EQUIVARIANT CHARACTERISTIC CLASSES OF EXTERNAL AND SYMMETRIC PRODUCTS OF VARIETIES

Transcription

1 EQUIVARIANT CHARACTERISTIC CLASSES OF EXTERNAL AND SYMMETRIC PRODUCTS OF VARIETIES LAURENȚIU MAXIM AND JÖRG SCHÜRMANN Abstact. We obtain efined geneating seies fomulae fo equivaiant chaacteistic classes of extenal and symmetic poducts of singula complex quasi-pojective vaieties. Moe concetely, we study equivaiant vesions of Todd, Chen and Hizebuch classes fo singula spaces, with values in delocalized Boel-Mooe homology of extenal and symmetic poducts. As a bypoduct, we ecove ou pevious chaacteistic class fomulae fo symmetic poducts and obtain new equivaiant genealizations of these esults, in paticula also in the context of twisting by epesentations of the symmetic goup. Contents 1. Intoduction Equivaiant chaacteistic classes Geneating seies fomulae Twisting by Σ n -epesentations 8 2. Delocalized equivaiant theoies Compatibilities with coss-poduct Geneating seies fo symmetic goup actions on extenal poducts Examples (Equivaiant) Pontjagin ings fo symmetic poducts Example: Contuctible functions and Obifold Chen classes Geneating seies fo (equivaiant) chaacteistic classes Chaacteistic classes of Lefschetz type Delocalized equivaiant chaacteistic classes Geneating seies fo (equivaiant) chaacteistic classes 28 Refeences Intoduction In this pape, we obtain efined geneating seies fomulae fo equivaiant chaacteistic classes of extenal and symmetic poducts of singula complex quasi-pojective vaieties, genealizing ou pevious esults fo symmetic poducts fom [9]. Date: Febuay 24, Mathematics Subject Classification. 55S15, 20C30, 57R20. Key wods and phases. chaacteistic classes, obifold classes, Hizebuch- and Lefschetz-Riemann-Roch, extenal and symmetic poducts of vaieties, geneating seies, epesentations of symmetic goups. 1

2 1.1. Equivaiant chaacteistic classes. All spaces in this pape ae assumed to be complex quasipojective, though many constuctions also apply to othe categoies of spaces with a finite goup action (e.g., compact complex analytic manifolds o vaieties ove any base field of chaacteistic zeo). Fo such a vaiety X, conside an algebaic action G X X by a finite goup G, with quotient map π : X X := X/G. Fo any g G, we let X g denote the coesponding fixed point set. We let cat G (X) be a categoy of G-equivaiant objects on X in the undelying categoy cat(x) (e.g., see [8, 20]), which in this pape efes to one of the following examples: coheent sheaves Coh(X ), algebaically constuctible sheaves of complex vecto spaces Const(X ), and (algebaic) mixed Hodge modules MHM (X) on X. We denote by K 0 (cat G (X)) the coesponding Gothendieck goups of these Q-linea abelian categoies. We will also wok with the elative Gothendieck goup K0 G (va/x) of G-equivaiant quasi-pojective vaieties ove X, defined by using the scisso elation as in [8]. Let H (X) denote the even degee Boel-Mooe homology Hev BM (X) R with coefficients in a commutative C-algeba R (esp. Q-algeba if G is a symmetic goup). Note that H ( ) is functoial fo all pope maps, with a compatible coss-poduct. Let cl ( ; g) : K 0 (cat G (X)) H (X g ) be one of the following equivaiant chaacteistic class tansfomation of Lefschetz type, see Section 5.1: (1) the Lefschetz-Riemann-Roch tansfomation of Baum-Fulton-Quat [4] and Moonen [23]: td ( ; g) : K 0 (Coh G (X )) H (X g ), with R = C (esp. R = Q if G is a symmetic goup). (2) the localized Chen class tansfomation [28]: c ( ; g) : K 0 (Const G (X )) H (X g ), with R = C (esp. R = Q if G is a symmetic goup). (3) the motivic vesion of the (un-nomalized) Atiyah-Singe class tansfomation [8]: T y ( ; g) : K G 0 (va/x) H (X g ), with R = C[y] (esp. R = Q[y] if G is a symmetic goup). (4) the mixed Hodge module vesion of the (un-nomalized) Atiyah-Singe class tansfomation [8]: T y ( ; g) : K 0 (MHM G (X)) H (X g ), with R = C[y ±1 ] (esp. R = Q[y ±1 ] if G is a symmetic goup). These class tansfomations ae covaiant functoial fo G-equivaiant pope maps and coss-poducts. Ove a point space, they educe to a cetain g-tace (as explained in Section 5.1). Fo a subgoup K of G, with g K, these tansfomations cl ( ; g) of Lefschetz type commute with the obvious estiction functo Res G K. Moeove, cl ( ; g) depends only on the action of the cyclic subgoup geneated by g. In paticula, if g = id G is the identity of G, we can take K the identity subgoup {id G } with Res G K the fogetful functo Fo : K 0 (cat G (X)) K 0 (cat(x)), so that cl ( ; id G ) = cl ( ) fits with a coesponding non-equivaiant chaacteistic class, which in the above examples ae: 2

3 (1) the Todd class tansfomation td of Baum-Fulton-MacPheson [3] appeaing in the Riemann- Roch theoem fo singula vaieties. (2) the MacPheson-Chen class tansfomation c [18]. (3) the motivic vesion of the (un-nomalized) Hizebuch class tansfomation T y of [5]. (4) the mixed Hodge module vesion of the (un-nomalized) Hizebuch class tansfomation T y of [5], see also [29]. The disjoint union g G Xg admits an induced G-action by h : X g X hgh 1, such that the canonical map i : X g X g G defined by the inclusions of fixed point sets becomes G-equivaiant. Theefoe, G acts in a natual way on g G H (X g ) by conjugation. Definition 1.1. The (delocalized) G-equivaiant homology of X is the G-invaiant subgoup (1) H G (X) := H (X g ) g G G. This theoy is functoial fo pope G-maps and induced coss-poducts. This notion is diffeent (except fo fee actions) fom the equivaiant Boel-Mooe homology HBM,2 G (X) R defined by the Boel constuction. In fact, since G is finite and R is a Q-algeba, one has (2) HBM,2 (X) G R ( H2 BM (X) R ) G H BM 2 (X/G) R, which is just a diect summand of H G (X) coesponding to the identity element of G, denoted by Hid, G (X). Fo example, if G acts tivially on X (e.g., X is a point), then H G (X) = H (X) C(G), whee C(G) denotes the fee abelian goup of Z-valued class functions on G (i.e., functions which ae constant on the conjugacy classes of G). Definition 1.2. Fo any of the Lefschetz-type chaacteistic class tansfomations cl ( ; g) consideed above, we define a coesponding G-equivaiant class tansfomation by: cl G : K 0 (cat G (X)) H G (X) cl G ( ) := cl ( ; g) H (X g ) g G g G G. The G-invaiance of the class cl G ( ) is a consequence of conjugacy invaiance of the Lefschetztype chaacteistic class cl ( ; g), see [8][Sect.5.3]. Note that the summand cl ( ; id) (H (X)) G coesponding to the identity element of G is just the non-equivaiant chaacteistic class, which fo equivaiant coefficients is invaiant unde the G-action by functoiality. Unde the identification (2), this class also agees (fo ou finite goup G) with the coesponding (naive) equivaiant chaacteistic 3

