A Decomposition of Twisted Equivariant K -Theory
aa r X i v : . [ m a t h . A T ] J a n A DECOMPOSITION OF TWISTED EQUIVARIANT K-THEORY
JOS´E MANUEL G ´OMEZ AND JOHANA RAMREZ
Abstract.
For G a finite group, a normalized 2-cocycle α ∈ Z ( G, S ) and X a G -spaceon which a normal subgroup A acts trivially, we show that the α -twisted G -equivariant K -theory of X decomposes as a direct sum of twisted equivariant K -theories of X parametrizedby the orbits of an action of G on the set of irreducible α -projective representations of A .This generalizes the decomposition obtained in [4] for equivariant K -theory. We also exploresome examples of this decomposition for the particular case of the dihedral groups D n with n ≥ Introduction
In the past few years there has been a growing interest in studying twisted K -theorymotivated by its appearance in string theory and also due to the celebrated theorem ofFreed, Hokpins and Teleman (see [3, Theorem 1]). In this article we study G -equivarianttwisted K -theory when G is a finite group. Our main goal is to show that, under suitablehypothesis, the canonical decomposition theorem for projective representations can be usedto obtain a decomposition for twisted G -equivariant K -theory as a direct sum of other twistedequivariant K -theories, thus generalizing the work in [4] where a similar decomposition wasobtained for equivariant K -theory.Suppose that we have a short exact sequence of finite groups1 → A → G π → Q → . Let X be a compact and Hausdorff G -space on which A acts trivially and fix α a normalized2-cocycle on G with values in S . Associated to the cocycle α we have a central extension1 → S → e G α → G → G on X can be extended to an action of e G α on X in such a way thatthe central factor S acts trivially. The α -twisted G -equivariant K -theory of X , α K ∗ G ( X ),is constructed using e G α -equivariant vector bundles on which the central factor S acts bymultiplication of scalars. The main object of study in this work are the twisted K -groups α K ∗ G ( X ).Representations of the group e G α on which the central factor S acts by multiplicationof scalars are in a one to one correspondence with α -projective representations of G . Viathis correspondence we can use the classical tools of projective representations to study thetwisted K -groups α K ∗ G ( X ). To this end we show in Section 2 that, if Irr α ( A ) denotes the set The first author acknowledges and thanks the financial support provided by COLCIENCIAS throughgrant number FP44842-013-2018 of the Fondo Nacional de Financiamiento para la Ciencia, la Tecnolog´ıa yla Innovaci´on. The second author acknowledges and thanks the financial support provided by COLCIEN-CIAS through grant number 727 of the program Doctorados nacionales 2015 of the Fondo Nacional deFinanciamiento para la Ciencia, la Tecnolog´ıa y la Innovaci´on. f isomorphism classes of α -projective representations of A , then there is an action of G onIrr α ( A ). and this action factors through an action of Q on Irr α ( A ). Given an isomorphismclass [ τ ] ∈ Irr α ( A ) let Q [ τ ] denote the isotropy subgroup at [ τ ]. Using Lemma 2.7 we showthat we can associate to each [ τ ] ∈ Irr α ( A ) a normalized 2-cocycle defined on Q [ τ ] with valuesin S . This is precisely the data needed to construct a twisted version of Q [ τ ] -equivariant K -theory. With this in mind, the main result of this article is the following theorem. Theorem 1.1.
Suppose that A is a normal subgroup of a finite group G . Let α be a normal-ized 2-cocycle on G with values in S and X a compact G -space on which A acts trivially.Then there is a natural isomorphism Φ X : α K ∗ G ( X ) → M [ τ ] ∈ G \ Irr α ( A ) β τ,α K ∗ Q [ τ ] ( X )[ E ] M [ τ ] ∈ G \ Irr α ( A ) (cid:2) Hom e A α ( V τ , E ) (cid:3) . When α is the trivial cocycle α K ∗ G ( X ) agrees with K ∗ G ( X ) and the previous decompositionagrees with [4, Theorem 3.2] in this case. Therefore Theorem 1.1 generalizes [4, Theorem3.2]. In fact, Theorem 1.1 is proved using ideas similar to the ones used to prove [4, Theorem3.2].We remark that Theorem 1.1 could be proved using the work of Freed, Hopkins andTeleman [3]. However, we have chosen to prove it directly as to obtain an explicit descriptionof this decomposition. Also, we chose to work with finite groups to obtain explicit formulasfor the cocycles used to twist equivariant K -theory in this decomposition. Theorem 1.1 alsoholds in general for compact Lie groups, a proof in this context can be obtained generalizingthe work in [2].The outline of this article is as follows. In Section 2 we review some basic definitions ofprojective representations that we use throughout this article. Section 3 is the main part ofthe article, Theorem 1.1 is proved there. Finally, in Section 4 we explore some examples ofTheorem 1.1 for the particular case of the dihedral group D n with n ≥ Projective representations
In this section we recall some basic definitions and properties of projective representationsthat will be used throughout this article.2.1.
Basic definitions.Definition 2.1.
Let G be a finite group and V a finite dimensional complex vector space. Amap ρ : G → GL ( V ) is called a projective representation of G if there exists a function α : G × G → C ∗ such that(1) ρ ( g ) ρ ( h ) = α ( g, h ) ρ ( gh )for all g, h ∈ G and ρ (1) = Id V .Note that if ρ satisfies Equation (1) then the function α defines a C ∗ -valued normalized2-cocycle on G ; that is, for all g, h, k ∈ G we have: ( gh, k ) α ( g, h ) = α ( g, hk ) α ( h, k ) ,α ( g,
1) = α (1 , g ) = 1 . To stress the dependence of ρ on V and α , we shall often refer to ρ as an α -representationof G on the space V or, simply as an α -representation of G , if V is not pertinent to thediscussion. Remark 2.2. If α is the trivial cocycle; that is, if α ( g, h ) = 1 for all g, h ∈ G , then α -representations of G are simply ordinary representations of G . Definition 2.3.
