TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000–000 S 0002-9947(XX)0000-0

ORTHOGONAL, SYMPLECTIC AND UNITARY REPRESENTATIONS OF FINITE GROUPS

CARL R. RIEHM

Abstract. Let K be a field, G a finite group, and ρ : G → GL(V) a linear representation on the finite dimensional K-space V. The principal problems considered are: I Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms h : V × V → K which are G-invariant. II If h is such a form, enumerate the equivalence classes of representations of G into the corresponding group (orthogonal, symplectic or unitary group). III Determine conditions on G or K under which two orthogonal, symplectic or unitary representations of G are equivalent if and only if they are equivalent as linear representations and their underlying forms are “isotypically” equivalent. This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) KG-module components of their spaces are equivalent. We assume throughout that the characteristic of K does not divide 2|G|. Solutions to I and II are given when K is a finite or local field, or when K is a global field and the representation is “split”. The results for III are strongest when the degrees of the absolutely irreducible representations of G are odd – for example if G has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index – and, in the case that K is a local or global field, when the representations are split.

Let G be a finite group of order g, K a field of characteristic relatively prime to 2g with an involution (possibly the identity) a ↦ ā, and ρ : G → GL(V) a linear representation of G on the finite dimensional K-vector space V.

The group algebra KG has an involution ¯ extending that on K and inverting the group elements. Since KG is a semisimple algebra, the involution algebra (KG, ¯ ) is a direct sum of simple involution algebras

(KG, ¯ ) = (A_{1}, ¯ ) ⊕ (A_{2}, ¯ ) ⊕ ⋯ ⊕ (A_{r}, ¯ )

1991 Mathematics Subject Classification. Primary 20C99, 11E57; Secondary 20F29, 11E08, 11E12.

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

where each A_{i} is either a simple algebra stabilized by the involution or is a “simple hyperbolic algebra” – a direct sum of two simple algebras interchanged by the involution. Now let

h : V × V → K

be a symmetric, skew symmetric or Hermitian form on V (we assume throughout that all forms are nonsingular) which is G-invariant. The decomposition

V = A_{1}V ⊕ A_{2}V ⊕ ⋯ ⊕ A_{r}V.

of the KG-module V into “isotypic” components is an orthogonal direct sum. (A_{i}V is the direct sum of two isotypic components in the usual sense if A_{i} is hyperbolic). The restrictions of h to these isotypic components are called the isotypic components of h, and two forms are called isotypically equivalent if their corresponding isotypic components are equivalent.

To say that h is G-invariant is the same as saying that ρ is an orthogonal representation G → O(V, h), a symplectic representation G → Sp(V, h) or a unitary representation G → U(V, h), according to the type of h. We refer to such representations as equivariant representations.

Two equivariant representations ρ and ρ′ are (equivariantly) equivalent if there is an isometry between their forms which is also KG-linear. There are therefore two obvious necessary conditions for ρ and ρ′ to be equivalent, namely

(i) ρ and ρ′ are equivalent as linear representations, and (ii) their underlying forms are isotypically equivalent.

This allows us to restrict our attention by and large to the isotypic case, that is to say when V = A_{i}V for some i. (In this case we say that V, or the associated representation of G, is A_{i}-isotypic or isotypic of type A_{i}.) Furthermore if the involution algebra (A_{i}, ¯ ) is hyperbolic, linear equivalence implies equivariant equivalence, and so we can assume in addition that A_{i} is a simple algebra.

Problems I, II and III were solved earlier for K algebraically closed or real closed.

In the algebraically closed case, Frobenius showed that two orthogonal or two symplectic representations are equivalent if and only if they are equivalent as linear representations ([13], pp. 184-186); furthermore Frobenius and Schur characterized linear representations which are equivalent to an orthogonal or symplectic representation (cf.[22], §13.2), that is to say, for which there is a G-invariant symmetric or skew symmetric form.

In the case of a real closed field K, Fröhlich and McEvett showed ([9], Prop. 4.9) that two unitary (over K(\sqrt{−1})) or orthogonal representations are equivalent if and only if they are linearly equivalent and their underlying forms are isotypically equivalent. They also showed that an invariant symmetric or Hermitian form on an irreducible KG-module is definite. Of course the existence of an invariant positive definite symmetric or Hermitian form on an arbitrary KG-module V is well known when K is formally real – one such form is ∑_{s∈G} f(sx, sy) where f is any positive definite symmetric or Hermitian form on V.

Recently, G. Nebe has given a method in [16] for calculating, under certain circumstances, the determinant or the Hasse invariant of an invariant symmetric form of a representation of a perfect group over a totally real number field, and in [17] she gives a recursion formula for the invariant symmetric form of certain irreducible representations of a symmetric group. This form is also determined, using different methods, in [10] – we give an example in §6.

Another result relevant here is the following (cf. §1.3): if ρ : G → GL(V) is a given linear representation, there is at least one G-invariant form h of a given type (symmetric, skew symmetric or Hermitian) on V if and only if the following holds:

1. In the symmetric case, len_{A}AV is even if (A, ¯ ) is a simple symplectic component of (KG, ¯ ) of index 1.

2. In the skew symmetric case, len_{A}AV is even if (A, ¯ ) is a simple orthogonal component of (KG, ¯ ) of index 1.

3. dim_{K} AV = dim_{K} ĀV for all simple algebra components A of KG. This is an easy consequence of results of McEvett [14].

Before summarizing our results, it is necessary to introduce some additional notions and notation. Unexplained terms can be found in §1.

Let A be a simple K-algebra, say A ≅ M(n, D) for the division algebra D with center L. If ¯ is an involution on A stabilizing K, K_{0} denotes the subfield of elements of K fixed by the involution, and L_{0} that of L.

An arbitrary linear representation is called split if it is the sum of absolutely irreducible representations. It is said to be of odd type if AV = 0 for each simple summand A of KG which has even degree and which is stable under the canonical involution. We shall say that G has odd representation type (with respect to K and its involution) if all nonhyperbolic simple involution components of KG are quasisplit and of odd degree. This is the case, for example, if G has odd order.

In general, we ignore the case of a unitary representation with respect to a skew Hermitian form since any such form is a scalar multiple of an Hermitian form.

Since skew symmetric forms are equivalent if and only if they have the same rank, symplectic representations are particularly easy to deal with. In particular since a space V has at most one equivalence class of skew symmetric forms, invariant or not, problem I is trivial.

Symplectic representations. (cf. §2)

II Let A be a simple algebra component of KG, stable under ¯ , and let m be a positive integer.

(A, ¯ ) orthogonal. Suppose that A has index 1 or that K is finite, real closed, local or global. Then there is exactly one equivalence class of A-isotypic symplectic representations G → Sp(V) of length m (assuming that there is at least one such symplectic representation – i.e. that m is even if the index of A is 1).

(A, ¯ ) symplectic. The equivalence classes of A-isotypic symplectic representations G → Sp(V) of length m are in bijective correspondence with the equivalence classes of symmetric bilinear forms over L respectively skew Hermitian forms over D of rank m, if A has index 1 respectively > 1. It follows that the number of equivalence classes of such symplectic representations G → Sp(V) is 2 if K is finite, (L^{∗} : L^{∗2}) resp. 2(L^{∗} : L^{∗2}) if K is local, A has index 1, and

m = 1 respectively m > 1,

(L^{∗} : L^{∗2}) − δ_{1m} if K is local and A has index > 1, 1 if K is real closed, ∞ if K is global.

(A, ¯ ) unitary. The equivalence classes of A-isotypic symplectic representations G → Sp(V) of length m are in bijective correspondence with the skew Hermitian forms of rank m over D. Thus the number of equivalence classes of such symplectic representations G → Sp(V) is 1 if K is finite, 2 if K is local, m + 1 if K is real closed, ∞ if K is global.

III Let ρ and ρ′ be symplectic representations of G. Suppose that at least one of the following conditions holds:

1. G is a symmetric group.

2. K is finite.

3. ρ is split.

4. If (A, ¯ ) is a unitary simple involution component of KG such that AV ≠ 0 under the action provided by ρ, then (A, ¯ ) is hyperbolic.

Assume in addition that ρ is of odd type in cases 2., 3. and 4.; then ρ and ρ′ are equivalent if and only if they are linearly equivalent.

From this we get a conjugacy result: Let G and G′ two subgroups of Sp(V, h) of order relatively prime to char K such that either

1. G is a symmetric group, or 2. G and G′ are of odd representation type, and K is either finite or a

splitting field for G. Then G and G′ are conjugate in Sp(V, h) if and only if they are conjugate in GL(V).

For the determinant det(¯ ) of an involution, see §1.2.

Unitary representations. (cf. Theorem 3.1, Theorem 4.1, Corollary 4.4, Theorem 5.2).

K finite.

I, II and III There is only one equivalence class of Hermitian forms of a given rank, invariant or not, and two arbitrary (not necessarily isotypic) unitary representations of G are equivalent if and only if they are linearly equivalent.

K local.

I and II Let A be a simple algebra component of KG, stable under ¯ , V an A-module of length m, and h an Hermitian form on V. There is a G-invariant Hermitian form on V equivalent to h if and only if n is odd or det h = N_{L/K}(det(¯ ))^{m}. If h′ is an invariant form, there is exactly one class of unitary representations G → U(V, h′) if n is odd; if n is even, there are two. III Two unitary representations of G of odd type are equivalent if and only if they are equivalent as linear representations and their underlying forms are isotypically equivalent.

K global. Assume that A is a simple component of KG which is split and is stable under ¯ . Let V be an A-module of length m and h an Hermitian form on V. Let ℘ be the set of real primes of K_{0} which do not split in K.

I There is an invariant form on V equivalent to h if and only if the positive and negative indices of h are divisible by n at each 𝔭 ∈ ℘ and, when n is even, det h = (det(¯ ))^{m}. II Let h′ be an invariant form on V.

(i) If n is odd, there is exactly one equivalence class of unitary representations G → U(V, h′) for which V is A-isotypic. (ii) If n is even, the equivalence classes of unitary representations G → U(V, h′) for which V is A-isotypic are in bijective correspondence with those norm classes in K_{0}^{∗}/N_{K/K_{0}}K^{∗} which are positive at each 𝔭 ∈ ℘.

III Two split unitary representations of G of odd type are equivalent if and only if they are equivalent as linear representations and their forms are isotypically equivalent.

Witt equivalence between symmetric forms is denoted by h ∼_{W} g.

Orthogonal representations. (cf. Theorem 3.1, Theorem 4.1, Corollary 4.4, Theorem 4.9, Theorem 5.3). Let A be a simple algebra component of KG, stable under ¯ , and let m be a positive integer.

K finite. I, II Let V be an A-isotypic module of length m.

(A, ¯ ) symplectic. V supports an invariant symmetric form if and only if m is even. If V does have an invariant symmetric form, every invariant form is hyperbolic and there is exactly one equivalence class G → O(V) of A-isotypic orthogonal representations. (A, ¯ ) orthogonal. When n is odd, there are invariant forms of both determinants on V and there is exactly one equivalence class of orthogonal representations for each of these forms. When n is even, an invariant form h on V has determinant N_{L/K}(det(¯ ))^{m}, and there are two equivalence classes of orthogonal representations G → O(V, h). (A, ¯ ) unitary. An invariant form on V has determinant (−disc(L/L_{0}))^{mn|L_{0}^{∗}|/|K^{∗}|} and supports exactly one equivalence class of orthogonal representations.

III Let ρ : G → O(V, h) be an orthogonal representation such that BV = 0 for every component (B, ¯ ) of (KG, ¯ ) which is orthogonal and of even degree. If ρ′ is another orthogonal representation on V, then ρ and ρ′ are equivalent if and only if they are linearly equivalent and their forms are isotypically equivalent. In particular if G has odd representation type, two orthogonal representations are equivalent if and only if they are linearly equivalent and their forms are isotypically equivalent.

K local. I, II See Theorems 4.1 and 4.9.

III Let ρ and ρ′ be orthogonal representations of G of odd type, and assume that either ρ is split or G is Abelian. Then ρ and ρ′ are equivalent if and only if they are linearly equivalent and their forms are isotypically equivalent.

K global. Assume that A is split.

(A, ¯ ) symplectic I, II If h is an invariant symmetric form on the A-isotypic module V, it is hyperbolic and the orthogonal representation it affords is unique up to equivalence.

(A, ¯ ) orthogonal I See Theorem 5.3.

II If h is an invariant symmetric form on the A-isotypic module V, there is (up to equivalence) only one orthogonal representation G → O(V, h) if n is odd, and there are an infinite number of equivalence classes of orthogonal representations G → O(V, h) if n is even.

III Two split orthogonal representations of odd type are equivalent if and only if they are equivalent as linear representations and their underlying forms are isotypically equivalent.

The case K local, (A, ¯ ) orthogonal, D = L ≠ K, n odd, and m = 1 is not completely resolved; its solution depends on the calculation of “trace forms” in the extension L/K. This is carried out in Theorem 4.9 when K is nondyadic (which means, given our assumptions on K, that K is not a finite extension of the 2-adic numbers Q_{2}). The dyadic case is unresolved but a partial solution (for the trace form problem) can be found in [6].

The proofs, when V = AV with A simple, are based on the determination of h in terms of the matrix h_{0} (cf. §1.2) and the discriminant matrix \underline{h̆} of the form h̆ over D which corresponds to ρ under Hermitian Morita equivalence (cf. §1.4). This computation appears in its simplest form when A is split: then the discriminant matrix of h is simply \frac{n}{g}(ϵ_{0}\underline{h̆} ⊗ h_{0}^{t}) where h_{0}^{∗} = ϵ_{0}h_{0}.

There is an extensive theory calculating the Witt group of equivariant representations, even in the integral case – cf. [23], [4]. But this work does not concern itself with the questions dealt with in this paper.

If K is a global field, we give a proof of the Hasse Principle (Theorem 5.6) for equivariant representations of finite groups over K. It describes the precise conditions under which the equivalence of two equivariant representations over a global field K is implied by their equivalence over all completions of K.

This theorem is well-known in the sense that anyone familiar with the Fröhlich-McEvett theory [9] and the Hasse Principle for ϵ-Hermitian forms over division algebras over global fields (cf. [21], p. 347), is aware that such a result exists. But to my knowledge, it is not documented in the literature.

The Hasse Principle always holds for unitary representations, but its validity in the orthogonal and symplectic cases depends on the nature of the representations as linear representations – see Corollary 5.10.

1. Preliminaries

1.1. Sesquilinear forms over fields and division algebras. If h : V × V → D is a form over D, we denote by det h ∈ L^{∗}/N_{L/L_{0}}L^{∗} the “norm class” of the reduced norm nrd_{B/L}\underline{h}, where \underline{h} ∈ B = M(k, D) is the matrix of h with respect to some basis of V over D and k = dim_{D} V. We interpret N_{L/L_{0}}L^{∗} to be = L^{∗2} if the involution is the identity on L, i.e. if the involution on D is of the first kind. We note that det h ∈ L_{0}^{∗}/N_{L/L_{0}}L^{∗} if h is Hermitian.