4 class defined in tems of the Boel constuction, e.g., fo cl = td, this is the equivaiant Riemann- Roch-Tansfomation of Edidin-Gaham [11]; and fo cl = c, this is the equivaiant Chen class tansfomation of Ohmoto [25, 26]. The above tansfomation cl G ( ) has the same popeties as the Lefschetz-type tansfomations cl ( ; g), e.g., functoiality fo pope push-downs, estictions to subgoups, and multiplicativity fo exteio poducts Geneating seies fomulae. Let now Z be a quasi-pojective vaiety, and denote by Z (n) := Z n /Σ n its n-th symmetic poduct (i.e., the quotient of Z n by the natual pemutation action of the symmetic goup Σ n on n elements), with π n : Z n Z (n) the natual pojection map. The standad appoach fo computing invaiants of the symmetic poducts Z (n) is to collect the espective invaiants of all symmetic poducts in a geneating seies, and then compute the latte solely in tems of invaiants of Z, e.g., see [9] and the efeences theein. In this pape, we obtain genealizations of esults of [9], fomulated in tems of equivaiant chaacteistic classes of extenal poducts and esp., symmetic poducts of vaieties. To a given object F cat(z) in a categoy as above, i.e., coheent o constuctible sheaves, o mixed Hodge modules on Z (esp., mophisms f : Y Z in the motivic context), we attach new objects as follows (see Section 5.3 fo details): (a) the Σ n -equivaiant object F n cat Σn (Z n ) on the catesian poduct Z n (e.g., f n : Y n Z n in the motivic context). (b) the Σ n -equivaiant object π n F n cat Σn (Z (n) ) on the symmetic poduct Z (n) (e.g., the Σ n -equivaiant map Y n Z (n) in the motivic context). (c) the following non-equivaiant objects in cat(z (n) ): (1) the n-th symmetic powe object F (n) := ( π n F n) Σ n on Z (n), defined by using the pojecto ( ) Σn onto the Σ n -invaiant pat (espectively, the map f (n) : Y (n) Z (n) induced by dividing out the Σ n -action in the motivic context). (2) the n-th altenating powe object F {n} := ( π n F n) sign Σ n on Z (n), defined by using the altenating pojecto ( ) sign Σn onto the sign-invaiant pat. (This constuction does not apply in the motivic context.) (3) Fo(π n F n ), obtained by fogetting the Σ n -action on π n F n cat Σn (Z (n) ) (e.g., the induced map Y n Z (n) in the motivic context). These constuctions and all of the following esults also apply to suitable bounded complexes (e.g., the constant Hodge module complex Q H Z ), see Remak 5.11 fo details. The main goal of this pape is to compute geneating seies fomulae fo the (equivaiant) chaacteistic classes of these new coefficients only in tems of the oiginal chaacteistic class cl (F). These geneating seies take values in a coesponding commutative gaded Q-algeba H Σ (Z), PH Σ (Z) and esp. PH (Z), and ae fomulated with the help of cetain opeatos which tanspot homology classes fom Z into these coesponding commutative gaded Q-algebas. In each of thee situations (a)-(c) above, these algebas of Pontjagin type and opeatos ae descibed explicitely as follows: (a) H Σ (Z) := H Σn (Z n ) t n, 4

5 (b) with ceation opeato a defined by: if σ = () is an -cycle in Σ, then a is the composition: a : H (Z) H (Z) = H ((Z ) σ ) Z Σ (σ) H Σ (Z ), whee σ = Z Σ (σ ) acts tivially on H ((Z ) σ ). Note that the diect summand H Σ id, (Z) := H Σn id, (Zn ) t n H Σ (Z) coesponding to the identity component is a subing, and the pojection of H Σ (Z) onto the subing H Σ id, (Z) kills all the ceation opeatos except a 1 = id H (Z). PH Σ (Z) := H Σn (Z (n) ) t n ( ) H (Z (n) ) C(Σ n ) t n PH (Z) Q[p i i 1], (c) with coesponding opeato p d : H (Z) H (Z () ) Q[p i, i 1], whee d := π : Z Z () is the composition of the natual pojection π : Z Z () with the diagonal embedding : Z Z. The algeba inclusion above is induced fom the Fobenius chaacte ch F : C(Σ) Q := n C(Σ n ) Q Q[p i, i 1] =: Λ Q to the gaded ing of Q-valued symmetic functions in infinitely many vaiables x m (m N), with p i := m xi m the i-th powe sum function, see [16][Ch.I,Sect.7], and PH (Z) defined as below. PH (Z) := H (Z (n) ) t n, with coesponding opeato d : H (Z) H (Z () ). Moeove, the ceation opeato a satisfies the identity: justifying the multiplication by in its definition. π a = p d, The main chaacteistic class fomula of this pape is contained in: Theoem 1.3. The following geneating seies fomula holds in the commutative gaded Q-algeba H Σ (Z) := HΣn (Z n ) t n, if cl is eithe td, c o T y : (3) cl Σn (F n ) t n = exp a (Ψ (cl (F)) t, 1 whee Ψ denotes the homological Adams opeation defined by id Ψ = 1 i 1 i if cl = c on H2i BM (Z) Q if cl = td on H BM 2i (Z) Q, and y y if cl = T y. 5

6 In paticula, by pojecting onto the identity component, we get (4) cl (F n ; id) t n = exp (t cl (F)) H Σ id, (Z). Fo the est of this Intoduction, cl denotes any of the classes td, c o T y. The poof of Theoem 1.3 is puely fomal, based on the multiplicativity and conjugacy invaiance of the Lefschetz-type claacteistic classes cl ( ; g), togethe with the following key localization fomula fom [9][Lemma 3.3, Lemma 3.6, Lemma 3.10]: cl (F ; σ ) = Ψ cl (F), unde the identication (Z ) σ Z. Fo this localization fomula in the context of Hizebuch classes, it is impotant to wok with the paamete y and the un-nomalized vesions of Hizebuch classes and thei espective equivaiant analogues, see [9]. In fact, fomula (3) is a special case of an abstact geneating seies fomula (35), which holds fo any functo H (covaiant fo isomophisms) with a compatible commutative, associative coss-poduct, with a unit 1 pt H(pt). The above-mentioned abstact fomula (35) codifies the combinatoics of the action of the symmetic goups Σ n, and it should be egaded as a fa-eaching genealization of the well-known identity of symmetic functions (e.g., see the poof of [16][(2.14)]): h n t n = exp p t, 1 with h n the n-th complete symmetic function. Othe applications of the abstact geneating seies fomula (35) in the famewok of obifold cohomology and, esp., localized K-theoy ae explained in Section 3.1. In this way, we epove and genealize some esults fom [27] and, esp., [30]. Moeove, in Section 4, we give anothe application of (35) to canonical constuctible functions and obifold-type Chen classes of symmetic poducts, epoving some esults of Ohmoto [26]. Remak 1.4. In the motivic context, the exponentiation map (5) K 0 (va/z) K0 Σn (va/zn ) t n, [f : X Z] [f n : X n Z n ] t n is well-defined as in [6], and should be egaded as an equivaiant analogue ot the (elative) Kapanov zeta function used in [9, 20]. In fact, the latte can be ecoveed fom (5) by pushing down to the symmetic poducts (esp. to a point), and taking the quotients by the Σ n -action. By pushing fomula (3) down to the symmetic poducts, we obtain by functoialy the following esult: Coollay 1.5. The following geneating seies fomula holds in the commutative gaded Q-algeba PH Σ (Z) := HΣn (Z (n) ) t n PH (Z) Q[p i, i 1]: (6) cl Σn (π n F n ) t n = exp p d (ψ (cl (F)) t. 1 This should be egaded as a chaacteistic class vesion of [12][Pop.5.4]. In paticula, if Z is pojective, then by taking degees, we get in Section 5.3 geneating seies fomulae fo the chaactes 6

