On an Equivariant Version of the Zeta Function of a Transformation

Many topological invariants have equivariant versions for spaces with actions of a group $G$, say, a finite one. For example, in [Verdier1973], an equivariant version of the Euler characteristic is an element of the Grothendieck ring of ${\mathbb{Z}}[G]$- or ${\mathbb{Q}}[G]$-modules. In tom Dieck ( [tom Dieck1979] , Section 5.4) it is defined as an element of the Burnside ring of the group $G$ (that is of the Grothendieck ring $K_{0}({\mbox{f.$G$-s.}})$ of finite $G$-sets). Applying these concepts to the Milnor fibre, one gets an equivariant version of the Milnor number of a $G$-invariant function-germ. For example, in [Wall1980] it is an element of the ring of virtual representations of the group $G$.

An important invariant of a germ of a holomorphic function (on $({\mathbb{C}}^{n},0)$ or on a germ of a complex analytic variety) is its monodromy and its corresponding zeta function, see e.g. [Arnold et al.1988]. It is defined as the zeta function of the classical monodromy transformation on the Milnor fibre. A number of statements have natural formulations in terms of monodromy zeta functions. As an example one can indicate the well-known monodromy conjecture: see, e.g., [Denef and Loeser1992]. The monodromy zeta function is connected with a number of other invariants, topological and analytic ones. For example, in [Gusein-Zade et al.1999], it was shown that, for an irreducible plane curve singularity, the monodromy zeta function of the corresponding function-germ coincides with the Poincaré series of the natural filtration on the local ring defined by the curve valuation. There are generalizations of this fact to some other situations [(see, e.g., a survey in [Gusein-Zade2010]]. In all these cases one has no intrinsic explanation of the relation. The relation is obtained by independent computation of the right and left hand sides of it in the same terms and comparison of the obtained results.

Generalizations of relations of this sort to equivariant settings could help to understand the general framework. For example, in [Ebeling and Gusein-Zade2012b] an equivariant version of a relation obtained earlier gave a better understanding of the role of the Saito duality in it. This leads to the desire to define equivariant analogues of monodromy zeta functions and of the Poincaré series of filtrations. These problem is not trivial and equivariant analogues are not unique. Moreover, up to now there were no definitions of equivariant analogues of monodromy zeta functions and of the Poincaré series which were elements of the same rings. For example, in [Campillo et al.2007]; [Campillo et al.2013], there were offered different approaches to equivariant Poincaré series. In [Campillo et al.2007] it is a power series with coefficients from the ring of one-dimensional representation of a group. In [Campillo et al.2013] it is an element of the Grothendieck ring of “locally finite” $G$-sets with an additional structure. In [Gusein-Zade et al.2008], there was given an equivariant version of the monodromy zeta function as a power series with the coefficients from $K_{0}({\mbox{f.$G$-s.}})\otimes{\mathbb{Q}}$. The fact that it was defined only after tensoring by the field ${\mathbb{Q}}$ of rational numbers makes it less reasonable, in particular, to compare it with the equivariant versions of the Poincaré series which were defined over integers. In [Gusein-Zade2013] an equivariant version of the monodromy zeta function was defined as an element of a generalization of the Burnside ring different from that in [Campillo et al.2013]. Just recently one gave a definition of an equivariant version of the Poincaré series as a power series with the coefficients from the Burnside ring of the group: [Campillo et al.2014]. (In that paper initially the Poincaré series is defined as a power series with the coefficients from a certain modification of the Burnside ring. A simple reductions sends this modification to the usual Burnside ring.)

One of the main ingredients of the definition of the equivariant version of the monodromy zeta function in [Gusein-Zade et al.2008] was the definition of the equivariant Lefschetz number of a transformation from [Lück and Rosenberg2003]. The definition from [Lück and Rosenberg2003] is rather natural. Moreover, one can say that it is the only possible definition possessing some reasonable properties. However the fact that it leads to a “non-integer” definition of the (monodromy) zeta function gives a hint that this definition is not absolutely adequate to this purpose.

There is certain freedom in a definition of an equivariant version of the Lefschetz number of a transformation connected with the question whether it should count the fixed points of the transformation or the fixed $G$-orbits of it. Here we use the second approach to the definition of the equivariant version of the Lefschetz number. This definition was introduced in [Dzedzej2001]. We describe the corresponding equivariant version of the zeta function of a transformation. This zeta-function is a power series with the coefficients from the ring $K_{0}({\mbox{f.$G$-s.}})$. We give an A’Campo type formula for the equivariant monodromy zeta function of a function germ in terms of a resolution.

