{'fraction_non_alphanumeric': 0.044118947796706116, 'fraction_numerical': 0.009181128270445258, 'mean_word_length': 3.800849256900212, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 40, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Selected results on Lie supergroups and their radial operators

Foundational material on complex Lie supergroups and their radial operators is presented. In particular, Berezin's recursion formula for describing the radial parts of fundamental operators in general linear and ortho-symplectic cases is proved. Local versions of results which are suitable for applications for computing characters which are only defined on proper subdomains or covering spaces thereof are established.

Using Howe-duality in a Fock space context one observes that certain integrals of physical importance can be interpreted as numerical parts of characters of holomorphic semigroup representations of pieces of Lie supergroups which appear as Howe partners of classical symmetry ( [CFZ, HPZ]). The characters are holomorphic superfunction of a parameter which is varying in a covering space of a domain in the base complex reductive group of the Lie supergroup at hand. Such covering spaces, which arise, e.g., due to the involvement of the metaplectic representation, can be regarded as the domains of definition of the semigroup representations. They contain pieces of maximal tori so that at least in a local sense one has the appropriate notions of radial functions and operators. In fact the restrictions χ of the character to this torus piece is the function, i.e., the integral, which is of interest.
The character property implies that the integrals are eigenfunctions of the radial parts of Laplace-Casimir operators. In fact, in the cases considered in [CFZ, HPZ] the eigenvalues are zero! Thus in those cases every element D of the center universal enveloping algebra yields a differential equation which is of the formḊ(χ) = 0 whereḊ is the associated radial part of D.
The above is conceptually pleasing, but to complete the task of obtaining an explicit formula for the correlation functions χ one needs more concrete information on the radial partsḊ of the Laplace-Casimir operators. Although all of the necessary information is contained in the fundamental work of F. Berezin, gleaning it from ( [B]) and adapting it to the local setting indicated above requires a serious effort. Originally the first author of the present paper had hoped that it would be possible to provide at least a roadmap of [B] and include this in [HPZ]. However, this proved to be an unsatisfactory solution, in particular due to the additional length. Furthermore, we felt that more than a roadmap is needed. Thus our work developed into the thesis project [Ka1], the second author's paper [Ka2] and our work here.
Let us now briefly summarize this paper. The first section is primarily devoted to an explanation of the method of Grassmann analytic continuation and its application to defining a Lie supergroup associated to a given Lie superalgebra g = g 0 + g 1 . We do discuss Berezin's definitions of the morphisms of multiplication, inverse and the unit. However, since we are primarily interested in the differential operators, we pay most attention to the representations of g as left-and right-invariant derivations on the structure sheaf.
The remainder of the paper is devoted to considerations of radial operators, primarily in the cases of gl and osp which are of interest for the above mentioned Fock space applications. The first general goal is to describe the radial part of a Laplace-Casimir operator D byḊ = J −1 P D J where P D is a constant coefficient polynomial differential operator on the given maximal torus of the Lie group G associated to g 0 . A number of assumptions are needed for this, in particular that the function J should be an eigenfunction of the second order Laplace-Casimir operator which is defined in the usual way by an invariant nondegenerate supersymmetric bilinear form. This function appears as the square-root of the superdeterminant of a Jacobian of a coordinate chart which identifies a neighborhood A of a regular point in a maximal torus with a product A × B where B is a subsupermanifold which is defined as a local orbit of the Lie supergroup by conjugation (see the Appendix).
These assumptions are satisfied in the two cases of interest mentioned above. Furthermore, in both cases there is an important infinite series {F ℓ } of elements of the center of the universal enveloping algebra which defines a series of {D ℓ } Laplace-Casimir operators for which the constant coefficient operators P D ℓ can be described via a certain recursive procedure. For example, in the case of osp, if we use the standard basis for the standard Cartan algebra, then the polynomials F ℓ are defined by F ℓ = ϕ 2ℓ i + (−1) ℓ φ 2ℓ j . Following Berezin we writeḊ ℓ = J −1 T (F ℓ )J. The goal is then to understand the map F ℓ → T (F ℓ ). Identifying the polynomials F ℓ with the constant coefficient operators which they define, the main result of Berizin is that T (F ℓ ) = F ℓ + Q ℓ where Q ℓ is a polynomial in F 1 , . . . , F ℓ−1 . The proof of this result is discussed here in substantial detail in §2.
The last paragraphs of §2 are devoted to local versions of the formula mentioned above. First, we show that the global results of Berezin apply to give the same results on the local product neighborhoods A × B. These then lift to the covering spaces mentioned at the outset to give global results there by applying the identity principle.
Finally, in the appendix we outline the proof ofḊ = J −1 P D J on A×B which is given in detail in ([Ka1], Chapter 4) by methods which are similar to those used by Helgason in the classical case. Of course this is only valid under the same conditions as Berezin's global result.

Grassmann envelope
The first step for the construction is to consider the graded tensor product g ⊗ Λ with an arbitrary (finite-dimensional) Grassmann algebra. We assume throughout that Λ is isomorphic to the Grassmann algebra of an N-dimensional vector space with N ≥ n := dim(g 1 ) and speak of Λ as being generated by N (odd) independent elements.
For homogeneous elements X, Y ∈ g and α, β ∈ Λ one defines Extending by linearity, this defines a Lie superalgebra structure on the tensor product. Equipped with this structure, the Grassmann envelope g(Λ) := (g ⊗ Λ) 0 is a usual Lie algebra. Note that in addition to being a Lie algebra, g(Λ) is a Λ 0 -module. For notational convenience we write the multiplication by elements of Λ on the left, i.e., X ⊗α =: (−1) |X||α| αX. Although we regard Λ as a variable, we suppress it notationally by lettingg := g(Λ).

Decomposition ofg
The Lie algebra g 0 is a subalgebra ofg which has a complementary ideal k which is generated by the homogeneous elements αX where deg(α) ≥ 1. We writeg = g 0 ⋉ k as a semidirect sum. Letg 0 denote the Lie subalgebra of g which is generated by the homogeneous elements αX where |α| = 0, i.e., where α is even. It follows thatg is the subalgebra of consisting even elements in the nilpotent Lie algebra k.
Observe that the linear subspace k 1 of odd elements of k is stabilized byg 0 , i.e., [g 0 , k 1 ] ⊂ k 1 .

