \\ 0 & r & 0 & r \end{matrix} ) , z ∈ U(1), r > 0.

Note that although K_{0} is not compact, its adjoint image in GL(𝔭_{¯0,ℝ}) is. This explains why there is no contradiction in the fact that the negative of the supertrace form is actually positive on 𝔭_{¯0,ℝ}, so that (G_{0}, K_{0}) is a Riemannian symmetric pair. The only choice of Cartan subalgebra is 𝔞 := 𝔭¯0, since 𝔭¯0 is Abelian and consists of semisimple elements. Let

h_{1} := ( \begin{matrix}1 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , h_{2} := ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 1 & 0 & −1 \end{matrix} )

be a basis of 𝔞. An easy calculation shows that there are only the roots ±α, with α defined by

α(a_{1}h_{1} + a_{2}h_{2}) := a_{1} − a_{2}.

We have 𝔪 = 𝔷_{𝔨}(𝔞) = 𝔨. The general elements of 𝔤^{α} and 𝔤^{−α} are respectively

( \begin{matrix}0 & B & 0 & 0 \\ 0 & 0 & G & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & 0 & F \\ C & 0 & 0 & 0 \end{matrix} ) .

We let Σ^{+} := {α} be our positive system, so 𝔫 = 𝔤^{α}, 𝔫̄ = 𝔤^{−α}, m_{α} = −2, and ϱ = −α. Since G_{0} admits a global Iwasawa decomposition, this is also the case for G. Indeed, every element in G_{0} can be decomposed uniquely as

( \begin{matrix}z & 0 & z̄^{−1} & 0 \\ 0 & r & 0 & s \end{matrix} ) = ( \begin{matrix}\frac{z}{|z|} & 0 & ∗ & ∗ \\ 0 & \sqrt{rs} & ∗ & ∗ \end{matrix} ) ( \begin{matrix}log |z| & 0 & ∗ & ∗ \\ 0 & \frac{1}{2} log(\frac{r}{s}) & ∗ & ∗ \end{matrix} ) ,

where the first factor is in K_{0} and the second in A_{0}. Note that N_{0} = 1. For any cs manifold T, the T -valued points of N and N̄ are of the form

n_{B,G} = ( \begin{matrix}1 & B & 1 & 0 \\ 0 & 1 & G & 1 \end{matrix} ) , n̄_{C,F} = ( \begin{matrix}1 & 0 & 1 & F \\ C & 1 & 0 & 1 \end{matrix} ) , B, C, F, G ∈ Γ(O_{T,¯1}).

Lemma 4.15. The restriction to N̄ of the morphism H : G → 𝔸(𝔞_{ℝ}) is given by

H(n̄) = \frac{1}{2}CF h^{+}, n̄ = n̄_{C,F} ∈_{T} N̄,

where we set h^{±} := h_{1} ± h_{2}.

Proof. Again, H: N̄ ⟶ 𝔞 will be derived from the equation

θ(n̄_{C,F})^{−1}n̄_{C,F} = θ(n_{B,G})^{−1} exp(2H(n̄_{C,F}))n_{B,G},  n̄_{C,F} ∈_{T} N̄, n_{B,G} ∈_{T} N.

The left-hand side equals

( \begin{matrix}1 & −F & 1 & 0 \\ 0 & 1 & −C & 1 \end{matrix} ) ( \begin{matrix}1 & 0 & 1 & F \\ C & 1 & 0 & 1 \end{matrix} ) = ( \begin{matrix}1 − F C & −F & 1 & F \\ C & 1 & −C & 1 − CF \end{matrix} ).

Upon writing H(n̄_{C,F}) = h_{1}t_{1} + h_{2}t_{2} for some t_{1}, t_{2} ∈ Γ(O_{T,¯0}), the right-hand side takes the form

( \begin{matrix}1 & 0 \\ −G & 1 \end{matrix} \begin{matrix}1 & −B \\ 0 & 1 \end{matrix} ) ( \begin{matrix}e^{2t_{1}} & 0 & e^{−2t_{1}} & 0 \\ 0 & e^{2t_{2}} & 0 & e^{−2t_{2}} \end{matrix} ) ( \begin{matrix}1 & B & 1 & 0 \\ 0 & 1 & G & 1 \end{matrix} )

= ( \begin{matrix}e^{2t_{1}} & Be^{2t_{1}} & e^{−2t_{1}} − BGe^{−2t_{2}} & −Be^{−2t_{2}} \\ −Ge^{2t_{1}} & e^{2t_{2}} − GBe^{2t_{1}} & Ge^{−2t_{2}} & e^{−2t_{2}} \end{matrix} ),

Therefore, e^{2t_{1}} = 1 − F C and e^{−2t_{2}} = 1 − CF = 1 + F C, so that

t_{1} = t_{2} = −\frac{1}{2}F C = \frac{1}{2}CF,

which proves the lemma. □

We find the following formula for ϕ_{λ}.