Two α -representations ρ i : G → GL ( V i ) with ( i = 1 ,
2) are said to be linearly equivalent or isomorphic if there exists a vector space isomorphism f : V → V such that ρ ( g ) = f ρ ( g ) f − for all g ∈ G .Just as in the case of the ordinary theory of representations of groups we have similarnotions for projective representations such as irreducible representations and unitary repre-sentations. Given a C ∗ -valued normalized 2-cocycle α on G we denote by Irr α ( G ) the setof isomorphism classes of complex irreducible α -representations of G . If ρ : G → GL ( V ) isan irreducible α -representation of G then [ ρ ] ∈ Irr α ( G ) denotes the corresponding isomor-phism class. We remark that the classical results of representation theory such as completereducibility and Schur’s lemma also hold for the case of projective representations. We referthe reader to [5] for the basic theory of projective representations. Example 2.4.
Consider the dihedral group D n of order 2 n defined by D n = h a, b | a n = b = 1 , bab = a − i . For such groups we have H ( D n , S ) ∼ = ( n is odd, Z / n is even.In this example we only consider the case where n is even as otherwise we will obtain usualrepresentations. Let n ≥ ǫ a primitive n -th root of unity in C and let α : D n × D n → S be the function defined by α ( a j , a k b l ) = 1 and α ( a j b, a k b l ) = ǫ k for 0 ≤ j, k ≤ n − l = 0 , . The function α defines a normalized 2-cocycle on D n with values in S whose correspondingcohomology class is a generator in H ( D n , S ) ∼ = Z /
2. For each i ∈ { , , . . . , n/ } put A i = (cid:18) ǫ i ǫ − i (cid:19) and B i = (cid:18) (cid:19) . Consider the map τ i : D n → GL ( C )defined by τ i ( a k b l ) = A ki B li for 0 ≤ k ≤ n − l = 0 , . These assignments determine the irreducible, non-equivalent α -representations of D n so thatIrr α ( D n ) = { [ τ ] , . . . , [ τ n/ ] } . (See for example [6, Chapter 5 Theorem 7.1]). key feature of projective representations is that we also have the following canonicaldecomposition whose proof can be obtained in a similar way as in the case of regular repre-sentations. Theorem 2.5. ( Canonical Decomposition ) Suppose that α is a normalized -cocycle ofa finite group G with values in S . Let W be a finite-dimensional α -representation. Thenthe assignment γ : M [ ρ ] ∈ Irr α ( G ) V ρ ⊗ Hom G ( V ρ , W ) → Wv ⊗ f f ( v ) defines an isomorphism of α -representations. Suppose now that α is a normalized 2-cocycle on G with values in S . We can associateto α a central extension of G by S in the following way. As a set define(2) e G α = { ( g, z ) | g ∈ G, z ∈ S } . The product structure in e G α is given by the assignment( g , z )( g , z ) := ( g g , α ( g , g ) z z ) . This way e G α a compact Lie group that fits into a central extension1 → S → e G α → G → . Let ρ : G → GL ( V ) be an α -representation of G . If we define ˜ ρ : e G α → GL ( V ) by˜ ρ ( g, z ) = zρ ( g ) then ˜ ρ defines a representation of e G α on which the central factor S acts bymultiplication of scalars. Conversely, if ˜ ρ : e G α → GL ( V ) is a representation of e G α on whichthe central factor S acts by multiplication of scalars then the function ρ : G → GL ( V )given by ρ ( g ) = ˜ ρ ( g,
1) defines an α -representation of G . The above assignment defines aone to one correspondence between α -representations of G and representations of e G α onwhich the central factor S acts by multiplication of scalars. Via this correspondence we willswitch back and forth between α -representations of G and representations of e G α on whichthe central factor S acts by multiplication of scalars without explicitly mentioning it.2.2. Cocycles and projective representations.
Suppose now that A is a normal sub-group of a finite group G so that we have a short exact sequence1 → A → G π → Q → . Assume that α is a normalized 2-cocycle of G with values in S , by restriction we can also see α as a cocycle defined on A . Let us define an action of G on the set Irr α ( A ) in the followingway. Given ρ : A → U ( V ρ ) an α -representation and g ∈ G we define g · ρ : A → U ( V ρ ) sothat if a ∈ A we have:(3) g · ρ ( a ) = α ( g − a, g ) α ( g, g − a ) − ρ ( g − ag ) ∈ U ( V ρ ) . Proposition 2.6.
The above assignment defines a left action of G on Irr α ( A ) . Furthermore,for all b ∈ A , we have that b · ρ ∼ = ρ so that the action of G on Irr α ( A ) factors to an actionof Q = G/A on Irr α ( A ) . roof. First we show that g · ρ is an α -representation of A . Indeed, for all g ∈ G and a, b ∈ A we have( g · ρ ( a ))( g · ρ ( b )) = ( α ( g − a, g ) α ( g, g − a ) − ρ ( g − ag ))( α ( g − b, g ) α ( g, g − b ) − ρ ( g − bg ))= α ( g − a, g ) α ( g, g − a ) − α ( g − b, g ) α ( g, g − b ) − α ( g − ag, g − bg ) ρ ( g − abg )= α ( g, g − a ) − α ( g − ab, g ) α ( g − a, b ) ρ ( g − abg )= α ( a, b ) α ( g − ab, g ) α ( g, g − ab ) − ρ ( g − abg )= α ( a, b ) ( g · ρ ( ab )) . The above equalities are obtained using the 2-cocycle equation for α . Now, we show thatthis definition satisfies the axioms of an action. If