Decomposition at the group level
Let K be the simply connected Lie group associated to the Lie algebra k.
Recall that exp : k → K is a diffeomorphism. If we embed k as a Lie algebra of matrices, exp is even polynomially defined with a polynomial inverse.
In particular it is bianalytic or biholomorphic, depending on the setting at hand. Let G 0 be a (connected) Lie group associated to g 0 which integrates the representation of g 0 on g 1 . It follows that G 0 acts on K andG := G 0 ⋉ K is associated tog. Since G 0 acts on the Lie group K 0 := exp(k 0 ), the semidirect productG 0 := G 0 ⋉ K 0 is a closed Lie subgroup ofG. The linear space k 1 of odd elements of k defines a submanifold K 1 := exp(k 1 ) of K which is stabilized byG 0 -conjugation. We writeG =G 0 K 1 which is in a certain sense also a semidirect product.

Grassmann canonical coordinates
Here, starting with canonical coordinates on G 0 , we construct Grassmann canonical coordinates onG.

Canonical coordinates on G 0
Let U be an open neighborhood of 0 ∈ g 0 so that exp : we have the open neighborhood V.g which is identified with the coordinate chart U via the (bianalytic/biholomorphic) diffeomorphism exp : U → V . We shrink U to U 1 so that products of two elements as well as inverses of elements in V 1 = exp(U 1 ) are still contained in V and can still be identified with elements of U. To simplify notation, the possibly smaller sets U 1 and V 1 are still denoted by U and V , respectively. We cover G 0 by neighborhoods of this form which satisfy the further condition that if V g 1 and V g 2 have nonempty intersection, then g 1 g −1 2 ∈ V .
In order to obtain concrete coordinates we let {X 1 , . . . X m } be a basis of g 0 and to X = x i X i ∈ U associate the m-tuple x = (x 1 , . . . , x m ). If g ∈ V g 1 ∩ V g 2 has coordinate x with respect to the chart V g 1 and z with respect to V g 2 , and g 1 g −1 2 = exp(X), then change of coordinates is computed by applying the Campbell-Baker-Hausdorff formula (CBH) to Grassmann coordinates onG 0 Recall that exp : k 0 → K 0 is a diffeomorphism. Thus for U as above we can view U × k 0 as a coordinate neighborhood of Id: Let x = (x 1 , . . . , x m ) be as above and h = (h 1 , . . . , h m ) be an m-tuple of elements h i ∈ Λ. Then coordinates on V × K 0 are given by the diffeomorphism Using the following remark, we will express these coordinates in a more convenient way.
Proposition 1.1. The restriction of theG 0 -exponential map to U × k 0 is a diffeomorphism onto its image V K 0 .
Proof. For (A, B) ∈ U × k 0 with B sufficiently small, i.e., sufficiently near 0 ∈ k 0 , we apply CBH to obtain a mapping B → β(A, B) defined by exp(A + B) = exp(A)exp(β(A, B)) Now the definition of β involves brackets where B appears in a given (higher order) bracket at most N times, where Λ is the Grassmann algebra of a vector space of dimension N. If we use the coordinate h to describe B, this means that for fixed A the mapping β is polynomial in h. So we fix A and regard β as a map β : k 0 → k 0 . Analogously we define α : k 0 → k 0 by exp(A)exp(C) = exp(A + α(C)) in the range where CBH applies and then extend by the polynomial property. In the range where CBH applies we have exp(A + B) = exp(A + αβ (B)) and therefore αβ ( implies that βα(C) = C for C sufficiently small and it follows from the polynomial property that βα = Id. Hence (A, B) → exp(A+B) is a bijective map U ×k 0 → V K 0 and the desired result follows from its analyticity. In fact β = β(A, B) is a polynomial map in B with coefficients analytic in A.
Now we cover G 0 with neighborhoods V g as before and as a result have coordinates (x, h) on Note that in all considerations the Grassmann variables z i = x i + h i behave as scalars.

Canonical coordinates onG
Now let {Ξ 1 , . . . , Ξ n } be a basis of g 1 . An n-tuple ξ = (ξ 1 , . . . , ξ n ) with ξ j ∈ Λ 1 defines ξ j Ξ j ∈ k 1 and, allowing ξ to be arbitrary, we view it as a coordinate on k 1 . The exact same argument as that above shows that (x, h, ξ) → exp( (x i + h i )X i + ξ j Ξ j ) defines a diffeomorphism (analytic/biholomorphic) U × k → V K. Again covering G 0 as above, we coverG with neighborhoods V Kg on which we have the coordinates (x, h, ξ) → exp( (x i + h i )X i + ξ j Ξ j )g. These are the promised Grassmann canonical coordinates.
For X a homogeneous element of g, let α be a homogeneous element of Λ with α = 1 if |X| = 0 and |α| = 1 if |X| = 1 so that in particular αX| = 0, and consider the 1-parameter subgroup t → exp(tαX) in the Lie groupG.
For t sufficiently small we use CBH to express its action by left multiplication in Grassmann canonical coordinates: For this recall that the CBH computation that produces the righthand side of this equation involves higher order brackets where in theory αX can appear any number of times. However, using the definition of the Lie bracket, in particular that the Grassmann variables are behaving as scalars, along with the fact that α 2 = 0, it follows that the only nonzero terms are those where αX appears exactly once. Now, for homogeneous elements Y, Z ∈ g 0 and γ, δ ∈ Λ, the bracket [γY, δZ] is defined as (−1) |Y ||δ| γδ [Y, Z]. But sinceg is the even part of g ⊗ Λ, it follows that |δ| = |Z|. Thus, when transporting tα to the front of a bracket of homogeneous elements which occurs above, one only pays the price of a sign which does not depend on α. Therefore, for a fixed coordinate chart V K.g, the element η ∈ g ⊗ Λ is well-defined independent of α. Now let L αX be the Lie derivative defined by left multiplication by the 1-parameter group exp(tαX).
Proof. Let the covering be denoted by {V Kg ℓ }. As we just observed there exist operators M ℓ X on functions on V Kg ℓ which are defined independent of α so that L ℓ αX = αM ℓ X . Here L ℓ αX denotes the expression of the globally defined Lie derivative in the local coordinates at hand. Regarding this as a vector field it obeys the change of coordinates rule L k αX = J kℓ L ℓ αX , where J kℓ is the Jacobian of the change of coordinates. Therefore α(M k X − J kℓ M ℓ X ) = 0 for every α ∈ Λ. Now we embed Λ in a larger Grassmann algebra Λ ′ so that if β ∈ Λ is such that αβ = 0 for all α in Λ ′ , then it follows that β = 0. We carry out the above construction for Λ ′ but, with the exception of α, all computations are made with elements ofg = g(Λ). It follows that applying the difference M k X − J kℓ M ℓ X to an element of OG ⊗ Λ results in a function with values β ∈ Λ with αβ = 0 for all α ∈ Λ ′ . Consequently, as desired M k X = J kℓ M ℓ X as operators on OG ⊗ Λ.

