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Semiinfinite symmetric powers

We develop a theory of measures, differential forms and Fourier tramsforms on some infinite-dimensional real vector spaces by generalizing the following two constructions: (a) The construction of the semiinfinite wedge power of a Tate vector space V. Recall that it is obtained as a certain double inductive limit of the exterior algebras of finite-dimensional subquotients of V. (b) The construction of the space of measures on a nonarchimedean local field K with maximal ideal M as a double projective limit of the spaces of measures (=functions) on finite subquotients M^i/M^j of K.

(a) The construction of the semiinfinite wedge power of a space V equipped with a class of commensurable subspaces [ADK] [KP] [S]. Recall that it is obtained as a certain double direct limit of the exterior algebras of finite-dimensional subquotients of V .
(b) The construction of the space of measures on a nonarchimedean local field K with maximal ideal m as a double inverse limit of the spaces of measures (which in this case are the same as functions) on finite subquotients m i /m j of K.
The importance of generalizing the construction (b) was emphasized by A.N. Parshin [P2-3]. Among other things, he pointed out that all the 4 types of possible combinations of inductive and projective limits of the spaces of measures on m i /m j have very transparent analytic meaning (Schwartz-Bruhat functions vs. distributions, compact support vs. arbitrary support) and called for a generalization of this approach to higher local fields. This note can be seen as step in that direction, treating 2-dimensional local fields such as R((t)), C((t)). More general 2-dimensional local fields and adeles (see [P1] for background) will be considered in a subsequent paper.
(0.2) Let us describe our constructions more precisely. Our setting is exactly the same as the one needed in the theory of the "Japanese group GL(∞)" and the semiinfinite Grassmannians which encompasses (a). Though this is not universally known, the relevant class of structured infinite-dimensional spaces was introduced in the 1942 book of S. Lefschetz [L] under the name of locally linearly compact spaces. We adopt this framework which predates by 40 years the relatively recent interest in "semiinfinite" structures.
For a locally linearly compact R-vector space V we introduce the space M (V ) of "smooth measures" on V as a double projective limit of the spaces of smooth measures on the finite-dimensional subquotients U 1 /U 2 where U i run over open, linearly compact subspaces of V . So the procedure is quite similar to (b). Here, however, one encounters the difficulty similar to that familiar in (a), namely that smooth measures can be restricted onto a subspace only after tensoring with some 1-dimensional space of Haar measures, and this leads to the fact that M (V ) will be only a projective representation of the group GL(V ) of continuous automorphisms of V . This can be pinned down more precisely by saying that we have a version M h (V ) of M (V ) for each "Haar theory" h on V , an object of a certain gerbe, see (2.3).
The category L of locally linearly compact spaces possesses a perfect duality V → V ∨ (the Lefschetz-Chevalley generalization of Pontryagin duality) and we construct a Fourier transform from M h (V ) to M h ∨ (V ∨ ) where h ∨ is a Haar theory on V ∨ naturally associated to h.
Further, we consider the de Rham complexes of forms on the subquotients U 1 /U 2 as before, and by taking their double projective limit we construct the semiinfinite de Rham complex Ω • (V ). Its "anomaly" (i.e., the central extension of GL(V ) arising from the projective action) is much simpler than that of M (V ): it reduces to a Z/2-central extension coming from the orientation, and in the case of C-spaces vanishes altogether. The complex Ω • (V ) is similar to Ω ch , the chiral de Rham complex of Malikov, Schechtman and Vaintrob [MSV]. A treatment of Ω ch itself from a similar standpoint will be given in the forthcoming paper [KV]. (0. 3) The construction of (0.1)(a) can be included in our framework as a particular case, if we extend it to encompass super-vector spaces, see [M]. In particular, for a purely odd finite-dimensional super-vector space W the space of measures is the exterior algebra of W , and the integration is understood in the sense of Berezin. If we apply our approach to finite-dimensional subqotients of a purely odd locally linearly compact supervector space V , we get precisely the semiinfinite wedge space. Therefore it is natural to call our space M (V ) (for any V ) the semiinfinite symmetric power of V . Unless we are in the purely odd case, it is of a pronounced analytic flavor. If V is equipped with a positive definite quadratic form q, one can specify a more algebraic subspace AG(V, q) ⊂ M (V ) invariant under the orthogonal group O(q) by considering measures which are almost Gaussian, i.e., are product of a polynomial and a Gaussian measure.