Proposition 4.16. For a suitable normalisation of the invariant density on K/M, we have for any λ ∈ 𝔞^{∗}

ϕ_{λ}(e^{h}) = −λ(h^{+})e^{λ(h)} sinh α(h), h ∈_{T} 𝔸(𝔞_{ℝ}).

Remarkably, the function ϕ_{λ} is not invariant under the symmetry h ⟼ −h. This is due to the fact that there are no even restricted roots in the case under consideration, so that the even Weyl group is trivial.

Proof of Proposition 4.16. We introduce coordinates ξ_{1}, ξ_{2} on N̄ by

ξ_{1}(n̄_{C,F}) = C, ξ_{2}(n̄_{C,F}) = F.

In terms of these coordinates, we may take |Dn̄| = |Dξ| = |D(ξ_{1}, ξ_{2})|. The spaces K/M and N̄ being of purely odd dimension, they both admit only one retraction. Thus, we may by Theorem 2.18 and Corollary 2.19 pull back the defining integral of ϕ_{λ}, leading to

ϕ_{λ}(e^{h})e^{−(λ−ϱ)(h)} = ∫N̄|Dn̄|e^{(λ−ρ)(H(e^{h}n̄e^{−h}))}e^{−(λ+ϱ)(H(n̄))} (4.25)

for h ∈_{T} 𝔸(𝔞_{ℝ}), by Proposition 3.17 and Equation (3.6). Clearly, we have by Corollary 2.11 that

e^{h}n̄_{C,F} e^{−h} = n̄_{e^{−2α(h)}C,e^{−2α(h)F}} ,

and ϱ(h^{+}) = −α(h^{+}) = 0. Furthermore, Lemma 4.15 gives

e^{λ(H(n̄_{C,F}))} = e^{\frac{1}{2} λ(h^{+})CF} = 1 + \frac{1}{2}λ(h^{+})CF.

Inserting these findings into Equation (4.25), we arrive at

ϕ_{λ}(e^{h})e^{−(λ−ϱ)(h)} = ∫N̄|Dξ| (1 + \frac{1}{2}λ(h^{+})e^{−2α(h)}ξ_{1}ξ_{2}) (1 − \frac{1}{2}λ(h^{+})ξ_{1}ξ_{2})
= \frac{1}{2}λ(h^{+})(e^{−2α(h)} − 1) = −λ(h^{+})e^{ϱ(h)} sinh α(h),

proving the assertion. □

We immediately obtain the following expression for the c-function.

Corollary 4.17. Let λ ∈ 𝔞^{∗} be arbitrary. Using h_{0} = c_{+}h^{+} + c_{−}h^{−} in Equation (4.1), c(λ) exists if and only if c_{−} = 0 or ℜc_{−} > 0, and then

c(λ) = \begin{cases}0, & c_{−} = 0, \\ −\frac{1}{2}λ(h^{+}), & ℜc_{−} > 0. \end{cases}

Proof. Since α(h^{+}) = 0 and α(h^{−}) = 2, we have

e^{ϱ(th_{0})} sinh α(th_{0}) = e^{−2c_{−}t} sinh 2c_{−}t = \frac{1}{2}(1 − e^{−4c_{−}t}).

Thus, the limit for t → ∞ exists if and only if c_{−} = 0 or ℜc_{−} > 0. In these cases, it equals 0 and \frac{1}{2}, respectively. □

Thus, in particular, c(λ) can be defined independently of a choice of h_{0} with α(h_{0}) = 2c_{−} > 0. Since α(h) = \frac{1}{2}⟨h^{+}, h⟩, the above result for c(λ) can be rewritten in the form stated in Theorem 4.2.

5. Determination of the spherical functions

In this subsection, we derive a convergent series expansion for the function ϕ_{λ}. The standard procedure to do so is to solve a differential equation by a pertubation ansatz, cf. [23, Chapter IV, 5]. We follow the same procedure. In the following, suppose (G, K) is one of the two following symmetric pairs

(5.1)

(U_{cs}(1, 1 + p|2q), U(1) × U_{cs}(1 + p|2q)),
(SOSp_{cs}^{+}(1, 1 + p|2q), SOSp_{cs}(1 + p|2q)).

