Dimensions of Prym Varieties

Given a tame Galois branched cover of curves pi: X ->Y with any finite Galois group G whose representations are rational, we compute the dimension of the (generalized) Prym variety corresponding to any irreducible representation \rho of G. This formula can be applied to the study of algebraic integrable systems using Lax pairs, in particular systems associated with Seiberg-Witten theory. However, the formula is much more general and its computation and proof are entirely algebraic.


Introduction
The most familiar Prym variety arises from a (possibly branched) double cover π : X → Y of curves. In this situation, there is a surjective norm map Nm : Jac(X) → Jac(Y ), and the Prym (another abelian variety) is a connected component of its kernel. Another way to think of this is that the involution σ of the double cover induces an action of Z/2Z on the vector space H 0 (X, ω X ), which can then be decomposed as a representation of Z/2Z. The Jacobian of the base curve Y and the Prym correspond to the trivial and sign representations, respectively. The Prym variety can be defined as the component containing the identity of (Jac(X) ⊗ Z ε) σ , where ε denotes the sign representation of Z/2Z.
The generalization of this construction that we will study in this paper is as follows. Let G be a finite group, and π : X → Y be a tame Galois branched cover, with Galois group G, of smooth projective curves over an algebraically closed field. The action of G on X induces an action on the vector space of differentials H 0 (X, ω X ), and on the Jacobian Jac(X). For any representation ρ of G, we define Prym ρ (X) to be the connected component containing the identity of (Jac(X) ⊗ Z ρ * ) G . The vector space H 0 (X, ω X ) will decompose as a Z[G]-module into a direct sum of isotypic pieces where ρ 1 , . . . , ρ N are the irreducible representations of G. If G is such that all of its representations are rational, then the Jacobian will also decompose, up to isogeny, into a direct sum of Pryms [D2]: In particular, if G is the Weyl group of a semisimple Lie algebra, then it will satisfy this property.
The goal of this paper is to compute the dimension of such a Prym variety. This formula is given in section 2, with a proof that uses only the Riemann-Hurwitz theorem and some character theory. Special cases of this formula relevant to integrable systems have appeared previously [A, Me, S, MS].
One motivation for this work comes from the study of algebraically integrable systems. An algebraically integrable system is a Hamiltonian system of ordinary differential equations, where the phase space is an algebraic variety with an algebraic (holomorphic, over C) symplectic structure. The complete integrability of the system means that there are n commuting Hamiltonian functions on the 2n-dimensional phase space. For an algebraically integrable system, these functions should be algebraic, in which case they define a morphism to an n-dimensional space of states for the system. The flow of the system will be linearized on the fibers of this morphism, which, if they are compact, will be n-dimensional abelian varieties.
Many such systems can be solved by expressing the system as a Lax pair depending on a parameter z. The equations can be written in the form d dt A = [A, B], where A and B are elements of a Lie algebra g, and depend both on time t and on a parameter z, which is thought of as a coordinate on a curve Y . In this case, the flow of the system is linearized on a subtorus of the Jacobian of a Galois cover of Y . If it can be shown that this subtorus is isogenous to a Prym of the correct dimension, then the system is completely integrable.
In section 3, we will briefly discuss two examples of such systems, the periodic Toda lattice and Hitchin systems. Both of these are important in Seiberg-Witten theory, providing solutions to N = 2 supersymmetric Yang-Mills gauge theory in four dimensions.
This work appeared as part of a Ph.D. thesis at the University of Pennsylvania. The author would like to thank her thesis advisor, Ron Donagi, for suggesting this project and for many helpful discussions. Thanks are also due to David Harbater, Eyal Markman, and Leon Takhtajan.

Dimensions
We can start by using the Riemann-Hurwitz formula to find the genus g X of X, which will be the dimension of the whole space H 0 (X, ω X ) and of Jac(X). Since π : X → Y is a cover of degree |G|, we get where g is the genus of the base curve Y and R is the ramification divisor.
The first isotypic piece we can find the dimension of is V 1 , corresponding to the trivial representation. The subspace where G acts trivially is the subspace of differentials which are pullbacks by π of differentials on Y . This tells us that dim In the case of classical Pryms, where G = Z/2, there is only one other isotypic piece, V ε corresponding to the sign representation ε. Thus we have For larger groups G, there are more isotypic pieces, but we also have more information: we can look at intermediate curves, i.e. quotients of X by subgroups H of G. Differentials on X/H pull back to differentials on X where H acts trivially. Thus The map π H : X/H → Y will be a cover of degree |G| |H| , so Riemann-Hurwitz gives us the following formula for the genus g H of X/H, which is the dimension of H 0 (X/H, ω X/H ): where again R H is the ramification divisor.
We can further analyze the ramification divisor, by classifying the branch points according to their inertial groups. Since π : X → Y is a Galois cover of curves over C, all of the inertial groups must be cyclic.
