On the signature of biquotients

We generalize Hirzebruch's computation of the signature of equal rank homogeneous spaces to a large class of biquotients.


Introduction
The signature of a homogeneous space G/H, where H ⊂ G are compact Lie groups of equal rank, is explicitly computable from the root systems of G and H. This was shown by Hirzebruch [10], as a corollary of a more general result for compact oriented manifolds on which a circle acts with finite fixed point set, see Theorem 2.6 below.
In this note we generalize Hirzebruch's computation to a large class of equal rank biquotients, i.e., quotients of a compact Lie group G by the free action of a subgroup H ⊂ G × G with rk H = rk G by left and right multiplication. In this way we continue the topological study of biquotients by extending methods from homogeneous spaces, which already lead to an understanding of the Euler characteristic [13], cohomology [3], and rational homotopy [11] of biquotients.
Biquotients were originally considered by Eschenburg [4] in the context of Riemannian geometry, but also appear naturally in other geometries, such as symplectic [5] or Sasakian geometry [2]. In all these considerations, symmetries play an essential role. We will use the fact that any Lie subgroup of G × G that commutes with H naturally acts on G//H, yielding in particular circle actions on many such biquotients. Our main result, Theorem 4.1, is applicable to any such circle action with finite fixed point set. The main difference to the homogeneous setting is the fact that because we do not have a transitive action on the space at our disposal, we need to keep track of orientations, see Definition 3.11 below. To illustrate this issue, we have included a detailed example, see Section 4.1.
Acknowledgements. The results of this paper are contained in the master thesis of the second named author, written at the Philipps University of Marburg under the supervision of the first named author.

Actions on Homogeneous Spaces
In this section we present the known results on homogeneous spaces from [10].
Consider G a compact, connected Lie group and H ⊂ G a subgroup with rk(H) = rk(G). Fix a shared maximal torus T ⊂ H ⊂ G. Left multiplication with elements of the torus induces a well-defined action of T on the homogeneous space G/H by t · gH := (tg)H. The fixed point set of this action is well-known and in particular finite: Proof. See e.g. [7,Proposition 2.2] We now want to understand the weights of the isotropy representation in the fixed points. Denote by π : G −→ G/H the natural projection. Then: Then for any t ∈ T and v ∈ T gH G/H we have where X ∈ g satisfies dπ g (X g ) = v and w −1 (t) = g −1 tg.
Proof. For such a fixed point we define w −1 (t) := g −1 tg ∈ T . Then: Remark 2.3. Let ∆ H ⊂ ∆ G be the root systems of H and G with respect to T . The former proposition tells us that the weights of the isotropy representation in each fixed point gH, where g ∈ N G (T ), are the roots ∆ G \ ∆ H , up to sign, twisted by a representative of the fixed point, i.e. {Ad * g −1 α | α ∈ ∆ G \ ∆ H }. See also [7], where even more information was obtained, in form of the GKM graph of the T -action on G/H. Let us assume that H is connected. A choice of positive roots ∆ + G ⊂ ∆ G induces an orientation of G/H as follows: the weight space decomposition of G yields a decomposition which is the same as the decomposition of T eH G/H into the irreducible submodules of the isotropy representation of T at eH. Each g C α is one-dimensional and g C −α = g C α . Hence, when choosing basis vectors , the choice of ±α as positive corresponds to the choice of a real basis {X, ±Y } of (g C α ⊕ g C −α ) ∩ g and therefore gives an orientation of this two-dimensional real vector space. In total this induces an orientation of the vector space T eH G/H, and since G acts transitively on G/H by left multiplication, we get an orientation of the homogeneous space G/H (This will not work analogously for biquotients). It is convenient to consider 1 i α for every root α whenever we make use of the roots as real functionals on the Lie algebra of the maximal torus, because α has purely imaginary values on the Lie algebra of maximal torus as simultaneous eigenvalue of skew-symmetric endomorphisms.
This data is now sufficient to understand the signature of these spaces, defined by Definition 2.4. Let M be a compact, connected, orientable manifold of dimension 4n. By Poincaré duality, multiplication in the middle cohomology defines a bilinear, symmetric, non-degenerate product We define the signature σ(M) of M to be the signature of this inner product. We set the signature of manifolds whose dimension is not divisible by four to zero. Hirzebruch computed this (oriented-homotopy) invariant using the famous Atiyah-Singer-Index Theorem [9, p. 63-72]. For the special case of S 1 -manifolds with finite fixed point set he obtained in [10, Section 1.7.b)]: Theorem 2.6. Take M a compact, oriented, 2n-dimensional manifold on which S 1 acts with isolated fixed points. Denote by V (m i ) ∼ = C the oriented real S 1 -module defined by such that the orientations on the V (m i ) induce the given orientation on T p M. Then these m i are well-defined up to an even number of sign changes and If we feed in the results on the canonical torus action on equal rank homogeneous spaces, restrict our torus action to a circle which has the same fixed points as the torus, and fix sets of positive roots ∆ + G on G and ∆ H ⊂ ∆ G on H which induce an orientation on G/H as described in Remark 2.3, Hirzebruch's fomula yields [10, Theorem 2.5.]: This formula is then used in numerous papers (e.g. [1,14]) to compute the signature of homogeneous spaces. In the following sections we will generalize this result to a large class of biquotients.