Representation theoretical properties of M
Define M : g → End(O G 0 ⊗ Λ) by X → M X . Although the notation may not indicate it, M X is an operator at the sheaf level. Observe that if X is homogeneous, then so is M X and |M X | = |X|. The main properties of M can be summarized as follows.
Proposition 1.3. The mapping M is a representation of the Lie superalgebra g in the space Der(OG ⊗ Λ) of derivations of OG ⊗ Λ.
Proof. In the discussion α and β are chosen to be homogeneous elements in some larger Grassmann algebra Λ ′ with the property that αβγ = 0 implies that γ = 0 for every γ ∈ Λ. For X, Y homogeneous elements ing = g(Λ), we carry out the above construction for αX, βY ∈ g(Λ ′ ). Since |α| = |X| and |β| = |Y |, it follows immediately from the definition of the Lie bracket that On the other hand, and therefore Applying this identity to f ∈ OG ⊗ Λ and using the cancellation property of αβ yields the representation property of M .
In order to show that M X is a derivation, we choose α ∈ Λ ′ with the cancellation property and note that for f, g ∈ OG ⊗ Λ and the desired derivation property follows by cancellation.
Before closing this section, we should note that the same discussion as above with left multiplication by the 1-parameter groups t → exp(tαX) being replaced by right multiplication leads to analogous representations of g in Der(OG ⊗ Λ). If it is necessary to differentiate between the representations defined by left multiplication and those defined by right multiplication, we denote the former by X → L X and the latter by X → R X . Of course these representations commute in the graded sense, i.e.,L X R Y = (−1) |X||Y | R Y L X and the representation which corresponds to conjugation in the classical Lie group case is given by X → L X + R X .
Above we have constructed g-representations on the function algebra of the Grassman envelopeG. Here we turn to the main task of this section which is to construct the representations of g on the structure sheaf F = O G 0 ⊗ Λ which correspond to the representations by invariant vector fields of a Lie algebra on the structure sheaf of an associated Lie group. The key idea is to extend (analytic/holomorphic) functions from G 0 to functions on the special Grassmann envelopeG where Λ = ∧g * 1 and apply the representations constructed above.

Grassmann analytic continuation
Let Λ = ∧g * 1 and consider a Λ-valued analytic or holomorphic function f on an open subset V of G 0 . We (analytically) continue f to a function Ψ(f ) on the open subset V K of the Grassmann envelopeG. Since the construction does not depend on the nature of V , we suppress it in the discussion and only discuss functions defined on the full group G 0 .
As a first step we will (analytically) continue f to a function onG 0 . For this we cover G 0 as usual by open sets of the form V g where V is an open neighborhood of Id with exp : U → V a diffeomorphism. Letting {X 1 , . . . , X m } be a basis of g 0 we have Grassmann coordinates on V K 0 g given by V K 0 and then composing by multiplication on V K 0 g. Let g k and g ℓ be such that the coordinate neighborhoods V g k and V g ℓ have nonempty intersection. Since by assumption f is analytic (or holomorphic), in the respective coordinate charts it has convergent power series representation f k (x) and f ℓ (x). Now a given point where g = g k or g = g ℓ , and the change of variables at the level of G 0 , which is computed by CBH, is given by x k = A kℓ (x ℓ ), then the change of variables for sum x + h is given by the same rule, i.e..
Now, since the elements h of the Grassmann algebra are nilpotent, the power series f k (x + h) and f ℓ (x + h) converge on U × k 0 , and since the transformation rule for the variable x is the same as that for x + h, the resulting locally defined functions agree on the intersection V K 0 g k ∩V K 0 g ℓ . Thus we have defined the basic first step of Grassmann analytic continuation which is a continuous morphism of sheaves of algebras The next step, i.e., the continuation to OG ⊗ Λ, is formal. For this we observe thatG is the productG 0 × k 1 . Using a basis {Ξ 1 , . . . , Ξ m } for g 1 , an element of g 1 ⊗ Λ 1 is written as σ j Ξ j . Thus the product structure is given by then an arbitrary function f ∈ OG ⊗ Λ can be expressed as where the coefficient functions are arbitrary analytic/holomorphic functions oñ G 0 .
Using the above description, the continuation of a function fromG 0 is formal.
Its continuation f ∈ OG ⊗ Λ is simply defined as re-interpreting the elements Ξ j ∈ g * 1 of the dual basis as odd coordinate functions onG. The Grassmann analytic continuation from O G 0 ⊗ Λ is then defined by The following summarizes the construction. Recall that for this Λ := ∧g * 1 .
Proposition 1.4. Grassmann analytic continuation Ψ : It should be remarked that a function F = f I Ξ * I is in the image A if and only if the expansions of the coefficents f I in the local coordinates (x, h) are power series in x + h. In other words, from the point of view of the coefficent functions the Grassmann variables x + h are scalars.
The representations of g on the structure sheaf Recall that at the beginning we deemed the structure sheaf F to be the sheaf of germs of holomorphic maps from G 0 to ∧g * 1 which we denote here by Λ. By definition the numerical functions are just the analytic/holomorphic functions O G 0 with the projection num : F → O G 0 being the obvious one. Since O G 0 is canonically embedded in F with ker(num) as complement, in the language of supergroups the structure is split.
We complete the task of constructing the representations of g on F by noting the following.
Proposition 1.5. The image A of Grassmann analytic continuation is invariant under the representations L and R of g on OG ⊗ Λ.
Proof. We must show that if X is a homogeneous element of g and in Grassmann analytic coordinates. For example, in a neighborhood of Id we must use CBH to convert the product to the form of the Grassmann coordinates. Here z = x + h is the Grassmann variable. The result is complicated, but is of the form The key point is that the coefficientsz i (z, σ) = J c J i (z)σ J are polynomials in σ with coefficients which are power series in z. Now recall that applying F to such an expression yields Using the power series representations of the f I when differentiating one obtains polynomials in σ with coeffiecients which are power series in the Grassmann variables z. Of course factoring out α doesn't change this structure and it follows that the resulting function is of the desired form, i.e., where the coefficients f I are power series in Grassmann analytic coordinates. Note that although we have only verified this in the coordinates at the identity, the verification in a general neighborhood V K 1 g only differs from that at the identity by conjugation of K 1 by g ∈ G 0 .
As a consequence we have the main result of this section.
Theorem 1.6. Restricting the representations L and R of g on OG⊗∧g * 1 to the image A of Grassmann analytic continuation Ψ : F ∼ = → A defines the representations of g on the structure sheaf F. Restricted to g 0 these are given by differentiating elements of F by the natural action of 1-parameter subgroups of G 0 on F.