Lemma 1. Let G be a finite group all of whose characters are defined over Q. If two elements x, y ∈ G generate conjugate cyclic subgroups, then they are conjugate.
Proof (adapted from [BZ]): We want to show that for any character χ of G, χ(x) = χ(y). Then the properties of characters will tell us that x and y must be in the same conjugacy class.
Since the values of χ are rational, this element will act trivially, so χ(y) = χ(x).
From now on, we will suppose that G is such that all of its characters are rational. (This will be true, for instance, if G is a Weyl group). Pick representative elements h 1 . . . h N for each conjugacy class in G, and let H 1 . . . H N be the cyclic groups that each of them generates. By Lemma 1, this will be the whole set (up to conjugacy) of cyclic subgroups of G. We can partially order this set of cyclic subgroups by their size, so that H 1 is the trivial subgroup. Now we can classify the branch points: let R k , k = 2 . . . N be the degree of the branch locus with inertial group conjugate to H k (ignoring the trivial group). Over each point of the branch locus where the inertial group is conjugate to H k , there will be |G|/|H k | points in the fiber. Thus the degree of the ramification divisor R of π : X → Y will be For each quotient curve X/H, each point in the fiber of π H : X/H → Y over a point with inertial group H k will correspond to a double coset H k \G/H. Thus the degree of the ramification divisor R H will be Combining these formulas with the earlier Riemann-Hurwitz computations, we get: Since the genera g H are exactly the dimensions dim H 0 (X/H, ω X/H ), we also have For each subgroup H, this is a linear equation for the unknown dimensions dim V j in terms of the genus g H . Thus by taking quotients by the set of all cyclic subgroups H 1 . . . H N , we get a system of N equations. We wish to invert the matrix dim ρ H i j and find the N unknowns dim V j .
Lemma 2. The matrix dim ρ H i j is invertible. Proof: We show that the rows of the matrix are linearly independent, using the fact that rows of the character table are linearly independent. First, note that dim ρ H i j , the dimension of the subspace of ρ j invariant under H i , is equal to the inner product of characters Res G H i ρ j , 1 , which we can read off from the character table of G as Compare this matrix to the matrix of the character table χ ρ j (a i ). From (12) we see that each row is a sum of multiples of rows of the character table. Since each element of a subgroup has order less than or equal to the order of the subgroup, the rows of the character table being added to get row i appear at or below row i in the character table. Thus if we write the matrix dim ρ H i j in terms of the basis of the character table, we will get a lower triangular matrix with non-zero entries on the diagonal. By row reduction, we see that the linear independence of the rows of dim ρ H i j is equivalent to the linear independence of the rows of the character table.
Theorem 1. For each nontrivial irreducible representation ρ j of G, V j has dimension Proof: Since the matrix dim ρ H i j is invertible, there is a unique solution to the system of equations (11), so we only need to show that this is a solution. Namely, given this formula for dim V j , and combining (10) and (11), we wish to show that for each cyclic subgroup H i , 4 Note that on the left side we are summing over all representations, not just the nontrivial ones, so our notation will be simpler if we write dim V 1 = g in a similar form to (11). For the trivial representation ρ 1 , (dim ρ 1 ) − (dim ρ H k 1 ) = 0 (since ρ 1 is fixed by any subgroup H k ), so The sum on the left hand side of (14) will be Let us look at the (g − 1) term and the R H k terms separately. For the (g − 1) coefficient, we can write both dim ρ H i j and dim ρ j in terms of characters of G (as in (12)) and exchange the order of summation to get where e is the identity element of G. The inner sum amounts to taking the inner product of two columns of the character table of G. The orthogonality of characters tells us that this inner product will be zero unless the two columns are the same, in this case if a i = e. Thus the sum over elements in H i disappears, and we get the sum of the squares of the dimensions of the characters: which is what we want.
The R H k term looks like We can distribute and rearrange the sums to get: As in (17) and (18), the first term becomes |G| |H i | . The second term is also the inner product of columns of the character table: This will be zero unless a i and a k are conjugate, in which case χ ρ j (a i ) = χ ρ j (a k ) and character theory tells us (see for example [FH],p. 18 where c(a i ) is the number of elements in the conjugacy class of a i . Now the second term has become where the sum is taken over pairs of elements a i ∈ H i , a k ∈ H k such that a i and a k are conjugate. This is exactly the number of double cosets # (H k Adding up all of the terms, the sum on the left hand side becomes which is exactly the right hand side. Corollary 1. For each nontrivial irreducible representation ρ j of G, Prym ρ j (X) has dimension 3. Integrable Systems.