7 of vitual Σ n -epesentations of H (Z n ; F n ), that is, (7) t Σn (Z n ; F n ) t n = exp p χ(h (Z, F)) t Q[p i, i 1][[t]], 1 fo F a coheent o constuctible sheaf and with χ denoting the coesponding Eule chaacteistic, espectively, (8) t Σn (Z n ; M n ) t n = exp p χ y (H (Z, M)) t Q[y ±1, p i, i 1][[t]], 1 fo M a mixed Hodge module on Z, and with χ y (H (Z, M)) the coesponding χ y -polynomial. By specializing all the p i s to the value 1 (which coesponds to the use of the pojectos ( ) Σn ), fomula (6) educes to the main esult of [9], namely: Coollay 1.6. The following geneating seies fomula holds in the Pontjagin ing PH (Z) := H (Z (n) ) t n : (9) cl (F (n) ) t n = exp d (ψ (cl (F)) t. 1 In paticula, if Z is pojective, we ecove the degee fomulae fom [9], which can now be also deived fom (7) and (8) by specializing all p i s to 1. Coollay 1.5 also has othe impotant applications. Fo example, by specializing the p i s to the value sign(σ i ) = ( 1) i 1 (which coesponds to the use of the altenating pojectos ( ) sign Σn ), fomula (6) educes to: Coollay 1.7. The following geneating seies fomula holds in the Pontjagin ing PH (Z): (10) cl (F {n} ) t n = exp d (ψ (cl (F)) ( t). 1 In paticula, if Z is pojective, we ecove special cases of the main fomulae fom [20][Co.1.5], which can now be also deived fom (7) and (8) by specializing the p i s to ( 1) i 1. Fo example, if cl = T y and F = Q H Z, we ecove the geneating seies fomula fo the degees deg(t y (Q H {n} Z )) = χ y ([Hc (B(Z, n), ɛ n )]), whee B(Z, n) Z (n) is the configuation space of unodeed n-tuples of distinct points in Z, and ɛ n is the ank-one local system on B(Z, n) coesponding to a sign epesentation of π 1 (B(Z, n)) as in [20][p.293], compae also with [13][Ex.3b] and [12][Co.5.7]. Note also that the specialization p 1 1 and p i 0 fo all i 2, coesponds to the evaluation homomophisms (fo all n N) 1 n! ev id = 1 n! ResΣn id : HΣn (Z (n) ) H (Z (n) ). Then Coollay 1.5 specializes to the following esult: 7

8 Coollay 1.8. The following exponential geneating seies fomula holds in the Pontjagin ing PH (Z): (11) cl (Fo(π n F n )) t n ( ) n! = exp t cl (F). The above coollay also follows fom fomula (4), afte a suitable enomalization of the poduct stuctue of H Σ id, (Z) in ode to make the pushfowad π := π n, : H Σ id, (Z) PH (Z) into a ing homomophism, see Section 4 fo details. In paticula, if Z is pojective, by taking degees we get exponential geneating seies fomulae fo the Eule chaacteistic and esp. χ y -polynomial of H (Z n, F ). Fo example, if cl = T y and F = M is a mixed Hodge module on Z, we get: (12) χ y (Z n, M n ) tn ( ) n! = exp t χ y (Z, M), which also follows diectly fom the Künneth fomula, e.g., see [19] Twisting by Σ n -epesentations. Additionally, fo a fixed n, one can conside the coefficient of t n in the geneating seies (3) fo the (equivaiant) chaacteistic classes of all exteio powes F n cat Σn (Z n ). Moeove, in this case, one can twist the equivaiant coefficients F n by a (finitedimensional) ational Σ n -epesentation V, and compute the coesponding equivaiant chaacteistic classes of Lefschetz-type (see Remak 5.3) (13) cl (V F n ; σ) = tace σ (V ) cl (F n ; σ), fo σ Σ n. By pushing down to the symmetic poduct Z (n) along the natual map π n : Z n Z (n), we then get by the pojection fomula the following identity in H Σn (Z (n) ) = H (Z (n) ) C(Σ n ) H (Z (n) ) Q[p i, i 1] : (14) cl Σn (π n (V F n p λ )) = χ λ (V ) (d (ψ (cl (F)))) k. z λ λ=(k 1,k 2, ) n Hee, fo a patition λ = (k 1, k 2, ) of n coesponding to a conjugacy class of an element σ Σ n (i.e., k = n), we denote by z λ := 1 k k! the ode of the stabilize of σ, by χ λ (V ) = tace σ (V ) the coesponding tace, and we set p λ := 1 pk. If Z is pojective, by taking the degee in fomula (14), we have the following chaacte fomulae genealizing (7) and (8): (i) fo F a coheent o constuctible sheaf, we get (15) t Σn (H (Z n ; V F n )) = p λ χ λ (V ) χ(h (Z; F)) l(λ), z λ λ n with χ denoting the coesponding Eule chaacteistic, and fo a patition λ = (k 1, k 2, ) of n we let l(λ) := k 1 + k 2 + be the length of λ; (ii) fo M a mixed Hodge module on Z, we get: (16) t Σn (H (Z n ; V M n p λ )) = χ λ (V ) (χ y (H (Z; M))) k, z λ λ=(k 1,k 2, ) n with χ y (H (Z, M)) the coesponding χ y -polynomial

9 Fomula (14) is a genealization of Coollay 1.5, which one gets back fo V the tivial epesentation. Futhemoe, by specializing all the p i s in equation (14) to the value 1 (which coesponds to the use of the pojectos ( ) Σn ), one obtains the following identity in H (Z (n) ): (17) cl ((π n (V F n 1 )) Σn ) = χ λ (V ) (d (ψ (cl (F))) k. z λ λ=(k 1,k 2, ) n Note that by letting V be the tivial (esp. sign) epesentation, fomula (17) educes to Coollay 1.6 (esp. Coollay 1.7). Anothe impotant special case of (17) is obtained by choosing V = Ind Σn K (tiv), the epesentation induced fom the tivial epesentation of a subgoup K of Σ n, with (π n (V F n )) Σn (Ind Σn K (tiv) π n (F n )) Σn (π n (F n )) K π ( (π (F n )) K). Hee π : Z n Z n /K and π : Z n /K Z (n) ae the pojections factoing π n. In this case, fomula (17) calculates the chaacteistic class 1 cl ((π n (F n )) K ) = π cl ((π (F n )) K ). In paticula, if Z is pojective and we conside the constant Hodge module F = Q H Z, we get at the degee level the following fomula fo the χ y -polynomial of the quotient Z n /K. (18) χ y (Z n 1 /K) = χ λ (Ind Σn K z (tiv)) χ y (Z) k. λ λ=(k 1,k 2, ) n The coesponding Eule chaacteistic fomula, obtained fo y = 1, is also a special case of Macdonald s fomula fo the coesponding Poincaé polynomial ([17][p.567]). Finally, by letting V = V µ Vµ be the (self-dual) ieducible epesentation of Σ n coesponding to a patition µ of n, the coefficients 1 (π n (V µ F n )) Σn (V µ π n (F n )) Σn =: S µ (π n F n ) of the left-hand side of (17) calculate the coesponding Schu functo of π n F n cat Σn (Z (n) ), with (19) π n F n µ n V µ S µ (π n F n ) cat Σn (Z (n) ), e.g., see Remak These Schu functos genealize the symmetic and altenating powes of F, which coespond to the tivial and esp. sign epesentation. Note that, by using (19), we get an altenative desciption of the equivaiant classes cl Σn (π n F n ) H Σn (Z (n) ) = H (Z (n) ) C(Σ n ) H (Z (n) ) Q[p i, i 1] in tems of the Schu functions s µ := ch F (V µ ) Q[p i, i 1], see [16][Ch.1, Sect.3 and Sect.7]: (20) cl Σn (π n F n ) = µ n s µ cl (S µ (π n F n )), with cl (S µ (π n F n )) computed as in (17). As a concete example, fo Z pue dimensional with coefficients given by the intesection cohomology Hodge module ICZ H on Z, the coesponding Schu functo S µ of π n ICZ H n is given by the twisted intesection cohomology Hodge module S µ (π n ICZ H n) = ICH (V Z (n) µ ) with twisted coefficients coesponding to the local system on the configuation space B(Z, n) of unodeed n-tuples of distinct points in Z, induced fom V µ by the goup homomophism π 1 (B(Z, n)) Σ n (compae [20][p.293] and [22][Pop.3.5]). Fo Z pojective and pue-dimensional, by taking the degees in (17) fo the 9