Comments on Lie supergroups
A Lie supergroup is more than a structure sheaf F with representations L and R which extend the natural representations of g 0 . The appropriate additional structure can be formulated in terms of a triple (µ, ι, ε) of maps at the sheaf level which correspond to multiplication, inverse and the identity. For example, in the classical case the multiplication map µ : Certain compatibility conditions must be fulfilled (see [B], p.247). Once this triple is defined, the representations L and R can be computed. Using Grassmann analytic continuation as above, Berezin does indeed define such a triple and then constructs L and R as above.
Berezin's construction of the Lie supergroup triple uses a notion of a superfunction which is slightly different from that above. In simple terms this is the difference between the notion of a variable and the evaluation of the variable. For example, a complex polynomial P in several commuting variables z 1 , . . . , z m is an element of the ring C[z 1 , . . . , z n ]. If R is any commutative C-algebra, e.g., of numbers, functions, operators, etc., P can be evaluated to define a function P R : R×· · ·×R → R. In the setting at hand ξ 1 , . . . , ξ n are anticommuting variables and Berezin defines a superfunction on G 0 to be an element f ∈ O G 0 [ξ 1 , . . . , ξ m ]. This is a function f = f (g, ξ) in two blocks of variables. The first variables g can be regarded as the commuting variables defined by the coordinates in the manifold G 0 , e.g., in some power series ring. The second variables are the anticommuting variables which generate a Grassmann algebra Λ(ξ 1 , . . . , ξ m ).
Continuing with the simple example, if a 1 , . . . a n are algebraically independent elements of a commutative ring R, then any algebraic combination c I a I defines a unique polynomial P ∈ R[z 1 , . . . , z n ]. In our context where n := dim(g 1 ), one considers O G 0 ⊗ Λ where Λ is a Grassmann algebra generated by N elements with N ≥ m. If a 1 , . . . , a m are independent elements of Λ, then there is a canonical isomorphism between the subalgebra spanned by these elements and the algebra of As stated at the outset we wish to regard the structure sheaf F as the sheaf of germs of holomorphic functions on G 0 with values in ∧g * 1 . This may seem different from Berezin's sheaf, but if one exchanges variables for evaluated variables, it is not. The main point in this regard is that Berezin's construction uses a basis {Ξ 1 , . . . , Ξ n } for g 1 . Thus evaluating a Berezin superfuction f = f (g, ξ) ∈ O G 0 [ξ 1 , . . . , ξ n ] on Ξ * 1 , . . . , Ξ * n defines such a function. Conversely, starting with a holomorphic function f : G 0 → ∧g * 1 and the basis {Ξ 1 , . . . , Ξ n } one obtains a Berizin superfunction by replacing the Ξ * j by the variables ξ j .
When defining the Lie supergroup triple (µ, ι, ε) the variable viewpoint is very useful. For example, the standard multiplication morphism µ : G 0 → G 0 × G 0 must be lifted to a map ϕ : F G 0 → F G 0 →G 0 of sheaves. This map can be defined as follows. We regard a function in F G 0 as a function of one variable (g, ξ). Here g is already Grassmann continued toG 0 where Λ is generated by N odd elements with N ≥ 2n. As usual ξ denotes an n-tuple of independent Grassmann variables. The functions in F G 0 ×G 0 are functions of two such variables ((g 1 , ξ 1 ), (g 2 , ξ 2 )). Now we evaluate these variables at points in the groupG by using 2n independent odd elements a 1 , . . . , a n , a n+1 , . . . , a 2n in Λ. The pair of variables defines a pair of group elements g 1 exp( a j Ξ j ), g 2 exp( a n+j Ξ j ) ∈G .
Using the decompositionG =G 0 K 1 , the product of these group elements is written as gexp( η j Ξ j ). Now we change back to the variable standpoint and regard g as a Grassmann analytically continued variable inG 0 and η = (η 1 , . . . , η n ) as a variable which (as can be checked) consists of n independent odd Grassmann variables.
Since the construction of manipulating the product to the form g exp( j η j Ξ j ) only uses the variable properties of the elments a j , the product variable (g, η) does not depend on the choice of the a j . The multiplication morphism at the sheaf level is then defined in the natural way: .
Lifting the inverse mapping g → g −1 to the sheaf level is defined analogously. The evaluation at the identity is given by ε(f )(g, ξ) = f (Id, 0). Since these operations are defined by the group structure ofG, it can be directly checked that they have the desired compatibility properties. As indicated at the outset of this paragraph, if we regard F as the sheaf of ∧g * 1 -valued holomorphic functions, then the Lie supergroup structure is defined at that level by replacing a function f I (g)(Ξ * ) I by a Berezin function f = f (g, ξ) of two variables defined by f (g, ξ) = I f I (g)ξ I . Multiplication is then defined as above at the level of variables and then one returns to ∧g * 1 -valued holomorphic functions by evaluating the variables at Ξ * 1 , . . . , Ξ * n .
The representations L and R of g on F are in fact by derivations which are invariant under the group structure. For example, invariance by right-multiplication means that for X ∈ g. This is immediate for the derivations defined by even elements, because the supergroup multiplication by elements ofG 0 is just usual multiplication. For odd elements X ∈ g 1 one must check that the formula analogous to 1 with M X replaced by the Lie derivative L αX inG makes sense and is valid. Then one cancels α in the same way that was done for the definition of M X .
There are various other methods of constructing a Lie supergroup associated to g, (see [K] and [Ka2] for constructions using Lie-Hopf algebras and [Ko] for the dual construction at the level of the structure sheaf). But it turns out that, even in the more delicate holomorphic setting, these are all equivalent (see [V]). For further discussion of this matter, e.g., for a detailed comparison of the various definitions, see [Ka1].

Radial operators
Our concrete goal here is to describe certain series of radial differential operators in sufficiently concrete form for computational applications in [CFZ] and [HPZ]. As in the previous section, the main results were originally proved by Berezin.

Basic definitions
Here we recall be the basic objects for the study of radial differential operators on Lie supergroups.

Universal enveloping algebra
The universal enveloping algebra of a Lie superalgebra g is the quotient U (g) of the full tensor algebra T (g) by the ideal generated by ( where X, Y are homogeneous elements of g regarded as elements of (g). If X 1 ⊗ X 2 · · · ⊗ X k is a monomial in T (g), then its image in U (g) is equipped with the sign (−1) |X 1 |+...+|X k | . This defines a Z 2 -grading on U (g) for which the induced bracket defines a Lie superalgebra structure. Here we shall primarily be concerned with the center Z(g) of U (g). This is the subalgebra of U (g) consisting of those elements X with ad(X) = [X, ·] = 0.