ρ : A → U ( V ρ ) is an α -representation, as α is a normalized cocycle, 1 · ρ ( a ) = α ( a, α (1 , a ) − ρ ( a ) = ρ ( a ) . Moreover, given g, h ∈ G and a ∈ A , we have g · ( h · ρ )( a ) = α ( g − a, g ) α ( g, g − a ) − ( h · ρ )( g − ag )= α ( g − a, g ) α ( g, g − a ) − α ( h − g − ag, h ) α ( h, h − g − ag ) − ρ ( h − g − agh )and ( gh ) · ρ ( a ) = α ( h − g − a, gh ) α ( gh, h − g − a ) − ρ (( gh ) − a ( gh ))= α ( h − g − a, gh ) α ( gh, h − g − a ) − ρ ( h − g − agh ) . Manipulating the cocycle equation for α it can be proved that α ( g − a, g ) α ( g, g − a ) − α ( h − g − ag, h ) α ( h, h − g − ag ) − = α ( h − g − a, gh ) α ( gh, h − g − a ) − . This implies that for all a ∈ A g · ( h · ρ )( a ) = ( gh ) · ρ ( a ) . Finally, for a, b ∈ A expanding and using the cocycle equation we obtain b · ρ ( a ) = ρ ( b ) − ρ ( a ) ρ ( b )and thus b · ρ ∼ = ρ as α -representations. (cid:3) As above assume that A is a normal subgroup of a group G and Q = G/A . Fix anassignment σ : Q → G such that π ( σ ( q )) = q for all q ∈ Q with σ (1) = 1. We remark thatthe map σ is only a set theoretical map so in particular it does not necessarily have to be agroup homomorphism.Suppose that ρ : A → U ( V ρ ) is a complex irreducible α -representation with the propertythat g · ρ is isomorphic to ρ for every g ∈ G (under the action defined in Proposition 3).Under this assumption, as σ ( q ) · ρ ∼ = ρ we can find an element M q ∈ U ( V ρ ) for each q ∈ Q such that(4) σ ( q ) · ρ ( a ) = M − q ρ ( a ) M q . This means that α ( σ ( q ) − a, σ ( q )) α ( σ ( q ) , σ ( q ) − a ) − ρ ( σ ( q ) − aσ ( q )) = M − q ρ ( a ) M q or all a ∈ A . We can choose M = 1 as σ (1) = 1 and σ (1) · ρ = ρ . Let e A α be the centralextension of A by S associated to the cocycle α and ˜ ρ : e A α → U ( V ρ ) the correspondingrepresentation. Remember that ˜ ρ ( a, z ) = zρ ( a ) so we can define σ ( q ) · ˜ ρ ( a, z ) := z ( σ ( q ) · ρ ( a )) . Therefore σ ( q ) · ˜ ρ ( a, z ) = z ( M − q ρ ( a ) M q )= M − q zρ ( a ) M q = M − q ˜ ρ ( a, z ) M q . (5)Define χ : Q × Q → A by the equation(6) χ ( q , q ) = σ ( q q ) − σ ( q ) σ ( q ) . Note that χ ( q , q ) belongs to A since π ( χ ( q , q )) = 1 and the map χ is normalized in thesense that χ ( q , q ) = 1 whenever q = 1 or q = 1. In addition, define τ : Q × Q → S by τ ( q , q ) = α ( σ ( q q ) − , σ ( q ) σ ( q )) α ( σ ( q q ) , σ ( q q ) − ) − α ( σ ( q ) , σ ( q ))= α ( σ ( q q ) , χ ( q , q )) − α ( σ ( q ) , σ ( q )) . (7)Now, for q , q in Q we notice that the element˜ ρ ( χ ( q , q ) , τ ( q , q )) M − q M − q M q q = τ ( q , q ) ρ ( χ ( q , q )) M − q M − q M q q belongs to the center Z ( U ( V ρ )) ∼ = S . Define the map β ρ,α : Q × Q → S by the equation(8) β ρ,α ( q , q ) := ˜ ρ ( χ ( q , q ) , τ ( q , q )) M − q M − q M q q . Lemma 2.7. If ρ : A → U ( V ρ ) is a complex irreducible α -representation such that g · ρ ∼ = ρ for every g ∈ G then the map β ρ,α : Q × Q → S defines a normalized 2-cocycle on Q withvalues in S .Proof. If either q = 1 or q = 1 we have that τ ( q , q ) = 1 as α is normalized. Also, as χ is normalized we have χ ( q , q ) = 1. Since we are choosing σ (1) = 1 and M = 1 it followsthat β ρ,α ( q , q ) = 1 if either q = 1 or q = 1. Therefore either β ρ,α is normalized. To finishwe need to prove that for every q , q , q ∈ Q we have β ρ,α ( q , q q ) β ρ,α ( q , q ) = β ρ,α ( q q , q ) β ρ,α ( q , q )To see this note that as β ρ,α ( q , q ) belongs to S we have that M q q = β ρ,α ( q , q ) M q M q ˜ ρ ( χ ( q , q ) , τ ( q , q )) − . Therefore, for q , q and q in Q we have, M q q q = M q ( q q ) = β ρ,α ( q , q q ) M q M q q ˜ ρ ( χ ( q , q q ) , τ ( q , q q )) − = β ρ,α ( q , q q ) M q β ρ,α ( q , q ) M q M q ˜ ρ ( χ ( q , q ) , τ ( q , q )) − ˜ ρ ( χ ( q , q q ) , τ ( q , q q )) − = β ρ,α ( q , q q ) β ρ,α ( q , q ) M q M q M q (cid:2) ˜ ρ ( χ ( q , q q ) , τ ( q , q q )) ˜ ρ ( χ ( q , q ) , τ ( q , q )) (cid:3) − . In a similar way, writing M q q q = M ( q q ) q and expanding we obtain M q q q = β ρ,α ( q q , q ) β ρ,α ( q , q ) M q M q M q (cid:2) ˜ ρ ( χ ( q , q q ) , τ ( q , q q )) ˜ ρ ( χ ( q , q ) , τ ( q , q )) (cid:3) − . his implies that β ρ,α ( q , q q ) β ρ,α ( q , q ) = β ρ,α ( q q , q ) β ρ,α ( q , q )as we wanted to prove. (cid:3) Decomposition of twisted equivariant K -theory In this section we use the canonical decomposition of vector bundles to show that undersome hypothesis the α -twisted equivariant K -theory α K ∗ G ( X ) of a G -space X can be decom-posed as a direct sum of twisted equivariant K-theories parametrized by the orbits of theaction of G on Irr α ( A ) constructed on the previous section.We start by recalling the definition of α -twisted equivariant K -theory that we will use.We follow the treatment used in [1, Section 7.2]. Assume that G is a finite group acting ona compact and Hausdoff space X . Let α be a normalized 2-cocycle on G with values in S .Consider 1 → S → e G α → G → G on X can be extended toan action of e G α in such a way that the central factor S acts trivially. Definition 3.1.