For (G, K) = ((GL × GL)_{ℂ|ℝ}^{+}(1|1), UGL_{ℝ}^{+}(1|1)), a closed expression for ϕ_{λ} was derived above in Proposition 4.16. Moreover, in the orthosymplectic case, a closed expression was obtained for p = 0, in Proposition 4.12. This section aims to remove such restrictions. Let h_{0} ∈ 𝔞_{ℝ} be determined uniquely by α(h_{0}) = 1. By the use of the bases h_{0} and α of 𝔞 and 𝔞^{∗}, respectively, we will identify these spaces with ℂ where convenient.

Proposition 5.1. Let Δ(L) be the differential operator on A given by

Δ(L)(f)(e^{th_{0}}) := (∂_{t}^{2} + (m_{α} coth(t) + 2m_{2α} coth(2t))∂_{t})f(e^{th_{0}}).

Then, for any λ ∈ 𝔞^{∗}, ϕ_{λ} is an eigenfunction of Δ(L). More precisely,

Δ(L)ϕ_{λ} = (λ^{2} − ϱ^{2})ϕ_{λ}. (5.2)

Proof. One follows the standard procedure, showing that Δ(L) is the N-radial part of the Laplacian on G/K, cf. [20, 4.2]. This can also be seen by a computation in 𝔘(𝔤) along the lines of the proof of [38, Proposition 9.1.2.11]. The eigenvalue equation is derived as in Ref. [1]. □

Since e^{(λ−ϱ)t} is an eigenfunction of ∂_{t}(∂_{t} + 2ϱ) for the correct eigenvalue, we make the following perturbation ansatz:

Φ_{λ}(e^{th_{0}}) = e^{(λ−ϱ)t} ∑_{ℓ=0}^{∞} γ_{ℓ}(λ)e^{−2ℓt}, e^{th_{0}} ∈ A^{+}, (5.3)

where A^{+} is the open subset of A_{0} = A defined by A^{+} := exp((0, ∞)h_{0}). For ε > 0, we denote by A_{ε}^{+} the subset of A_{0} defined by A_{ε}^{+} := exp([ε, ∞)h_{0}).

Proposition 5.2. Let λ ∈ 𝔞^{∗}, λ ≡ λ(h_{0}) ∉ ℕ. Assuming that the series Φ_{λ} in Equation (5.3) converges to a solution of Equation (5.2), absolutely on A_{ε}^{+} for any ε > 0, the coefficients γ_{ℓ}(λ) are determined by the choice of γ_{0}(λ) and the relation

\begin{aligned}ℓ(ℓ − λ)γ_{ℓ}(λ) = & \frac{m_{α}}{2} (ϱ − λ + 2(ℓ − 1))γ_{ℓ−1}(λ) \\ & + (ϱ − λ + ℓ − 2)(ϱ + ℓ − 2)γ_{ℓ−2}(λ). \end{aligned} ; (5.4)

Conversely, define γ_{ℓ}(λ) by the above equation. Then the series Φ_{λ} converges to a solution of Equation (5.2), absolutely on A_{ε}^{+} for all ε > 0.

Proof. In view of Proposition 5.1, we may follow the standard procedure from Ref. [23, Chapter IV, § 5]. We obtain the recursion relation

(5.5)  ℓ(ℓ − λ)γ_{ℓ}(λ) = ∑_{n=1}^{2} \frac{nm_{nα}}{2} ∑_{k⩾1,ℓ⩾nk}(ϱ − λ + 2(ℓ − nk))γ_{ℓ−nk}(λ).

On the right-hand side, the contribution of the three terms corresponding to (n, k) = (1, 1), (1, 2), (2, 1) is

\frac{m_{α}}{2} (ϱ − λ + 2(ℓ − 1))γ_{ℓ−1}(λ) + ϱ(ϱ − λ + 2(ℓ − 2))γ_{ℓ−2}(λ).

On applying the recursion relation to the remaining terms, we see that the remainder amounts to

(ℓ − 2)(ℓ − 2 − λ)γ_{ℓ−2}(λ).