10 pesent choice of coefficients ICZ H and epesentation V µ, we obtain the following identity fo the χ y -polynomial of the twisted intesection cohomology: (21) χ y (H (Z (n) ; IC H 1 (V Z (n) µ )) = χ λ (V µ ) χ y (H (Z; ICZ H )) k. z λ λ=(k 1,k 2, ) n 1 In futue woks, the techniques and esults obtained in this pape will be applied to: (a) cohomology epesentations of extenal and symmetic poducts, genealizing pevious esults of the authos fom [20], see [21]; (b) geneating seies fomulae fo the singula Todd classes td (F [n] ) of tautological sheaves F [n] (associated to a given F Coh(Z )) on the Hilbet scheme Z [n] of n points on a smooth quasi-pojective algebaic suface Z. Acknowledgements. The authos thank Tou Ohmoto fo eading a fist vesion of this pape and fo pointing out connections to his wok on obifold Chen classes of symmetic poducts. L. Maxim was patially suppoted by gants fom NSF, NSA, by a gant of the Ministy of National Education, CNCS-UEFISCDI poject numbe PN-II-ID-PCE , and by a fellowship fom the Max-Planck-Institut fü Mathematik, Bonn. J. Schümann was suppoted by the SFB 878 goups, geomety and actions". 2. Delocalized equivaiant theoies In this section, we intoduce the notion of delocalized equivaiant theoy of a G-space X (with G a finite goup), associated to a covaiant functo H with compatible coss-poduct. We also descibe the coesponding estiction and induction functos, which will play an essential ole in the subsequent sections of the pape. Fo simplicity, all spaces in this pape ae assumed to be complex quasi-pojective, though many constuctions in this section apply to othe categoies of spaces with a finite goups action (e.g., topological spaces o vaieties ove any base field). Fo such a vaiety X, conside an algebaic action G X X by a finite goup G. Fo any g G, we let X g denote the coesponding fixed point set. Let H be a covaiant (with espect to isomophisms) functo to abelian goups, with a compatible coss-poduct (Z-linea in each vaiable), which is commutative, associative and with a unit 1 pt H(pt). As main examples used in this pape, we conside the following, with R a commutative ing with unit (e.g., R = Z, Q o C): (1) the even degee Boel-Mooe homology H BM ev (X) R of X with coefficients in R. (2) Chow goups CH (X) R with R-coefficients. (3) Gothendieck goup of coheent sheaves K 0 (Coh(X )) R with R-coefficients. Othe possible choice would be: usual R-homology in even degees H ev (X) R. Since in this section we only need functoiality with espect to isomophisms, we could also wok with cohomological theoies, such as the even degee (compactly suppoted) R-cohomology H(c) ev (X) R o the Gothendieck goup of algebaic vecto bundles K 0 (X) R with R-coefficients, as used in [27, 31]. In this case, the coesponding covaiant tansfomation g, as used in this pape, is given by (g ) 1, the invese of the induced pullback unde g. If X is smooth, this fits with the folowing Poincaé duality isomophisms: (22) H ev (X) R = H ev c (X) R, H BM ev (X) R = H ev (X) R, K 0 (Coh(X )) R K 0 (X) R. 10

11 The disjoint union g G Xg admits an induced G-action by h : X g X hgh 1, such that the canonical map i : X g X g G defined by the inclusions of fixed point sets becomes G-equivaiant. Theefoe, G acts in a natual way on g G H(Xg ). Definition 2.1. The delocalized G-equivaiant theoy of X associated to H is the G-invaiant subgoup of g G H(Xg ), namely, (23) H G (X) := H(X g ) g G This theoy is functoial fo pope G-maps (esp. G-equivaiant isomophisms). Remak 2.2. An equivalent intepetation of this delocalized G-equivaiant theoy H G (X) of a G- space X can be obtained by beaking the summation on the ight-hand side of (23) into conjugacy classes, i.e., (24) H G (X) = ( h H(X g ) Z G(g) ) = H(X g ) ZG(g), (g) [h] G/Z G (g) (g) whee (g) is the conjugacy class and Z G (g) is the centalize of g G. Remak 2.3. If X is smooth, then also all fixed-point sets X g ae smooth, so the classical Poincaé duality isomophisms (22) induce simila duality isomophisms H G (X) = H G(X) between the coesponding delocalized equivaiant (co)homology theoies. Remak 2.4. If G acts tivially on X (e.g., X is a point), then (25) H G (X) = H(X) C(G), whee C(G) denotes the fee abelian goup of Z-valued class functions on G (i.e., functions which ae constant on the conjugacy classes of G). Remak 2.5. If G is an abelian goup, then: (26) H G (X) = g G H(X g ) G. G. Let us next descibe two functos which will be used late. Definition 2.6. (Restiction functo) Let X be a G-space, as befoe. Fo a subgoup K of G, the estiction functo fom G to K, Res G K, is the goup homomophism Res G K : H G (X) H K (X) induced by esticting to the G-invaiant pat the pojection H(X g ) H(X g ). g G g K 11