The α -twisted G -equivariant K -theory of X , denoted by α K G ( X ), is definedas the Grothendieck group of the set of isomorphism classes of e G α -equivariant vector bundlesover X on which S acts by multiplication of scalars on the fibers. For n > α K nG ( X ) are defined as α ˜ K G (Σ n X + ), where as usual X + denotes the space X withan added base point.To obtain the desired decomposition of twisted equivariant we are going to construct anequivalent formulation for such twisted K-groups following the work in [4]. For this supposethat we have a short exact sequence of finite groups1 → A → G π → Q → . Assume that G acts on a compact and Hausdorff space X in such a way that A acts trivially.Fix α ∈ Z ( G, S ) a normalized 2-cocycle. Notice that by restriction we can see any e G α -equivariant vector bundle over X as an e A α -equivariant vector bundle, where e A α denotes thecentral extension associated to the cocycle α seen as a cocycle defined on A . Also, recall thatassociated to an α -representation ρ : A → U ( V ρ ) we have a representation ˜ ρ : e A α → U ( V ρ ). Definition 3.2.
Suppose that ρ : A → U ( V ρ ) is a complex irreducible α -representation. A( e G α , ρ )-equivariant vector bundle over X is a e G α -vector bundle on which the central factor S acts by multiplication of scalars on E and the map γ : V ρ ⊗ Hom e A α ( V ρ , E ) → Ev ⊗ f f ( v )is an isomorphism of e A α -vector bundles on which S acts by multiplication of scalars.In the above definition, if ρ is an α -representation of A then V ρ denotes the trivial e A α -vector bundle over π : X × V ρ → X . Observe that if p : E → X is a e G α -vector bundleon which the central factor S acts by multiplication then, as we are assuming that A acts rivially on X , it follows that for every x ∈ X the fiber E x can be seen as a representation of e A α on which the central factor acts by scalar multiplication. Thus for every x ∈ X the fiber E x can be seen as an α -representation of A . With this point of view, a ( e G α , ρ )-equivariantvector bundle is a e G α -equivariant vector bundle p : E → X such that for every x ∈ X thefiber E x is an α -representation of A isomorphic to a direct sum of the α -representation ρ .Let Vect e G α ,ρ ( X ) denote the set of isomorphism classes of ( e G α , ρ )-equivariant vector bundles,where two ( e G α , ρ )-equivariant vector bundles are isomorphic if they are isomorphic as e G α -vector bundles. Notice that if E and E are two ( e G α , ρ )-equivariant vector bundles then sois E ⊕ E . Therefore Vect e G α ,ρ ( X ) is a semigroup. Following [4, Definition 2.2] we have thenext definition. Definition 3.3.
Assume that G acts on a compact space X in such a way that A actstrivially on X and let α be a normalized 2-cocycle α ∈ Z ( G, S ). We define K e G α ,ρ ( X ), the( e G α , ρ )-equivariant K -theory of X , as the Grothendieck construction applied to Vect e G α ,ρ ( X ).For n > K n e G α ,ρ ( X ) is defined as ˜ K e G α ,ρ (Σ n X + ).As our next step we show that the previous definition can be described using the usualdefinition of twisted equivariant K -theory provided in Definition 3.1. For this suppose that α is a normalized 2-cocycle of G with values in S . As above assume that A is a normalsubgroup of G and let Q = G/A . Let ρ be an irreducible α -representation such that g · ρ ∼ = ρ for all g ∈ G . Fix a set theoretical section σ : Q → G such that σ (1) = 1 as in the previoussection. We can extend σ to obtain a map ˜ σ : Q → e G α by defining ˜ σ ( q ) = ( σ ( q ) , ∈ e G α .Let β ρ,α be the 2-cocycle defined on Q with values in S constructed in Equation (8). Withthis cocycle we can consider the central extension1 → S → e Q β ρ,α → Q → . With this in mind we have the following generalization of [4, Theorem 2.1].
Theorem 3.4.
Suppose that ρ is an irreducible α -representation such that g · ρ ∼ = ρ for all g ∈ G . Let X be a G -space such that A acts trivially on X . If p : E → X is a ( e G α , ρ ) -equivariant vector bundle, then Hom e A α ( V ρ , E ) has the structure of a e Q β ρ,α -vector bundle onwhich the central factor S acts by multiplication of scalars. Moreover, the assignment [ E ] → [Hom e A α ( V ρ , E )] is a natural one to one correspondence between isomorphism classes of ( e G α , ρ ) -equivariantvector bundles over X and isomorphism classes of e Q β ρ,α -equivariant vector bundles over X for which the central factor S acts by multiplication of scalars.Proof. We are only going to provide a sketch of the proof as it follows the same steps usedin the proof of [4, Theorem 2.1].Suppose that p : E → X is a e G α -vector bundle. Then Hom e A α ( V ρ , E ) is a non-equivariantvector bundle over X . Next we give Hom e A α ( V ρ , E ) an action of e Q β ρ,α . Suppose that f ∈ Hom e A α ( V ρ , E ) x . If q ∈ Q we define q • f ∈ Hom e A α ( V ρ , E ) q · x by( q • f )( v ) = ˜ σ ( q ) · f ( M − q v ) = ( σ ( q ) , · f ( M − q v ) , here M q ∈ U ( V ρ ) is the element chosen in the Section 2.2. It is easy to see that q • f is e A