Thus, we obtain Equation (5.4). For the convergence, we follow the procedure of [23, Chapter IV, §5, Lemma 5.3]: Since we have

\frac{ℓ(ℓ − λ)}{ℓ^{2}} ⟶ 1, \frac{ϱ − λ + 2ℓ}{ℓ + 1} ⟶ 2

for ℓ → ∞, there are constants 0 < c < \frac{1}{2} and 2 < C < ∞ such that

|ϱ − λ + 2ℓ| ⩽ C(ℓ + 1), |ℓ(ℓ − λ)| ⩾ 2cℓ^{2} for all ℓ ∈ ℕ. For any ε > 0, there is some ℓ_{0} ∈ ℕ, ℓ_{0} ⩾ 2, such that

∑_{n=1}^{2} \frac{nm_{nα}}{4} (coth(\frac{nt}{2}) − 1) ⩽ \frac{cℓ_{0}}{C} , ∀t : t ⩾ ε.

Choose K = K_{ε} > 0 such that

|γ_{ℓ}(λ)| ⩽ Ke^{ℓt}, ∀ℓ, t : ℓ ⩽ ℓ_{0}, t ⩾ ε.

We claim that the ‘Gangolli estimate’ |γ_{ℓ}(λ)| ⩽ Ke^{ℓt} holds for all ℓ ∈ ℕ and t ⩾ ε. Indeed, assume that this holds for all ℓ′ < ℓ_{1}, where ℓ_{1} > ℓ_{0}. Then Equation (5.5) and the assumption imply

|γ_{ℓ_{1}}(λ)| ⩽ \frac{C(ℓ_{1} + 1)}{2cℓ_{1}^{2}} ∑_{n=1}^{2} \frac{n|m_{nα}|}{2} ∑_{k⩾1,ℓ⩾nk}|γ_{ℓ−nk}(λ)|

⩽ \frac{Ke^{ℓt}C}{ℓ_{0}c} ∑_{n=1}^{2} \frac{n|m_{nα}|}{4}(coth(\frac{nt}{2}) − 1) ⩽ Ke^{ℓt}

because ℓ_{0}(ℓ_{1} +1) ⩽ 2ℓ_{1}^{2} and coth(\frac{t}{2})−1 = 2 ∑_{k=1}^{∞} e^{−kt}. This proves our claim. In particular, we have

∑_{ℓ=0}^{∞}|γ_{ℓ}(λ)e^{−2ℓt}| ⩽ K ∑_{ℓ=0}^{∞} e^{−ℓt} ⩽ \frac{K}{1 − e^{−ε}} < ∞

and the series converges absolutely on A_{ε}^{+}. This justifies the term-by-term differentiation and implies that the limit Φ_{λ} is a solution of Equation (5.2). □

Remark 5.3. When m_{2α} = 0 (i.e. in the orthosymplectic case), the two-term recursion for γ_{ℓ}(λ) simplifies, and we easily obtain the following closed expression

\begin{aligned}γ_{ℓ}(λ) & = γ_{0}(λ) ∏_{m=0}^{ℓ−1} \frac{(m + ϱ)(m + ϱ − λ)}{(m + 1)(m + 1 − λ)} \\ & = γ_{0}(λ)c(−λ)(−1)^{l}\binom{−ϱ}{ℓ}\frac{−λ}{(ℓ − λ)c(ℓ − λ)}. \end{aligned} ; (5.6)

Using this formula, the convergence of the Harish-Chandra series follows from a simple-minded application of the ratio test.

From now on, will make the choice

(5.7)  γ_{0}(λ) := c(λ).

Contrary to the classical situation of Riemannian symmetric spaces of noncompact type, the Harish-Chandra series may be finite.

Corollary 5.4. If G = SOSp_{cs}(1, 1 + p|2q) and m_{α} = p − 2q ≤ 0 is even, then the series Φ_{λ} is finite. More precisely, we have

Φ_{λ}(e^{th_{0}}) = c_{1}e^{(λ−ϱ)t} ∑_{ℓ=0}^{−ϱ} \binom{λ − ϱ}{ℓ}\binom{−λ − ϱ}{−ϱ − ℓ}e^{−2ℓt}.

for some constant c_{1}. In particular, Φ_{λ} is well-defined on A.

Proof. In view of Equation (5.6), we have γ_{ℓ}(λ) = 0 for ℓ > −ϱ. Recall from Proposition 4.11 and Corollary 4.13 that

(5.8)  c(λ) = c′_{0} \frac{Γ(λ)}{Γ(λ + ϱ)} = c′_{0} ∏_{k=1}^{−ϱ}(λ − k).