12 Clealy, Res G K is tansitive with espect to subgoups, with ResG G the identity homomophism. In tems of fixed-point sets of conjugacy classes, i.e., with espect to the isomophisms: H G (X) = (g) H(X g ) Z G(g), H K (X) = (k) H(X k ) Z K(k), the estiction facto can be descibed explicitly as follows (compae with [31][p.4]): if g G is not conjugate by elements in G to any element in K, then Res G K H(X g ) Z G (g) = 0; othewise, assume that g is conjugate by elements in G to k 1,, k s K which have mutually diffeent conjugacy classes in K: then H(X g ) Z G(g) = H(X k i ) Z G(k i ) fo i = 1,, s, and Res G K H(X g ) Z G (g) is given by the diect sum of inclusions H(X k i ) Z G(k i ) H(X k i ) Z K(k i ). The following induction functo will be used in the Section 4 in the definition of Pontjagin-type poducts. Definition 2.7. (Induction functo) Fo a G-space X as befoe, and K a subgoup of G, the induction fom K to G, Ind G K, is the goup homomophism (compae with [27][p.9]): (27) Ind G K = g ( ) : H K (X) H G (X), [g] G/K whee the summation is ove K-cosets of G. In paticula, on a G-invaiant class (i.e., in the image of the estiction functo Res G K ) this induction map is just multiplication by the index [G : K] of K in G. Note that Ind G K is tansitive fo subgoups of G, with IndG G the identity homomophism. In tems of fixed-point sets of conjugacy classes, this induction is given as follows (compae with [31][p.4]): fo any conjugacy class (k) in K which intesects the conjugacy class (g) in G, we have (28) Ind G K = h ( ) : H(X k ) ZK(k) H(X k ) Z G(k) = H(X g ) ZG(g), [h] Z G (k)/z K (k) so on a G-invaiant class this is just multiplication by the index [Z G (k) : Z K (k)]. Remak 2.8. In tems of the above induction functos, the identification (24) is given by: Ind G Z G (g) : H(X g ) ZG(g) H G (X), (g) (g) whee Ind G Z G (g) : H(Xg ) Z G(g) H G (X) is the estiction of Ind G Z G (g) to the diect summand H(X g ) Z G(g) H Z G(g) (X) coming fom the Z G (g)-equivaiant diect summand H(X g ) h Z G (g) H (X h ). Remak 2.9. Ove a point space, the above functos educe in many cases to the classical estiction and induction functos fom the epesentation theoy of finite goups Compatibilities with coss-poduct. Assume G acts on X, with g G and K G a subgoup, and similaly fo G acting on X, with g G and K G a subgoup. Then (X X ) g g = X g X g, Z G G (g g ) = Z G (g) Z G (g ), as well as G G /K K = G/K G /K, and similaly fo the quotient of centalizes as above. 12

13 Then all poducts : H(X g ) H(X g ) H(X g X g ) induce by the functoiality of a coesponding commutative and associative coss-poduct : H G (X) H G (X ) H G G (X X ) (with unit 1 pt H {id} (pt), fo {id} denoting the tivial goup). Moeove, this poduct is compatible with the estiction and induction functos, i.e., (29) Ind G G K K ( ) = Ind G K( ) Ind G K ( ) and (30) Res G G K K ( ) = Res G K( ) Res G K ( ). by Finally, the above facts about coss-poduct and estiction functos can be used to define a paiing: H G (pt) H G (X) C(G) H G (X) H G (X) H G G (pt X) Res H G (X), with pt X = X, and Res denoting the estiction functo fo the diagonal subgoup G G G. Remak The distinguished unit element id G gives the diect summand H G id (X) := H(X)G H G (X), i.e., the G-invaiant subgoup H(X) G of H(X). This diect summand is compatible with estiction, induction and induced coss-poducts. If the functo H is also covaiantly functoial fo closed embeddings, we get a pushfowad fo the closed fixed point inclusions i g : X g X, i.e., and a goup homomoohism sum G := g i g : H(X g ) H(X), i g : H G (X) H G id (X) = H(X)G H(X). Note that this homomophism commutes with induction and coss-poducts. Remak If X is smooth, the induction-estiction functos, as well as thei compatibilities with coss-poducts ae also compatible with Poincaé duality fo (co)homology as in Remak Geneating seies fo symmetic goup actions on extenal poducts In this section, we descibe a vey geneal geneating seies fomula fo symmetic goup actions on extenal poducts, which should be egaded as a fa-eaching genealization of a well-known identity of symmetic functions. In section 3.1, we give applications of this abstact geneating seies fomula in the context of obifold cohomology and esp. localized K-theoy. Let Z be a quasi-pojective vaiety, with the symmetic goup Σ n acting on the catesian poduct Z n of n 0 copies of Z by the natual pemutation action. Fo ou geneating seies fomula, it is impotant to look at all goups H Σn (Z n ) simultaneously. Let (31) H Σ (Z) := 13 H Σn (Z n ) t n

14 be the commutative gaded Z-algeba (with unit) with poduct := Ind Σ n+m Σ n Σ m ( ) induced fom the extenal poduct by induction. Hee, HΣn (Z n ) becomes a commutative gaded ing with poduct, and we view the completion H Σ (Z) as a subing of the fomal powe seies ing HΣn (Z n )[[t]]. The algeba H Σ (Z) is, in addition, endowed with ceation opeatos a : H(Z) H Σ (Z ), 1, which allow us to tanspot elements fom H(Z) to the delocalized goups H Σ (Z ). These ae defined as follows: if σ = () is an -cycle in Σ, then a is the composition a : H(Z) H(Z) = H((Z ) σ ) Z Σ (σ) H Σ (Z ), whee σ = Z Σ (σ ) acts tivially on (Z ) σ, and theefoe also on H((Z ) σ ). The ole of multiplication by in the definition of ceation opeato will become clea late on, e.g., in the poof of Theoem 3.1 below. The ceation opeato a can be e-witten as with a := Ind Σ σ i, i : H(Z) H((Z ) σ ) σ H <σ> (Z ). Hee, the last inclusion is just a diect summand, because σ is abelian. In the following we omit to mention i explicitly. Let σ Σ n have cycle patition λ = (k 1, k 2, ), i.e., k is the numbe of length cycles in σ and n = k. Then (Z n ) σ ((Z ) σ ) k (Z) k Z k 1+k 2 +, whee σ denotes as above a cycle of length in Σ n, and (Z) is the diagonal in Z, i.e., the image of the diagonal map : Z Z. Let us now choose a sequence b = (b 1, b 2, ) of elements b H(Z), 1, and associate to a conjugacy class epesented by σ Σ n of type (k 1, k 2, ) the element b (σ) H Σn (X n ) coesponding to (b ) k H( Z k ) H((Z n ) σ ), as it will be explained below. Recall that Z Σn (σ) is a poduct ove of semidiect poducts of Σ k with σ k, that is, (32) Z Σn (σ) = Σ k Z k (with σ denoting as befoe an -cycle). The goup Z k = σ k acts tivially on Z k, wheeas Σ k pemutes the coesponding Z-factos of Z k (compae [31][p.8]). By commutativity and associativity of the coss-poduct, it follows that (b ) k is invaiant unde Z Σn (σ), so it indeed defines an element ) (33) b (σ) = Ind Σn Z Σn (σ) ( (b ) k H Σn (Z n ), 14

15 with induction defined as in Remak 2.8. Moeove, fo σ Σ n and σ Σ m, we have: (34) b (σ) b (σ ) = b (σ σ ) H Σ n+m (Z n+m ). In what follows, we assume that the functo H takes values in R-modules, with R a commutative Q-algeba (othewise, wok with H Σ (Z) R). It follows that H Σ (Z) is also a commutative gaded Q-algeba. Note that one can also switch between covaiant and contavaiant notions, e.g. between homology and cohomology by Poincaé duality, if X is smooth. The main esult of this section is the following geneating seies fomula: Theoem 3.1. With the above notations, the following geneating seies fomula holds in the Q- algeba H Σ (Z): (35) t n = exp a (b ) t, 1 (σ) (Σ n) b (σ) whee (Σ n ) denotes the set of conjugacy classes of Σ n. Poof. We have the following sting of equalities in the Q-algeba (H Σ (Z), ): ( ) t ( t ) exp x = exp x =1 =1 ( t ) k 1 = x k! (36) =1 k =0 = N 0 = N 0 = = m=0 m=0 k 1,,k N x k 1,,k N x t m t m k 1 1 xk N N k 1! k N! k 1 1 xk N N k 1! k N! k 1 +2k 2 + +Nk N =m N ( t =1 k 1 +2k 2 + +Nk N =m =1 ) k t k 1+2k 2 + +Nk N 1 k 1 N k N x k 1 1 xk N N k 1! k N!1 k 1 N k N N x k k! k Note that the sum ove k 1 + 2k 2 + Nk N = m coesponds to a summation ove the cycle classes (σ) in Σ m given by σ k, fo σ = () an -cycle in Σ. In ou case, we take: x = a (b ) = Ind Σ σ ( b ). All poducts (and powes) above ae with espect to the multiplication in H Σ (Z), which is defined via coss-poduct and induction. In paticula, N (x ) k / k = Ind Σm ( N =1(x ) k / k ). (Σ)k =1 15