Actions on Biquotients
In the following G will always denote a compact, connected Lie group, with maximal torus T max ⊂ G. Furthermore T shall denote a torus in T max × T max of dimension equal to the rank of G. We fix a complementary torus Let H ⊂ G × G be a closed, connected subgroup containing T with rk G = rk H. We assume that H (or, equivalently, T ) acts freely on G by (h 1 , h 2 ) · g = h 1 gh −1 2 , and we denote the H-orbit space by G//H. It is called a biquotient. We assume that H commutes with a subtorusT ⊂ T ′ , so that we get a well-defined action ofT on the biquotient G//H via (t 1 , t 2 )Hg = H(t 1 gt −1 2 ). The aim of this section is to understand the weights of the isotropy representation of this action in the fixed points. In the biquotient setting as above, in the special case H = T andT = T ′ , a similar statement is true. Let π : G → G//T be the projection. The preimage π −1 ((G//T ) T ′ ) is equal to the set of elements g ∈ G for which T max gT max is of minimal possible dimension, or equivalently equal to T g. This set clearly contains the normalizer N G (T max ). On the other hand, if g is in this set, then both T max g and gT max are equal to T max gT max , which implies that g ∈ N G (T max ). This implies The normalizer N G (T max ) acts on this finite set, because for all g, g ′ ∈ N G (T max ) we have g · T g ′ = g · (g ′ T max ) = (gg ′ )T max = T gg ′ . The subaction of T max is trivial, because for g ′ ∈ N G (T max ) and t ∈ T max , we have tg ′ ∈ T max g ′ = T g ′ . This implies that we obtain a free and transitive action of the Weyl group W (G) on (G//T ) T ′ . Proof. Take (g, g) ∈ H ∩ ∆G. Then (g, g)e = geg −1 = e and therefore (g, g) ∈ H e , so g equals e according to the freeness of the action.
Now we are able to compute the isotropy representation of this action in a fixed point.
Let g ∈ G be such that Hg ∈ (G//H)T . Then, because H acts freely on G, for each Invoking the defining equation of (s 1 , s 2 ) we compute using Lemma 3.3: Remark 3.6. The homomorphism ψ g depends on the choice of g, i.e. some representative of Hg.
Proof. As observed above, the freeness of the H-action implies that ψ and ψ g are welldefined. Let for (t 1 , t 2 ), (t 1 ,t 2 ) ∈T be (s 1 , s 2 ), (ŝ 1 ,ŝ 2 ) ∈ H as above. Then where we used thatT and H commute. It is clear that ψ and ψ g are continuous. But every continous homomorphism of Lie groups is differentiable.
For later purposes we need to determine the differential of ψ g .
Lemma 3.7. Denote by τ i :t → g and π i : h → g the respective projections to the i-th factor. Furthermore we consider the maps α :t −→ g given by α(X, X ′ ) = X − X ′ and β : h −→ g given by β(Y, Y ′ ) = Y − Y ′ . Then Proof. Writing (s 1 , s 2 ) = ψ(t 1 , t 2 ), we have , and differentiating this we obtain for (X 1 , X 2 ) ∈t which we can express as We note that β is injective, since h ∩ ∆g = ker(β) = 0, its image contains t max and α has image contained in t max . Therefore we have Now we can use this to differentiate the homomorphism ψ g , which was given by which completes our proof.
Corollary 3.9. If we fix an auxiliary biinvariant Riemannian metric on G and denote bŷ ∆ g the set of weights of the restriction of the adjoint representation of G on g to the subspace d(l g −1 ) e (ker dπ g ) ⊥ and the subtorus Im(ψ g ), the set of weights of the isotropy representation in the fixed point Hg is ∆ g := {d(ψ g ) * λ|λ ∈∆ g }.
Proof. In Proposition 3.4 we proved the commutativity of the following diagram: In order to get isomorphic representations we fix a biinvariant Riemannian metric on G, restrict to appropriate subspaces and finally achieve the following diagram: (d(l g −1 ) g (ker dπ g )) ⊥ (ker dπ g ) ⊥ T Hg (G//H) (d(l g −1 ) g (ker dπ g )) ⊥ (ker dπ g ) ⊥ T Hg (G//H). The weights of the above twisted adjoint representation are then the twisted weights {d(ψ g ) * λ|λ ∈∆ Hg }.
Remark 3.10. The most convenient situation occurs, whenT lies in T max × T max and for each fixed point Hg ∈ (G//H)T there exists a representative g ∈ N G (T max ). Then Im(ψ g ) lies in T max and the weights are pulled back roots associated to the maximal torus T max .
Definition 3.11. The weights are only well-defined up to sign. If we fix an orientation on G//H, we denote by ∆ + g the set of weights ∆ g with fixed signs, such that the oriented weight space decomposition where T Hg (G//H) α is the weight space corresponding to the weight α, induces the set orientation on T Hg (G//H).