α -equivariant. Also, if q , q ∈ Q then q • ( q • f ) is such that for v ∈ V ρ q • ( q • f )( v ) = ˜ σ ( q ) · ( q • f )( M − q v ) = ˜ σ ( q )˜ σ ( q ) · f ( M − q M − q v )= ( σ ( q q ) , χ ( q , q ) , τ ( q , q )) · f ( M − q M − q v )= ˜ σ ( q q ) f (( χ ( q , q ) , τ ( q , q )) · M − q M − q v )= ˜ σ ( q q ) f ( β ρ,α ( q , q ) M − q q v )= β ρ,α ( q , q )˜ σ ( q q ) f ( M − q q v ) = β ρ,α ( q , q )( q q • f ( v )) . We conclude that q • ( q • f )( v ) = β ρ,α ( q , q )( q q • f ( v )) . The last equation allows us to define an action of e Q β ρ,α on Hom e A α ( V ρ , E ) as follows. If( q, z ) ∈ Q β ρ,α and f ∈ Hom e A α ( V ρ , E ) x define( q, z ) · f := z ( q • f ) . Thus if v ∈ V ρ then(( q, z ) · f )( v ) = z (˜ σ ( q ) · f ( M − q v )) = ˜ σ ( q ) · ( zf ( M − q v )) . Unraveling the definitions, it is easy to see that this way Hom e A α ( V ρ , E ) has the structureof a e Q β ρ,α -equivariant vector bundle such that the central factor S acts by multiplication ofscalars.Suppose now that p : F → X is a e Q β ρ,α -equivariant vector bundle over X for which thecentral S acts by multiplication of scalars. Given ( g, z ) ∈ e G α , f ∈ F x and v ∈ V ρ define(9) ( g, z ) · ( v ⊗ f ) := M π ( g ) ˜ ρ (˜ σ ( π ( g )) − ( g, z )) v ⊗ (( π ( g ) , · f ) ∈ ( V ρ ⊗ F ) π ( g ) · x . The above assignment defines an action of e G α on V ρ ⊗ F and V ρ ⊗ F becomes a e G α vectorbundle. Moreover for ( a, z ) ∈ e A α , as M = 1, we have( a, z ) · ( v ⊗ f ) = M ˜ ρ (˜ σ (1) − ( a, z )) v ⊗ (1 , · f = ˜ ρ ( a, z ) v ⊗ f so that e A α acts on V ρ ⊗ F by the representation ˜ ρ ; that is, p : V ρ ⊗ F → X is a ( e G α , ρ )-equivariant vector bundle over X .Finally, assume that p : E → X is a ( e G α , ρ )-equivariant vector bundle over X . Bydefinition the map γ : V ρ ⊗ Hom e A α ( V ρ , E ) → E ( v, f ) f ( v ) s an isomorphism of e A α -vector bundle. We may endow V ρ ⊗ Hom e A α ( V ρ , E ) with structureof a e G α -vector bundle. That the map γ is an isomorphism of vector bundles and its e G α -equivariance follows from next equations. For ( g, z ) ∈ e G α we have γ (( g, z ) · ( v ⊗ f )) = γ ( M π ( g ) ˜ ρ (˜ σ ( π ( g )) − ( g, z )) v ⊗ ( π ( g ) , · f = ( π ( g ) , · f ( M π ( g ) ˜ ρ (˜ σ ( π ( g )) − ( g, z )) v )= π ( g ) • f ( M π ( g ) ˜ ρ (˜ σ ( π ( g )) − ( g, z )) v )= ˜ σ ( π ( g )) · f ( ˜ ρ (˜ σ ( π ( g )) − ( g, z )) v )= ( g, z ) · f ( v ) = ( g, z ) · γ ( v ⊗ f ) . The previous argument shows that γ : V ρ ⊗ Hom e A α ( V ρ , E ) → E is an isomorphism of e G α -vector bundles.Now, if p : F → X is a e Q β ρ,α -equivariant vector bundle over X for which the central S acts by multiplication of scalars, then by Equation (9) we know that V ρ ⊗ F is a ( e G α , ρ )-equivariant vector bundle. The canonical isomorphism of vector bundles F → Hom e A α ( V ρ , V ρ ⊗ F ) x f x : v v ⊗ x is in fact e Q β ρ,α -equivariant, and therefore the vector bundles F and Hom e A α ( V ρ , V ρ ⊗ F ) areisomorphic as e Q β ρ,α -equivariant vector bundles.We conclude that the inverse map of the assignment [ E ] [Hom e A α ( V ρ , E )] is preciselythe map defined by the assignment [ F ] [ V ρ ⊗ F ]. (cid:3) As an immediate corollary of Theorem 3.4 we obtain the following identification of the( e G α , ρ )-equivariant K -theory groups of Definition 3.3 with the β ρ,α -twisted Q -equivariant K -theory groups provided in Definition 3.1. Corollary 3.5.
Let X be a G -space such that A acts trivially on X . Assume that ρ is an α -representation of A such that g · ρ ∼ = ρ for every g ∈ G . Then the assignment K n e G α ,ρ ( X ) ∼ = → β ρ,α K ∗ Q ( X )[ E ] [Hom e A α ( V ρ , E )] defines a natural isomorphism. Suppose now that α is a normalized 2-cocycle on G with values in S . Consider the actionof G on Irr α ( A ) constructed in Section 2.2. Given [ τ ] ∈ Irr α ( A ) let G [ τ ] = { g ∈ G | g · τ ∼ = τ } denote the isotropy subgroup of the action of G at [ τ ]. The group G [ τ ] fits into the shortexact sequence 1 → A → G [ τ ] π → Q [ τ ] → Q [ τ ] = G [ τ ] /A agrees with the isotropy of the group Q at [ τ ]. Let ( e G [ τ ] ) α be the centralextension corresponding to the cocylce α seen as a cocycle defined on G [ τ ] . Therefore we ave the following commutative diagram of central extensions1 −−−→ S −−−→ ( e G [ τ ] ) α −−−→ G [ τ ] −−−→ id y y y −−−→ S −−−→ e G α −−−→ G −−−→ . Assume that X is a compact and Hausdorff G -space on which A acts trivially. As before wecan extend the action of G on X to an action of e G α on X in such a way that S acts trivially.Let p : E → X be a e G α -equivariant vector bundle on which S acts by scalar multiplicationon the fibers. As e A α acts trivially on X each fiber of E can be seen as an α -representationof A . Using fiberwise the canonical decomposition theorem for α -representations (Theorem2.5) we see that the assignment γ : M [ τ ] ∈ Irr α ( A ) V τ ⊗ Hom e A α ( V τ , E ) → Ev ⊗ f f ( v )defines an isomorphism of e A α -equivariant vector bundles. Using this decomposition we obtainthe following theorem. Theorem 3.6.