We also define the discriminant of a symmetric bilinear form h (when D = L) to be disc h = (−1)^{k(k−1)/2} det h, and that of a skew Hermitian form over L to be disc h = det \sqrt{λ_{0}}h ∈ L_{0}^{∗}/N_{L/L_{0}}L^{∗} where L = L_{0}(\sqrt{λ_{0}}). Of course \sqrt{λ_{0}}h is an Hermitian form. We put disc h = det h if h is Hermitian.

Lemma 1.1. Let L/K be a finite separable extension such that if the involution on L is ≠ the identity, then it is also ≠ the identity on K. Suppose that h : V × V → L is either a nonsingular symmetric bilinear form or a nonsingular Hermitian or skew Hermitian form of rank k. Then the transfer Tr_{L/K}h : V × V → K has determinant

det Tr_{L/K}h = (disc(L/K))^{k}N_{L/K}(det h),

where disc(L/K) is the field discriminant. Similarly

disc Tr_{L/K}h = ((−1)^{l(l−1)/2}disc(L/K))^{k}N_{L/K}(disc h),

where l = (L : K). If h is Hermitian, we also have

det Tr_{L/K}h = (disc(L_{0}/K_{0}))^{k}N_{L_{0}/K_{0}}(det h).

A proof for the bilinear case can be found, e.g., in [15], Lemma 2.2, and a similar proof applies to the Hermitian and skew Hermitian cases if one carries out the calculation using a basis of L_{0} over K_{0}. □

The following result of Milnor ([15], pp. 89 and 91) is critical in the study of equivariant representations over local fields:

Lemma 1.2. If L/K is a finite separable extension of local fields, the transfers from L to K of two inequivalent nonsingular symmetric forms with the same determinant are inequivalent over K. Similarly if L has a nonidentity involution which is the identity on K, the transfers of two inequivalent Hermitian forms from L to K remain inequivalent, in fact have the same determinant but different Hasse symbols.

Let b = ⟨a_{1}, …, a_{m}⟩ be a diagonalization of the symmetric bilinear form b. We use the definition S_{K}(b) = S(b) = ∏_{i<j}(a_{i}, a_{j})_{K} of the Hasse-Witt invariant, where (a_{i}, a_{j})_{K} stands for the Brauer class of the quaternion algebra

– which we also denote by (a_{i}, a_{j})_{K}.

Lemma 1.3. (i) If η ∈ K^{∗},

S(ηb) = S(b)(η, −1)_{K}^{m(m−1)/2}(η, det b)_{K}^{m−1}.

(ii) If b_{1}, …, b_{r} are symmetric forms over K,

S(b_{1} ⊥ … ⊥ b_{r}) = ∏_{i} S(b_{i}) ∏_{i<j}(det b_{i}, det b_{j})_{K}.

(iii) If η_{1}, …, η_{n} ∈ K^{∗},

S(⟨det η_{1}b, …, det η_{n}b⟩) = S(⟨η_{1}^{m}, …, η_{n}^{m}⟩)(det b, −1)_{K}^{n(n−1)/2}(det b, η_{1} ⋯ η_{n})_{K}^{m(n−1)}.

(iv) Let h_{0} be another symmetric form, say of rank n. Then

S(b ⊗ h_{0}) = S(b)^{n}S(h_{0})^{m}(det b, −1)_{K}^{n(n−1)/2}(det b, det h_{0})_{K}^{mn−1}(det h_{0}, −1)_{K}^{m(m−1)/2}.

(v) If b is hyperbolic of dimension m = 2k, then

S(b) = (−1, −1)_{K}^{k(k−1)/2}.

These are either well known or easily checked. □

The equivalence theory of forms over finite, real closed, local (nonArchimedean) and global fields is summarized in the following table. It is derived from [21], p. 347. In it, “1^{st} k.” refers to the involution being of the first kind (cf. §1.2), and “Hasse” means that the Hasse principle holds.

Table 1

Type of form  finite  real  local  global
symmetric  dim, det dim, sgn dim, det, S  Hasse
skew-symmetric  dim  dim  dim  dim
Herm/quat.(1^{st} k.)  ∄  dim, sgn  dim  Hasse
skew-Herm/quat.  ∄  dim  dim, det
Herm/field  dim dim, sgn dim, det dim, det, sgn_{𝔭}
Herm/quat.(2^{nd} k.)  ∄  ∄  ∄  dim, det, sgn_{𝔭}

1.2. Simple involution algebras and representations of finite groups.

Lemma 1.4. The center L of any simple direct summand A of KG is Abelian over K. And if A is stable under the canonical involution ¯ of KG, L/K_{0} is also Abelian.

Proof. If K is a finite field, L/K is of course Abelian, and the same is true for an arbitrary field K of characteristic ≠ 0 by extension of scalars from the finite case. If the characteristic of K is 0, it is well known that L is a subfield of a cyclotomic extension of K (cf. [3], §70) – in fact it is generated by the character values of any absolutely irreducible representation ρ arising from A – and so is Abelian.

Suppose then that A is stable under the involution of KG and that K ≠ K_{0}. Let A_{1} be the simple direct summand of K_{0}G from which A arises by extension of scalars. If L_{1} is the center of A_{1}, K ⊗_{K_{0}} L_{1} is the center of K ⊗_{K_{0}} A_{1} ⊂ KG. By [1], Prop. 3, K ⊗_{K_{0}} L_{1} is the direct sum of the composites of K and L_{1} over K_{0}, and therefore either K ⊗_{K_{0}} L_{1} is a field, namely L, or K ⊗_{K_{0}} L_{1} ≅ L_{1} ⊕ L_{1}. In the first case, the group generated by the automorphisms ¯ ⊗ id_{L_{1}} and id_{K} ⊗ σ (σ ∈ Gal(L_{1}/K_{0})) is Abelian and has order 2[L_{1} : K_{0}] and so is the Galois group of L/K_{0}. In the second case L_{1} = L.

By Brauer’s Theorem ([22], p. 24), every representation of G over K is split (i.e. K is a splitting field of G) if the exponent of G divides the order |μ(K)| of the group of roots of unity of K, or if it divides the order of the multiplicative group of the residue class field when K is local.

We note that G has odd representation type if it has odd order or is Abelian, or more generally if it has a normal Abelian subgroup of odd index. See [22], Corollary, p. 61, for the case of characteristic 0. The case of prime characteristic p follows from the fact that any absolutely irreducible representation over a field of characteristic p which contains a primitive g^{th} root of unity, is the reduction “mod p” of an absolutely irreducible representation in characteristic 0.

The algebra A ≅ M(n, D) is called split if D = K, quasisplit or of (Schur) index 1 if D = L. Otherwise A is “not quasisplit” or it is “of index > 1”. The index of A is d and the degree of A is nd where dim_{L} D = d^{2}.

An involution on the simple algebra A is of the first kind if the involution is the identity on the center of A, otherwise it is of the second kind, or unitary. An involution of the first kind is orthogonal if its 1-eigenspace has dimension \frac{1}{2}nd(nd + 1) (where nd is the degree of A) over the center of A, otherwise is symplectic – in which case the 1-eigenspace has dimension \frac{1}{2}nd(nd − 1).

A hyperbolic involution component of KG is also considered to be of the second kind.

If the simple algebra A ≅ M(n, D) has an involution, then the division algebra D has one of the same kind. If D is not a field, it can be chosen to be a symplectic involution; in this case we denote it by ^{∗}.

If D is a quaternion algebra over K, ^{∗} will be assumed to be the standard conjugation on D. In this case D_{0} denotes the skew conjugate elements, the “pure” quaternions. If ¯ is an orthogonal involution on D, d̄ = j^{−1}d^{∗}j for j ∈ D_{0} uniquely determined up to a nonzero element of K.

Note that if K is a local field, a quaternion division algebra D does not admit an involution of the second kind ([21], Theorem 2.2 (ii), p. 353).

The symbol a^{∗} will also denote the “transpose conjugate” of a matrix a ∈ M(n, D), or more generally of a rectangular matrix with entries in D. If A is identified with M(n, D), there is a matrix h_{0} ∈ A such that ā = h_{0}^{−1}a^{∗}h_{0}, satisfying h_{0}^{∗} = ϵ_{0}h_{0} for ϵ_{0} = ±1; we can and shall suppose that ϵ_{0} = 1 if the involution on A is unitary.

Suppose that the involution ¯ on A is of the first kind. If D = L, the transpose on A is an orthogonal involution, and ¯ is an orthogonal respectively symplectic involution if h_{0} is a symmetric respectively skew symmetric matrix. If D is a quaternion algebra, transpose conjugate is a symplectic involution, and the involution ¯ on A is a symplectic respectively orthogonal involution if h_{0} is an Hermitian respectively skew Hermitian matrix (with respect to ^{∗}).

If the involution ¯ on A is orthogonal and A has even degree nd, its determinant det(¯ ) ∈ L^{∗}/L^{∗2} and discriminant disc(¯ ) = (−1)^{nd/2} det(¯ ) are defined. If D = L, det( ¯ ) = det h_{0} and if D is a quaternion algebra, disc(¯ ) = nrd_{A/L}h_{0}. See [11], Prop. 7.3(2),(3), p. 81.

It is useful to define as well the determinant of an involution of the second kind on a central simple algebra A of even degree. In this case, with the assumption ϵ_{0} = 1, we define det(¯ ) = nrd_{A/K}h_{0} ∈ L_{0}^{∗}/N_{L/L_{0}}L^{∗}. It is not difficult to check that it is well-defined since A has even degree.

We note that it follows from [14], 2.5, that a simple module V over A supports a nonsingular Hermitian form respectively skew Hermitian form h : V × V → A if the simple involution algebra (A, ¯ ) is orthogonal respectively symplectic – and if A is has index > 1 or if (A, ¯ ) is unitary, it supports both. In the exceptional case when A has index 1 and the involution is of the first kind, it supports only an Hermitian form if the involution is orthogonal and only a skew Hermitian form if the involution is symplectic.

Every orthogonally indecomposable nonsingular Hermitian or skew Hermitian space (V, h) over a simple hyperbolic involution algebra is hyperbolic (of length 2), and is uniquely determined up to equivalence.

Now let K be a real closed field, and denote by H the unique quaternion division algebra (−1, −1)_{K} over K. It is known that all simple components of KG are stable under the canonical involution (which is the identity on K), and that for each of the three possibilities A ≅ M(n, D), D = K, K(\sqrt{−1}), H, the induced involution is conjugate transpose; it is orthogonal, unitary or symplectic respectively. See for example [9] or [21], 8.13.3, p. 323.

1.3. Reformulation of the equivalence of equivariant representations. The canonical trace Tr_{KG/K} = Tr : KG → K is the K-linear functional

Tr(∑_{s∈G} α_{s}s) = α_{1}.

It is clear that this is the same thing as \frac{1}{g}Tr_{reg} where Tr_{reg} is the “algebra trace” arising from the regular representation.

If A_{1} is a separable K-algebra, recall that trd = trd_{A_{1}/K} : A_{1} → K denotes the reduced trace.

Theorem 1.5. If A ≅ M(n, D) is an arbitrary simple component of KG, then for all a ∈ A

Tr_{KG/K}a = \frac{nd}{g}trd_{A/K}a. (1.1)

Proof. The regular trace of A is the restriction of the regular trace of KG, and is also nd trd_{A/K}.

Let h : V × V → K be a (nonsingular, as always) ϵ-Hermitian form with ϵ = ±1 – if the involution on K is the identity, we interpret this to mean that h is symmetric or skew symmetric – and ρ : G → I(V, h) an equivariant representation, I = O, Sp, or U. The equivalence class of ρ is determined by the equivalence class of the nonsingular ϵ-Hermitian form

ĥ : V × V → KG, ĥ(u, v) = ∑_{s} h(ρ(s^{−1})u, v)s (1.2)

over the group algebra KG with the canonical involution (cf. [8], §7). Note that

Tr_{KG/K} ∘ ĥ = h, (1.3)

and that ĥ is the unique ϵ-Hermitian form on V satisfying (1.3).

Let (KG, ¯ ) = ⊕_{i=1}^{r}(A_{i}, ¯ ) be the decomposition of (KG, ¯ ) into simple involution components. The associated orthogonal decomposition

V = A_{1}V ⊥ A_{2}V ⊥ ⋯ ⊥ A_{r}V

with respect to h is also an orthogonal decomposition with respect to the form ĥ. The restriction ĥ_{i} of ĥ to the isotypic component A_{i}V takes its values in A_{i} and so can be considered as an ϵ-Hermitian form ĥ_{i} : A_{i}V ×A_{i}V → A_{i}. Since any isomorphism of KG-modules is an isomorphism on corresponding isotypic components, the equivalence class of the equivariant representation ρ is therefore determined by the equivalence classes of the ĥ_{i}. If A_{i} is a simple hyperbolic involution algebra, the equivalence class of ĥ_{i} is completely determined by the length of the A_{i}-module A_{i}V (see [14], §2, especially 2.3), and hence by the equivalence class of ρ as a linear representation.

This means that ρ is determined up to equivalence by its linear equivalence class and its isotypic subrepresentations G → I(A_{i}V) where A_{i} runs over the simple algebras fixed by the involution of KG. Suppose that these simple algebras are A_{1}, …, A_{k}.

We can now apply the Hermitian Morita theory ([8], §8, or §1.4 of this paper) to ĥ_{1}, …, ĥ_{k}. This yields forms h̆_{1}, …, h̆_{k} where h̆_{i} is a symmetric, skew symmetric, Hermitian or skew Hermitian form over the division algebra which is Brauer equivalent to A_{i}. Once again the equivalence classes of the h̆_{i} characterize the equivalence class of ρ. Moreover if ρ is orthogonal respectively symplectic and A_{i} has index 1 with ¯ symplectic respectively orthogonal, then h̆_{i} is a skew symmetric form and so is completely determined by dim_{K} A_{i}V. Thus we can ignore such A_{i} as well.

Therefore let A_{1}, …, A_{l} be the simple summands of KG such that Ā_{i} = A_{i} and such that if ρ is orthogonal respectively symplectic and A_{i} has index 1, then the involution ¯ on A_{i} is orthogonal respectively symplectic. Then the equivalence class of ρ is determined by its equivalence class as a linear representation and by the equivalence classes of h̆_{1}, …, h̆_{l}.

Another simple consequence of this procedure is that if ρ : G → GL(V) is a given linear representation, there is always a G-invariant form h of a prescribed type on V if and only if the following holds for each simple component A of KG:

1. If A is stable under the involution of KG and has index 1, then len_{A}AV is even if either

(i) (A, ¯ ) is symplectic and h is to be symmetric, or (ii) (A, ¯ ) is orthogonal and h is to be skew symmetric.

2. dim_{K} AV = dim_{K} ĀV. Condition 1. arises from the fact that if D is a division algebra with an involution, there is a nonsingular form h̆ of any type (symmetric, skew symmetric, ...) and any given rank m ≥ 1 over D unless D is a field with trivial involution, h̆ is skew symmetric, and m is odd. Condition 2. results from the fact that any nonsingular ϵ-Hermitian form over a hyperbolic algebra is hyperbolic (§2, [14]).