Another natural candidate for a "semiinfinite symmetric power" of V would be an irreducible module over the Heisenberg algebra of V ⊕ V ∨ . For example any open linearly compact subspace U ⊂ V gives a "vacuum module". But unless V is purely odd (so that we have a Clifford algebra) these vacuum modules are not isomorphic to each other, so there is no preferred one. We show that our M (V ) (or, rather, the dual space D(V ) formed by distributions) contains naturally all such vacuum modules.
(0.4) It is perhaps worth emphasizing the difference of our approach with the more traditional one of probability theory and cylindric measures, cf. e.g., [K]. Namely, the latter amounts to viewing an infinite-dimensional (say linear) space V as a pro-object, a certain completion of a projective limit of finite-dimensional spaces W i . The approach of cylindric measures is (in modern language) an elaboration of the naive idea of formally putting where Meas stands for the space of measures. What we do is similar, except that we consider our spaces not as pro-objects but rather as ind-pro-objects which is the natural structure present on local fields.
(0.5) This note was finished when the author was staying at the Max-Planck-Institut für Mathematik in summer 2001. The author's research was also partially supported by grants from NSF and NSERC. §1. Locally linearly compact spaces (1.1) Definitions. We recall some classical concepts, due to Lefschetz and Chevalley [L]. Let k be a field, considered with discrete topology. A topological k-vector space is called linearly topological if it has a basis of neighborhoods of 0 formed by linear subspaces. All linearly topological vector spaces will be assumed Hausdorff. A linearly topological vector space V is called linearly compact, if any family A i ⊂ V , i ∈ I of closed affine subspaces such that i∈J A i = ∅ for any finite J ⊂ I, has i∈I A i = ∅. More generally, one says that V is locally linearly compact, if it has a basis of neighborhoods of 0 formed by linearly compact open (automatically closed) subspaces. We will drop "linearly topological" when speaking about (locally) linearly compact spaces.
(1.1.1) Examples. (a) Any finite-dimensional k-vector space with discrete topology is linearly compact. The space k [[t]] of formal Taylor series with the t-adic topology is linearly compact. In fact, any product of linearly compact spaces is linearly compact.
(b) The space k((t)) of formal Laurent series is locally linearly compact.
(1.1.2) Remark. From the modern point of view, all the above categories can be obtained from Vect 0 (k) by the purely algebraic constructions of passing to the categories of ind-and pro-objects, see, e.g., [AM]. More precisely, in the notations of loc. cit. Vect(k) = Ind(Vect 0 (k)) and Vect(k) = Pro(Vect 0 (k)). An embedding in Vect(k) is open iff it is induced (in the sense of forming a Cartesian square) from an embedding in Vect 0 (k). Finally, L(k) is identified with the full subcategory in Ind(Pro(Vect 0 (k))) formed by inductive systems over Pro(Vect 0 (k)) consisting of open (in the above sense) embeddings. General definitions of locally compact indpro-objects (with Vect 0 (k) replaced by more general categories) were proposed by Beilinson [Be] and earlier by Kato [Ka]. We tried to keep this paper as short and elementary as possible and thus avoided any systematic use of ind/pro-objects.