Inserting γ_{0}(λ) = c(λ) into Equation (5.6), we obtain

\begin{aligned}γ_{ℓ}(λ) & = c′_{0}(−1)^{ℓ}\binom{−ϱ}{ℓ} ∏_{k=ℓ+1}^{−ϱ}(λ − k) ∏_{m=0}^{ℓ−1}(λ − ϱ − m) \\ & = c′_{0}(−ϱ)!\binom{−λ − ϱ}{−ϱ − ℓ}\binom{λ − ϱ}{ℓ}. \end{aligned} ; (5.9)

By the definition of Φ_{λ}, this proves the claim. □

Combining the above results, we arrive at the following expression for the spherical superfunctions.

Theorem 5.5. Let (G, K) be one of the symmetric pairs in Equation (5.1). For ℜλ > 0 and λ ∉ \frac{1}{2}ℤ, we have

ϕ_{λ}|_{A^{+}} = ∑_{w∈W_{0}} Φ_{wλ},

where Φ_{λ} is defined by Equations (5.3) and (5.7). Moreover, W_{0} = {±1} unless G = SOSp_{cs}(1, 1|2q), in which case W_{0} = {1}.

Proof. Firstly, Equation (4.1), Theorem 4.2, and Proposition 5.2 lead to

(5.10)  lim_{t→∞} ϕ_{λ}(e^{th_{0}})e^{−(λ−ϱ)t} = c(λ) = lim_{t→∞} Φ_{λ}(e^{th_{0}})e^{−(λ−ϱ)t}.

Therefore, in case W_{0} is trivial, then Proposition 4.12 and Corollary 5.4 combine to prove ϕ_{λ} = Φ_{λ}. It remains to prove the assertion in case W_{0} is non-trivial. We observe that by Proposition 5.2 again, the functions Φ_{±λ} are solutions of Equation (5.2). This is a differential equation of order two and Φ_{±λ} are visibly linearly independent, so

ϕ_{λ} = b_{1}Φ_{λ} + b_{2}Φ_{−λ}, b_{1}, b_{2} ∈ ℂ.

Let w_{0} ∈ M′_{0} be a representative of the non-trivial Weyl group element. Using Corollary 3.20 and the K × K-invariance of ϕ_{λ}, we derive for any a ∈ A^{+} that

ϕ_{λ}(a) = ϕ_{−λ}(a^{−1}) = ϕ_{−λ}(waw^{−1}) = ϕ_{−λ}(a).

This implies b_{1} = b_{2}. Using Equation (5.10), and

lim_{t→∞} Φ_{−λ}(e^{th_{0}})e^{−(λ−ϱ)t} = 0,

which is derived easily from Proposition 5.2, we conclude that b_{1} = 1. □

The result of Theorem 5.5 can be given a more explicit form in the cases where the Harish-Chandra series is finite.

Corollary 5.6. Let G = SOSp_{cs}(1, 1 + p|2q) and m_{α} = p − 2q ≤ 0 be even. Then

ϕ_{λ}(e^{th_{0}}) = c_{1}e^{(λ−ϱ)t}P_{−ϱ}^{(−λ,2ϱ−1)}(1 − 2e^{−2t}),

where the P_{n}^{(a,b)} denote the Jacobi polynomials.

Proof. By [18, §10.8, (16)], we have

P_{n}^{(a,b)}(x) = \binom{n + a}{n}_{2}F_{1}(−n, 1 + a + b + n; a + 1; \frac{x − 1}{2}),

where _{2}F_{1} is the Gaussian hypergeometric function. Inserting the hypergeometric series, we obtain easily that

P_{n}^{(a,b)}(x) = \frac{(a + n)!}{n!(a + b + n)!} ∑_{ℓ=0}^{n} \binom{n}{ℓ}\frac{(a + b + n + ℓ)!}{(a + ℓ)!}(\frac{x − 1}{2})^{ℓ},

where we write y! = Γ(y + 1). Let

n := −ϱ = q − \frac{p}{2}, a := −λ, b := 2ϱ − 1 = p − 2q − 1.

On applying Equation (5.9) and Corollary 5.4, we obtain

Φ_{λ}(e^{th_{0}}) = c′_{0}(−1)^{−ϱ}P_{n}^{(a,b)}(x)

for x = 1 − 2e^{−2t}, c′_{0} denoting the constant from Equation (5.8). □

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Alexander Alldridge

Universität zu Köln

Mathematisches Institut Weyertal 86-90 50931 Köln Germany alldridg@math.uni-koeln.de Wolfgang Palzer Universität zu Köln Mathematisches Institut Weyertal 86-90 50931 Köln Germany palzer@math.uni-koeln.de