Laplace-Casimir operators
Recall that a representation of a Lie superalgebra g is a superalgebra morphism ρ : g → End (V ) to the superalgebra of linear maps of a graded vector space V = V 0 ⊕V 1 . Such a representation extends to a representation ρ : U (g) → End (V ). The examples of main importance here are the representations L and R of g constructed in the previous section on the structure sheaf F of superfunctions of the associated Lie supergroup. In particular, we consider the representation X → L X and extend it to a representation of U (g) by differential operators. The Laplace-Casimir operators are those in the image of the center Z(g).

Radial functions
A superfunction f on G 0 is said to be radial if it is annihilated by the representation X → L X + R X of g. Let R G 0 denote the sheaf of radial holomorphic functions on G 0 . Observe that if D is a Laplace-Casimir operator, then D|R G 0 : We regard R G 0 as the natural domain of definition of these operators.
Note that if f is globally defined on G 0 , then the condition L X f + R X f = 0 for all X ∈ g 0 just means that f is conjugation invariant, i.e., f (g 0 gg −1 0 ) = f (g) for all g 0 ∈ G 0 . If G 0 is reductive, which we assume from now on, it follows that globally defined conjugation invariant superfunctions are completely determined by their restrictions to any given maximal torus H. We fix such a maximal torus and let h denote its Lie algebra in g 0 . Let us also assume, as will be the case in all applications, that h is a Cartan algebra of the full Lie superalgebra g.

Restriction theorem at the Lie superalgebra level
At the infinitesimal level we are interested in understanding the image of the restriction map from the space S(g * ) g of ad g -invariant (super) polynomials on g to the space S(h * ). We view an element P ∈ S(g * ) as a (holomorphic) polynomial map P : g 0 → ∧g * 1 . If U is a sufficiently small neighborhood of 0 ∈ g 0 which is identified by the exponential map with a neighborhood V of Id in G 0 , then P ∈ S(g * ) g if and only if the resulting function on V is radial.
Since it has been assumed that h is an even subspace of g, polynomials in S(h * ) are just standard (numerically valued) polynomials. Since P ∈ S(g * ) g is invariant by the adjoint representation of G 0 , it is immediate that R(P ) is invariant under the Weyl group W = W (g 0 , h 0 ). Thus we regard R as a map R : S(g * ) g → S(h * ) W .
A great deal is known about the restriction morphism R. In particular, it is always injective. In our cases of interest, the basic results are proved in [B] and [S]. Let us quote Berezin's Theorem 3.1.
Theorem 2.1. Let G 0 be reductive and assume that h is a Cartan algebra of g which is contained in g 0 . Assume further that g is endowed with a nondegenerate invariant scalar product and that its odd root spaces are 1-dimensional. Then a W -invariant polynomial is in the image of R if and only if for every odd root β (with dual root β * ) it follows that where Q = Q(h) is a polynomial on h.
We say that the extendible polynomials are exactly those which are W -invariant and satisfy the divisibility condition (2). Since our work here is aimed at understanding properties of radial functions and operators in the cases of gl and osp, it should be emphasized that the conditions of Berezin's theorem are fulfilled in those cases.
For the statement of the version of Berizin's extension theorem for holomorphic functions we say that a holomorphic superfunction on g 0 is radial if and only if it is annihilated by the all ad g -derivations. The divisibility condition for holomorphic functions is the same as that for polynomials.
Theorem 2.2. Under the assumptions of Theorem 2.1 it follows that a Winvariant function f ∈ O(h) can be (uniquely) extended to a radial holomorphic superfunction on g 0 if and only if it satisfies the divisibility condition.
Sketch of Proof. For the neccessity, i.e., that the divisibility condition is really needed for extension, one replaces the polynomial P in Berezin's proof by the convergent power series representation of the given function f at 0 ∈ h. For the sufficiency Berezin uses the fact that the given polynomial P can be extended to a unique Ad G 0 -invariant polynomial on g 0 and then proceeds by using generalities on superfunctions. Since W -invariant holomorphic functions on h extend to Ad G 0invariant holomorphic functions on g 0 , the same proof can be carried out in the holomorphic case.

Restriction theorem at the group level
Now let us turn to the Lie supergroup associated to g equipped with its sheaf F of holomorphic superfunctions, i.e., the sheaf of germs of holomorphic maps with values in ∧g * 1 . We assume that g satisfies the assumptions of Theorem 2.1 andlet H be the maximal (complex) torus in G 0 associated to h. In this situation we wish to determine, e.g., the image of the restriction map R : R G 0 (G 0 ) → O(H) W from the globally defined holomorphic radial functions on G 0 in the algebra of W -invariant holomorphic functions on H.
For this the divisibility condition must be transferred to the group level: A holomorphic function f ∈ O(H) is said to satisfy the divisibility condition if and only if its pull-back f • exp satisfies the divisibility condition on h. The following is an immediate consequence of the results in the previous paragraph. The extension theorem at the group level is stated as expected. Proof. To prove the sufficiency of the divisibility condition we let f be a Winvariant (numerical) holomorphic function on H which satisfies the divisibility condition andf = f • exp. Sincef satisfies the divisibility condition on h, it is the restriction of a (unique) holomorphic radial superfunction E(f ) on g 0 . This means thatf is the numerical part of E(f )|h. The uniqueness of the extension E(f ) implies that E(f )|h is periodic with respect to the discrete additive subgroup of h which is the kernel of exp : Observe that since E(f ) is invariant by conjugation by elements of the normalizer of H, it follows that its restriction to h is W -invariant and thus f s is W -invariant. Classical invariant theory then implies that f s is the restriction of a unique conju- Thus E(f )•exp is the radial extension off . In particular, if U is a neighborhood of 0 ∈ g 0 such that exp : U → V is biholomorphic, then the fact that the operators L X + R X annihilate E(f )|U for all X ∈ g implies that they annihilate E(f )|V for all X ∈ g. Hence the identity principle implies that (L X + R X )E(f ) = 0 for all X ∈ g and consequently E(f ) is the desired radial extension of f .
For the necessity of the divisibility condition, we just reverse the argument: If E(f ) is the radial extension of f , then E(f ) • exp|U is annihilated by the operators L X + R X and the identity principle implies that E(f ) • exp is the radial extension off . Consequentlyf satisfies the divisibility condition which by definition is the divisibility condition for f .