Difficulties to compare equivariant versions of monodromy zeta functions and of the Poincaré series being elements of different nature (e.g. those described above) leads to the idea to compare their “integer valued reductions”. These reductions can be made with the help of the usual Euler characteristic and also with the help of the orbifold Euler characteristic. In the light of this, we also discuss possible orbifold versions of the zeta function of a transformation.

A finite $G$-set is a finite set with an action (say a left one) of the group $G$. Isomorphism classes of irreducible $G$-sets (i.e. those which consist of exactly one orbit) are in one-to-one correspondence with the set $\mbox{Consub}(G)$ of conjugacy classes of subgroups of $G$: to the conjugacy class containing a subgroup $H\subset G$ one associates the isomorphism class $[G/H]$ of the $G$-set $G/H$. The Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$ of finite $G$-sets (also called the Burnside ring of $G$) is the group generated by isomorphism classes of finite $G$-sets with the relation $[A\coprod B]=[A]+[B]$ and with the multiplication defined by the cartesian product. As an abelian group $K_{0}({\mbox{ f.$G$-s.}})$ is freely generated by the isomorphism classes $[G/H]$ of irreducible $G$-sets. The element 1 in the ring $K_{0}({\mbox{ f.$G$-s.}})$ is represented by the $G$-set consisting of one point (with the trivial $G$-action).

Recall that given a subgroup $H$ of $G$ there are two natural maps $\mbox{Res}_{H}^{G}:K_{0}(\mbox{f.}{G}\mbox{-sets})\to K_{0}(\mbox{f.}{H}% \mbox{-sets})$ and $\mbox{Ind}_{H}^{G}:K_{0}(\mbox{f.}{H}\mbox{-sets})\to K_{0}(\mbox{f.}{G}% \mbox{-sets})$. The restriction map $\mbox{Res}_{H}^{G}$ sends a $G$-set X to the same set considered with the $H$-action. The induction map $\mbox{Ind}_{H}^{G}$ sends an $H$-set $X$ to the product $G\times X$ factorized by the natural equivalence: $(g_{1},x_{1})\sim(g_{2},x_{2})$ if there exists $g\in H$ such that $g_{2}=g_{1}g$, $x_{2}=g^{-1}x_{1}$ with the natural (left) $G$-action. The induction map $\mbox{Ind}_{H}^{G}$ sends the class $[H/H^{\prime}]$ ($H^{\prime}$ is a subgroup of $H$) to the class $[G/H^{\prime}]$. Both maps are group homomorphisms, however the induction map $\mbox{Ind}_{H}^{G}$ is not a ring homomorphism.

In some places, say, in [tom Dieck1979], [Lück and Rosenberg2003] and [Gusein-Zade et al.2008], the equivariant Euler characteristic of a $G$-space is considered as an element of the Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$. For a relatively good $G$-space $X$ (say, for a quasiprojective variety) the equivariant Euler characteristic $\chi^{G}(X)\in K_{0}({\mbox{ f.$G$-s.}})$ can be defined in the following way. For a point $x\in X$, let $G_{x}=\{g\in G:\,g\cdot x=x\}$ be the isotropy subgroup of the point $x$. For a conjugacy class ${\mathcal{H}}\in\mbox{ Consub}(G)$, set $X^{{\mathcal{H}}}=\{x\in X:x\mbox{ is a fixed point of a subgroup }H\in{\mathcal{H}}\}$ and let $X^{({\mathcal{H}})}=\{x\in X:G_{x}\in{\mathcal{H}}\}$ be the set of points with the isotropy subgroups from ${\mathcal{H}}$. One can see that in the natural sense $X^{({\mathcal{H}})}=X^{{\mathcal{H}}}{\setminus}X^{>{\mathcal{H}}}$, where $X^{>{\mathcal{H}}}=\bigcup\nolimits_{{\mathcal{H}}^{\prime}>{\mathcal{H}}}X^{{% \mathcal{H}}^{\prime}}$.

Then the equivariant Euler characteristic of the $G$-space $X$ is defined as

$\chi^{G}(X):=\sum_{{\mathcal{H}}\in{ Consub}(G)}\frac{\chi(X^{({\mathcal{H}% })})\,|H|}{|G|}[G/H]=\sum_{{\mathcal{H}}\in{ Consub}(G)}\chi(X^{({\mathcal{% H}})}/G)[G/H],$

(1)

where $H$ is a representative of the conjugacy class ${\mathcal{H}}$.

The Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$ has a natural pre-$\lambda$-ring structure defined by the series

$\sigma_{X}(t)=1+[X]\,t+[S^{2}X]\,t^{2}+[S^{3}X]\,t^{3}+\cdots,$

where $S^{k}X=X^{k}/S_{k}$ is the $k$-th symmetric power of the $G$-set $X$ with the natural $G$-action. This pre-$\lambda$-ring structure induces a power structure over the Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$: see [Gusein-Zade et al.2006]. This means that for a power series $A(t)\in 1+t\cdot K_{0}({\mbox{ f.$G$-s.}})[[t]]$ and $m\in K_{0}({\mbox{ f.$G$-s.}})$ there is defined a series $\left(A(t)\right)^{m}\in 1+t\cdot K_{0}({\mbox{ f.$G$-s.}})[[t]]$ so that all the properties of the exponential function hold. In these notations $\sigma_{X}(t)=(1-t)^{-[X]}$. The geometric description of the natural power structure over the Grothendieck ring of quasiprojective varieties given in [Gusein-Zade et al.2006] using graded spaces is also valid for the power structure over $K_{0}({\mbox{ f.$G$-s.}})$ as well.

Some examples of computation of the series $(1-t)^{-[G/H]}$ for $G$ being the cyclic group ${\mathbb{Z}}_{6}$ of order 6 and the group ${\mathcal{S}}_{3}$ of permutations on three elements can be found in [Gusein-Zade et al.2008] (with some misprints). For example

$$ \renewcommand\Z{\mathbb Z}\begin{align} (1-t)^{-[{\mathcal S}_3/\langle e\rangle]}=&\frac{1}{1-t^6}[1]+\frac{t^3}{(1-t^3)(1-t^6)}[{\mathcal S}_3/\Z_3]\\ &+\frac{3t^2}{(1-t^2)^2(1-t^6)}[{\mathcal S}_3/\Z_2]\\ &+\frac{t(1+4t^2+t^3+4t^4-2t^5+3t^6+t^7)}{(1-t^2)^2(1-t^3)(1-t^6)(1-t)^2}[{\mathcal S}_3/\langle e\rangle].\end{align} $$

There is a natural homomorphism from the Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$ to the ring $R(G)$ of virtual representations of the group $G$ which sends the class $[G/H]\in K_{0}({\mbox{ f.$G$-s.}})$ to the representation $i^{G}_{H}[1_{H}]$ induced from the trivial one-dimensional representation $1_{H}$ of the subgroup $H$. (A virtual representation of the group $G$ is an element of the Grothendieck ring of representations, i.e. a formal difference of two representations.) This homomorphism is a homomorphism of pre-$\lambda$-rings ([Knutson1973]).

Let us show that for any subgroup $H$ of $G$ the series $(1-t)^{-[G/H]}$ represents a rational function with the denominator equal to a product of the binomials of the form $(1-t^{m})$, $m\in{\mathbb{Z}}_{\geq 1}$. Since irreducible $G$-sets are in one-to-one correspondence with the set $\mbox{ Consub}(G)$ of conjugacy classes of subgroups of $G$ and $K_{0}({\mbox{ f.$G$-s.}})$ is freely generated by isomorphism classes $[G/H]$ of irreducible $G$-sets then

$(1-t)^{-[G/H]}=\sum_{{\mathcal{F}}\in{ Consub}(G)}{\mathcal{A}}_{H,{% \mathcal{F}}}(t)[G/F]$

(2)

where $F$ is a representative of the conjugacy class ${\mathcal{F}}$ and ${\mathcal{A}}_{H,{\mathcal{F}}}(t)\in{\mathbb{Z}}[[t]]$.

Let ${\mathcal{F}}$ be a conjugacy class of subgroups of $G$ and let $F$ be a representative of it. The subgroup $F$ acts on the $G$-space $G/H$. Let $F\backslash G/H$ be the quotient of $G/H$ by this action and let $p:G/H\to F\backslash G/H$ be the quotient map. For $m=1,2,\ldots,$ let $Y_{m}$ be the set of points of $F\backslash G/H$ with $m$ preimages in $G/H$ and let ${\ell}^{{\mathcal{F}}}_{m}=|Y_{m}|$. (The numbers ${\ell}^{{\mathcal{F}}}_{m}$ depend only on the conjugacy class ${\mathcal{F}}$.) For an abelian $G$, ${\ell}^{{\mathcal{F}}}_{m}$ is different from zero if and only if $m=\frac{|F|}{|F\cap H|}$ and in this case ${\ell}^{{\mathcal{F}}}_{m}=|G|/|F+H|$.