(1.2) Dim and Det. We start by reformulating, in the form needed for us, the basic constructions of [ADK]. Let V be a locally linearly compact k-vector space. We denote by G(V ) and call the semiinfinite Grassmannian of V , the set of open linearly compact subspaces in V . If U 1 , U 2 ∈ G(V ) and U 1 ⊂ U 2 , then (U 2 /U 1 ) is finite-dimensional. Moreover, for arbitrary U 1 , U 2 ∈ G(V ) both U 1 ∩U 2 and U 1 +U 2 are in G(V ). By a dimension theory on V we mean a map d : G(V ) → Z such that, whenever U 1 , U 2 ∈ G(V ) and U 1 ⊂ U 2 , we have d(U 2 ) = d(U 1 ) + dim(U 2 /U 1 ). The set of dimension theories will be denoted Dim(V ). The group Z acts on Dim(V ) by adding constant functions. It is clear from the above that this makes Dim(V ) into a Z-torsor.
For W ∈ Vect 0 (k) let det(W ) be the top exterior power of W . Recall that for any short exact sequence and these identifications are associative for any filtration W 1 ⊂ W 2 ⊂ W of length 2.
(1.2.2) Definition. Let V be a locally linearly compact k-vector space. A determinantal theory on V is a rule ∆ which associates to each U ∈ G(V ) a 1-dimensional k-vector space ∆(U ), to each embedded pair U 1 ⊂ U 2 , U i ∈ G(V ), an isomorphism so that for any nested triple U 1 ⊂ U 2 ⊂ U 3 the obvious diagram is commutative.
We denote by Det(V ) the category (groupoid) formed by all determinantal theories on V and their isomorphisms (in the obvious sense). If φ : ∆ → ∆ ′ is an isomorphism of determinantal theories and λ ∈ k * , then λφ is also an isomorphism. One easily sees that: (1.2.3) Proposition. The above action of k * on the morphisms makes Det(V ) into a k * -gerbe [Bre] [Bry], i.e., each Hom Det(V ) (∆, ∆ ′ ) becomes a k * -torsor and the composition is bilinear.
(1.2.4) Example. To compare our approach with that of [ADK], note that any U ∈ G(V ) defines a unique dimension theory d U such that d U (U ) = 0. The difference (in the Z-torsor Dim(V )) of two such elements d U , d U ′ is the integer denoted in [ADK] by [U ′ : U ] (the relative dimension of U and U ′ ).
Similarly, a choice of U gives rise to a canonical determinantal theory ∆ U such that ∆ U (U ) = k and the values of ∆ U on other elements of G(V ) is recovered uniquely (that is, up to unique isomorphism) from the axioms of Definition 1.2.2. The Hom-torsor (in the k * -gerbe Det(V )) between two such theories ∆ U , ∆ U ′ is the k * -torsor corresponding to the 1-dimensional vector space denoted in [ADK] by [U ′ |U ] (the relative determinant of U ′ and U ). (1. 3) The group GL(∞). Let V be a locally linearly compact k-vector space. We write GL(V ) for Aut L(k) (V ), i.e., for the group of continuous automorphisms of V . When V = k((t)), this is the so-called "Japanese group GL(∞)".
The group GL(V ) acts on the k * -gerbe Det(V ) and any object ∆ ∈ Det(V ) gives, in a standard way, a central extension More precisely, see, e.g., [Bry], an element of . Let us recall its standard properties. (1.3.3) Examples. (a) A choice of a reference subspace U ∈ G(V ) produces an object ∆ U ∈ Det(V ) and hence a particular central extension. ( is the standard central extension of the Japanese group [ADK]. (c) If V is discrete, than 0 ∈ G(V ) so ∆ 0 is an equivariant object of Det(V ) and the extension splits. Similarly, if V is linearly compact, then V ∈ G(V ) and the extension splits.
(1.4) Further properties of Dim and Det. The category L(k) possesses the following extra structures: (1.4.1) Duality, which is an antiequivalence V → V ∨ = Hom(V, k) (the space of continuous functionals). The space V ∨ is again locally linearly compact and G(V ) and G(V ∨ ) are in order-reversing bijection U → U ⊥ (the orthogonal complement).
(1.4.2) The structure of an exact category in the sense of Quillen [Q], i.e., a class of admissible short exact sequences More precisely, one calls a sequence (1.4.3) admissible, if it is exact as a sequence of algebraic vector spaces, if V ′ is closed in V and the topology on V ′′ is the quotient one. These structures are compatible in the obvious sense: the dual of an admissible short exact sequence is again admissible. As in [Q], we will also speak about admissible filtrations etc.