Under the above assumptions there is a natural isomorphism Ψ X : α K ∗ G ( X ) → M [ τ ] ∈ G \ Irr α ( A ) K ∗ ( e G [ τ ] ) α ,τ ( X )[ E ] M [ τ ] ∈ G \ Irr α ( A ) (cid:2) V τ ⊗ Hom e A α ( V τ , E ) (cid:3) . This isomorphism is functorial on maps X → Y of G -spaces on which A acts trivially.Proof. The proof of this theorem follows the same lines of the proof of [4, Theorem 3.1] sowe only provide an outline of the proof.Let us show first that the map Ψ X is well defined. To see this we have to show that if ρ isan α -representation of A then V ρ ⊗ Hom e A α ( V ρ , E ) has the structure of a (( e G [ ρ ] ) α , ρ )-vectorbundle. Following the notation of Section 2.2 fix a set theoretical section σ : Q [ ρ ] → G [ ρ ] forthe projection map π : G [ ρ ] → Q [ ρ ] in such a way that σ (1) = 1. Also, for every q ∈ Q [ ρ ] fixan element M q ∈ U ( V ρ ) such that σ ( q ) · ρ ( a ) = α ( σ ( q ) − a, σ ( q )) α ( σ ( q ) , σ ( q ) − a ) − ρ ( σ ( q ) − aσ ( q )) = M − q ρ ( a ) M q . This is possible as σ ( q ) ∈ G [ ρ ] so that we have σ ( q ) · ρ ∼ = ρ . We can choose M = 1 as σ (1) = 1.Now if ( h, z ) ∈ ( e G [ ρ ] ) α and v ⊗ f ∈ V ρ ⊗ Hom e A α ( V ρ , E ) we define M ( h,z ) ∈ U ( V ρ ) by M ( h,z ) := M π ( h ) ˜ ρ (˜ σ ( π ( h )) − ( h, z )) , where ˜ ρ and ˜ σ are defined in a similar way as in Theorem 3.4. Observe that M ( a,z ) = ˜ ρ ( a, z )for all ( a, z ) ∈ e A α . Moreover, given ( h, z ) ∈ ( e G [ ρ ] ) α and v ⊗ f ∈ V ρ ⊗ Hom e A α ( V ρ , E ) wedefine ( h, z ) ⋆ ( v ⊗ f ) = M ( h,z ) v ⊗ ( h, z ) • f, here ( h, z ) • f ( w ) = ( h, z ) f ( M − h,z ) w ). Unraveling the definitions it can be seen that thisdefines an action of ( e G [ ρ ] ) α on V ρ ⊗ Hom e A α ( V ρ , E ) in such a way that the central factor S actsby multiplication of scalars. This way V ρ ⊗ Hom e A α ( V ρ , E ) has the structure of a ( e G [ ρ ] ) α -vectorbundle and A acts by the α -representation ρ on the fibers so that [ V ρ ⊗ Hom e A α ( V ρ , E )] ∈ K ∗ ( e G [ ρ ] ) α ,ρ ( X ). This shows that Ψ X is well defined.Next we show that Ψ X is an isomorphism. For this write Irr α ( A ) = A ⊔ A ⊔ · · · ⊔ A k ,where A , A , . . . , A k are the different G -orbits of the action of G on Irr α ( A ) defined in theEquation (3). For every 1 ≤ i ≤ k define E A i = M [ τ ] ∈A i V τ ⊗ Hom e A α ( V τ , E ) . Note that V τ ⊗ Hom e A α ( V τ , E ) is an e A α -equivariant vector bundle over X , so each E A i isalso an e A α -equivariant vector bundle over X and the map γ : k M i =1 E A i = M [ τ ] ∈ Irr α ( A ) V τ ⊗ Hom e A α ( V τ , E ) → E defines an isomorphism of e A α -vector bundles. We are going to show that each E A i is a e G α -vector bundle and that the map γ is e G α -equivariant. For this fix an index 1 ≤ i ≤ k andan irreducible α -representation ρ : A → U ( V ρ ) such that [ ρ ] ∈ A i . The elements in A i canbe written in the form [ g · ρ ] , . . . , [ g r i · ρ ] for some elements g = 1 , g , . . . , g r i ∈ G . Therefore E A i = r i M j =1 V g j · ρ ⊗ Hom e A α ( V g j · ρ , E ) . We can give a structure of e G α -space on E A i in the following way. Suppose that ( g, s ) ∈ e G α and that v ⊗ f ∈ V ρ ⊗ Hom e A α ( V ρ , E ) x . Decompose gg j in the form gg j = g l h , where1 ≤ l ≤ r i and h ∈ G [ ρ ] . In other words g l is the representative chosen for the coset ( gg j ) G [ ρ ] and h = g − l gg j . Then ( g, s )( g j ,
1) = ( g l , h, z )where z = sα ( g, g j ) α ( g l , h ) − . We define( g, s ) ⋆ ( v ⊗ f ) := M ( h,z ) v ⊗ ( g, s ) • f ∈ (cid:0) V g l · ρ ⊗ Hom e A α ( V g l · ρ , E ) (cid:1) ( g,s ) · x where ( g, s ) • f ( w ) = ( g, s ) f (cid:16) M − h,z ) w (cid:17) = ( g, s )( h, z ) − ˜ σ ( π ( h )) f (cid:16) M − π ( h ) w (cid:17) . It can be seen that this defines an action of e G α ∈ E A i . for each 1 ≤ i ≤ k making thevector bundle E A i into a e G α -vector bundle in such a way that the central factor S acts bymultiplication of scalars. Furthermore, the map γ : k M i =1 E A i → E s an isomorphism of e A α -vector and the map γ is e G α -equivariant so that γ is an isomorphismof e G α -vector bundles. Now, the desired map Ψ X can be seen as the direct sum ⊕ ki =1 Ψ iX choosing for each A i a representation [ τ i ] ∈ A i . Where each Ψ iX is given byΨ iX : α K ∗ G ( X ) → K ∗ ( e G [ τi ] ) α ,τ i ( X )[ E ] (cid:2) V τ i ⊗ Hom e A α ( V τ i , E ) (cid:3) . In what follows we will construct the map ζ i : K ∗ ( e G [ τi ] ) α ,τ i ( X ) → α K ∗ G ( X ) which will be theright inverse of Ψ iX . Take ρ = τ i and consider a vector bundle F ∈ Vect ( e G [ ρ ] ) α ,ρ ( X ). Fix g = 1 , g , . . . , g r representatives for the different cosets in G/G [ ρ ] . Let L F := r M j =1 [( g j , − ] ∗ F. Using ideas similar to the ones used in [4, Theorem 3.1] we can endow L F with the structureof a e G α -vector bundle on which the central factor S acts by multiplication of scalars in sucha way that the action of ( e G [ ρ ] ) α on [( g , − ] ∗ F ∼ = F agrees with the given action of ( e G [ ρ ] ) α on F . We define ζ i : K ∗ ( e G [ τi ] ) α ,τ i ( X ) → α K ∗ G ( X )[ F ] [ L F ] . Now, since we have the isomorphism V ρ ⊗ Hom e A α ( V ρ , ⊕ rj =1 [( g j , − ] ∗ F ) ∼ = F as ( e G [ ρ ]) α vector bundles. We obtain at the level of K -theory that ζ i is a right inverse for Ψ i so that ζ = ⊕ ri =1 is a right inverse for Ψ X . In a similar way, using the work given above it can beseen that ζ is also a left inverse so that the map Ψ X is indeed an isomorphism.To finish, we observe that functoriality follows from the fact that if τ is an α -representationof A then the bundles V τ ⊗ Hom e A α ( V τ , f ∗ E ) and f ∗ (cid:0) V τ ⊗ Hom e A α ( V τ , E ) (cid:1) are canonicallyisomorphic as (( e G [ τ ] ) α , τ )-equivariant bundles whenever f : Y → X is a G -equivariant mapfrom spaces on which A acts trivially. (cid:3) As a result of Theorem 3.6 and Corollary 3.5 we obtain the following theorem that is themain result of this article.