Periodic Toda lattice. The periodic Toda system is a Hamiltonian system of differential equations with Hamiltonian where p and q are elements of the Cartan subalgebra t of a semisimple Lie algebra g, and the sum is over the simple roots of g plus the highest root. This system can be expressed in Lax form [AvM] B], where A and B are elements of the loop algebra g (1) . and can be thought of as elements of g which depend on a parameter z ∈ P 1 . For sl(n), A is of the form  For any representation ̺ of g, the spectral curve S ̺ defined by the equation det(̺(A(z) − λI) = 0 is independent of time (i.e. is a conserved quantity of the system). The spectral curve is a finite cover of Y which for generic z parametrizes the eigenvalues of ̺(A(z)). While the eigenvalues are conserved by the system, the eigenvectors are not. The eigenvectors of ̺(A) determine a line bundle on the spectral cover, so an element of Jac(S ̺ ). The flow of the system is linearized on this Jacobean. Since the original system of equations didn't depend on a choice of representation ̺, the flow is actually linearized on an abelian variety which is a subvariety of Jac(S ̺ ) for every ̺.
In fact, instead of considering each spectral cover we can look at the cameral cover X → P 1 . This is constructed as a pullback to P 1 of the cover t → t/G, where G is the Weyl group of g. This cover is pulled back by the rational map P 1 t/G defined by the class of A(z) under the adjoint action of the corresponding Lie group. (For A(z) a regular semisimple element of sl(n), this map sends z to the unordered set of eigenvalues of A(z).) Thus the cameral cover is a finite Galois cover of P 1 whose Galois group G is the Weyl group of g. The flow of the Toda system is linearized on the Prym of this cover corresponding to the representation of G on t * . This is an r-dimensional representation, where r is the rank, so the dimension of this Prym is The ramification of this cover has been analyzed in [D1] and [MS]. There are 2r branch points where the inertial group H is Z/2Z generated by one reflection, so for each of these dim t H is r − 1. There are also two points (z = 0 and ∞) where the inertial group H is generated by the Coxeter element, the product of the reflections corresponding to the simple roots. This element of G doesn't fix any element of t, so for these two points dim t H = 0. Thus the dimension of the Prym is −r + (r − (r − 1)) 2r 2 + (r − 0) 2 2 = r.
Since the original system of equations had a 2r-dimensional phase space, this is the answer that we want.
Hitchin systems. Hitchin showed [H] that the cotangent bundle to the moduli space of semistable vector bundles on a curve Y has the structure of an algebraically completely integrable system. His proof, later extended to principal G bundles with any reductive Lie group G [F,S], uses the fact that this moduli space is equivalent (by deformation theory) to the space of Higgs pairs, pairs (P, φ) of a principal bundle and an endomorphism φ ∈ H 0 (Y, ad(P ) ⊗ ω Y ). As in the case of the Toda system, the key construction is of a cameral cover of Y . The eigenvalues of φ, which are sections of the line bundle ω Y , determine a spectral cover of Y in the total space bundle. The eigenvectors determine a line bundle on this spectral cover. The Hitchin map sends a Higgs pair (P, φ) to the set of coefficients of the characteristic polynomial. Each coefficient is a section of a power of ω Y , so the image of the Hitchin map is B : where the d i are the degrees of the basic invariant polynomials of the Lie algebra g.
Again, we can consider instead the cameral cover X b → Y , which is obtained as a pullback to Y vi φ of t ⊗ ω Y → t ⊗ ω Y /G. The generic fiber of the Hitchin map is isogenous to Prym ≈ (X), which has dimension By looking at the generic fiber, we can restrict our attention to cameral covers where the only ramification is of order two, with inertial group H generated by one reflection. The last piece of information we need to compute the dimension is the degree of the branch divisor of X → Y . The cover t ⊗ ω Y → t ⊗ ω Y /G is ramified where any of the roots, or their product, is equal to zero. There are (dim G −r) roots, so this defines a hypersurface of degree (dim G −r) in the total space of ω Y . The ramification divisor of X → Y is the intersection of this hypersurface with the section φ, which is the divisor corresponding to the line bundle ω . Thus the degree of the branch divisor will be (dim G − r)(2g − 2).
Combining all of this information, we see that the dimension of the Prym is By comparison, the dimension of the base space is The sum of the degrees d i of the basic invariant polynomials of g is the dimension of a Borel subalgebra, (dim G + r)/2. For g > 1, Riemann-Roch gives Which, as Hitchin said, "somewhat miraculously" turns out to be the same thing. Markman [Ma] and Bottacin [B]  Markman showed that the generic fiber of this system is again isogenous to Prym t (X), where X is a cameral cover of the base curve Y . The construction of the cameral cover is similar to the case of the Hitchin system, except that φ is a section of ad(P ) ⊗ ω Y (D). Thus the ramification divisor is (ω Y (D)) ⊗(dim G−r) , and the dimension is dim Prym t (X) = r(g − 1) + (dim G − r)(2g − 2 + deg D) 2 = dim G(g − 1) + (dim G − r) 2 deg D.
Again, this is the same dimension as the base of the system.