On the other hand, for an abelian group A the category of A-torsors is a symmetric monoidal category with duality, see [Bre]. The monoidal operation (tensor product of torsors over A) will be denoted ⊗ for A = k * and ⊙ for A = Z. The dual of a torsor T will be denoted T ∨ = Hom(T, A). For t ∈ T let t ∨ ∈ T ∨ be the unique morphism taking t to the unit element of A. ( and these identifications are associative in any admissible filtration of length 2. Proof: (a) For a dimension theory d on V we have a dimension theory We leave the details to the reader.
For any k * -gerbes G ′ , G ′′ we denote G ′ ⊠ G ′′ the gerbe whose class of objects is Ob(G ′ ) × Ob(G ′′ ) and We also denote by G op the gerbe opposite to G. We think of it as having for objects formal symbols and natural transformation of "associativity" for these equivalences for any admissible filtration of V 1 ⊂ V 2 ⊂ V of length 2: The transformations ε V 1 ⊂V 2 ⊂V fit into a commutative cube for any admissible length 3 filtration.
The (somewhat lengthy) checking of details as well as the construction of the ε V 1 ⊂V 2 ⊂V and verification of their properties, are left to the reader. §2. Measures on locally linearly compact spaces (2.0) Orientation issues. From now on we take k = R and write Vect 0 for Vect 0 (R), as well as L for L(R) etc. Recall that 1-dimensional R-vector spaces are essentially the same as R * -torsors. If L is such a space, we denote by |L| the 1-dimensional R-vector space whose corresponding torsor is induced from that of L by the homomorphism Alternatively, |L| can be identified with the space of functions s : For any W ∈ Vect 0 we have a canonical identification which expresses the fact that a volume form in the presense of an orientation gives a measure.
(2.1) The finite-dimensional case. Let W ∈ Vect 0 be a finite-dimensional R-vector space. We introduce the following function spaces: S(W ): the Schwartz space of smooth rapidly decreasing functions W → R, see [H]. If β : W → W ′′ is a surjection in Vect 0 , we have the direct image map (integration along the fibers): If α : W ′ → W is an injection in Vect 0 , then the restriction of functions induces a map The following is then straightforward.
(2.2) Linearly compact case. Let now U be a linearly compact space. We define the space of measures on U as where U 1 runs over open subspaces of U (so that U/U 1 ∈ Vect 0 ) and the limit is taken with respect to the maps (2.1.1) associated to the surjections U/U 1 → U/U 2 , U 2 ⊂ U 1 . In a dual fashion, we define the space of distributions on U to be where limit is now taken with respect to the maps obtained by tensoring (2.1.5). Let now α : U ′ → U be an open embedding of linearly compact spaces. For an open U 1 ⊂ U let U ′ 1 = α −1 (U ). Then we have an embedding α 1 : Cartesian, so satisfies the assumptions of Proposition 2.1.3(c). We get, therefore: (2.3) Locally linearly compact case. Let now V be a locally linearly compact R-vector space and G(V ) the set of its open linearly compact subspaces. By a Haar theory on V we will understand a rule h which to each U ∈ G(V ) associates a 1-dimensional R-vector space h(U ) and to each pair U 1 ⊂ U 2 an isomorphism h(U 1 ) ⊗ | det(U 2 /U 1 ) * | → h(U 2 ) satisfying a condition for nested triples similar to one given in Definition 1.2.2. It is clear that Haar theories form an R * -gerb, in particular, any such theory h gives rise to a central extension GL h (V ) of GL(V ) by R * . It is also clear that each determinantal theory ∆ on V gives a Haar theory |∆ * |, and GL |∆| * (V ) is just the extension induced from GL ∆ (V ) via the automorphism x → |x| −1 of R * .