Theorem 3.7.
Suppose that A is a normal subgroup of a finite group G . Let α be a normal-ized 2-cocycle on G with values in S and X a compact G -space on which A acts trivially.Then there is a natural isomorphism Φ X : α K ∗ G ( X ) → M [ τ ] ∈ G \ Irr α ( A ) β τ,α K ∗ Q [ τ ] ( X )[ E ] M [ τ ] ∈ G \ Irr α ( A ) (cid:2) Hom e A α ( V τ , E ) (cid:3) . This isomorphism is functorial on maps X → Y of G -spaces on which A acts trivially. . Examples
In this section we explore some examples of Theorems 3.6 and 3.7 for the dihedral groups D n where n ≥ G = D . The group D is generatedby the elements a, b subject to the relations a = b = 1 and bab = a . Let α : D × D → S be the 2-cocycle defined by α ( a l , a j b k ) = 1 and α ( a l b, a j b k ) = i j for 0 ≤ j, l ≤ k = 0 , . Note that α is a nontrivial normalized 2-cocycle such that its corresponding cohomologyclass defines a generator of H ( D ; S ) ∼ = Z /
2. By Example 2.4, taking n = 4, we know thatup to isomorphism D has two irreducible projective α -representations τ and τ defined by τ l ( a j b k ) = A jl B kl for 0 ≤ j, k ≤ l = 0 , . In the above definition we have A = (cid:18) i
00 1 (cid:19) , A = (cid:18) − − i (cid:19) and B = B = (cid:18) (cid:19) . With this in mind we are going to explore the following examples of Theorems 3.6 and 3.7.
Example 4.1.
Suppose first that G = D and A = Z / h a i . Therefore Q = G/A = { [1] , [ b ] } ∼ = Z / . Let us take X to be the space with only one point ∗ equipped with the trivial D -action. Inthis case α K ∗ D ( ∗ ) = R α ( D ), where R α ( D ) denotes the α -twisted representation ring of D .As pointed out above τ and τ are the only irreducible α -representations of D and thus wehave an isomorphism of abelian groups α K ∗ D ( ∗ ) ∼ = R α ( D ) = Z τ ⊕ Z τ . On the other hand, observe that α | A is trivial so thatIrr α ( A ) = Irr( A ) = { [1] , [ ρ ] , [ ρ ] , [ ρ ] } , where ρ : A → C is the irreducible representation defined by ρ ( a ) = i . For the action of D on Irr α ( A ) we have b · ρ ( a ) = α ( ba, b ) α ( b, ba ) − ρ ( bab )= α ( a b, b ) α ( b, a b ) − ρ ( a )= ( i ) − i = 1therefore b · ρ = 1. Moreover, b · ρ ( a ) = α ( ba, b ) α ( b, ba ) − ρ ( bab )= α ( a b, b ) α ( b, a b ) − ρ ( a )= ( i ) − ( ρ ( a )) = − i o that b · ρ = ρ . We conclude that orbits of the D action on Irr α ( A ) are { [1] , [ ρ ] } and { [ ρ ] , [ ρ ] } . Thus we can choose [1] and [ ρ ] as representatives for the elements in D \ Irr α ( A )and G [1] = G [ ρ ] = A . In this case Theorem 3.6 gives us an isomorphismΨ : R α ( D ) = α K ∗ D ( ∗ ) ∼ = → K ∗ ( e G [1] ) α , ( ∗ ) ⊕ K ∗ ( e G [ ρ ) α ,ρ . As G [1] = G [ ρ ] = A and α restricted to A is trivial we have that ( e G [1] ) α = ( e G [ ρ ] ) α = A × S . Therefore K ∗ ( e G [1] ) α , ( ∗ ) ∼ = Z ˜1 and K ∗ ( e G [ ρ ) α ,ρ ∼ = Z ˜ ρ . (Recall that ˜ ρ denotes therepresentation of ( e G [ ρ ] ) α on which S acts by multiplication of scalars corresponding to ρ and similarly for ˜1). For the representations τ and τ we have τ ( a ) = i
00 1 ! and τ ( a ) = − − i . Thus as A -representations τ is isomorphic to 1 ⊕ ρ and τ is isomorphic to ρ ⊕ ρ . Moreover,in the isomorphism given by Theorem 3.6 we haveΨ : R α ( D ) ∼ = Z τ ⊕ Z τ ∼ = → K ∗ ( e G [1] ) α , ( ∗ ) ⊕ K ∗ ( e G [ ρ ) α ,ρ ∼ = Z ˜1 ⊕ Z ˜ ρ τ ˜1 τ ˜ ρ . On the other hand, by Theorem 3.7 we have an isomorphismΦ : R α ( D ) ∼ = → β ,α K ∗ Q [1] ( ∗ ) ⊕ β ρ ,α K ∗ Q [ ρ ( ∗ ) . As G [1] = G [ ρ ] = A we have that Q [1] = Q [ ρ ] = { } is the trivial group. Therefore β ,α and β ρ ,α are the trivial cocycles and Theorem 3.7 gives us the isomorphismΦ : R α ( D ) ∼ = Z τ ⊕ Z τ ∼ = → K ∗{ } ( ∗ ) ⊕ K ∗{ } ( ∗ ) ∼ = Z ⊕ Z . Example 4.2.