1.4. Explicit Hermitian Morita theory and transfer theory. Suppose that D is a division algebra over K, with an involution ^{∗} which is compatible with the involution on K. Let A = M(n, D) and denote the conjugate transpose of a ∈ A by a^{∗}. Let h_{0} ∈ GL(n, D) be ϵ_{0}-Hermitian, h_{0}^{∗} = ϵ_{0}h_{0} with ϵ_{0} = ±1. Suppose that A is given the involution ā = h_{0}^{−1}a^{∗}h_{0}.

Denote by D^{k×l} the k × l matrices over D, and let ^{∗} again stand for conjugate transpose D^{k×l} → D^{l×k}. D^{n×k} is a semisimple left module over A of length k, and every (finitely generated) left module over A is of this form, up to isomorphism.

Lemma 1.6. (a) The map D^{m×k} → Hom_{A}(D^{n×m}, D^{n×k}) given by right multiplication, is an isomorphism of M(k, D)-modules.

(b) Multiplication D^{m×n}⊗_{A}D^{n×k} → D^{m×k} is an isomorphism of (M(m, D), M(k, D))- bimodules.

Proof. (a) follows at once by looking at the homomorphisms D^{n×m} → D^{n×k} as matrices with respect to the direct sum decompositions of D^{n×m} and D^{n×k} into their “columns”.

(b) Every element of D^{1×n} ⊗_{A} D^{n×1} is of the form e_{11} ⊗e′_{11}d where d ∈ D and e_{11} and e′_{11} are matrix units. It follows that D^{1×n} ⊗_{A} D^{n×1} is a 1dimensional vector space over D, and then (b) follows easily.

Now suppose that V = D^{n×k}. By the foregoing lemma, V^{∗} = Hom_{A}(V, A) = D^{k×n}. Let f : V × V → (A, ¯ ) be a sesquilinear form, and consider f_{r} : V → V^{∗}, f_{r}(v) = f_{r}(·, v). If we compose this map with the map V^{∗} → V given by x ↦ h_{0}^{−1}x^{∗}, the result is an A-linear homomorphism V → V, and so is given by right multiplication by a matrix b ∈ D^{k×k}. It follows that f(·, v) = ϵ_{0}b^{∗}v^{∗}h_{0}, and so if we set ϵ_{0}b^{∗} = \underline{f} ∈ M(k, D), we get the formula

f(u, v) = u\underline{f}v^{∗}h_{0}. (1.4)

Thus \underline{f} is the generalization of the notion of “discriminant matrix” (sometimes also called the “Gram matrix”) of a sesquilinear form over a division ring. It is of fundamental importance in this paper.

We note that f is nonsingular if and only \underline{f} is an invertible matrix, and f is ϵ-Hermitian if and only if \underline{f}^{∗} = ϵ_{0}ϵ\underline{f}.

We give next a matrix version of Hermitian Morita theory for A = M(n, D). We refer to Chapter 4 of [20] for the “ordinary” Morita theory (between categories of modules) and to §8 of [8] for the Hermitian Morita theory.

Let B = M(m, D). We have perfect pairings

V × V^{∗} → A, V^{∗} × V → B,

given respectively by

⟨v, x⟩ = vx, [x, v] = xv,

satisfying the “associative laws”

⟨v, x⟩w = v[x, w], and x⟨v, y⟩ = [x, v]y.

These data constitute a Morita context which provides a Morita equivalence between the categories _{A}M of left A-modules and _{B}M of left B-modules. The equivalence functors are U ⇝ V^{∗} ⊗_{A} U of _{A}M into _{B}M and X ⇝ V ⊗_{B} X in the opposite direction.

If we restrict the Morita equivalence to the “skeletal” subcategory of AM of modules of the form D^{n×k}, it follows from Lemma 1.6, (b), that the resulting functor is isomorphic to D^{n×k} ⇝ D^{m×k}, with the morphism D^{n×k} → D^{n×l} in _{A}M given by right multiplication by a matrix in D^{k×l}, being transformed into the morphism D^{m×k} → D^{m×l} in _{B}M given by right multiplication by the same matrix.

Suppose that h_{1} is an ϵ_{1}-Hermitian form over A on a simple A-module where ϵ_{1} = ±1 – for example one can always find such an h_{1} if ϵ_{1} = 1 unless D = L and the involution ¯ on A is symplectic, i.e. h_{0} is skew symmetric, in which case h_{1} exists for ϵ_{1} = −1 (cf. §1.2 – this also follows easily by use of the discriminant matrix). Then an ϵ-Hermitian form ĥ over A is Hermitian Morita equivalent to an ϵ_{1}ϵ-Hermitian form h̆ over D where D has the adjoint involution of h_{1}. See [8], §8. The Hermitian Morita theory tells us that ĥ ≅ ĝ if and only if h̆ ≅ ğ.

Hermitian Morita equivalence in the case at hand is very easily described in terms of the discriminant matrix; we give it here only in the case of interest to us, when h_{1} is defined on the simple module V_{1} = D^{n×1}.

Lemma 1.7. A sesquilinear form f over (A, ¯ ) on D^{n×k} with discriminant matrix \underline{f} ∈ M(k, D) corresponds, via the Hermitian Morita equivalence effected by h_{1}, to the sesquilinear form over (D, ¯ ) with the discriminant matrix ϵ_{1}\underline{f}. The involution ¯ on D is the adjoint with respect to h_{1}.

Proof. The discriminant matrix \underline{h}_{1} of h_{1} is an ϵ_{0}ϵ_{1}-Hermitian element of (D,^{∗}), i.e. \underline{h}_{1}^{∗} = ϵ_{0}ϵ_{1}\underline{h}_{1}. The adjoint involution on D with respect to h_{1} is d̄= \underline{h}_{1}d^{∗}\underline{h}_{1}^{−1}.

Let V̄_{1} be the (A, D)-bimodule V_{1} twisted into a (D, A)-bimodule by the involution ¯ on both A and D. The ϵ_{1}-Hermitian form H : V̄ × V̄ → (D, ¯ ) which effects the Morita equivalence between the categories of sesquilinear forms over (A, ¯ ) and (D, ¯ ) respectively is defined by

h_{1}(v_{1}, v_{2})v_{3} = v_{1}H(v̄_{2}, v̄_{3}) for all v_{1}, v_{2}, v_{3} ∈ V

(cf. (8.3), [8]) where v̄_{i} stands for the vector v_{i} ∈ V considered as a vector of V̄_{1}. Since h_{1}(v_{1}, v_{2}) = v_{1}\underline{h}_{1}v_{2}^{∗}h_{0}, this implies that

H(v̄_{2}, v̄_{3}) = \underline{h}_{1}v_{2}^{∗}h_{0}v_{3}. (1.5)

This is the analogue of (1.4) for a form on a right module instead of a left module. Thus the discriminant matrix of H is h_{0}.

We identify the twisted bimodule V̄_{1} with the A-dual V_{1}^{∗} = D^{1×n} via the (A, D)-bimodule isomorphism h_{1r} which takes v̄ to h_{1}(·, v) = \underline{h}_{1}v^{∗}h_{0}.

Now let (U, f) be a sesquilinear form over (A, ¯ ), where U = D^{n×k}, and suppose that the discriminant matrix of f is \underline{f}. We wish to find the discriminant matrix of the sesquilinear form (V̄_{1}⊗_{A}U, Hf) over D which corresponds to (U, f) under the Hermitian Morita equivalence effected by h_{1} and H. Both of the mappings

V̄_{1} ⊗_{A} U ⟶^{h_{1r}⊗id_{U}} V_{1}^{∗} ⊗_{A} U ⟶^{mult.} D^{1×k}.

are isomorphisms of left D-spaces and their composition takes v̄ ⊗ u to \underline{h}_{1}v^{∗}h_{0}u ∈ D^{1×k}. We identify V̄ ⊗_{A} U and D^{1×k} by this isomorphism.

Now by definition of the product form Hf,

Hf(v̄_{1} ⊗ u_{1}, v̄_{2} ⊗ u_{2}) = H(v̄_{1}f(u_{1}, u_{2}), v̄_{2}) = H(\overline{\overline{f(u_{1}, u_{2})}v_{1}}, v̄_{2}). (1.6)

A straightforward computation shows that

Hf(v̄_{1} ⊗ u_{1}, v̄_{2} ⊗ u_{2}) = ϵ_{1}(v̄_{1} ⊗ u_{1})\underline{f}(v̄_{2} ⊗ u_{2})^{∗}h_{1}^{−1}

which implies that the discriminant matrix of Hf is ϵ_{1}\underline{f}.

This explicit version of the Hermitian Morita theory underlies our computation of the equivalence classes of equivariant representations in the rest of the paper, and we now describe the version of it which will be used.

If the involution on A = M(n, D) is of the first kind and D is a quaternion algebra, we choose it to be conjugation ^{∗} .

Now choose h_{1} : V_{1} × V_{1} → A to be the form h_{1}(u_{1}, v_{1}) = u_{1}v_{1}^{∗}h_{0}. Then h_{1} is ϵ_{0}-Hermitian with respect to ¯ and its adjoint involution on D is the involution ¯ chosen above; in particular it is conjugation if it is of the first kind and D is a quaternion algebra.

We now apply Lemma 1.7 to get:

Lemma 1.8. With the foregoing notation, the ϵ-Hermitian form

ĥ : D^{n×m} × D^{n×m} → A

with discriminant matrix \underline{ĥ} ∈ D^{m×m} (which is ϵ_{0}ϵ-Hermitian),

ĥ(u, v) = u\underline{ĥ}v^{∗}h_{0},

corresponds via the Hermitian Morita theory arising from h_{1} to the ϵ_{0}ϵ-Hermitian form

h̆ : D^{1×m} × D^{1×m} → D

which has discriminant matrix ϵ_{0}\underline{ĥ},

h̆(u, v) = uϵ_{0}\underline{ĥ}v^{∗}.

And if D is a quaternion algebra and the involution on A is of the first kind, the involution on D is conjugation.

Remark 1.9. In the exceptional case when (A, ¯ ) is symplectic of index 1, the correspondence between forms over A and over L = cen A under which the form over A with matrix f corresponds to the form over L with matrix f is a category equivalence, since for example the functor (W, g) ⇝ (W, −g) is certainly an automorphism of the category of bilinear forms over L. But this correspondence does not arise from a Morita equivalence.

We now use the discriminant matrix to compute the transfer trd_{A/L}ĥ of the ϵ-Hermitian form ĥ over A; the transfer is an ϵ-Hermitian form over L (symmetric or skew symmetric if ¯ is the identity on K).

We note that if a = (a_{ij}) ∈ A,

trd_{A/K}a = ∑_{i} trd_{D/K}a_{ii} = trd_{D/K} ∑_{i} a_{ii}.

Define trd_{A/D}a = ∑_{i} a_{ii}. This depends on the identification A = M(n, D), but is nevertheless useful. The above formula becomes

trd_{A/K} = trd_{D/K} ∘ trd_{A/D}.

A straightforward matrix computation proves that:

Lemma 1.10. Suppose that D = L. The discriminant matrix of trd_{A/L}ĥ (in a suitable basis) is

ˆ\underline{h} ⊗ h_{0}^{t}, (1.7)

and its determinant is

(det ˆ \underline{h})^{n}(det h_{0})^{m}. (1.8)

Now consider the case where A has index > 1, i.e. D is not commutative. We can suppose that h_{0} and \underline{ĥ} are diagonal matrices, say h_{0} = ⟨η_{01}, …, η_{0n}⟩ and \underline{ĥ} = ⟨η_{1}, …, η_{m}⟩. Note that for all i and j, η_{0i}^{∗} = ϵ_{0}η_{0i} and η_{j}^{∗} = ϵ_{0}ϵη_{j}. The one dimensional form

H_{ij} : D × D → D, H_{ij}(u, v) = uη_{j}v^{∗}η_{0i}, 1 ≤ i ≤ n, 1 ≤ j ≤ m (1.9) is an ϵ-Hermitian form with respect to the involution

d ↦ η_{0i}^{−1}d^{∗}η_{0i} (1.10)

of D. Another matrix computation proves:

Lemma 1.11. (a) The transfer trd_{A/L}ĥ : D^{n×m} × D^{n×m} → L is an ϵ-Hermitian form and is the orthogonal direct sum of the transfers from D to L of the nm forms H_{ij} in (1.9), where H_{ij} is a one dimensional ϵ-Hermitian form with respect to the involution (1.10).

(b) Suppose that h_{0} = η_{0}I_{n} (for example h_{0} = I_{n}). Then the transfer

H = trd_{A/D}ĥ : D^{n×m} × D^{n×m} → D

is a nonsingular ϵ-Hermitian form with respect to the involution

d̄= η_{0}^{−1}d^{∗}η_{0},

and its transfer to L is the transfer of ĥ to L,

trd_{D/L}H = trd_{A/L}ĥ.

Its discriminant matrix \underline{H} ∈ D^{nm×nm}, defined by H(u, v) = u\underline{H}v^{∗}η_{0}, u, v ∈ D^{1×nm}, is given by

\underline{H} = \underline{ĥ} ⊗ h_{0}.

If K is a nonArchimedean local field and D is the unique quaternion algebra over K, with conjugation ^{∗} as its involution, it is always possible to achieve h_{0} = ηI_{n} except in the case n even and h_{0}^{∗} = −h_{0}. See the table on p. 9.

The next lemma is useful in calculating the transfers of the H_{ij}.

Let D be a central quaternion division algebra over K with an orthogonal involution ¯ . There is a pure quaternion j such that d̄ = j^{−1}d^{∗}j for all d ∈ D, where ^{∗} denotes conjugation in D (cf. §1.2).

Thus if h : D × D → (D, ¯ ) is a rank 1 Hermitian form, there is another pure quaternion η such that

h(d_{1}, d_{2}) = d_{1}ηd_{2}^{∗}j = d_{1}ηjd̄_{2}.

With this notation, we have:

Lemma 1.12. The quaternary quadratic form trd_{D/K}h over K has determinant 1, and is isotropic (hence hyperbolic) if and only if (η^{2}, j^{2})_{K} ≅ D.

Proof. Conjugation by j of the space of pure quaternions D_{0} is an involutory linear transformation, and it follows, by consideration of the eigenspaces, that we can find another pure quaternion i ≠ 0 satisfying η = θj + δi with θ ∈ K, ij = −ji, and δ = 0 or 1.

Let i^{2} = α, j^{2} = β, so D = (α, β)_{K}. And

η^{2} = θ^{2}β + δ^{2}α.

The discriminant matrix of trd_{D/L}h in the basis 1, k = ij, i, j of D is

2β ( \begin{matrix}θ & −αδ & 0 & 0 \\ −αδ & −θαβ & 0 & 0 \\ 0 & 0 & θα & −αδ \\ 0 & 0 & −αδ & −θβ \end{matrix} ) .