Jacobian formula
Recall that we have regarded the Laplace-Casimir operators as being differential operators D : R → R on the sheaf of radial holomorphic superfunctions on the Lie supergroup associated to a Lie superalgebra g. Here we restrict our considerations to the setting of Theorem 2.1 so that the algebra of global radial functions is decribed by the divisibility condition along a given maximal torus H in G 0 . Note that divisibility at the group level means that for every odd root β the directional derivative X β (f ) is divisible by (r β − 1) where r β is the character associated to β.
Denote by D H the image in O(H) W of the restriction map R and E : D H → R(G 0 ) the extension map which is R −1 . The associated radial part of a Laplace-Casimir operator D on H is defined asḊ := RDE : D H → D H . Our goal in this paragraph is to describe a basic result of Berezin which, at least for the Lie superalgebras and operators of main interest for the applications in [CFZ] and [HPZ], shows that the study of the radial operatorsḊ can be reduced to analyzing certain constant coefficient polynomial differential operators on H. This is proved by introducing a sort of change of variables along H so that generically along H one has a local product decomposition in the H-direction and the transversal direction of the supergroup action. As a consequence, a Jacobian J appears and therefore we refer to the result as the Jacobian formula.
The Jacobian J is defined as follows as a meromorphic function on H: where ∆ + = ∆ + 0 ∪ ∆ + 1 is a system of even and odd positive roots. Under certain restrictive conditions the Jacobian formula states that given a Laplace-Casimir operator D there is a uniquely defined polynomial operator with constant coefficients P D on H so thatḊ = J −1 P D J. It should be remarked that in the classical case of Lie groups the analogous formula (without the odd roots in the Jacobian) holds in great generality. In the Lie supergroup setting we state it in the cases of g = gl(m, n) and osp(2m, 2n). In [B] the latter Lie superalgebra is denoted by C(m.n). From now on g is restricted to be one of these (complex) Lie superalgebras equipped with the nongenerate bilinear form (X, Y ) = STr(XY ).
In the case of gl(m, n) we choose h to be the Cartan algebra of diagonal matrices with coordinates h = ϕ 0 0 ψ .
One checks that the polynomial function satisfies the divisibility condition and therefore is extendible to an ad-invariant element of S(g * ), i.e., to an element of Z(g).
In the case of osp(2m, 2n) we recall that g 0 = so 2m ⊕ sp 2n and as above choose h to be in diagonal form with coordiantes with the full Cartan algebra given by h = Diag(h so , h sp ). In this case one defines the extendible polynomials In the case of gl(m, n) the algebra of extendable polynomials consists of polynomials with constant coefficients in the F k . In the case of osp(2m, 2n) one requires one additional generator which is most conveniently chosen as L = ϕ 1 · . . . · ϕ m R where R is the product of the odd positive root functions. For the sake of brevity of notation we letF be the constant coefficient differential operator defined by F k . A simplified version of Berezin's Jacobian theorem (see Theorem 3.2 on p. 302 of [B]) can be stated as follows.
Theorem 2.5. Let g be either gl(m.n) or osp(2m, 2n) and P be an extendible polynomial on h which defines the radial differential operatorḊ P byḊ P (f ) = RD P E(f ) for f ∈ D H a W -invariant function satisfying the divisibility condition. Then there exists a uniquely determined polynomial function T (P ) on h with associated constant coefficient differential operator denoted by T (P )( 1 i ∂ ∂t ) so thaṫ Furthermore, if P = F k , then T (P ) is of degree k with top degree term F k .
It should be underlined that the partial derivative operators are defined by the coordinates (ϕ, ψ) of h which were introduced above.
Remark. For applications in [HPZ] the first author was originally interested in a local version of Theorem 2.5 at a generic point of a maximal torus. Using techniques which are much closer to the methods of Helgason in the classical case, the second author has proved such a result in his thesis [Ka1]. As is shown in the sequel, Berezin's global result Theorem 2.5 implies the local result. Vice versa, application of the identity principle shows that the local result implies Berezin's global result. Thus the two results are equivalent. Nevertheless we feel that it is of interest to have new viewpoint on these matters. Hence, the statement and a sketch proof of the local theorem have been included as an appendix to this paper.
Now we turn to understanding the mapping P → T (P ). This will be discussed for the polynomials P = F k in both cases gl(m, n) and osp(2m, 2n).

Finite-dimensional representations
Here we explain how to compute the polynomials T (P ) in terms of the eigenvalues of the radial operatorsḊ P on characters of finite-dimensional representations. We restrict to the cases g = gl(m, n), osp(2m, 2n), but most of the discussion applies in a much more general setting, e.g., where g 0 is semisimple. For more details see Chapter 3.10 (p.307-311) of [B].

Character formula
Consider a finite-dimensional irreducible representation ρ of a complex Lie supergroup associated to one of the Lie superalgebras g = gl(m, n), osp(2m, 2n). This is by definition a homomorphism of the Lie supergroup associated to g to that associated to Lie superalgebra gl (V ). Such is defined by a holomorphic mapping G 0 → GL(V ) which lifts to the sheaf level as a mapping F GL(V ) → F G 0 which preserves the defining Lie supergroup triples (see [B], p.248). Taking a basis of homogeneous elements of V one interprets ρ as a holomorphic map of G 0 to matrices whose entries are superfunctions. The character of such a representation is defined by χ(g) := STr(ρ(g)). It is a radial superfunction on G 0 and we consider its restriction χ to a Cartan subgroup H. It is an eigenfunction of every radial operatorḊ. In other words there is a homomorphism λ defined on the space of radial differential operators with values in C so thatḊ(χ) = λ(Ḋ)χ.
Now apply the Jacobian formula,Ḋ P = JT (P )( 1 i ∂ ∂t )J −1 , defineχ = Jχ and observe that In other words, the eigenvalue homomorphism for the radial operatorḊ P on the character χ is the same as the eigenvalue homomorphism for the constant coefficient operator T (P )( 1 i ∂ ∂t ) on the functionχ. This simple remark leads to an exact description of T (P ) in terms of eigenvalues of irreducible representations.
For this note that J is defined on H so thatχ can be expanded in a Fourier series Applying T (P )( 1 i ∂ ∂t ) to both sides one shows that if a k = 0, then for every extendible polynomial P . Letting P range over all such polynomials one proves the following fact.
Proposition 2.6. The set of lattice elements k such that a k = 0 is a W -orbit W.k 0 .
It should be noted that since T (P ) is itself W -invariant the lack of uniqueness of the lattice element k is minimal.
Now χ is W -invariant. Furthermore, for σ in the Weyl group it follows that σ(J) = ε(σ)J where ε(σ) = det(σ) = ±1. Hence, up to a multiplicative constant for any fixed k 0 in the support ofχ. Now order the weight lattice so that the roots α and β which occur in the above products are positive and write where the m j are the weights of the representation ρ with Λ being the highest weight which occurs. Compare this expression for χ(t) with that above to obtain Equating the highest order terms on each side yields where k 0 is the highest of the elements in the W -orbit W.k 0 . Turning this around, we see that Theorem 2.7. Let ρ be a finite-dimensional representation of a Lie supergroup associated to one of the Lie superalgebras g = gl(m, n), osp(2m, 2n). Let P be an extendible polynomial on h and T (P ) be the polynomial which is defined bẏ If χ is the character of ρ with the homomorphism λ defined bẏ where Λ is the highest weight of ρ.
Proof. This follows immediately from (3) and the fact that our choice of k = k 0 in the W -orbit is k 0 = Λ + δ.