For conjugacy classes ${\mathcal{F}}$ and ${\mathcal{F}}^{\prime}$ from ${ Consub}(G)$, let $F$ and $F^{\prime}$ be their representatives, and let $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}}$ be the number of fixed points of the group $F$ on $G/F^{\prime}$. The integer $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}}$ is different from zero if and only if ${\mathcal{F}}^{\prime}\geq{\mathcal{F}}$ (i.e. there exist representatives $F^{\prime}$ of $F$ of them such that $F^{\prime}\supset F$. For an abelian $G$ and for ${\mathcal{F}}^{\prime}\geq{\mathcal{F}}$, one has $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}}=|G/F^{\prime}|$. For a non-abelian group the equation is more involved and $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}}$ depends on ${\mathcal{F}}$ as well.

Since $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}}$ is different from zero if and only if ${\mathcal{F}}\leq{\mathcal{F}}^{\prime}$ and $r_{{\mathcal{F}}^{\prime},{\mathcal{F}}^{\prime}}$ is different from zero, the system of Eq. (3) is a triangular one (with respect to the partial order on the set of conjugacy classes of subgroups of $G$). Together with the fact that the denominators of the left hand side of the Eq. (3) are product of the binomials of the form $(1-t^{m})$ this implies the following statement.

The natural homomorphism from the Grothendieck ring $K_{0}({\mbox{ f.$G$-s.}})$ to the ring $R(G)$ of virtual representations of the group $G$ sends the equivariant Euler characteristic $\chi^{G}(X)$ to the one used in [Wall1980]. Since this homomorphism is, generally speaking, neither injective, no surjective, the equivariant Euler characteristic as an element in $K_{0}({\mbox{ f.$G$-s.}})$ is a somewhat finer invariant than the one as an element of the ring $R(G)$.

Let $X$ be a relatively good topological space (say, a quasiprojective complex or real variety) with a $G$-action and let $\varphi:X\to X$ be a $G$-equivariant proper map. The usual (“non-equivariant”) Lefschetz number $L(\varphi)$ counts the fixed points of $\varphi$ (or rather of its generic perturbation). The equivariant version $L^{G}(\varphi)$ of the Lefschetz number from [Lück and Rosenberg2003] counts the fixed points of $\varphi$ as a (finite) $G$-set. This leads to the following equation for the equivariant Lefschetz number

$L^{G}(\varphi)=\sum\limits_{{\mathcal{H}}\in\mbox{Consub}(G)}\frac{L(\varphi_{% |(X^{{\mathcal{H}}},X^{>{\mathcal{H}}})})|H|}{|G|}[G/H],$

(4)

where $H$ is a representative of the class ${\mathcal{H}}$. If $\varphi$ is a $G$-homeomorphism (like the monodromy transformation, see Sect. , (5) and (6) , (7) with the two parts of the equation (1) .]

Just in the same way as in [Lück and Rosenberg2003] one can formulate the equivariant version of the Lefschetz fixed point theorem for ${\widetilde{L}}^{G}(\varphi)$ (an analogue of Theorem 2.1 in [Lück and Rosenberg2003]).

Let $\varphi:X\to X$ be as above. The usual (non-equivariant) zeta function of $\varphi$ is defined in terms of the action of $\varphi$ in the (co)homology groups of $X$ (in a way somewhat similar to the definition of the Lefschetz number). This definition is not convenient for a direct generalization to the equivariant case. It is more convenient to use the definition of the zeta function of the transformation $\varphi$ in terms of the Lefschetz numbers of the iterates of $\varphi$. One defines integers $s_{i}$, $i=1,2\ldots,$ recursively by the equation

$L(\varphi^{m})=\sum_{i|m}s_{i}.$

(8)

The number $s_{m}$ counts (with integer multiplicities) the points $x\in X$ with the $\varphi$-order equal to $m$ (i.e. $\varphi^{m}(x)=x$, $\varphi^{i}(x)\neq x$ for $0<i<m$). Together with each such point all its images under the iterates of $\varphi$ (there are exactly $m$ different ones) are of this sort. Therefore $s_{m}$ is divisible by $m$. One defines the zeta function $\zeta_{\varphi}(t)$ to be

$\zeta_{\varphi}(t):=\prod_{m\geq 1}(1-t^{m})^{-{s_{m}}/{m}}.$

(9)

In the equivariant version, let $s_{m}^{{{G}}}(\varphi)$ and ${\widetilde{s}}_{m}^{{{G}}}(\varphi)$ be defined through $L^{G}(\varphi^{i})$ and ${\widetilde{L}}^{G}(\varphi^{i})$ respectively by the analogues of the Eq. (8)

$L^{G}(\varphi^{m})=\sum_{i|m}s_{i}^{G}(\varphi),\quad{\widetilde{L}}^{G}(% \varphi^{m})=\sum_{i|m}{\widetilde{s}}_{i}^{G}(\varphi).$

(10)

The elements $s_{m}^{G}(\varphi)$ and ${\widetilde{s}}_{m}^{G}(\varphi)$ count the points in $X$ the $\varphi$-order of which is equal to $m$ in $X$ and in $X/G$ respectively.