Its determinant is 4β^{2}(−θ^{2}αβ − α^{2}δ^{2})^{2} = 1 in K^{∗}/K^{∗2}.

If θ = 0, a glance at the above matrix shows that trd_{D/K}h is isotropic; furthermore η = i so (η^{2}, j^{2})_{K} = D.

Suppose θ ≠ 0. The forms b_{1} and b_{2} with matrices

( \begin{matrix}θ & −αδ \\ −αδ & −θαβ \end{matrix} ) and ( \begin{matrix}θα & −αδ \\ −αδ & −θβ \end{matrix} )

have the same determinant, and so their sum b_{1} ⊥ b_{2} is isotropic if and only if b_{2} ≅ −b_{1}. This is equivalent to the form ⟨θ, −θ^{3}αβ −θα^{2}δ^{2}⟩ representing θβ, hence to ⟨1, −θ^{2}αβ − α^{2}δ^{2}⟩ representing β. Since binary forms are classified by their determinant and Hasse symbol, this in turn is equivalent with

(β, −θ^{2}αβ^{2} − α^{2}δ^{2}β)_{K} = (1, −θ^{2}αβ − α^{2}δ^{2})_{K} = 1

i.e. to (β, θ^{2}αβ + α^{2}δ^{2})_{K} = 1. Thus the condition is

1 = (β, α)_{K}(β, θ^{2}β + αδ^{2})_{K} = (β, α)_{K}(j^{2}, η^{2})_{K}.

i.e. D = (α, β)_{K} ≅ (η^{2}, j^{2})_{K}.

2. Symplectic Representations

Because of the fact that two nonsingular skew symmetric forms are equivalent if and only if they have the same rank, symplectic representations are considerably easier to deal with than are orthogonal or unitary representations and so we treat them separately here.

We now prove the statements on symplectic representations on pp. 3–4. We identify A = M(n, D) and V = D^{n×m}. The involution on A is of the

form ā = h_{0}^{−1}a^{∗}h_{0} – see §1.2.

If \underline{ĥ} is an m × m matrix, the form h(u, v) = Tr_{A/K}(u\underline{ĥ}v^{∗}h_{0}) on D^{n×m} will be skew symmetric if and only if ĥ(u, v) = u\underline{ĥ}v^{∗}h_{0} is skew Hermitian.

Suppose that (A, ¯ ) is orthogonal and of index 1. Then \underline{ĥ} must be skew symmetric (in order for h to be skew symmetric). This verifies II (in the orthogonal case) for any field K if (A, ¯ ) has index 1. This is always the case if K is finite or real closed – see §1.2 – which proves I for these fields (by Table 1).

Suppose next that (A, ¯ ) is orthogonal but has index > 1. Recall that we assume that the involution ^{∗} on D is symplectic when D ≠ L. Since h_{0} is then skew Hermitian, \underline{ĥ} must be an Hermitian matrix, so the equivalence classes of symplectic representations on V are in bijective correspondence with the equivalence classes of Hermitian forms of rank m over D by Hermitian Morita theory. There is only 1 when K is local, by Table 1.

Now suppose that K is global. We must count the number of Hermitian forms h̆ of rank m over D, a quaternion division algebra with the canonical involution. If D splits at the prime 𝔭 of its center L, h̆_{𝔭} is an Hermitian form over M(2, L_{𝔭}) with a symplectic involution. Thus we can assume that h̆_{𝔭}(u, v) = u\underline{ĥ}v^{t}h_{1} where h_{1} is a skew symmetric matrix of degree 2. Therefore as a 2m × 2m matrix over L_{𝔭}, \underline{ĥ} is skew symmetric and so is uniquely determined up to equivalence.

We note that D does split at every real prime 𝔭 of the center since (A, ¯ ) is orthogonal – cf. §1.2.

If D is not split at the finite prime 𝔭, h̆_{𝔭} is an Hermitian form over the local division algebra D_{𝔭} and so is again uniquely determined up to equivalence by its rank (cf. Table 1). By the Hasse Principle for Hermitian forms over D (cf. Table 1), this finishes the proof for (A, ¯ ) orthogonal.

Suppose now that (A, ¯ ) is symplectic. This means that h_{0} is skew symmetric if A has index 1 and is Hermitian otherwise. Then h(u, v) = trd_{A/K}(u\underline{ĥ}v^{∗}h_{0}) is an invariant skew symmetric form on D^{n×m} if and only if \underline{ĥ} is nonsingular and symmetric in the first case and skew Hermitian in the second. The cases of K finite, or K global, or K local and A of index 1 follow easily. If K is real closed, A is necessarily of index > 1 (§1.2); by Table 1, skew Hermitian forms over D are classified by their rank and so there is only 1 equivalence class of symplectic representations in this case. Finally the case of K local and (A, ¯ ) of index > 1 follows from [21], Theorem 10.3.6.

The case (A ¯ ) unitary is proved in a similar manner. Note that (noncommutative) division algebras over local or real closed fields do not admit involutions of the second kind – in the real closed case, A admits a unitary involution if and only if A is a full matrix algebra over the algebraic closure of K, again by §1.2.

Now III. If G is a symmetric group, it is well known that all of the simple components of KG are split and stable under the canonical involution, and that the induced involution in each case is orthogonal (cf. §6). This proves III in this case (by II). Now consider 2., 3. and 4. The assumption of odd representation type rules out both a symplectic involution and an orthogonal involution on an algebra of Schur index > 1. So the III is clear if K is finite. Otherwise conditions 3. and 4. both rule out a unitary involution which is not hyperbolic. Thus each simple component A of KG, stable under the canonical involution and for which AV ≠ 0, is orthogonal of index 1.

Finally consider the statement on conjugacy of subgroups of Sp(V, h). Since G and G′ are conjugate in GL(V), there is a ϕ ∈ GL(V) such that s ↦ ϕ^{−1}sϕ is an isomorphism G → G′. This map and the identity are symplectic representations which are linearly equivalent, so by III we can find ψ ∈ Sp(V, h) such that s → ψ^{−1}sψ is an isomorphism G → G′. □

3. Unitary and orthogonal representations

over finite fields

If K is a finite field, the simple algebra A is a matrix algebra M(n, L) over the field L.

Theorem 3.1. Let K be a finite field, A a simple K-algebra and V an A-module of length len_{A}V = m.

Let h : V × V → K be a symmetric, Hermitian or skew Hermitian form.

I, II Unitary representations. There is exactly 1 equivalence class of Aisotypic unitary representations G → U(V, h), in both the Hermitian and skew Hermitian cases.

Orthogonal representations. If h is symmetric, the number of equivalence classes of A-isotypic orthogonal representations G → O(V, h) is

(a) 1 if (A, ¯ ) is symplectic, m is even and h is hyperbolic,
(b) 1 if (A, ¯ ) is orthogonal and n is odd,
(c) 2 if (A, ¯ ) is orthogonal, n is even and det h = N_{L/K}(det(¯ ))^{m},
(d) 1 if (A, ¯ ) is unitary and det h = (−disc (L/L_{0}))^{mn|L_{0}^{∗}|/|K^{∗}|},

and is 0 otherwise.

III Unitary representations. Two unitary representations of G over K are equivalent if and only if they are linearly equivalent.

Orthogonal representations. Two orthogonal representations of G over K are equivalent if and only if they are linearly equivalent and their forms are isotypically equivalent, unless KG has a simple orthogonal component (A, ¯ ) of even degree for which AV ≠ 0.

If ρ : G → O(V, h) is an orthogonal representation and r is the number of orthogonal components (A ¯ ) of even degree for which AV ≠ 0, the number of equivalence classes of orthogonal representations G → O(V, h) which are linearly equivalent to ρ is 2^{r}. In particular two orthogonal representations of G of odd type are equivalent if and only if they are equivalent as linear representations and have isotypically equivalent forms.

Corollary 3.2. If K is a finite field, two subgroups of U(V, h) which have order relatively prime to char K are conjugate in U(V, h) if and only if they are conjugate in GL(V).

Proof. of the theorem. We view the succession of equivalences ρ ⇝ ĥ ⇝ h̆ (cf. §1.3) in the reverse direction. Thus we consider all possibilities for h̆ (up to equivalence) and (A, ¯ ), and for each such pair h̆, (A, ¯ ), determine the form h by first determining ĥ by Hermitian Morita theory using Lemma 1.8, and then h = \frac{n}{g}trd_{A/K} ĥ via the transfer theory using Lemma 1.10. The equivalence classes of h’s so obtained are those whose forms which admit an equivariant representation linearly equivalent to ρ, and the number of equivalence classes of h̆’s which lead to the equivalence class of h is the number of nonequivalent orthogonal representations G → O(V, h) or unitary representations G → U(V, h).

We use the notation of Lemma 1.8. The form h̆ is a symmetric, skew symmetric, Hermitian or skew Hermitian form over L on L^{m} with matrix ϵ_{0}\underline{ĥ}.

Suppose first that h̆ is skew symmetric – in particular, m is even and the involution on K is the identity. If the involution on A is orthogonal, the form ĥ is skew Hermitian and the form h must be skew symmetric. This case is already handled in §2.

If h̆ is skew symmetric and the involution on A is symplectic, ĥ is an Hermitian form over A on L^{n×m}, and its transfer h′ = trd_{A/L}ĥ is a symmetric form over L on L^{n×m} with matrix \underline{ĥ} ⊗ h_{0}^{t} (cf. §1.4) which, since h_{0} and \underline{ĥ} are both skew symmetric, is hyperbolic. Thus Tr_{L/K}(h′) = trd_{A/K} ĥ is also hyperbolic, and so also h = Tr ĥ = \frac{n}{g}trd_{A/K} ĥ. This is (a).

Now suppose that h̆ is symmetric. If the involution on A is symplectic, ĥ is skew Hermitian, so Tr ĥ = \frac{n}{g}trd_{A/K} ĥ is skew symmetric and can be disregarded.

If h̆ is symmetric and the involution is orthogonal, ĥ is Hermitian, and trd_{A/L}ĥ is symmetric; moreover by the determinant formula det (trd_{A/L}ĥ) = (det \underline{ĥ})^{n}(det h_{0})^{m} (cf. (1.8)), the two possibilities for h̆ (up to equivalence) give rise to two nonequivalent transfers trd_{A/L}ĥ if n is odd, while if n is even there is only one possibility for trd_{A/L}ĥ, up to equivalence. Moreover in the latter case, trd_{A/L}ĥ has determinant (det h_{0})^{m} = (det(¯ ))^{m}. Since the norm N_{L/K} preserves nonsquares, by Lemma 1.1 the same thing is true for trd_{A/K} ĥ = Tr_{L/K}trd_{A/L}ĥ = \frac{g}{n}Tr_{KG/K} ĥ = \frac{g}{n}h (namely two nonequivalent forms when n is odd, and one when n is even), so also for h, where h has determinant (\frac{n}{g})^{lmn}N_{L/K}(det h_{0})^{m} = N_{L/K}(det (¯ ))^{m} in K^{∗}/K^{∗2} when n is even. This gives (b) and (c).

Now suppose that the involution on A is of the second kind, so we can take h_{0} = I_{n} since Hermitian forms over a finite field are characterized by their dimension (cf. Table 1). Consider first the case where the involution on K is the identity. If h̆ is skew Hermitian, h is skew symmetric. If h̆ is Hermitian, we get one class of orthogonal representations for the form h since there is only one equivalence class of Hermitian forms of a given rank. Furthermore if L = L_{0}(\sqrt{λ_{0}}), h has discriminant

N_{L_{0}/K}(−λ_{0})^{mn} = (−λ_{0})^{|L_{0}^{∗}|/|K^{∗}|}

by Lemma 1.1, since trd_{A/L}ĥ has matrix I_{m} ⊗ I_{n} and the discriminant of the binary symmetric form Tr_{L/L_{0}}(xȳ) on L is −λ_{0}. This is (d).

Now suppose the involution on K is also not the identity. There is only one equivalence class of skew Hermitian forms on L^{m}, and it gives rise to a single equivalence class of unitary representations (with respect to a skew Hermitian form). Similarly we get a single equivalence class of unitary representations with respect to an Hermitian form from the unique equivalence class of Hermitian forms on L^{m}.

III is clear from I and II.

4. Unitary and orthogonal representations over local fields

If A is an Abelian group, σ_{2}(A) denotes the number of square classes (A : A^{2}).

In this section K is a local (nonArchimedean) field. It is “dyadic” if 2 is not a unit, “nondyadic” otherwise, and

σ_{2}(L^{∗}) = (L^{∗} : L^{∗2}) = \{ \begin{matrix}4 & if L is nondyadic, \\ 2^{2+(L:ℚ_{2})} & if L is dyadic, \end{matrix}

([21],p. 217). Let g_{K} denote the unique quaternary anisotropic form over K, and G the (Abelian) Galois group Gal(L/K).

The integer m′ in part (k) of the next theorem is

m′ = \{ \begin{matrix}2 & if m = 1 and either n is odd and h ∼_{W} 0, \\ & or n is even and h ∼_{W} g_{K}, \\ m & otherwise. \end{matrix}

Theorem 4.1. Let K be a local field, A ≅ M(n, D) a simple algebra component of KG stable under ¯ , and V an A-module of length len_{A}V = m.

Let h : V × V → K be a symmetric or Hermitian form.

Unitary representations. The number of equivalence classes of A-isotypic unitary representations G → U(V, h) is

type of # of equiv.  conditions
(A, ¯ )  classes
(a) unitary  1  n odd,
(b) unitary  2  n even and det h = N_{L/K}(det(¯ ))^{m},

and 0 otherwise.

Orthogonal representations. The number of equivalence classes of Aisotypic orthogonal representations G → O(V, h) is

type of # of equiv.  conditions
(A, ¯ )  classes
(c) unitary  1  n odd and det h = N_{L_{0}/K}(−disc(L/L_{0}))^{m},
(d) unitary  2  n even, and

h ∼_{W} 0 if (det(¯ ))^{m} = (−1)^{mn/2},

h ∼_{W} g_{K} otherwise,

(e) sympl.  1  h ∼_{W} g_{K} if mn is odd, otherwise h ∼_{W} 0 and,

if D = L, m is even,

(f) orthog.,  \frac{σ_{2}(G)σ_{2}(L^{∗})}{σ_{2}(K^{∗})}  n odd, m > 1, det h ∈ (disc(L/K))^{m}(N_{L/K}L^{∗})K^{∗2},

ind. = 1

(g) orthog.,  1 ≤ #  n odd, m = 1, h ≅ Tr_{L/K}⟨αh_{0}⟩, α ∈ L^{∗},

ind. = 1

≤ \frac{σ_{2}(G)σ_{2}(L^{∗})}{σ_{2}(K^{∗})}
(h) orthog.,  \frac{σ_{2}(L^{∗})}{1+δ_{1m}}  n even, disc(¯ ) ≠ 1, det h = N_{L/K}(det(¯ ))^{m},

ind. = 1

(i) orthog.,  \frac{2σ_{2}(L^{∗})}{1+δ_{1m}}  n even, disc(¯ ) = 1, h ∼_{W} 0 or g_{K},
ind. = 1  if mn/2 is even, h ∼_{W} 0 ⇔ S(h_{0})^{m} = (−1, −1)_{L}^{m^{2}n/4},
if mn/2 is odd, h ∼_{W} 0 ⇔ S(h_{0})^{m} = (−1, −1)_{L}^{(m^{2}n−2)/4},
(j) orthog., σ_{2}(L^{∗}) − δ_{1m} det(¯ ) = 1, h ∼_{W} 0 if mn is even,
ind. > 1  h ∼_{W} g_{K} if mn is odd,
(k) orthog., \frac{1}{2}σ_{2}(L^{∗}) − δ_{1m′} det(¯ ) ≠ 1, h ∼_{W} 0 or g_{K},

ind. > 1

and 0 otherwise.