Fix a Haar theory h on V . We define the spaces of measures and distributions on V associated to h to be It can be viewed as the algebra of polynomial differential operators on V . For any linearly compact open subspace U ⊂ V we denote by N U the vacuum representation of H(V ) corresponding to U . By definition, it is generated by one vector |U ("vacuum") subject to the relations: We can view it as the space of distributions on V which have support in U and which are smooth along U . For different U these modules are not isomorphic to each other. In particular, the group GL(V ) does not act, even projectively, in any of the N U (even though its Lie algebra does, via is embedding into a completion of H(V ) by vertex operators [KR]).
(2.4.3) Proposition. N U is naturally embeddeded into the space D |∆ U | * (V ) as an H(V )-submodule. Here ∆ U is the determinantal theory associated to U, see (1.2.4).
Proof: We exhibit an element δ U ("delta function along U ") in D |∆ U | * (V ) satisfying the relations (2.4.2). Then the embedding would be uniquely determined by sending Consider the term of this double inductive system corresponding to U 1 = U 2 = U . This term is canonically identified with R and we define δ U as the image of 1 ∈ R in the inductive limit. The relations (2.4.2) are verified straightforwardly. Q.E.D.
If h is some other Haar theory, then we have not a canonical embedding of N U into D h (V ) but a canonical 1-dimensional space formed by such embeddings.
(2.5) Bilinear and quadratic forms. Let V ∈ L be a locally linearly compact R-space. We want to compare two possible approaches to defining quadratic forms on V . First of all, by a bilinear form on V we will mean a morphism b : V → V ∨ , where V ∨ ∈ L is the topological dual of V . As usual, for such a b we have the transposed form b t : V → V * and b is called symmetric if b t = b. We denote by B(V ) the space of all symmetric bilinear forms on V . An element b ∈ B(V ) can be regarded as a continuous function b : V × V → R and we denote by Consider now the case of finite-dimensional vector spaces W ∈ Vect 0 . In this case the above concepts have their usual meaning. Let α : W ′ → W be an injection in Vect 0 and b ∈ B(W ). Then we have the restriction α * b ∈ B(W ′ ). Note that for a nondegenerate b the form α * b may be degenerate, but if b is positive definite, then so is α * b (and, in particular, α * b is nondegenerate). Further, let β : W → W ′′ be a surjection in Vect 0 and b be a positive definite symmetric bilinear form on W . Then we define the direct image β * (b) ∈ B(W ′′ ) as follows. Consider the injection β t : (W ′′ ) * → W * . dual to β and define Here we used the nondegeneracy of the positive definite forms b and (β t ) * (b −1 ). The following is then elementary.
(2.5.2) Proposition. Let q b be the quadratic form corresponding to b and similarly for β * (b). Then for w ′′ ∈ W ′′ we have (2.5.3) Corollary. (a) The operations α * and β * on positive definitite symmetric bilinear forms (on finite-dimensional spaces) is compatible with the composition of injections (resp. surjections).
Let now V ∈ L be a locally linearly compact R-space. We define where the limits are taken with respect to the operations α * and β * on quadratic forms. Elements of Q(V ) will be called (positive-definite) quadratic forms on V .
Proof: Let b be a symmetric bilinear form on V . The topology in V is that of inductive limit of its linearly compact open subspaces U ⊂ V . Further, the basis of linearly compact neighborhoods of 0 in V ∨ is formed by the orthogonals U ⊥ where U is a linearly compact neighborhood of 0 in V . Accordingly, Note that in the above we can take U 2 ⊂ U 1 , and in this case a morphism from U 1 to U ⊥ 2 gives (after restriction to U 1 ) a bilinear form b U 1 /U 2 on the finite-dimensional space U 1 /U 2 . Assuming that b is nondegenerate and positive definite, we find that these forms on finite-dimensional subquotients are positive definite and compatible with respect to the restrictions and projections. In other words, the system of their associated quadratic forms is an element of the double inverse limit (2.5.4). The converse is similar.