16 Moeove, by using the compatibility of induction with, Z-lineaity and tansitivity, we have: Ind Σm (Σ)k ( N =1(x ) k / k ) = Ind Σm σ k ( (b ) k ) = Ind Σm Z Σm (σ) IndZ Σm (σ) σ k ( (b ) k ), whee, as befoe, σ is a epesentative of the cycle type (k 1, k 2, ). But, as aleady mentioned, Z Σm(σ) acts tivially on (b ) k, so Ind Z Σm(σ) is just multiplication by the index [Z σ k Σ m(σ) : σ k ] = k!. Altogethe, we get N (x ) k /k! k = b (σ), which finishes the poof. = Examples. Let us now explain some special cases of Theoem 3.1 in the cohomological language, which in some situations ae aleady available in the liteatue. Ou main applications, to equivaiant chaacteistic classes fo singula spaces, will be given late on, in Section 5.3, afte we develop the necessay backgound Obifold cohomology. Hee we wok with H(X) := H ev (X) Q, the (even degee) ational cohomology functo. Fo X smooth, ou notion of H G (X) coesponds to the even degee obifold cohomology Hob 2 (X/G), as used fo example in [27]. Fo Z a quasi-pojective complex vaiety, and fo a given γ H(Z), let b := γ, fo all 1. Then b (σ) coesponds to γ l(σ) H((Z n ) σ ) fo σ Σ n of cycle type (k 1, k 2, ), and l(σ) := k the length of the patition associated to σ. Following [27], we set: (37) η n (γ) := ( Ind Σn Z Σn(σ) γ l(σ)) = b (σ). (σ) (Σ n) (σ) (Σ n) Then ou fomula (35) specializes to the following esult: (38) η n (γ) t n = exp a (γ) t. 1 Fo X smooth, this fits with the fomula stated afte Definition 3.2 in [27]. Howeve, ou poof is puely fomal, so it applies to any topological space, as well as to algebaic vaieties with ational Chow goups fo H Localized K-theoy. Hee we wok with H(X) := K 0 (X) C, the complexified Gothendieck goup of algebaic vecto bundles on X. Fo a finite goup G acting algebaically on a quasi-pojective complex vaiety X, we define a localization map of Lefschetz-type L G : K 0 G(X) H G (X) on the Gothendieck goup of G-equivaiant algebaic vecto bundles as a diect sum of tansfomations: L(g) : [W ] K 0 G(X) [W X g] K 0 g (Xg ) K 0 (X g ) Rep C ( g ) K 0 (X g ) C, whee the last map is induced by taking the tace against g G. Hee, Rep C ( g ) is the Gothendieck goup of complex epesentations of the goup g, and the isomophism in the above definition holds since g acts tivially on the fixed-point X g (e.g., this fact follows fom [10][(1.3.4)], compae also [12]). Note that, by constuction, L G commutes with coss-poducts as in Section

17 Fo Z a quasi-pojective complex vaiety and an algebaic vecto bundle V on Z, we get the Σ n - equivaiant vecto bundle V n on Z n. Let σ Σ n be of cycle type (k 1, k 2, ). Then, by the multiplicativity of L(σ), we get: L(σ) = L(σ ) k. So it suffices to undestand the tansfomations L(σ ), fo all -cycles ( 1). Fo σ n = (n) an n-cycle, we have that (39) L(σ n )[V n ] = ψ n (V ) K 0 (Z) Q unde the identification (Z n ) σn = n (Z) Z, with n(v n ) = V n. Hee ψ n denotes the n-th Adams opeation defined by Atiyah in the topological context [1] and e.g., Noi [24][Lem.3.2] in the algebaic geometic context. Note that we can wok hee with ational coefficients, since chaactes of symmetic goups ae intege-valued. If we choose b := ψ (V ), fo all 1, ou main fomula (35) specializes to the following geneating seies identity: (40) L Σn (V n ) t n = exp a (ψ (V )) t, 1 since, by multiplicativity and conjugacy invaiance of L(σ), we have that L Σn (V n ) = ) Ind Σn Z Σn (σ) ( L(σ ) k = b (σ). (σ) (Σ n) (σ) (Σ n) The same poof applies in the topological context, fo topological K-theoy, in which case we obtain a special situation (fo G the identity goup) of [30][Pop.4]. Note that [30] uses the identification L G C : K 0 G (X) C HG (X). Similaly, one can wok with algebaic vaieties ove any base field of chaacteistic zeo, with L(g) the coesponding Lefschetz tansfomation of Baum-Fulton-Quat [4] Localized Gothendieck goups of constuctible sheaves. Hee we wok with H(X) := K 0 (Const(X )) C, the complexified Gothendieck goup of (algebaically) constuctible sheaves of complex vecto spaces. Fo a finite goup G acting algebaically on a quasi-pojective complex vaiety X, we define a localization map of Lefschetz-type L G : K G 0 (Const(X )) H G (X) on the Gothendieck goup of G-equivaiant (algebaically) constuctible sheaves of complex vecto spaces as a diect sum of tansfomations: L(g) : [F] K G 0 (Const(X )) [F X g] K g 0 (Const(X g )) K 0 (Const(X g )) Rep C ( g ) K 0 (Const(X g )) C, whee the last map is induced, as befoe, by taking the tace against g G. The isomophism in the above definition holds since g acts tivially on the fixed-point X g and Const(X g ) is an abelian C-linea categoy (e.g., see as befoe [10][(1.3.4)], compae also [12]). Note that, by constuction, L G commutes as befoe with coss-poducts as in Section