Remark 4.2. 1. In (f) and (g), σ_{2}(L^{∗}) = σ_{2}(K^{∗}) if K is nondyadic, σ_{2}(G) = 1 and (N_{L/K}L^{∗})K^{∗2} = K^{∗} if (L : K) is odd.

2. In (g), ⟨αh_{0}⟩ is the symmetric form of rank n and matrix αh_{0}. The determinant of its trace Tr_{L/K}⟨αh_{0}⟩ is disc(L/K)N_{L/K}(α det h_{0}). Its Hasse symbol can be calculated using [6], Lemma 1, 3., but the result unfortunately involves the unknown factor S_{K}(⟨α disc h_{0}⟩), the Hasse symbol of a “scaled trace form” of the extension L/K. It is possible however to give complete results when K is a nondyadic field; this is carried out in Theorem 4.9.

We note that the discriminants of the h in this case are (\frac{n}{g})^{l}(disc(L/K))(N_{L/K}L^{∗})K^{∗2} and, by Remark 1., the number of equivalence classes is 1 if l is odd and K is nondyadic, or if A is split.

3. In (i), S(h_{0}) is well-defined. In fact by Lemma 1.3 (i), if η ∈ L^{∗}

S(ηh_{0}) = S(h_{0})(η, −1)_{L}^{n(n−1)/2}(η, (−1)^{n(n−1)/2})_{L} = S(h_{0})

since disc h_{0} = 1 implies that det h_{0} = (−1)^{n(n−1)/2}.

4. The integer m can be calculated easily from a knowledge of h and (A, ¯ ). Namely the rank of h is mnd^{2}l.

Proof. The proof proceeds in the same way as that of Theorem 3.1, starting with the form

h̆ : D^{m} × D^{m} → D

and then “descending” first to

ĥ : D^{n×m} × D^{n×m} → M(n, D) = A

via Morita theory using Lemma 1.8, then to

trd_{A/D}ĥ : D^{nm} × D^{nm} → D

via a transfer from A to D, then to

trd_{D/L}trd_{A/D}ĥ = trd_{A/L}ĥ : L^{d^{2}nm} × L^{d^{2}nm} → L

(where d^{2} = dim_{L} D = 1 or 4) via a transfer from D to L, and then finally to

\frac{nd}{g}trd_{L/K}trd_{A/L}ĥ = \frac{nd}{g}trd_{A/K} ĥ = Tr_{KG/K} ĥ = h : K^{ld^{2}nm} × K^{ld^{2}nm} → K

via the transfer from L to K. The “reduced trace” trd_{A/D} is defined just before Lemma 1.10.

h̆ : L^{m} × L^{m} → L skew symmetric. This means that \underline{ĥ} is a skew symmetric matrix (Lemma 1.7), m is even, and A has index 1. If the involution on A is orthogonal, i.e. h_{0}^{∗} = h_{0} so ϵ_{0} = 1, then ĥ is skew Hermitian (cf. p. 14), and so trd_{L/K} ĥ = \frac{g}{n}h and h are skew symmetric. Since the skew symmetric case is handled in §2, we ignore it here.

If the involution on A is symplectic, h_{0} is skew symmetric and ĥ is Hermitian. The transfer trd_{A/L}ĥ is symmetric on L^{nm} with discriminant matrix −\underline{ĥ} ⊗ h_{0} by (1.7). This implies readily that trd_{A/L}ĥ is hyperbolic and so trd_{A/K} ĥ and h are also hyperbolic. Thus we get exactly one orthogonal representation, up to equivalence, and it is on a hyperbolic space. This is the index 1 case of (e).

h̆ : L^{m}×L^{m} → L symmetric. Thus \underline{ĥ} is a symmetric matrix. If the involution on A is symplectic, ĥ is skew Hermitian and h is skew symmetric, and this is already handled elsewhere.

Now suppose the involution on A is orthogonal. Then ĥ is Hermitian and the transfer trd_{A/L}ĥ is a symmetric form on L^{nm} with discriminant matrix \underline{ĥ} ⊗ h_{0} and determinant (det \underline{ĥ})^{n}(det h_{0})^{m} (cf. Lemma 1.10), and h is symmetric as well.

Consider first the case n odd. Then the determinant of b = trd_{A/L}ĥ is (det \underline{ĥ})(det h_{0})^{m} and so is arbitrary since det(\underline{ĥ}) = det h̆ is arbitrary. Assume also that m > 1. Then by Lemma 1.3 (iv), the Hasse symbol of b is also arbitrary, in other words, b runs over all equivalence classes of symmetric forms on L^{nm} as h̆ runs over all equivalence classes of symmetric forms on L^{m} – thus the correspondence is bijective. Now consider the transfer b_{K} = trd_{L/K}b. Since L/K is Abelian, l = [L : K] = [K^{∗} : N_{L/K}L^{∗}] by local class field theory, and it follows that, when l is odd, the map

N̄ : L^{∗}/L^{∗2} → K^{∗}/K^{∗2} (4.1)

induced by the norm is onto; thus it is an isomorphism when K is nondyadic, and in general has kernel of order σ_{2}(L^{∗})/σ_{2}(K^{∗}).This means there are exactly σ_{2}(L^{∗})/σ_{2}(K^{∗}) equivalence classes of orthogonal representations for each symmetric form on K^{lnm} by Lemmas 1.1 and 1.2. This gives (f) when l is odd.

Suppose that l is even. The image of N̄ is (N_{L/K}L^{∗})K^{∗2}/K^{∗2}, and by local class field theory, (N_{L/K}L^{∗})K^{∗2} consists of the norms from the compositum of the quadratic extensions of K contained in L; the degree of this compositum over K is σ_{2}(G) = (Gal(L/K) : Gal(L/K)^{2}). Thus since (L^{∗} : L^{∗2}) = |imN̄| · | ker N̄|, each b_{K} is, up to equivalence, the image of | ker N̄| of the b where

| ker N̄| = σ_{2}(L^{∗})σ_{2}(G)/σ_{2}(K^{∗}). (4.2)

Thus in this case the symmetric forms on K^{lmn} which support an orthogonal representation of G which is linearly equivalent to ρ are those with determinant in (disc(L/K))^{mn}(N_{L/K}L)^{∗}K^{∗2} (since (g/n)^{lmn} is a square), and each of them supports | ker N̄| different orthogonal equivalence classes of orthogonal representations. This completes the proof of (f).

Now suppose that n is odd and m = 1. If \underline{ĥ} = (α), b = trd_{A/L}ĥ = ⟨α h_{0}⟩ which has determinant α^{n} det h_{0}. Since n is odd, we get one form b over L for each possible discriminant α ∈ L^{∗}/L^{∗2}. By the same analysis as in the case m > 1, the discriminants which occur among the transfers b_{K} down to K are (disc(L/K))(N_{L/K}L^{∗})K^{∗2} and each discriminant receives \frac{σ_{2}(G)σ_{2}(L^{∗})}{σ_{2}(K^{∗})} of the b. This proves (g). We note also that this shows that the possible discriminants of h are (\frac{n}{g})^{l}(disc(L/K))(N_{L/K}L^{∗})K^{∗2} as mentioned in Remark 2. after the theorem.

Now consider the case n is even (and the involution of A is orthogonal). The determinant of the transfer b = trd_{A/L}ĥ down to L is the same for all h̆, namely (det h_{0})^{m}, and

(4.3) S(b) = S(\underline{ĥ} ⊗ h_{0}) = S(h_{0})^{m}(det \underline{ĥ}, disc h_{0})_{L}(det h_{0}, −1)_{L}^{m(m−1)/2}

by Lemma 1.3 (iv). If disc h_{0} ∉ L^{∗2}, this implies that we get both forms of determinant (det h_{0})^{m} over L and so by Lemmas 1.1 and 1.2, both forms of determinant (disc(L/K))^{mn}N_{L/K}(det h_{0})^{m} = N_{L/K}(det h_{0})^{m} over K via the transfer from L to K. Furthermore by (4.3) one of the forms arises from those h̆ whose determinants are in the kernel of canonical homomorphism L^{∗}/L^{∗2} → L^{∗}/N_{L(\sqrt{disc h_{0}})/L}L(\sqrt{disc h_{0}})^{∗} ≅ ℤ/2, so the number of equivalence classes of orthogonal representations for each of the two forms is σ_{2}(L^{∗}) if m > 1, \frac{1}{2}σ_{2}(L^{∗}) if m = 1. This is (h).

On the other hand disc h_{0} ∈ L^{∗2} means that det h_{0} = (−1)^{n(n−1)/2} = (−1)^{n/2} and det b = (−1)^{mn/2}; in this case we get only one form over L as a transfer – that with determinant (−1)^{mn/2} and Hasse symbol

S(b) = S(h_{0})^{m}(−1, −1)_{L}^{mn(m−1)/4}.

Therefore there we get only one G-invariant symmetric form b_{K} over K. Its determinant is (disc(L/K))^{mn}N_{L/K}(−1)^{mn/2} = (−1)^{lmn/2}, and we now determine its equivalence class.

Suppose first that mn/2 is even. Then det b = 1 and dim b = mn ≡ 0 mod 4, so b ∼_{W} 0 or b ∼_{W} g_{L}. Moreover by Lemma 1.3(v)

b ∼_{W} 0 ⇔ S(b) = (−1, −1)_{L}^{mn/4}
⇔ S(h_{0})^{m}(−1, −1)_{L}^{mn(m−1)/4} = (−1, −1)_{L}^{mn/4}
⇔ S(h_{0})^{m} = (−1, −1)_{L}^{m^{2}n/4}.

On the other hand if mn/2 is odd, then det b = −1 and dim b ≡ 2 mod 4, so if c is a hyperbolic plane, det(b ⊥ c) = 1 and dim(b ⊥ c) ≡ 0 mod 4, and so since b ∼_{W} b ⊥ c, we see again that b ∼_{W} 0 or b ∼_{W} g_{L}. Now S(b ⊥ c) = S(b)(−1, −1)_{L} by Lemma 1.3 (ii), and by applying the same kind of argument to b ⊥ c as we did above to b, we get (i).

h̆ : L^{m} × L^{m} → L skew Hermitian. The involution on A is unitary so we are assuming that h_{0} is an Hermitian matrix, h_{0}^{∗} = h_{0}. The form ĥ and its discriminant matrix \underline{ĥ} are skew Hermitian. If the involution is the identity on K, h is skew symmetric and we can ignore it.

Suppose therefore that the involution is not the identity on K, say K = K_{0}(\sqrt{λ_{0}}); then also L = L_{0}(\sqrt{λ_{0}}). The transfer trd_{A/L}ĥ is a skew Hermitian form over L of rank mn and determinant (det \underline{ĥ})^{n}(det h_{0})^{m}. Write \underline{ĥ} = \sqrt{λ_{0}}\underline{ĥ}_{1} where \underline{ĥ}_{1} is an Hermitian matrix. Then the determinant of trd_{A/L}ĥ is \sqrt{λ_{0}}^{mn}(det \underline{ĥ}_{1})^{n}(det h_{0})^{m}, and det \underline{ĥ}_{1} and det h_{0} are both in L_{0}^{∗}.

If n is even, we get but one form (up to equivalence) as a transfer over L — it has determinant (λ0^{\frac{n}{2}} det h_{0})^{m} — and so only one transfer down to K, of determinant N_{L/K}(λ0^{\frac{n}{2}} det h_{0})^{m} by Lemma 1.1. This form gives rise, then, to two inequivalent unitary representations with respect to this unique skew Hermitian form. This is the skew Hermitian version of (b) (cf. definition of disc h in §1.1).

Suppose now that n is odd. Then there are two transfers (up to equivalence) down to L, with their determinants representing the two possible classes in L^{∗}/N_{L/L_{0}}L^{∗}. Since the restriction of N_{L/K} to L_{0} is N_{L_{0}/K_{0}}, the following lemma and Lemma 1.1 imply that their transfers down to K are also distinct, and so we have 2 distinct skew Hermitian forms over K each with a single unitary representation. This is the skew Hermitian version of (a).

Lemma 4.3. The homomorphism N̄ : L_{0}^{∗}/N_{L/L_{0}}L^{∗} → K_{0}^{∗}/N_{K/K_{0}}K^{∗} induced by N_{L_{0}/K_{0}}, is onto.

Proof. Let Gal_{L_{0}}^{ab} be the Galois group of the Abelian closure L_{0}^{ab} of L_{0} in some separable closure. Define Gal_{K_{0}}^{ab} similarly, using the same separable closure. Let Gal_{L_{0}} and Gal_{K_{0}} be the absolute Galois groups of L_{0} and K_{0}. We may assume Gal_{L_{0}} ⊂ Gal_{K_{0}} and this inclusion induces a homomorphism Gal_{L_{0}}^{ab} = Gal_{L_{0}}/Gal′_{L_{0}} →^{i} Gal_{K_{0}}^{ab} = Gal_{K_{0}}/Gal′_{K_{0}}, σGal′_{L_{0}} ↦ σGal′_{K_{0}}. This fits into the commutative diagram

L_{0}^{∗} ⟶ Gal_{L_{0}}^{ab}
N_{L_{0}/K_{0}} ↓  ↓^{i}
K_{0}^{∗} ⟶ Gal_{K_{0}}^{ab}

where the horizontal maps are given by the norm residue symbol – cf. [2], p.141. This implies a commutative diagram

L_{0}^{∗}/N_{L/L_{0}}L^{∗} ⟶ Gal_{L/L_{0}}
N̄↓  ↓
K_{0}^{∗}/N_{K/K_{0}}K^{∗} ⟶ Gal_{K/K_{0}}

where the right vertical map is restriction – and is an isomorphism. Since the horizontal maps are isomorphisms by local class field theory, N̄ is an isomorphism as well.

h̆ : L^{m} × L^{m} → L Hermitian. Again h_{0} is Hermitian and the involution is unitary. The transfer trd_{A/L}ĥ of ĥ down to L is Hermitian with determinant (det \underline{ĥ})^{n}(det h_{0})^{m}. Thus if n is even, we get only one form over L (up to equivalence) while if n is odd we get two.