(2.6) Gaussian measures. Let W ∈ Vect 0 . Any q ∈ Q(W ). q ∈ Q(W ) gives rise to a Haar measure dVol q ∈ | det(W * )| (the measure of a q-orthocube is 1). We get therefore the Gaussian measure (2.6.1) Recall the standard properties of Gaussian integrals.
Let now V ∈ L and q ∈ Q(V ). Note that q trivializes the space | det(U 2 /U 1 ) * | for any open, linearly comapct U 2 ⊂ U 2 ⊂ V . This is because U 2 /U 1 is identified with the q-orthogonal complement of U 1 in U 2 and the latter comes equipped with the Haar measure dVol q . If U 1 ⊂ U 2 ⊂ U 3 ⊂ V , then these trivializations are compatible with the exact sequence It follows that for any two open, linearly compact U 1 , U 2 ⊂ V the Haar theories |∆ U 1 | * , |∆ U 2 | * are canonically identified. Further, Proposition 2.6.2 implies the following fact.
(2.6.6) Definition. Let V ∈ L and q ∈ Q(V ). We define AG(V, q), the space of almost Gaussian measures on V (with respect to q) to be where q U/U ′ is the quadratic form induced by q on U/U ′ , see (2.5.4) and the limits are taken with respect to the maps (2.6.4-5).
By construction, AG(V, q) is a representation of the group O(q). It can be regarded as a kind of "algebraic semiinfinite symmetric power" of V . Let now V ∈ L and ∆ be a determinantal theory on V . Let also h ∨ be the dual determinantal theory on V ∨ , defined as in (1.4.5). Then, the maps (2.7.1) for finite-dimensional subquotients of V are assembled together into an isomorphism which we will also call the Fourier transform. The following is an immediate consequence of the standard properties of the Fourier transform. (b) If q ∈ Q(V ) and q −1 ∈ Q(V ∨ ) corresponds to the inverse bilinear form, then (c) In the situation of (b), the Fourier transform takes the space AG(V, q) ⊗ C into AG(V ∨ , q −1 ) ⊗ C. §3. Forms on locally linearly compact spaces.
(3.1) The finite-dimensional case. Let W ∈ Vect 0 be a finite-dimensional R-vector space. Let Ω • (W ) be the de Rham complex of differential forms on W whose components (with respect to some linear coordinate system) are Schwartz functions. We denote Then for each surjection β : W → W ′′ in Vect 0 we have the direct image map (integration of forms along the fibers in the presense of orientation) Here [d] means the shift of grading by d. Similarly, for an injection α : W ′ → W we have the restriction map (3.1.4) Remark. If W is a finite-dimensional C-vector space (considered as an R-vector space) then OR(W ) is canonically trivialized. Accordingly, for surjections or injections in Vect 0 (C) we have functorialities β * , α * sithout any twist (but with a shift of grading for β * ).
(3.1.5) Proposition. (a) Each β * and α * is a morphism of complexes. (b) The direct and inverse images of forms commute with compositions of surjections (resp. injections).
( 3.2) The locally linearly compact case. Let now V ∈ L and G(V ), as in (1.2), denote the set of open linearly compact subspaces in V . We call an orientation theory on V a rule O which to any U ∈ G(V ) associates a {±1}-torsor O(U ) and to any pair U 1 ⊂ U 2 an isomorphism U (U 1 ) ⊗ OR(U 2 /U 1 ) → O(U 2 ) so that for any nested triple U 1 ⊂ U 2 ⊂ U 3 the diagram analogous to (1.2.2) commutes. Every determinantal theory ∆ on V defines an orientation theory OR(∆) via the change of structure groups given by sgn : R * → {±1}.
(3.2.1) Example. Let V ∈ L(C) be a locally linearly compact C-space and V R ∈ L be V considered as an R-space. Then any U ∈ G(V R ) contains a C-subspace U ′ ∈ G(V ) so that dim R (U/U ′ ) < ∞. We set C(U ) = OR(U/U ′ ). A different'choice of U ′ leads to a canonically isomorphic {±1}-torsor because a finite-dimensional Cspace has a canonical orientation. It is clear that C : U → C(U ) is an orientation theory on V R .