Signature
Just as in the homogeneous case we can now invoke Hirzebruch's signature formula to prove a result on the signature of biquotients. Proof. Since G//H is compact, the fixed point set is finite. Fixing an orientation on G//H, while having Corollary 3.9 and Definition 3.11 in mind, carries us directly to the situation of Theorem 2.6. We can apply Hirzebruch's Theorem 2.6 for oriented S 1 -manifolds which implies the announced formula.
Remark 4.2. By [13, Corollary 3.4. and Property 1.7.] G//H is orientable whenever G and H are connected. In that case, we can orient G//H as follows. By introducing a bi-invariant auxiliary Riemannian metric on G we can make the following identifications: which gives us a splitting Therefore fixing orientations of G and H we get an orientation of each orbit H · g and an induced orientation of its normal space ν(H · g), which is by the previous considerations isomorphic to T Hg G//H. Note that the orientation of the orbit is the blockwise embedding and SU (5) is embedded in the upper left corner. Let T ⊂ H be the maximal torus given by diagonal matrices in both components. We will compute the signature of the biquotient G//H, in order to illustrate our formula. This will not be a new result; as G//H = ∆ 3 (SU(2))\ SU(6)/ SU(5) ∼ = ∆ 3 (SU(2))\S 11 ∼ = HP 2 , the signature is well-known to be ±1. The first step is to find a subtorus of G × G which commutes with H and acts with finite fixed point set on G//H, and determine the weights of the isotropy representation in each fixed point. Such a torus is for example given byT = {diag(λ, λ, λ −1 , λ −1 , 1, 1)|λ ∈ S 1 } × {1}. We note thatT is contained in the flipped torus T ′ = {(t 1 , t 2 ) | (t 2 , t 1 ) ∈ T }. It is easily seen that the action ofT on G//H ∼ = HP 2 is given by λ · [q 1 : q 2 : q 3 ] = [λq 1 : λ −1 q 2 : q 3 ] because the diffeomorphism SU(6)/ SU(5) ∼ = S 11 is just projection on the last column.
We define g 1 , g 2 , g 3 as the above representatives of the fixed points. Note that we are in the situation of Remark 3.10.
Throughout this example, we denote by V jk ⊂ su(6), where j, k = 1, . . . , 6, j = k, the span of E ij − E ji and i(E ij + E ji ). This is the root space of the adjoint representation of the standard maximal torus on su(6) of the root ±(e i − e j ). By choosing the set of positive roots {e i − e j | i < j} we induce an orientiation on V ij , with respect to which the above fixed basis is positively oriented. We thus obtain an orientation on su(6) = t max ⊕ i<j V ij by declaring the basis {i(E 11 − E 66 ), . . . , i(E 55 − E 66 )} of t max to be positively oriented.
Using the Frobenius inner product or equivalently the Killing form on SU(6) we can determine the complements ker(dπ) ⊥ g i ∼ = T Hg i G//H. We obtain (6), By Corollary 3.9, the weights of theT -isotropy representation in the three fixed points are where we now denote by 1 i (e i − e j ) the restrictions of the realifications of the usual roots to the tori Im(ψ g k ). We now have to choose appropriate signs of these weights, i.e., define compatible sets of weights ∆ + g k as in Definition 3.11. For every k, the subspace (6) is the sum of the Lie algebra of the maximal torus of su(6) and certain root spaces, and hence oriented by our conventions above. Using the bases above, and taking into account the embeddings of su(2) and su(5) into su (6), in order to define ∆ + g k we have to determine if the natural maps (6).