We consider first the case when the involution is the identity on K (so h is symmetric) and n is even. Thus trd_{A/L}ĥ has determinant (det h_{0})^{m}. The transfer of a rank 1 Hermitian form with matrix (η) over L = L_{0}(\sqrt{λ_{0}}) down to L_{0} is the symmetric form ⟨2η, −2ηλ_{0}⟩, so the transfer of ĥ down to L_{0} is the symmetric form of rank 2mn with matrix trd_{A/L}(ĥ ⊗ ⟨2, −2λ_{0}⟩), under the assumption that trd_{A/L}ĥ is diagonal – in fact we can assume that trd_{A/L}ĥ = ⟨1, …, 1, (det h_{0})^{m}⟩. The determinant of this form over L_{0} is

(det trd_{A/L}ĥ)^{2}(−λ_{0})^{mn} = 1,

and by Lemma 1.3 (iv) its Hasse invariant is

(−1, −1)_{L_{0}}^{mn/2}((−1)^{n/2} det h_{0}, λ_{0})_{L_{0}}^{m}.

Since its rank is divisible by 4, it is ∼_{W} 0 or g_{L_{0}}, and is ∼_{W} 0 if and only if (−1, −1)_{L_{0}}^{mn/2}((−1)^{n/2} det h_{0}, λ_{0})_{L_{0}}^{m} = (−1, −1)_{L_{0}}^{mn/2} by Lemma 1.3; this condition is equivalent to (−1)^{mn/2}(det h_{0})^{m} ∈ N_{L/L_{0}}L^{∗}. By Lemma 1.2, the same situation obtains when we take the transfer down to K, and in either case we get a single symmetric form h supporting 2 nonequivalent orthogonal representations. This is (d).

Now suppose that the involution is the identity on K and n is odd. As in the case n even, the two transfers to L_{0} have matrix (trd_{A/L}ĥ) ⊗ ⟨2, −2λ_{0}⟩. Their determinants are both (−λ_{0})^{mn} = (−disc(L/L_{0}))^{m}, and their Hasse invariants are given by

((−1)^{m(mn−1)/2}(2 det h_{0})^{m} det \underline{ĥ}, λ_{0})_{L_{0}} (−1, −1)_{L_{0}}^{m(mn−1)/2} .

These Hasse invariants are distinct since det \underline{ĥ} represents both norm residue classes in L_{0}^{∗}/N_{L/L_{0}}L^{∗}, and so the transfers to L_{0} remain distinct. Both transfers to K have determinant N_{L_{0}/K}(−disc(L/L_{0}))^{m}, and they remain inequivalent by Lemma 1.2. This is (c).

Now suppose the involution is not the identity on K. As before, the transfer trd_{A/L}ĥ is an Hermitian form over L with determinant (det ĥ)^{n}(det h_{0})^{m}. If n is even, we get but one form (up to equivalence) as a transfer over L — it has determinant (det h_{0})^{m} and rank mn — and so also only one transfer down to K, of determinant N_{L_{0}/K_{0}}(det h_{0})^{m} by Lemma 1.1 and rank lmn. Thus this form supports two inequivalent unitary representations. This is (b) in the Hermitian case.

If n is odd, there are two distinct transfers (up to equivalence) down to L, with determinants det ĥ(det h_{0})^{m} representing the two norm classes in L_{0}^{∗}/N_{L/L_{0}}L^{∗}. By Lemmas 1.1 and 4.3, their transfers down to K are also inequivalent, and so we have two inequivalent Hermitian forms over K, each with a single unitary representation. This is (a) in the Hermitian case.

The remaining cases involve A of index > 1. Thus A = M(n, D) where D is a quaternion division algebra over K with conjugation ^{∗} as its involution. Since an Hermitian or skew Hermitian form over D has an orthogonal basis, we can suppose that h_{0} is diagonal, say h_{0} = ⟨η_{01}, …, η_{0n}⟩ where η_{0j}^{∗} = ϵ_{0}η_{0j} for all j. If ϵ_{0} = 1, we can assume that h_{0} = I_{n} (cf. Table 1).

Similarly h̆ is either Hermitian or skew Hermitian and we can assume that its matrix ϵ_{0}\underline{ĥ} is diagonalized, say \underline{ĥ} = ⟨η_{1}, …, η_{m}⟩, and that \underline{ĥ} = I_{m} if h̆ is Hermitian.

h̆ : D^{m} ×D^{m} → D Hermitian. Suppose first that h_{0} is Hermitian (i.e. (A, ¯ ) is symplectic of index > 1 – cf. §1.2), so h_{0} = I_{n}, \underline{ĥ} = I_{m}, and ĥ(u, v) = uv^{∗}. By Lemma 1.11,(b), if we define the “trace” trd_{A/D}(a_{ij}) = ∑_{i} a_{ii}, then trd_{A/D}ĥ is the nonsingular Hermitian form on D^{nm} with respect to conjugation, with discriminant matrix I_{nm}. Furthermore trd_{A/L}ĥ = trd_{D/L}trd_{A/D}ĥ and so it follows that trd_{A/L}ĥ is the orthogonal direct sum of nm copies of the norm form g_{L} of D. Since g_{L} ⊥ g_{L} is hyperbolic, trd_{A/L}ĥ is hyperbolic of rank 4nm if n or m is even – in which case trd_{A/K} ĥ is also hyperbolic, of rank 4lnm. Otherwise trd_{A/L}ĥ ∼_{W} g_{L}, and by Lemma 1.2, trd_{A/K} ĥ ∼_{W} g_{K}.

In each of these two cases, nm even or odd, we get a unique orthogonal representation linearly equivalent to ρ. This is (e) in the nonsplit case.

If h_{0} is skew Hermitian ((A, ¯ ) is orthogonal with index > 1), so ĥ is also skew Hermitian. It follows that trd_{A/K} ĥ is skew symmetric, so we can ignore it.

h̆ : D^{m} × D^{m} → D skew Hermitian. If h_{0} is Hermitian ((A, ¯ ) symplectic with index > 1), then ĥ is skew Hermitian and the transfer trd_{A/K} ĥ is a skew symmetric form (on K^{4lnm}).

Suppose then that (A, ¯ ) is orthogonal, so h_{0} is skew Hermitian, ĥ is Hermitian and \underline{ĥ} is a skew Hermitian matrix. By Lemma 1.11(a), trd_{A/L}ĥ is the orthogonal direct sum of the mn transfers

h_{ij}(u, v) = trd_{D/L}(uη_{i}v^{∗}η_{0j}) (1 ≤ i ≤ m, 1 ≤ j ≤ n)

of one dimensional forms over D. By Lemma 1.12, h_{ij} is a quaternary symmetric form of determinant 1, and is hyperbolic if and only if (η_{i}^{2}, η_{0j}^{2})_{L} = −1. It follows that trd_{A/L}ĥ ∼_{W} 0 if and only if

∏_{i,j}(η_{i}^{2}, η_{0j}^{2})_{L} = (−1)^{mn},

and otherwise it is ∼_{W} g_{L}. Now ∏_{i} η_{i}^{2} = ∏_{i}(−η_{i}η_{i}^{∗}) = ∏_{i}(−nrd_{D/L}η_{i}) = (−1)^{m}nrd_{B/L}\underline{ĥ} (B = M(m, D)), so trd_{A/L}ĥ ∼_{W} 0 if and only if

((−1)^{m}nrd_{B/L}\underline{ĥ}, (−1)^{n}nrd_{A/L}h_{0})_{L} = (−1)^{mn}.

Parts (j) and (k) now follow if one takes into account the following facts, along with Lemma 1.2:

(i) Two nonsingular skew Hermitian forms of the same rank over D are equivalent if and only if their determinants are equal (in L^{∗}/L^{∗2}). (ii) For any fixed rank r and square class λ ∈ L^{∗}/L^{∗2}, there is a skew Hermitian form of rank r and determinant λ unless r = 1 and λ = −1. See [21], Theorem 3.6, p. 363.

This finishes the proof of Theorem 4.1.

Corollary 4.4. We use the same notation and assumptions as in Theorem 4.1, in particular K is a local nonArchimedean field and ρ is isotypic. Assume in addition that ρ has odd type. Unitary representations. There is exactly 1 equivalence class of unitary representations G → U(V, h) linearly equivalent to ρ.

Thus two unitary representations of odd type are equivalent if and only if they are linearly equivalent and their underlying forms are isotypically equivalent.

Orthogonal representations. The number of equivalence classes of orthogonal representations G → O(V, h) which are linearly equivalent to ρ is as follows:

kind of # of equiv.  conditions
(A, ¯ )  classes
first  \frac{σ_{2}(G)σ_{2}(L^{∗})}{σ_{2}(K^{∗})}  m > 1 and det h ∈ (disc(L/K))^{m}(N_{L/K}L^{∗})K^{∗2},
first  1 ≤ #  m = 1 and h ≅ Tr_{L/K}⟨αh_{0}⟩, α ∈ L^{∗},
≤ \frac{σ_{2}(G)σ_{2}(L^{∗})}{σ_{2}(K^{∗})}
second 1 det h = N_{L_{0}/K}(−disc(L/L_{0}))^{m},

and 0 otherwise. When G is Abelian, we get

kind of (A, ¯ ) # of equiv. classes  conditions
first  1
second 1 det h = N_{L_{0}/K}(−disc(L/L_{0}))^{m},

and 0 otherwise.

Two not necessarily isotypic orthogonal representations of odd type, which we assume to be split if G is not Abelian, are equivalent if and only if they are linearly equivalent and their underlying forms are isotypically equivalent.

Remark 4.5. When G is Abelian and the involution on K is the identity, the restriction of the involution of KG to any direct summand L must be nontrivial if L ≠ K (i.e the involution is unitary), since L is generated over K by the images of G under the projection KG → L.

Proof. . The assumption that ρ has odd type means that A has index 1, and that n is odd. The statements about the number of equivalence classes in each case follow by Theorem 4.1 – and by the above Remark in the case of an orthogonal representation of an Abelian group.

The statements about the equivalence of two not necessarily isotypic representations follow since the conditions stated for orthogonal representations rule out the cases where the number of equivalence classes of isotypic representations is > 1.

We now treat in detail the nondyadic case in Theorem 4.1, (g), for orthogonal representations, and we begin with two auxiliary lemmas. We assume that K_{nr} is the unramified closure of K in L, with (K_{nr} : K) = f, and that K_{tm} is the largest tamely ramified subextension of L/K, with (K_{tm} : K_{nr}) = e. Let (L : K_{tm}) = q, which is a power of the residue class field characteristic p and so is odd.

Recall that G = Gal(L/K). Note that in the case when K is nondyadic, σ_{2}(G) = (G : G^{2}) = 1, 2, or 4 (since, for example, it is the order of the kernel of the map L^{∗}/L^{∗2} → K^{∗}/K^{∗2} induced by the norm, according to the proof of (f) of Theorem 4.1).

Lemma 4.6. (a) The Galois group Gal(K_{tm}/K) is isomorphic to the direct product of Gal(K_{nr}/K) and Gal(K_{tm}/K_{nr}).

(b) If K is nondyadic or L/K is tamely ramified,

σ_{2}(G) = \{ \begin{matrix}1 & if and only if e and f are odd, \\ 2 & if and only if e and f have different parity, \\ 4 & if and only if e and f are both even. \end{matrix}

Proof. (a) is a special case of [18], aufgabe 1, p. 185^{1}, according to which Gal(L/K) ≅ Gal(L/K_{nr}) ⋊ Gal(K_{nr}/K) for any tamely ramified Galois extension L/K of a Henselian field K. (b) follows at once from (a) since the wild ramification index (L : K_{tm}) is odd.

Let c_{K}(h) denote the Witt invariant of the symmetric form h. The following lemma is easily checked.

Lemma 4.7. (a) If the rank of h is odd, then c_{K}(αh) = c_{K}(h) for any α ∈ K^{∗}.

(b) If the rank of h is even and K is nondyadic, then c_{K}(αh) = c_{K}(h) for any unit α ∈ K^{∗}.

If M/K is any finite separable extension, ⟨M⟩ stands for the symmetric form Tr_{M/K}(xy) (x, y ∈ M), in other words ⟨M⟩ is the transfer of the symmetric bilinear form over M with matrix (1). We also denote by ord_{K}α the order of α ∈ K^{∗} with respect to a prime element of the K.

Let h be a symmetric form of rank n over L. Define

r = ord_{K_{tm}}disc(Tr_{L/K_{tm}}h)
= ord_{K_{tm}} ((disc L/K_{tm})^{n}N_{L/K_{tm}}(disc h))
= n ord_{K_{tm}}(disc L/K_{tm}) + ord_{L}(disc h).

Lemma 4.8. Suppose that K is a nondyadic field and that the rank n of h is odd.

(a) If (L : K) is odd,

c_{K}(Tr_{L/K}h) = c_{L}h. (4.4)

(b) If (L : K) is even,

c_{K}(Tr_{L/K}h) = c_{K}⟨K_{tm}⟩^{r+1}(π, disc(Tr_{L/K}h))_{K}^{er}(Π, disc(Tr_{L/K_{tm}}h))_{K_{tm}}^{(e+1)(r+1)}c_{L}h, and

1I am grateful to M. Kolster for providing this reference.

c_{K}⟨K_{tm}⟩ = \{ \begin{matrix}1 & if e is odd, \\ (π, (−1)^{(f−1)/2}e)_{K} & if e is even and f is odd, \\ −(π, (−1)^{f/2}discL/K)_{K} & if e and f are both even. \end{matrix}

Here π and Π are prime elements of K and K_{tm} respectively, arbitrary unless f is odd, in which case they are chosen to satisfy Π^{e} ≡ −π mod K_{nr}^{∗2}.

Proof. The first formula (4.4) follows from [5], Theorem 4.4, 3. The second formula follows from [5], Theorem 5.6, after a straightforward computation. It uses the fact that if α ∈ L, (π, α)_{L} = (π, N_{L/K}α)_{K} (cf. [12], Theorem 2.14, 8.), and that c_{K_{tm}}(Tr_{L/K_{tm}}h) = c_{L}h (which is the special case of (4.4) with K = K_{tm}). The formula for c_{K}⟨K_{tm}⟩ follows from [5], Theorems 4.2, 5.3, 5.4, and 5.5.

Theorem 4.9. Let K be a nondyadic local field, and suppose that (A, ¯ ) is orthogonal of odd degree n with center L and index 1. We denote by f, e, and q respectively the index of inertia, the tame ramification index, and the wild ramification index of L/K.

Suppose that ρ is a linear representation of type A on the vector space V of length 1, and that h is a symmetric form on V.

1. e and f odd. Then there is an invariant form equivalent to h if and only if c_{K}(h) = c_{L}(h_{0}), and each invariant form admits exactly one orthogonal representation (up to equivalence, of course).