Generating functions
Here we fix g as one of the Lie superalgebras gl(m, n) or osp(2m, 2n) and letḊ ℓ be the radial operator defined by the particular extendible polynomial F ℓ . Using the Fourier series development of characters of representation, it was shown above that the value λ(Ḋ ℓ ) of the eigenvalue homomorphism onḊ ℓ for the character of an irreducible representation ρ of highest weight Λ of the associated Lie supergroup is the value of the polynomial T (P ) on lattice point k 0 = Λ + δ. Letting ρ range through all such representations, we see that T (P ) is the unique polynomial with this property. If we think of such a point k 0 as a weight, then it is in h * ; so we reformulate the result as follows: There is a uniquely determined polynomial function R ℓ on h * with R ℓ (Λ + δ) = λ(Ḋ ℓ ) on every irreducible representation of highest weight Λ.
Associated to the sequence {R ℓ } of polynomials one has the generating function which is computed in closed form in [B] (see Lemma 4.3, pages 327-329, for the case of gl(m, n) and Lemma 4.4, pages 335-341, for osp(2m, 2n)). The resulting formulas for the polynomials R ℓ are derived after the proofs of these lemmas. Using the identificatiion R ℓ = T (F ℓ ), one has the following consequence which we formulate simultaneously for both gl(m, n) and osp(2m, 2n).
Theorem 2.8. IfḊ ℓ is the radial differential operator defined by the extendible polynomial F ℓ withḊ is a polynomial of lower degree.

An application
In [CFZ] and [HPZ] characters χ of representations Lie supergroups on certain infinite-dimensional spaces play an important role. In [CFZ] the complex Lie superalgebra at hand is gl(m, n) and in [HPZ] it is osp(2n, 2n). In these situations one would hope to apply the above results on radial operators. However, this can not be directly done, because the characters are defined by supertrace and only converge on certain open domains H in G 0 or on finite covering spaces H of such domains. On the other hand, Laplace-Casimir operators are local and can therefore be applied to such characters and in the settings of [CFZ] and [HPZ] the characters χ which appear are annihilated by Laplace-Casimir operators D ℓ defined by the F ℓ .
In the domains H or the covering spaces H there are closed connected complex submanifolds T + which are either open subsets of a Cartan algebra H or lifts of such into the covering space. Now the radial operatorsḊ ℓ are differential operators which are apriori defined on the space D H of globally defined extendible W -invariant holomorphic functions and on that space we know how to compute them using the righthand side of (4). The restrictions of the characters χ to T + , which are by definition the numerical parts of χ|T + , are by definition extendible as radial superfunctions, but they are only defined on T + and not on H. Nevertheless we wish to show that they are annihlated by the operators which are described by the righthand side of (4). For this we prove a local version of (4) and obtain the desired result on T + by applying the identity principle.

Local formula forḊ ℓ
We refer to a point in H as being superregular if it is regular in the sense of Lie theory and is not contained in any of the odd root hypersurface {r β = 1}. Every a superregular point x has a basis of open neighborhoods V in H which are relatively compact in the set of superregular points in H with the property that σ(V ) ∩ V = ∅ for every σ ∈ W \ {Id}. Given such a V we thicken it as follows to an open neighborhood U in G 0 . Let ∆ be a polydisk in g 0 which is transversal to h and define U = {exp(ξ).x; ξ ∈ ∆, x ∈ V }. We choose ∆ small enough so that U ∼ = ∆ × V . For x ∈ V fixed we think of exp(∆).x as a local orbit of G 0 .
Proposition 2.9. Every superregular point x in H has a neighborhood basis of open sets V and U as above so that the restriction map R : Proof. Since holomorphic maps U → ∧g * 1 which are invariant by the local conjugation-action of G 0 are completely determined by their restrictions to V , it follows that R is injective. Surjectivity is proved by the following approximation argument.
First, in order to take care of W -invariance we consider the quotient π : H → Z = H/W . The restriction π|V maps V biholomorphically onto a domainṼ . A basic theorem of complex analysis states that we may chooseṼ (and accordingly V ) so that the restriction map O(Z) → O(Ṽ ) has dense image. Now let R be the product of the odd root functions on H andR be the associated function on Z. Definef be the function onṼ associated to a given holomorphic function f on V . Letf n be a sequence of holomorphic functions on Z which converge toR −2f in O(Ṽ ). It follows thath n :=R 2f n converges tof . The point of this construction is that the sequence {h n } of lifts defined by h n := π * (h n ) converge to f on V . In addition these are W -invariant and have the divisibility property. Thus we have the sequence {E(h n )} of radial extensions. Now the extension h n |V → E(h n )|V is such that the convergence of h n |V implies the convergence of E(h n )|V as a sequence of ∧g * 1 -valued holomorphic maps. Consequently the maps E(h n )|U which are constant along the local G 0 -orbits defined by ∆ also converge. If a sequence of holomorphic functions converges, then so does any induced sequence of derivatives. Thus the limit E(f ) of the sequence {E(h n )|U } is a radial holomorphic function whose (numerical) restriction to V is the given function f .
Having localized to the open sets V = U ∩ H of superregular elements of H and proved the above extension result, given a Laplace-Casimir operator D we define its radial part on U ∩ H in the same way as in the global case:Ḋ U ∩H (f ) := RDE(f ). Since the extension result was proved by taking limits of globally defined extendible functions and the global operatorḊ is continuous, it follows thatḊ U ∩H is just the restriction ofḊ to U ∩ H. Thus we have the following local version of Theorem 2.5.
Theorem 2.10. Under the assumptions of Theorem 2.5, let D be a Laplace-Casimir operator defined by an extendible polynomial P . If U is as above, then the domain of definition of the radial operatorḊ U ∩H is the full algebra of holomorphic functions O(U ∩ H) andḊ As a result we have the local version of Theorem 2.8 Theorem 2.11. Under the assumptions of Theorem 2.8, for U as above it follows thatḊ U ∩H = J −1 (F ℓ + Q(F 1 , . . .F ℓ−1 ))J where Q is a polynomial operator of lower degree thanF ℓ . Now let us return to the settings of [CFZ] and [HPZ] where we have an open piece T + of the Cartan algebra H contained as a closed submanifold of a domain H in G 0 or a finite-to-one covering space H of such a domain. The Weyl group W acts on these domains so that the above arguments apply: If x ∈ T + is superregular, then we setup U as above and prove the following result.
Theorem 2.12. Let D ℓ be the Laplace-Casimir operator defined on H or H by the extendible polynomial F ℓ . Then a holomorphi superfunction on such a domain is annihilated by D ℓ if and only if its (numerical) restriction to T + is annhilated by J −1 (F ℓ + Q(F 1 , . . .F ℓ−1 ))J.
Proof. Since D ℓ acts on the full sheaf of radial functions, we may regard it as acting on the restriction of given holomorphic superfunction to U . Thus its restriction to U ∩ H is annihilated by the associated radial operatorḊ U ∩H . By Theorem 2.11 the operator J −1 (F ℓ + Q(F 1 , . . . F ℓ−1 )J annihilates this restriction on U ∩ H and the desired result follows by the identity principle.