18 Fo Z a quasi-pojective complex vaiety and an (algebaically) constuctible sheaf F on Z, we get the Σ n -equivaiant (algebaically) constuctible sheaf F n on Z n. Fo σ n = (n) an n-cycle, we have that (41) L(σ n )[F n ] = ψ n (V ) K 0 (Const(Z )) Q unde the identification (Z n ) σn = n (Z) Z, with n(f n ) = F n. Hee ψ n denotes the n-th Adams opeation of the pe-lambda ing stuctue on K 0 (Const(Z )) induced fom the symmetic monoidal tenso poduct of constuctible sheaves, as in [20][Lem.2.1]. As befoe, we can wok hee with ational coefficients. If we choose b := ψ (F), fo all 1, ou main fomula (35) specializes as above to the following geneating seies identity: (42) L Σn (F n ) t n = exp a (ψ (F)) t Fobenius chaacte. Specializing to a point space X, the above localized theoies fo vecto bundles and esp. constuctible sheaves educe to the classical chaacte theoy of a finite goup G: t G : Rep C (G) C(G) C [V ] {tace g (V ), g G}, with Rep C (G) the Gothendieck goup of complex epesentations of G, and C(G) the fee abelian goup of Z-valued class functions on G. The tace tace g is of couse multiplicative and conjugacy invaiant. Fo symmetic goups, we can wok again with ational coefficients, and get an algeba homomophism t Σ : Rep C (Σ) := Rep C (Σ n ) C(Σ) Q := C(Σ n ) Q n n with espect to the classical induction poduct: := Ind Σ n+m Σ n Σ m ( ) fo epesentations and esp. chaactes, see e.g., [16][Ch.I,Sect.7]. This homomophism can be composed with the Fobenius chaacte ch F : C(Σ) Q := C(Σ n ) Q Q[p i, i 1] =: Λ Q n to the gaded ing of Q-valued symmetic functions in infinitely many vaiables x m (m N), with p i := m xi m the i-th powe sum function. On C(Σ n ) Q, ch F is defined by: (43) ch F (f) := 1 f(σ)ψ(σ), n! σ Σ n with ψ(σ) := p k fo σ of cycle type (k 1, k 2, ), e.g., see [16][Ch.I,Sect.7]. Fo example, if f is the indicato function of the conjugacy class of the n-cycle σ n in Σ n, then ch F (f) = 1 n p n since n = Z Σn (σ n ). In paticula, the ceation opeato a : Q Q[p i, i 1] = Λ Q is (up to the Fobenius isomophism) given 18

19 by multiplication with p, which also motivates the use of multiplication by in the definition of ou ceation opeato in Section 3. If we choose b := 1 Q, fo all 1, ou main fomula (35) specializes to the well-known identity of symmetic functions (e.g., see the poof of [16][(2.14)]): (44) H(t) := h n t n = exp p t, 1 with h n = ch F (1 Σn ) the n-th complete symmetic function (see [16][p.113]), and 1 Σn := t Σn (tiv n ) the identity chaacte of the tivial epesentation tiv n of Σ n. 4. (Equivaiant) Pontjagin ings fo symmetic poducts Let Z be a quasi-pojective vaiety, and denote by Z (n) its n-th symmetic poduct, i.e., the quotient of the poduct Z n of n copies of Z by the natual action of the symmetic goup on n elements, Σ n. Let π n : Z n Z (n) denote the natual pojection map. In this section, the functo H fom Section 2 is equied in addition to be covaiant (at least) fo finite maps such as π n o the closed embedding i σ : (Z n ) σ Z n, fo σ Σ n. We will cay ove the assumption that H takes values in R-modules, with R a commutative Q-algeba. Besides H Σ (Z) := HΣn (Z n ) t n, hee we conside othe stuctues of commutative gaded Q-algeba with units, defined in tems of symmetic and espectively extenal poducts of Z: (a) On (b) On PH(Z) := H(Z (n) ) t n = H(Z (n) ) thee is the Pontjagin ing stuctue, with multiplication induced fom the maps Z (n) Z (m) Z (m+n), see [9][Definition 1.1] fo moe details. Hee, H(Z(n) ) becomes a commutative gaded ing with poduct, and we view the completion PH(Z) as a subing of the fomal powe seies ing H(Z(n) )[[t]]. PH Σ (Z) := H Σn (Z (n) ) t n = ( ) H(Z (n) ) C(Σ n ) Q t n PH(Z) (C(Σ) Q) thee is a poduct induced fom that of the Pontjagin poduct in the H-facto and the induction poduct fo class functions. Via the Fobenius chaacte identification ch F : C(Σ) Q Q[p i, i 1], we can also view PH Σ (Z) as a gaded subalgeba of PH(Z) Q[p i, i 1], and with i-th powe sum p i egaded as a degee i vaiable. (c) By Remak 2.10, the diect summand H Σ id (Z) := H Σn id (Zn ) t n H Σ (Z) 19

20 coesponding to the identity component is a subing, so that sum Σn : H Σ (Z) H Σ id (Z) is a ing homomophism. With espect to the Fobenius homomophism, it is moe natual to use the aveaging homomophisms av n := 1 n! sum Σ n : H Σn (Z n ) H Σn id (Zn ). Then the gaded goup homomophism av := av n : H Σ (Z) H Σ id (Z) becomes a gaded algeba homomophism if we intoduce on H Σ id (Z) the twisted poduct := n!m! (n + m)! : HΣn id (Zn ) H Σm id (Zm ) H Σ n+m id (Z n+m ). With this twisted poduct, we also have a Fobenius-type ing homomophism given by 1 n! av F : H Σ (Z) H Σ id (Z) Q[p i, i 1] σ Σ n i σ ψ(σ) : H Σn (Z n ) H Σn id (Zn ) Q[p i, i 1], with ψ(σ) as in the Fobenius homomophism ch F of equation (43). These stuctues ae elated by homomophisms of commutative gaded Q-algebas, fitting into the following commutative diagam: (45) H Σ (Z) := HΣn (Z n ) t n av F H Σ id (Z) Q[p i, i 1] π = nπ n PH Σ (Z) := HΣn (Z (n) ) t n 1 n Σn σ evσ π id PH(Z) Q[p i, i 1] p i =1 PH(Z) := H(Z(n) ) t n = PH(Z) with ev σ : C(Σ n ) Q Q the evaluation map at σ Σ n. Let d := π be the composition d : Z Z Z () of the natual pojection π : Z Z () with the diagonal embedding : Z Z. Then the ceation opeato a satisfies the identities: (46) π a = p d, av F a = p, and av a =. This genealizes the coesponding elation between a and p discussed at the end of Section Remak 4.1. Unde the assumptions fom the beginning of this section, the commutative diagam (45) is functoial in Z fo finite maps. If, moeove, the functo H and the coss-poduct ae functoial fo pope mophisms, then (45) is also functoial fo such mophisms. In paticula, fo Z compact, we can push down ou geneating seies fomulae (such as (35)) to a point to obtain (equivaiant) degee fomulae. Finally, the diagam (45) is compatible with natual tansfomations of such functos. 20