(3.2.2) Definition. Let V ∈ L and O be an orientation theory on V . The (semiinfinite) de Rham complex of V associated to O is It is graded by the Z-torsor Dim(V ).
If V ∈ L(C) and O = C is the canonical orientation theory on V R then we can write the above more simply and denote it by where U, U 1 now run over open, linearly compact C-subspaces of V . This complex can be seen as an analog of the chiral de Rham complex of Malikov, Schechtman and Vaintrob [MSV]. §4. Measures on locally linearly compact superspaces and complexes (4.1) Superspaces and Berezin integration. Let k be a field. By SVect 0 (k) we denote the category of finite-dimensional k-supervector spaces [M]. Thus an object of SVect 0 (k) is a finite-dimensional k-vector space W together with a Z/2-grading W = W 0 ⊕W 1 ; here we denote elements of Z/2 by 0, 1. By Π we denote the functor of shift of Z/2-grading; (ΠW ) i = W i+1 . Similarly, we denote by SL(k) the category of locally linearly compact k-supervector spaces V = V 0 ⊕ V 1 . It can be identified with the category of locally compact indpro-objects (in the sense of [Be] [K]) of SVect 0 (k).
From now on we take k = R and write SVect 0 , SL etc. Let W ∈ SVect 0 . According to the general principles of super-analysis [M] we define S(W ), the Schwartz space of W as where S(W 0 ) is the usual Schwartz space and Λ(W 1 * ) is the exterior algebra. similarly, the space of distributions on W is defined as The 1-dimensional R-space of Haar measures on W is defined to be This is justified by the existence of the integration map (4.1.4) which on S(W 0 ) ⊗ | det(W 0 ) * | is the usual integration over W 0 and on Λ(W 1 * ) ⊗ det(W 1 ) is the "Berezin integration" [M] which is, algebraically, just the projection which satisfy the properties similar to those listed in Proposition 2.1.3.
(4.2) Measure on locally linearly compact superspaces. Let now V = V 0 ⊕V 1 ∈ SL. We denote by G(V ) the set of open linearly compact sub-superspaces . The concept of a Haar theory from (2.3) generalizes trivially to the case of a superspace such as V . Namely, a Haar theory on V is a rule h which associates to any U ∈ G(V ) a 1-dimensional R-vector space h(U ) and to any pair U 1 ⊂ U 2 an isomorphism h(U 1 ) ⊗ µ(U 2 /U 1 ) → h(U 2 ) such that the properties of Definition 1.2.2 hold. Given a Haar theory h on V , we define the spaces of measures and distributions on V , similarly to (2.3), as where the limits are taken with respect to the maps (4.1.6-7) and similar maps on distributions.
(4.2.3) Example. Let V = V 1 be a purely odd superspace and V = ΠV be the same space but considered as an even one. Then a Haar theory h on V is the same as a determinantal theory ∆ on V . It follows that the space of h-distributions on V has the form In other words, this is the semiinfinite exterior power of V corresponding to the determinantal theory ∆, see [ADK] [KP] [S]. The space M h (V ) is just the dual of Λ ∞ 2 +• ∆ (V ) (being a double projective limit). Note that Λ ∞ 2 +• ∆ (V ) is graded by the Z-torsor Dim(V ).
Therefore it is natural to think about D h (V ) (even for a purely even V ) as a "semiinfinite symmetric power" of V .
(4.3) Measures on complexes. Let V • be a bounded admissible complex over the exact category L = L(R). By assembling together terms of the same parity, we associate to V • a superspace V sup ∈ SL with (4.3.1) V j , I ∈ Z/2.
A Haar theory on V • is, by definition, a Haar theory on V sup . For example, given determinantal theories ∆ i on V i for each i, we have a Haar theory Let h be a Haar theory on V • . We define the spaces of measures and distributions on V • to be those on V sup . These spaces are naturally graded by the Z/2-torsor