2. e odd and f even. An invariant form has determinant disc L/K or ϵ disc L/K, where ϵ is any nonsquare unit of K. Assume that h has one of these as its determinant. If (π, disc h)_{K} = 1, there is a form equivalent to h which supports two nonequivalent orthogonal representations if c_{K}(h) = c_{L}(h_{0}), otherwise there is no invariant form equivalent to h. If (π, disc h)_{K} = −1, there is an invariant form equivalent to h and it supports one orthogonal representation.

3. e even and f odd. There is an invariant form equivalent to h if and only if det h ∈ (disc(L/K))(N_{L/K}L^{∗})K^{∗2} and, if ord_{K}(det h) is even, c_{K}(h) = c_{L}(h_{0}). In the latter case, an invariant form supports two inequivalent orthogonal representations. Each of the two inequivalent invariant forms with determinant of odd order supports a single orthogonal representation.

4. e and f both even. There is an invariant form equivalent to h if and only if det h = disc(L/K), and it admits three inequivalent orthogonal representations if c_{K}(h) = c_{L}(h_{0}), and one otherwise.

Proof. We know that the invariant forms on V are those of the form Tr_{L/K}αh_{0} for α ∈ L^{∗}. Since n is odd, we can assume that det h_{0} = 1. Then det Tr_{L/K}αh_{0} = (disc(L/K)N_{L/K}α by Lemma 1.1.

Suppose e and f are both odd. Then (L : K) is also odd, and by (4.4) and Lemma 4.7 (a), c_{K}Tr_{L/K}αh_{0} = c_{L}h_{0}. Since the map L^{∗}/L^{∗2} → K^{∗}/K^{∗2} is an isomorphism when (L : K) is odd, the theorem follows in this case.

Suppose e is odd and f is even. We first prove that the image of N̄ : L^{∗}/L^{∗2} → K^{∗}/K^{∗2} consists of the square classes K^{∗2} and ϵK^{∗2}, where ϵ is any nonsquare unit of K. (Note that (U_{K} : U_{K}^{2}) = 2 since K is nondyadic.) The image certainly is of order 2 since its kernel has order σ_{2}(G) = 2 by (4.2) and Lemma 4.6. The norm-induced map L^{∗}/L^{∗2} → K_{nr}^{∗}/K_{nr}^{∗2} is bijective since the degree of L/K_{nr} is odd. The norm of a nonsquare unit of K_{nr} in K is a nonsquare since

𝔬_{K_{nr}} ⟶ k_{nr}
↓  ↓
𝔬_{K} ⟶ k

is commutative, and the norm in a finite extension of finite fields is onto. (Here 𝔬 is a ring of integers, k its residue class field, and the vertical maps are norms). Thus the image of N̄ is as stated. It also follows that each of these images is the image of a square class of L containing a unit and of one containing a prime.

Thus det(Tr_{L/K}αh_{0}) = disc(L/K) N_{L/K}α = disc(L/K) or ϵ disc(L/K), so an invariant form must have one of these determinants. By Lemma 4.8, (b),

c_{K}(Tr_{L/K}αh_{0}) = (π, disc(Tr_{L/K}αh_{0}))_{K}^{r} c_{L}(h_{0})

where r = ord_{K_{tm}}disc(L/K_{tm}) + ord_{L}α. (We can drop the n from this expression since n is odd and r only appears in the exponents of powers of −1). Thus

c_{K}(Tr_{L/K}αh_{0}) = c_{L}(h_{0}) if ord_{L}α ≡ ord_{K_{tm}}disc(L/K_{tm}) mod 2,
= (π, disc(Tr_{L/K}αh_{0}))_{K} c_{L}(h_{0}) otherwise.

By local class field theory, N_{K(\sqrt{π})/K}K(\sqrt{π})^{∗} consists of two of the four square classes in K^{∗}/K^{∗2}, and it is clear they must be K^{∗2} and −πK^{∗2}. It follows that ϵ is not a norm from K(\sqrt{π}), and so (π, ϵ)_{K} = −1. Since an invariant form h satisfies c_{K}(h) = c_{L}(h_{0}) if it arises from an α satisfying ord_{L}α ≡ ord_{K_{tm}}disc(L/K_{tm}) mod 2, and otherwise satisfies c_{K}(h) = (π, disc(h))_{K} c_{L}(h_{0}), the statement in the theorem for e odd and f even follows readily.

Suppose next that e is even and f is odd. The norm from L to K_{tm} (on nonzero elements) is bijective mod squares since the degree is odd. It follows from Lemma 4.6, (b), that the norm from K_{tm} to K_{nr} takes units to square units, and all primes to a single square class containing primes, and so the same thing is true for the norm from L to K since (K_{nr} : K) = f is odd.

By Lemma 4.8,

(4.5) c_{K}(Tr_{L/K}αh_{0}) = c_{K}⟨K_{tm}⟩^{r+1}(Π, disc(Tr_{L/K_{tm}}αh_{0}))_{K_{tm}}^{r+1}c_{L}h_{0}.

Thus if ord_{L}α ≢ ord_{K_{tm}}disc(L/K_{tm}) mod 2, c_{K}(Tr_{L/K}αh_{0}) = c_{L}(h_{0}); since ord_{L}α ≡ ord_{L}α′ mod 2 implies that the norms of α and α′ down to K are equal mod squares, we get one invariant form supporting two inequivalent orthogonal representations.

We now show that the condition ord_{L}α ≢ ord_{K_{tm}}disc(L/K_{tm}) mod 2 is equivalent to ord_{K}(det Tr_{L/K}αh_{0}) being even. Note that ord_{L}α and ord_{K}N_{L/K}α have the same parity, and the same is true of ord_{K_{tm}}disc(L/K_{tm}) and ord_{K}N_{K_{tm}/K}disc(L/K_{tm}). Thus the condition is equivalent to

ord_{K}(disc(L/K)N_{L/K}α) ≢ ord_{K}(disc(L/K)N_{K_{tm}/K}disc(L/K_{tm}))
≡ ord_{K}(disc(K_{tm}/K)^{q}(N_{K_{tm}/K}disc(L/K_{tm}))^{2})
≡ ord_{K}(disc(K_{tm}/K)
= f(e − 1) by Dedekind’s discriminant theorem
≡ 1 mod 2

as desired.

Now suppose that ord_{L}α ≡ ord_{K_{tm}}disc(L/K_{tm}) mod 2. By a now familiar argument, (Π, N_{L/K_{tm}}α)_{K_{tm}} takes on both ±1 for the two possible square classes that α represents, and so we get two invariant forms, each supporting one orthogonal representation. This finishes the proof of the case e even and f odd.

Now suppose both e and f are even. Then det Tr_{L/K}αh_{0} = disc(L/K) by Lemma 4.6 (b). Furthermore (4.5) again holds, and so if ord_{L}α ≢ ord_{K_{tm}}disc(L/K_{tm}) mod 2, we get one invariant form of determinant disc(L/K) and Witt invariant c_{L}(h_{0}) with two inequivalent orthogonal representations. In the other case, we get two inequivalent invariant forms, again with determinant disc(L/K), with one orthogonal representation for each. This finishes the proof of the case e and f both even, and of the theorem as well.

5. Global fields and the Hasse Principle

In this section, K is a global field with an involution ¯ . Our principal goals are to consider problems I, II and III for split representations over K, and to prove the Hasse Principle for equivariant representations.

In the case of split representations, it is particularly easy to determine the equivariant representations of G: if A is simple with involution ā = h_{0}^{−1}a^{∗}h_{0}, the association \underline{ĥ} ⇝ \underline{ĥ} ⊗ h_{0} yields a bijective correspondence between the equivalence classes of ϵ-Hermitian forms of a given rank m and the equivalence classes of all equivariant representations of G of rank mn and type A with respect to ϵ_{0}ϵ-Hermitian forms. (It is understood that if the involution on K is the identity, then Hermitian means symmetric). As we already have seen, it is easy to calculate such things as the determinant, Hasse symbol and signatures from this formula.

The case of split symplectic representations has already been covered in §2, so we deal only with unitary and orthogonal represenatations.

If K_{0} is formally real and h_{0} is symmetric or Hermitian, we can assume that h_{0} is a positive definite matrix at each ordering of K_{0}.

If h is a symmetric bilinear form over R respectively an Hermitian form over C, the number of −1’s in any diagonalization ⟨1, …, 1, −1, …, −1⟩ is the negative index of h. Similarly the number of 1’s is the positive index. If h is a symmetric or Hermitian form over the number field K and 𝔭 is a real prime of K_{0} which does not split in K, the indices of h_{𝔭} are denoted by r_{𝔭}^{−}(h) and r_{𝔭}^{+}(h) respectively.

Lemma 5.1. Let δ ∈ K_{0}^{∗}. Let ℘ be the set of all real primes of K_{0} which do not split in K. Suppose that m is a positive integer and, for each 𝔭 ∈ ℘, that r_{𝔭} is an integer such that 0 ≤ r_{𝔭} ≤ m and δ(−1)^{r_{𝔭}} >_{𝔭} 0. Then there is an m × m Hermitian matrix \underline{ĥ} whose determinant is δ and whose negative index at each 𝔭 ∈ ℘ is r_{𝔭}.

Proof. If m = 1, take \underline{ĥ} = (δ). Suppose m > 1. By the weak approximation theorem, there is an α_{m} ∈ K_{0}^{∗} such that α_{m} >_{𝔭} 0 if r_{𝔭} < m, α_{m} <_{𝔭} 0 if r_{𝔭} = m. Put δ′ = δα_{m}, r′_{𝔭} = r_{𝔭} if α_{m} >_{𝔭} 0, = r_{𝔭} − 1 if α_{m} <_{𝔭} 0. Then δ′(−1)^{r′_{𝔭}} > 0 and so by induction there is an Hermitian matrix \underline{ĥ}′ of rank m − 1 with det \underline{ĥ}′ = δ′ and with negative indices r′_{𝔭}. Then \underline{ĥ} = \underline{ĥ}′ ⊥ (α_{m}) has the required properties.

Theorem 5.2. Split unitary representations. Suppose that the simple involution algebra component A of KG is ≅ M(n, K), and that K is a global field with a nontrivial involution. Let ℘ be the set of real primes of K_{0} which do not split in K. Let V be an A-module of length m and h an Hermitian form on V.

I There is an invariant form on V equivalent to h if and only if the positive and negative indices of h are divisible by n for each real prime 𝔭 ∈ ℘ and, when n is even, det h = (det h_{0})^{m}. II Suppose h is invariant.

(i) If n is odd, h supports exactly one unitary representation of type A (up to equivalence). (ii) If n is even, the equivalence classes of unitary representations supported by h are in bijective correspondence with the norm classes in K_{0}^{∗}/N_{K/K_{0}}K^{∗} which are positive at each 𝔭 ∈ ℘. (Note that the elements of N_{K/K_{0}}K^{∗} are positive at each 𝔭 ∈ ℘).

III Two split unitary representations of odd type are equivalent if and only if they are equivalent as linear representations and their forms are isotypically equivalent.

Proof. Most of this follows easily from the fact that there is a bijection between Hermitian forms over K and isotypic unitary representations, arising from \underline{ĥ} ⇝ \underline{ĥ} ⊗ h_{0}^{t}. The necessity of the condition in I that the indices be divisible by n is a consequence of the fact that the n×n matrix h_{0} is definite. Conversely suppose that the indices of h are nr_{𝔭}^{−} and nr_{𝔭}^{+}. If n is odd, put

δ = (\frac{n}{g})^{m}(det h_{0})^{m} det h.

Since det h_{0} is totally positive and (det h)(−1)^{nr_{𝔭}^{−}} = 1 for all 𝔭 ∈ ℘, we can apply the above lemma to find an Hermitian matrix \underline{ĥ} with determinant δ and negative indices r_{𝔭}. Since \frac{n}{g}(\underline{ĥ} ⊗ h_{0}) has the same determinant and indices as h, it is equivalent to h. Furthermore since the determinant and indices of \underline{ĥ} are uniquely determined by those of h (and h_{0}), we get II in the case of n odd as well.

Suppose n is even. The necessity of I is clear. Conversely it is easy to see by the foregoing lemma that there are Hermitian matrices \underline{ĥ} with indices equal to those of h divided by n, and then for any such matrix, \frac{n}{g}(\underline{ĥ} ⊗ h_{0}) is equivalent to h. This proves I in the case even. Furthermore any two such matrices \underline{ĥ} have determinants differing by an element of K_{0} which is positive at each 𝔭 ∈ ℘. Conversely if α_{0} ∈ K_{0} is positive at each 𝔭 ∈ ℘ and \underline{ĥ} is such an Hermitian matrix, then by the foregoing lemma, there is another such matrix with determinant α_{0} det \underline{ĥ}. This proves II in the case n even.

And III follows easily from II.

Theorem 5.3. Split orthogonal representations Suppose that K is a global field and that the simple involution algebra component A of KG is ≅ M(n, K). Let V be an A-module of length m.

(A, ¯ ) symplectic I, II Any invariant form on V is hyperbolic, and the orthogonal representation it affords is unique up to equivalence.

(A, ¯ ) orthogonal

I Suppose that h is a symmetric form on V.

If n is odd, there is an invariant form ≅ h on V if and only if (i) h ≅ ⟨αh_{0}⟩ for some α ∈ K^{∗} when m = 1, and (ii) n | sgn_{𝔭}h for all real primes 𝔭 of K when m > 1. If n is even, there is an invariant form ≅ h on V if and only if

(iii) det h = (det h_{0})^{m}, and
(iv) for all real primes 𝔭 of K, n|r_{𝔭}^{−}(h), and

(v) S_{𝔭}(h) = S_{𝔭}(h_{0})^{m}(−1, −1)𝔭^{\frac{nm(m−1)}{4}} for all 𝔭 at which

disc h_{0} = 1.

II If h is invariant, there is (up to equivalence) only one orthogonal representation G → O(V, h) if n is odd, and there are an infinite number if n is even.

III Two split orthogonal representations of odd type are equivalent if and only if they are equivalent as linear representations and their underlying forms are isotypically equivalent.

Remark 5.4. 1. In (v), S_{𝔭}(h_{0}) is well-defined since for any α ∈ K^{∗}, S_{𝔭}(αh_{0}) = S_{𝔭}(h_{0}) when n is even and disc h_{0} = 1 at 𝔭.

2. In III, “odd type” can be weakened to: AV = 0 for all simple orthogonal involution components (A, ¯ ) of even degree.

Proof. If (A, ¯ ) is symplectic, \underline{ĥ} must be skew symmetric and this case follows.

Suppose that (A, ¯ ) is orthogonal. If n is odd, the necessity of (i) follows since \underline{ĥ} = ⟨α⟩ implies \underline{ĥ} ⊗ h_{0} = αh_{0}. The necessity of (ii) is clear since h ≅ ⟨\frac{n}{g}(\underline{ĥ} ⊗ h_{0})⟩ for some \underline{ĥ} (and h_{0} is positive definite).