Suppose now that G = D and A = Z ( D ) = h a i ∼ = Z /
2. Therefore in thiscase we have Q = G/A = { [1] , [ b ] , [ a ] , [ ab ] } ∼ = Z / ⊕ Z / . As in the previous example α | A is trivial andIrr α ( A ) = Irr( A ) = { [1] , [ σ ] } , where σ : A → C is the representation defined by σ ( a ) = −
1. For the action of D onIrr α ( A ) we have b · σ ( a ) = α ( ba , b ) α ( b, ba ) − σ ( ba b )= α ( a b, b ) α ( b, a b ) − σ ( a )= ( − −
1) = 1therefore b · σ = 1. In particular the action of D on Irr α ( A ) is transitive and we canchoose [1] as a representative for the set D \ Irr α ( A ). For the representation 1 we have G [1] = h a i ∼ = Z /
4. If we take again X = ∗ then Theorem 3.6 gives us an isomorphismΨ : R α ( D ) ∼ = Z τ ⊕ Z τ ∼ = → K ∗ ( e G [1] ) α , ( ∗ ) . s α is trivial on h a i we have that ( e G [1] ) α = h a i× S . If ρ : h a i → C denotes the representationdefined by ρ ( a ) = i then ˜1 and ˜ ρ can be seen as (( e G [1] ) α , ∗ and K ∗ ( e G [1] ) α , ( ∗ ) ∼ = Z ˜1 ⊕ Z ˜ ρ . Observe that τ ( a ) = − and τ ( a ) = 1 00 − Therefore as A -representations we have τ ∼ = τ ∼ = 1 ⊕ σ . However, as h a i -representations τ is isomorphic to 1 ⊕ ρ and τ is isomorphic to ρ ⊕ ρ . It follows that in the isomorphismgiven by Theorem 3.6 we haveΨ : R α ( D ) ∼ = Z τ ⊕ Z τ ∼ = → K ∗ ( e G [1] ) α , ( ∗ ) ∼ = Z ˜1 ⊕ Z ˜ ρ .τ ˜1 τ ˜ ρ . On the other hand, by Theorem 3.7 we have an isomorphismΦ : R α ( D ) ∼ = → β ,α K ∗ Q [1] ( ∗ ) . In this case Q [1] = h a i / h a i ∼ = Z /
2. The cocycle β ,α is the trivial cocycle and Theorem 3.7gives us an isomorphismΦ : R α ( D ) ∼ = Z τ ⊕ Z τ ∼ = → K ∗ Z / ( ∗ ) = R ( Z / . For the group Z / R ( Z / ∼ = Z ⊕ Z s , where s denotes the sign representation. Theisomorphism Φ maps τ to 1 and τ to s . Example 4.3.
Example 4.1 can easily be generalized to the dihedral groups D n with n aneven number. Suppose then that n is an even number and let D n be the group generatedby the elements a, b subject to the relations a n = b = 1 and bab = a − . Fix ǫ a primitive n -th root of unity and let α : D n × D n → S be the function defined by α ( a j , a k b l ) = 1 and α ( a j b, a k b l ) = ǫ k for 0 ≤ j, k ≤ n − l = 0 , . The function α defines a normalized 2-cocycle on D n with values in S whose correspondingcohomology class is a generator in H ( D n , S ) ∼ = Z /
2. By Example 2.4 we know that theirreducible projective α -representations of D n are τ i : D n → GL ( C ) for i = 1 , . . . , n/ τ i ( a k b l ) = A ki B li for 0 ≤ k ≤ n − l = 0 , , where A i = (cid:18) ǫ i ǫ − i (cid:19) and B i = (cid:18) (cid:19) . If we take X = ∗ endowed with the trivial D n action we have α K ∗ D n ( ∗ ) ∼ = R α ( D n ) = Z τ ⊕ · · · ⊕ Z τ n/ . ake A = h a i ∼ = Z /n so that Q = D n /A = { [1] , [ b ] } ∼ = Z /
2. Observe that α | A is trivial andthus Irr α ( A ) = Irr( A ) = { [1] , [ ρ ] , [ ρ ] , . . . , [ ρ n − ] } , where ρ ( a ) = ǫ . For the action of D n on Irr α ( A ) we have that b · ρ, b · ρ = ρ n − , . . . , b · ρ n/ = ρ n/ so that orbits of the D n action on Irr α ( A ) are { [1] , [ ρ ] } , { [ ρ ] , [ ρ n − ] } ,..., { [ ρ n/ ] , [ ρ n/ ] } .We can choose [ ρ ] , [ ρ ] , . . . , [ ρ n/ ] as representatives for the set D n \ Irr α ( A ) and we have G [ ρ ] = G [ ρ ] = · · · = G [ ρ n/ ] = A . As α | A is trivial we have ( e G [ ρ i ] ) α = A × S for i = 1 , . . . , n/ R α ( D n ) = Z τ ⊕ · · · ⊕ Z τ n/ ∼ = → n/ M i =1 K ∗ ( e G [ ρi ] ) α ,ρ i ( ∗ ) ∼ = n/ M i =1 Z ˜ ρ i τ i ˜ ρ i . On the other hand, for i = 1 , . . . , n/ Q [ ρ i ] = { } and β ρ i ,α is the trivial cocycle. Inthis case Theorem 3.7 gives us an isomorphismΦ : R α ( D n ) ∼ = Z τ ⊕ · · · ⊕ Z τ n/ ∼ = → n/ M i =1 K ∗{ } ( ∗ ) ∼ = Z n/ . References [1] A. Adem and Y. Ruan. Twisted orbifold K -theory. Comm. Math. Phys, 237, (2003), 3, 533–556.[2] A. ´Angel, J. M. G´omez and B. Uribe. Equivariant complex bundles, fixed points and equivariant unitarybordism. Algebr. Geom. Topol. 18, (2018), No 7, 4001-4035.[3] D. S. Freed, M. J. Hopkins and C. Teleman. Loop groups and twisted K -theory. Journal of Topology,4, (2011), 737–798.[4] J. M. Gmez, B. Uribe. A descomposition of equivariant K-Theory in twisted equivariant K-Theories.International Journal of Mathematics. 28, (2017), No. 2.[5] G. Karpilovsky, Group Representations , Vol. 2, North-Holland, Amsterdam, (1993).[6] G. Karpilovsky,
Group Representations , Vol. 3, North-Holland, Amsterdam, (1994).
Escuela de Matem´aticas, Universidad Nacional de Colombia, Medell´ın, Colombia
E-mail address : [email protected] Escuela de Matem´aticas, Universidad Nacional de Colombia, Medell´ın, Colombia
E-mail address : [email protected]@unal.edu.co