21 4.1. Example: Contuctible functions and Obifold Chen classes. In this example, we explain how ou main esult of Theoem 3.1 can be used to epove Ohmoto s geneating seies identities fo canonical contuctible functions ([26][Pop.3.9]) and obifold Chen classes ([26][Thm.1.1]). These ae genealized class vesions of the celebated Hizebuch-Höfe (o Atiyah-Segal) fomula fo the obifold Eule chaacteistic of symmetic poducts. Let H be the functo F ( ) of Q-valued algebaically constuctible functions, which is covaiant fo all mophisms, and with a compatible coss-poduct. Following Ohmoto s notations, fo a fixed goup A, let j (A) be the numbe of index subgoups of A, which is assumed to be finite fo all. In the notations of Theoem 3.1, let b := j (A) 1 Z F (Z). Then ) b (σ) = Ind Σn Z Σn (σ) ( (j (A) 1 Z ) k F Σn (Z n ), and the element 1 A Z n ;Σ n := b (σ) F Σn (Z n ) (σ) (Σ n) appeaing on the left hand side of Equation (35) is a delocalized vesion of Ohmoto s canonical constuctible function 1 (A) Z n ;Σ n of [26][Defn.2.2], in the sense that av n ( 1 A Z n ;Σ n ) = 1 (A) Z n ;Σ n F Σn id (Zn ). This identification follows fom [26][Lem.3.4]. (Note that Ohmoto s poduct in loc.cit. coesponds to ou twisted poduct.) Then ou main Theoem 3.1 yields the following identity: (47) 1 A Z n ;Σ n t n = exp j (A) t a (1 Z ) F Σ (Z). 1 By applying to (47) the ing homomophism av : (F Σ (Z), ) (F Σ id (Z), ), we ecove by (46) Ohmoto s geneating seies fomula [26][Pop.3.9]: (48) 1 (A) Z n ;Σ n t n = exp j (A) t (1 Z ). 1 Recall now that MacPheson s Chen class tansfomation (with ational coefficients) c : F ( ) H ( ) := Hev BM ( ) Q commutes with pope pushfowad and coss-poducts. Applying c to equation (47) we obtain a new geneating seies: (49) c (1 A Z n ;Σ n ) t n = exp j (A) t a (c (1 Z )) F Σ (Z), 1 with c (1 A Z n ;Σ n ) H Σn (Z n ) a delocalized vesion of Ohmoto s obifold Chen class c (1 (A) Z n ;Σ n ) H Σn id (Zn ). Ohmoto s fomula [26][Thm.1.1] fo obifold Chen classes of symmetic poducts is obtained by applying c to (48), o equivalently, by applying av to (49). If Z is pojective, by taking the degees deg c (1 A Z n ;Σ n ) = deg c (1 (A) Z n ;Σ n ) of the above chaacteistic class fomulae one ecoves geneating seies fo obifold-type Eule chaacteistics, see [26] fo details and examples. 21

22 Finally, note that fo A = Z (hence j (A) = 1 fo all ), we ecove a special case (fo cl = c and F = Q Z ) of Theoem 1.3 fom the Intoduction, via the identification: c Σn (Q n Z ) = c (1 Z Z n ;Σ n ). Howeve, these obifold classes ae in geneal vey diffeent fom the delocalized chaacteistic classes of the next section. Remak 4.2. Instead of fixing the goup A and the coefficients b = j (A) 1 Z F (Z), one could also stat with b = 1 Z fo all. Applying the Fobenius-type homomophism av F to the coesponding identity deived fom ou Theoem 3.1, we ecove the above esults in a unifom way by specializing p, fo a given goup A, to p = j (A) fo all. 5. Geneating seies fo (equivaiant) chaacteistic classes In this section, we specialize ou abstact geneating seies fomula (35) in the famewok of chaacteistic classes of singula vaieties Chaacteistic classes of Lefschetz type. Fo a complex quasi-pojective vaiety X, with an algebaic action G X X of a finite goup G, let π : X X := X/G be the quotient map. We denote geneically by cat G (X) a categoy of G-equivaiant objects on X in the undelying categoy cat(x), see [20, 8]. Fom now on, H(X) := H (X) will be Hev BM (X) R, the even degee Boel- Mooe homology of X with R-coefficients, fo R a commutative C-algeba, esp. Q-algeba if G is a symmetic goup. Note that H( ) is functoial fo all pope maps, with a compatible coss-poduct (as used in the pevious section). Definition 5.1. An equivaiant chaacteistic class tansfomation of Lefschetz type is a tansfomation cl ( ; g) : K 0 (cat G (X)) H (X g ) so that the following popeties ae satisfied: (1) cl ( ; g) is covaiant functoial fo G-equivaiant pope maps. (2) cl ( ; g) is multiplicative unde coss-poduct. (3) if X is a point space and cat(pt) is an abelian C-linea categoy, then the categoy Vect C (G) of (finite-dimensional) complex G-epesentations is a subcategoy of cat G (pt) and cl ( ; g) is a cetain g-tace (as shall be explained late on), with cl ( ; g) = tace g on Rep C (G). (4) if G acts tivially on X and cat(x) is an abelian C-linea categoy, then K 0 (cat G (X)) K 0 (cat(x)) Rep C (G) via the Schu functo decomposition as in (19), and (50) cl ( ; g) = cl ( ) tace g, with cl ( ) the coesponding non-equivaiant chaacteistic class, as explained below. If G = Σ n is a symmetic goup, it is enough to assume that cat(x) is an abelian Q-linea categoy, with the categoy Vect Q (Σ n ) of ational Σ n -epesentations a subcategoy of cat Σn (pt). Remak 5.2. Fo a subgoup K of G, with g K, we assume that such a tansfomation cl ( ; g) of Lefschetz type commutes with the estiction functo Res G K. Then cl ( ; g) depends only on the action of the cyclic subgoup geneated by g. In paticula, if g = id G is the identity of G, we can take K the identity subgoup {id G } with Res G {id G } the fogetful functo Fo : K 0 (cat G (X)) K 0 (cat(x)), 22

23 so that cl ( ; id G ) = cl ( ) fits with a coesponding non-equivaiant chaacteistic class. Remak 5.3. The above assumptions about coss-poduct and estiction functos can be used to define a paiing: Vect C (G) cat G (X) cat G (X) by cat G (pt) cat G (X) cat G G (pt X) Res cat G (X), with pt X = X, and Res denoting the estiction functo fo the diagonal subgoup G G G. This induces a paiing Rep C (G) K 0 (cat G (X)) K 0 (cat G (X)) on the coesponding Gothendieck goups, such that (51) cl (V F; g) = tace g (V ) cl (F; g) fo V a G-epesentation and F cat G (X). If G is the symmetic goup, then the above holds also fo ational epesentations. Let us give some examples of equivaiant chaacteistic class tansfomations of Lefschetz type. Example 5.4. Todd classes Let X be a quasi-pojective G-vaiety, and denote by K 0 (Coh G (X )) the Gothendieck goup of the abelian categoy Coh G (X ) of G-equivaiant coheent algebaic sheaves on X. Fo each g G, the Lefschetz-Riemann-Roch tansfomation [4, 23] (52) td ( ; g) : K 0 (Coh G (X )) H (X g ) is of Lefschetz type with R := C (esp. R := Q if G is a symmetic goup). Moeove, td ( ; id G ) is the complexified (non-equivaiant) Todd class tansfomation td of Baum-Fulton-MacPheson [3]. Ove a point space, the tansfomation td ( ; g) educes to the g-tace on the coesponding G- equivaiant vecto space. In paticula, if X is pojective, by pushing down to a point we ecove the equivaiant holomophic Eule chaacteistic, i.e., fo F Coh G (X ) the following degee fomula holds: χ a (X, F; g) := ( 1) i tace ( g H i (X; F) ) = td ([F]; g). i [X g ] Example 5.5. Chen classes Let K 0 (Const G (X)) be the Gothendieck goup of the abelian categoy Const G (X) of algebaically constuctible G-equivaiant sheaves of complex vecto spaces on X. Then the localized Chen class tansfomation [28][Ex.1.3.2] c ( ; g) := c (t g ( X g)) : K 0 (Const G (X)) H (X g ) is of Lefschetz type with R := C (esp. R := Q if G is a symmetic goup). Hee c ( ) is the Chen-MacPheson class tansfomation [18], and t g ( X g) is the goup homomophism which, fo F Const G (X), assigns to [F] K 0 (Const G (X)) the constuctible function on X g defined by x tace(g F x ). (Note that fo x X g, g acts on the finite dimensional stalk F x fo a constuctible G-equivaiant sheaf F.) Fo the identity element, the tansfomation c ( ; g) educes to the complexification of MacPheson s Chen class tansfomation. It also follows by definition that if X is a point space, 23