For a $G$-variety $X$, its orbifold Euler characteristic $\chi^{orb}(X,G)\in{\mathbb{Z}}$ is defined, e.g., in [Atiyah and Segal1989] or [Hirzebruch and Höfer1990]. For a subgroup $H$ of $G$, let $X^{H}=\{x\in X:Hx=x\}$ be the fixed point set of $H$. The orbifold Euler characteristic $\chi^{orb}(X,G)$ of the $G$-space $X$ is defined, e.g., in [Atiyah and Segal1989] and [Hirzebruch and Höfer1990]:

$\chi^{orb}(X,G)=\sum_{[g]\in{ Consub}(G)}\chi(X^{\langle g\rangle}/C_{G}(g)),$

(13)

where $C_{G}(g)=\{h\in G:h^{-1}gh=g\}$ is the centralizer of $g$, and $\langle g\rangle$ the subgroups generated by $g$.

There is a natural homomorphism of abelian groups $\Phi:K_{0}({\mbox{ f.$G$-s.}})\to{\mathbb{Z}}$ which sends the generator $[G/H]$ of $K_{0}({\mbox{ f.$G$-s.}})$ to $\chi^{orb}(G/H,G)$ and therefore the equivariant Euler characteristic $\chi^{G}(X)\in K_{0}({\mbox{ f.$G$-s.}})$ to the orbifold Euler characteristic $\chi^{orb}(X,G)$. For an abelian $G$, $\Phi([G/H])=|H|$ and $\Phi$ is a ring homomorphism, but this is not the case in general.

The Lefschetz number is a sort of generalization of the Euler characteristic: the Euler characteristic is the Lefschetz number of the identity map. The definition of the orbifold Euler characteristic gives the hint that there can be the corresponding definition(s) of the orbifold Lefschetz number of a $G$-equivariant transformation. It can be expressed through an equivariant version of the Lefschetz number with values in the Burnside ring: the image of the Lefschetz number by the homomorphism $\Phi$. The two versions $L^{G}(\varphi)$ and ${\widetilde{L}}^{G}(\varphi)$ of the equivariant Lefschetz number [see (4) and (6) ] give two versions

$L^{orb}(\varphi)=\Phi(L^{G}(\varphi))\,\,\mbox{ and }\,\,{\widetilde{L}}^{orb}(\varphi)=\Phi({\widetilde{L}}^{G}(\varphi))$

of orbifold Lefschetz numbers.

The usual definition of the zeta function of a transformation [e.g. Eqs. (8) , (9) , (10) and (11) ] gives two orbifold versions of the zeta function of a $G$-equivariant transformation $\varphi:X\to X$:

${\zeta}_{\varphi}^{orb}(t)=\prod_{m\geq 1}(1-t^{m})^{-{{s}_{m}^{{orb}}}/{m}},% \,\,\mbox{ and }\,\,{\widetilde{\zeta}}_{\varphi}^{orb}(t)=\prod_{m\geq 1}(1-t% ^{m})^{-{{\widetilde{s}}_{m}^{{orb}}}/{m}},$

(14)

where $L^{orb}(\varphi^{m})=\sum_{i|m}s^{orb}_{i}(\varphi)$ and ${\widetilde{L}}^{orb}(\varphi^{m})=\sum_{i|m}{\widetilde{s}}^{orb}_{i}(\varphi)$.

The exponents $-{{\widetilde{s}}_{m}^{{orb}}}/{m}$ are integers and therefore the orbifold monodromy zeta function ${\widetilde{\zeta}}_{\varphi}^{orb}(t)$ is a rational function in $t$. The exponents $-{{s}_{m}^{{orb}}}/{m}$ are in general rational numbers.

For instance, for $f$ from Example 2 in Sect. 5 one has

${\widetilde{\zeta}}_{f}^{orb}(t)=(1-t^{6k})^{-1+2(6k-1)+(1-6k^{2})}\cdot(1-t^{% 3k})^{-1}\cdot(1-t^{2k})^{-1}.$

This follows from the fact that, for $G={\mathcal{S}}_{3}$ and for a subgroup $H$ of ${\mathcal{S}}_{3}$, $\chi^{orb}({\mathcal{S}}_{3}/H,{\mathcal{S}}_{3})=|H|$.