Appendix: local representation of radial operators
Here, referring to [Ka1,Ka2] for details, we outline a proof of a local version of Theorem 2.5. We do this in the setting of complex Lie supergroups. Let us begin with the preparation which is required to state the result.
Recall that if G is a reductive complex Lie group with maximal complex torus H, then h ∈ H is by definition regular if its centralizer in G is just H itself. The set of regular elements is open and dense in H and every h 0 ∈ H reg has an open neighborhood A in H such that G.A = {ghg −1 ; h ∈ A, g ∈ G} is G-equivariantly a product A × G/H. A local version of this which can be applied, e.g., in situations where G is only acting locally on a neighborhood of h 0 in G, can be stated as follows: There is an open neighborhood B of the identity in the submanifold of G which is defined to be the product of all H-root spaces so that B.A is an open neighborhood of h 0 which is (locally) B-equivariantly the product A × B.
It is shown in [Ka2] that this local product decomposition holds for Lie supergroups (G, F) of type I which are equipped with a nondegenerate invariant bilinear form b : Der(F) × Der(F) → F. Recall that "type I" means that Cartan algebras are even, i.e., are contained in g 0 and all root spaces are 1-dimensional. The complex manifolds A and B are defined exactly as above. The complex subsupermanifold structure on A is even, but nevertheless we denote it by A to emphasize that it is a subsupermanifold. The subsupermanifold structure on B is not even. It is analogously denoted by B.
Theorem 2.13. For h 0 a regular element of H there is an open (supermanifold) neighborhood B.A which is locally B-equivariantly isomorphic to the product A×B.
In this context a superfunction is radial if it is annihilated by all derivations in Der (B) which, using the product structure at hand, is regarded as being contained in the space of derivations on the product neighborhood A × B. A (super) differential operator D on this neighborhood is said to be radial if it maps radial functions to radial functions.
Using the product structure, one extends a holomorphic superfunction on A, which is by definition a numerical functions, to a unique holomorphic radial superfunction on A × B. Applying a radial operator D to such a function, we again obtain a radial function which is uniquely determined by a function on A. Thus we obtain a classical differential operatorḊ : O(A) → O(A) on numerical functions.
The main source of radial operators is the center of the universal enveloping algebra. The resulting operators on A × B are called Laplace-Casimir operators. The local version of Theorem 2.5 which is proved in detail in [Ka1], Chapter 4, is formulated below. As in Theorem 2.5 this requires not only type I and the existence of the supersymmetric invariant form, but also that the function J is an eigenfunction of a second order Laplacian which is constructed in the proof. Thus, to simplify the formulation, we have only stated it for the cases of relevance for the applications in [HPZ] and [CFZ] where these assumptions are satisfied.
Theorem 2.14. If g is either gl(m, n) or osp(2m, 2n), then for every Laplace-Casimir operator D on A × B there is a uniquely determined polynomial constant coefficient operator P on A so thatḊ = J −1 P J.
Here J is the same globally defined function as in Theorem 2.5.
Let us conclude this appendix by commenting on the proof of this local result. As in the classical case the main point is to define a Laplace operator which in end effect is a second order Laplace-Casimir operator and whose radial part is (by an explicit computation) of the form in the theorem. Using the invariant form b(x, y) = STr(xy), this operator is constructed as follows.
Fixing a splitting, one obtains a deRham complex of holomoprhic differential forms in ∧(T * 0 M ⊕ ΠT * 1 M ), where here M is the supermanifold A × B of dimension k|ℓ. The operation Π exchanges the even and odd parts of a graded vector space. From the point of view of integration theory the correct "top-dimensional" form is a nowhere vanishing section ω M (a "Berezinian") of ∧ k+ℓ (T * 0 M ⊕ ΠT 1 M ). Using b this is transformed to a top-dimensional holomorphic "volume form" ω M in ∧ k+ℓ (T * 0 M ⊕ ΠT * 1 M ). Again using b to define contraction of ω M with 1-forms to obtain a * -operation so that ∂ * ∂f is a multiple hω M and then defining * from topdimensional forms back to functions, one obtains a Laplace operator L M := * ∂ * ∂ on superfunctions (see [Ka1], 4.1.1-4.1.3).
In order to define L M one only needs a Lie supergroup of type I with a nondegenerate supersymmetric bilinear form. It is a radial operator and a formula for its radial part can be explicitly computed ( [Ka1], Lemma 4.4). In order to obtain a formula of the type in Theorem 2.14 the following nontrivial result is needed. Here J is the function defined above which is qualitatively described as the square-root of the superdeterminant of the Jacobian of the mapping which identifies A × B with an open subset of the Lie supergroup (G, F).
Theorem 2.15. If g is either osp(2m, 2n) or gl(m, n), then J is an eigenfunction of L M .
One way of seeing this is to first show that L M is (up to a constant multiple) in fact the operator defined by the second order element of the center of the universal enveloping algebra which is defined in the usual way by b ( [Ka1], Lemmas 4.7 and 4.8). Then one computes explicitly to show that J is indeed an eigenfunction. In general such computations are quite involved ( [B] Theorems 4.1 and 4.4), but in certain cases which are relevant for applications they are simple (see [HPZ], §4.3) Due to the fact that J is an eigenfunction one can express the radial part of the second order Laplacian as in Theorem 2.14. The general theorem follows immediately in the classical way using in addition the fact that the given Laplace-Casimir operator commutes with L M ([Ka1], Theorem 4.1).