The sufficiency of (i) is also clear. Suppose that n is odd and m > 1. By Lemmas 1.1 and 1.3 (iv), and the fact that sgn_{𝔭}(\underline{ĥ} ⊗ h_{0}) = n(sgn_{𝔭}\underline{ĥ}), the association \underline{ĥ} ⇝ \underline{ĥ} ⊗ h_{0} is injective (on equivalence classes of symmetric forms) since n is odd. We now show that its image consists of all equivalence classes of symmetric forms on V with signature at each real prime 𝔭 divisible by n. Suppose that h is such a form, with signatures nr_{𝔭}. We show the existence of a symmetric form \underline{ĥ} such that h ≅ ⟨\underline{ĥ} ⊗ h_{0}⟩. Its determinant satisfies det h = (det \underline{ĥ})^{n}(det h_{0})^{m}, so det \underline{ĥ} = (det h)(det h_{0})^{m}. Similarly its Hasse-Witt invariant is determined by Lemma 1.3 (iv):

S(\underline{ĥ}) = S(h)S(h_{0})^{m}(det \underline{ĥ}, −1)_{K}^{n(n−1)/2}(det \underline{ĥ}, det h_{0})_{K}^{mn−1}(det h_{0}, −1)_{K}^{m(m−1)/2}, (5.1)

and its real signatures are r_{𝔭}. We must check that the desired signatures are compatible with the determinant and Hasse-Witt invariant.

Let r_{𝔭}^{−} = (m − r_{𝔭})/2 be the negative index at 𝔭. At any real prime 𝔭, det h_{0} = 1 so

det \underline{ĥ} = det h = (−1)^{nr_{𝔭}^{−}} = (−1)^{r_{𝔭}^{−}}

as desired.

By (5.1) the Hasse invariant at 𝔭 is

S_{𝔭}(\underline{ĥ}) = S_{𝔭}(h)(det \underline{ĥ}, −1)_{𝔭}^{(n−1)/2}
= (−1)^{nr_{𝔭}^{−}(nr_{𝔭}^{−}−1)/2}(−1)^{(n−1)r_{𝔭}^{−}/2}.

It is easy to check that this is = (−1)^{r_{𝔭}^{−}(r_{𝔭}^{−}−1)/2} as desired.

The “local-global existence theorem” for quadratic forms ([19], Theorem 72:1) shows that \underline{ĥ} exists. The upshot is that every form h satisfying (i) or (ii) is equivalent to the trace trd_{A/K} ĥ of an Hermitian form ĥ over (A, ¯ ) and so is equivalent to an invariant form. The bijectivity implies II as well when n is odd and m > 1. And in the case m = 1, II follows from the fact that ⟨αh_{0}⟩ ≅ ⟨βh_{0}⟩ implies ⟨α⟩ ≅ ⟨β⟩.

Now assume n even. Suppose that h is invariant. Since h ≅ ⟨\frac{n}{g}(\underline{ĥ} ⊗ h_{0})⟩ for some m × m nonsingular symmetric matrix \underline{ĥ}, (iii) and (iv) follow at once. From Lemma 1.3 (i) and (iv),

S(h) = S(\frac{n}{g}(\underline{ĥ} ⊗ h_{0}))
(5.2)  = S(h_{0})^{m}(det \underline{ĥ}, disc h_{0})(det h_{0}, −1)^{m(m−1)/2}(\frac{n}{g}, disc h_{0})^{m}.

If disc h_{0} = 1 at 𝔭, det h_{0} = (−1)^{n/2} and (v) follows at once.

Conversely suppose that (iii), (iv) and (v) prevail. First we show there is δ ∈ K^{∗} satisfying

(a) (δ, disc h_{0}) = S(h)S(h_{0})^{m}(det h_{0}, −1)^{m(m−1)/2}(\frac{n}{g}, disc h_{0})^{m}, and

(b) the sign of δ at any real prime 𝔭 at which disc h_{0} = 1 is (−1)^{r_{𝔭}^{−}}, where r_{𝔭}^{−}(h) = nr_{𝔭}^{−} is the negative index of h. By Lemma 5.5 it is enough to find δ satisfying (a). If disc h_{0} is a square at 𝔭, the right side of (a) is 1 at 𝔭 by assumption (iv). By the Hilbert reciprocity law, and [19], 71:19 and 71:19a, there is a δ satisfying (a).

Now suppose 𝔭 is a real prime of K. Since h_{0} is positive definite at all real primes,

S(h_{0}) = 1, det h_{0} = 1, disc h_{0} = (−1)^{n/2} at 𝔭.

It follows from (a) that, also at 𝔭,

(δ, disc h_{0}) = (δ, −1)^{\frac{n}{2}}
= S(h) = (−1)^{\frac{nr^{−}_{𝔭}}{2}} = (−1)^{r_{𝔭}^{−}( \frac{n}{2} )}.

Let ĥ_{𝔭} be a symmetric bilinear form of rank m over K_{𝔭} with negative index r_{𝔭}^{−}. If disc h_{0} ≠ 1 at 𝔭, then n/2 must be odd and it follows that (δ, −1)_{𝔭} = (−1)^{r_{𝔭}^{−}} = det ĥ_{𝔭}. Since each of δ and det ĥ_{𝔭} can be considered to be ±1 at 𝔭, it follows that δ = det ĥ_{𝔭}. On the other hand if disc h_{0} is a square at 𝔭, we get δ = det ĥ_{𝔭} at 𝔭 by (b).

It follows once more from the local-global existence theorem that there is a nonsingular symmetric matrix \underline{ĥ} of rank m over K of determinant δ whose localizations at each real prime 𝔭 are the ĥ_{𝔭} just constructed. From (a), the expression for S(\frac{n}{g}(\underline{ĥ} ⊗ h_{0})) in (5.2), the fact that det \frac{n}{g}(\underline{ĥ} ⊗ h_{0}) = (det h_{0})^{m}, and r_{𝔭}^{−}(\underline{ĥ} ⊗ h_{0}) = nr_{𝔭}^{−}(\underline{ĥ}) = nr_{𝔭}^{−} = r_{𝔭}^{−}(h), it follows that Tr_{KG/K} ĥ ≅ h where ĥ is the Hermitian form over (A, ¯ ) on K^{n×m} with discriminant matrix \underline{ĥ}. Thus Tr_{KG/K} ĥ is an invariant form ≅ h.

In fact there are an infinite number of such \underline{ĥ} which are inequivalent, since we can choose the S_{𝔭}(\underline{ĥ}) for 𝔭 finite arbitrarily (up to the Hilbert reciprocity law). Thus we get II for n even: each invariant form h supports an infinite number of inequivalent orthogonal representations.

III is a consequence of II of course.

Lemma 5.5. Let γ ∈ K^{∗} and let (δ, γ) be a fixed quaternion Brauer class over the algebraic number field K. Then the signs of δ at the real primes 𝔭 at which γ is a square (i.e. for which γ >_{𝔭} 0) can be arbitrarily prescribed.

Proof. Let “signs” ϵ_{𝔭} = ±1 be given for each real prime at which γ is a square. For each such prime, choose α_{𝔭} and β_{𝔭} in K_{𝔭} so that (α_{𝔭}^{2}−β_{𝔭}^{2}γ)δϵ_{𝔭} >_{𝔭} 0. Apply the weak approximation theorem to find α and β in K such that (α^{2} − β^{2}γ)δϵ_{𝔭} >_{𝔭} 0 for each 𝔭 and set δ′ = (α^{2} − β^{2}γ)δ. Then

(δ′, γ) = (α^{2} − β^{2}γ, γ)(δ, γ) = (δ, γ)

since, if γ is not a square in K, α^{2}−β^{2}γ is a norm in K(\sqrt{γ})/K. Furthermore

δ′ϵ_{𝔭} = (α^{2} − β^{2}γ)δϵ_{𝔭} >_{𝔭} 0

for any real 𝔭 at which γ is a square.

The Hasse Principle. We now prove the Hasse Principle for equivariant representations.

The completion of the fixed field K_{0} of the involution at any discrete or Archimedean prime 𝔭 is denoted by K_{0𝔭}. Similarly V_{𝔭} = K_{0𝔭} ⊗_{K_{0}} V, A_{𝔭} = K_{0𝔭} ⊗_{K_{0}} A, … – these are all modules over K_{𝔭} = K_{0𝔭} ⊗_{K_{0}} K. The latter ring has a K-involution given by \overline{α ⊗ β} = α ⊗ β̄. If 𝔭 splits in K – which implies that the involution on K is not the identity – K_{𝔭} is the hyperbolic involution algebra K_{0𝔭} ⊕ K_{0𝔭}. Otherwise it is the completion of K with respect to the unique extension of 𝔭 to K. Similarly

(KG)_{𝔭} = K_{0𝔭} ⊗_{K_{0}} KG = K_{𝔭} ⊗_{K} KG = K_{𝔭}G

is also a hyperbolic involution algebra if 𝔭 splits in K. In this case any Hermitian form over K_{𝔭}G is hyperbolic and so is determined by the isomorphism class of the K_{𝔭}G-module V_{𝔭} ( [14], 2.3), and so by the isomorphism class of the KG-module V itself.

Of course if the involution on K is the identity, then K_{0} = K and K_{0𝔭} = K_{𝔭}.

Note that any sesquilinear form f : V × V → A over the K-involution algebra A has a unique extension to a sesquilinear form

f_{𝔭} : V_{𝔭} × V_{𝔭} → A_{𝔭}

determined by

f_{𝔭}(α ⊗ v, β ⊗ u) = αβ ⊗ f(v, u)

where α, β ∈ K_{0𝔭}.

We can also extend an equivariant representation ρ : G → I(V, h) (I = O, Sp, or U) to an equivariant representation ρ_{𝔭} : G → I(V_{𝔭}, h_{𝔭}) in the obvious way,

ρ_{𝔭}(s)(α ⊗ v) = α ⊗ ρ(s)v.

It is clear that ρ_{𝔭}(s) ∈ I(V_{𝔭}, h_{𝔭}) and that ρ_{𝔭} is a homomorphism.

Theorem 5.6. Hasse Principle for equivariant representations. Let G be an arbitrary finite group and K a global field. Let ρ : G → I(V, h) be an equivariant representation, and P the set of all equivariant representations on V which are linearly equivalent to ρ. Then the “Hasse Principle”

for all ρ′ ∈ P, ρ ≅ ρ′ if and only if ρ_{𝔭} ≅ ρ′_{𝔭} for all primes 𝔭 of K_{0}

holds if and only if AV = 0 for every simple direct summand A of KG which satisfies the following: (*) A is a matrix algebra over a quaternion algebra which is nonsplit at

more than 2 primes (of its center), and the restriction of the canonical involution on KG to A is an involution of orthogonal type if ρ is orthogonal, and of symplectic type if ρ is symplectic.

Remark 5.7. In particular this means that the Hasse Principle always holds for unitary representations.

Proof. Suppose that

ρ : G → I(V, h), ρ′ : G → I(V′, h′).

are two equivariant representations which are “of the same type” (h and h′ both symmetric, or both Hermitian, …) and that they are linearly equivalent. In fact we assume that they are equal (as linear representations). Let ϵ = 1 if h and h′ are symmetric or Hermitian, −1 if they are skew symmetric or skew Hermitian. We shall refer to h and h′ as ϵ-Hermitian in both cases.

According to §1.3 there are simple summands A_{1}, …, A_{k} of KG stable under ¯ and forms h̆_{1}, …, h̆_{k} over the division algebras D_{1}, …, D_{k}, D_{i} Brauer equivalent to A_{i}, whose (ordinary) equivalence classes characterize the equivariant equivalence classes of ρ. We now look more closely at the last step in this association, the Hermitian Morita equivalence of ĥ_{i} and h̆_{i}, using the “Hermitian Morita context” of Lemma 1.8.

There is an ϵ_{i}-Hermitian matrix \underline{h}_{0i} ∈ A_{i} = M(n_{i}, D_{i}) such that ā = \underline{h}_{0i}^{−1}a^{∗}\underline{h}_{0i} for all a ∈ A_{i}. Let V_{0i} be the simple A_{i}-module D_{i}^{n_{i}×1}. It is an (A_{i}, D_{i})-bimodule. By Lemma 1.8, the nonsingular ϵ_{i}-Hermitian form h_{0i} : V_{0i} × V_{0i} → A_{i} defined by h_{0i}(u, v) = uv^{∗}\underline{h}_{0i} is ϵ_{i}-Hermitian and, if D_{i} is a quaternion algebra and the involution on A_{i} is of the first kind, the adjoint involution ¯ on D_{i} is conjugation; in this latter case we note that ϵ_{i} = 1 respectively −1 if (A, ¯ ) is symplectic respectively orthogonal. Twist the action of A_{i} and D_{i} on V_{0i} using their involutions, and let V̄_{0i} denote the resulting (D_{i}, A_{i})-bimodule.

By the Hermitian Morita theory, there is a nonsingular ϵ_{i}-Hermitian form H_{0i} : V̄_{0i} × V̄_{0i} → D_{i} whose adjoint involution on A_{i} is ¯ and which satisfies the Hermitian Morita associative relationship

h_{0i}(u, v)w = uH_{0i}(v̄, w̄) for all u, v, w ∈ V_{0i}. (5.3)

It effects an equivalence between the category _{A_{i}}H_{ϵ} of ϵ-Hermitian forms over A_{i} and the category _{D_{i}}H_{ϵϵ_{i}} of ϵϵ_{i}-Hermitian forms over D_{i}, via the map ĝ ⇝ H_{0i}ĝ (product of forms - cf. [8], §2). This is the equivalence described in Lemma 1.8.

By Table 1 in §1.1 and 10.4.6, [21], the Hasse Principle applies to the forms H_{0i}ĝ unless they are skew Hermitian (ϵϵ_{i} = −1) and D_{i} is a quaternion division algebra which is nonsplit at more than 2 primes. These exceptional cases are precisely those arising from (*) and so in order to prove the sufficiency of the condition in the theorem, we assume that A_{i}V = 0 for any such exceptional i, 1 ≤ i ≤ k. Of course this means also that ĥ_{i} = 0.

It follows that ρ ≅ ρ′ if and only if (H_{0i}ĥ_{i})_{𝔓} ≅ (H_{0i}ĥ′_{i})_{𝔓} for 1 ≤ i ≤ k and all primes 𝔓 of L_{i0}, where L_{i} is the center of D_{i} and L_{i0} is the subfield of elements fixed by the involution of L_{i}. We now wish to show that this is equivalent with

ρ ≅ ρ′ if and only if (H_{0i}ĥ_{i})_{𝔭} ≅ (H_{0i}ĥ′_{i})_{𝔭} for 1 ≤ i ≤ k and all primes 𝔭 of K_{0}. (5.4)

In doing this we shall drop the index i from L_{i}, L_{i0}, D_{i}, etc. in order to simplify the notation.

By [2], p. 57, there is an isomorphism K_{0𝔭} ⊗_{K_{0}} L_{0} ≅ ⊕_{𝔓|𝔭}L_{0𝔓}, which implies an isomorphism

L_{𝔭} ≅ ⊕_{𝔓|𝔭}L_{𝔓} (L_{𝔭} = K_{0𝔭} ⊗_{K_{0}} L, L_{𝔓} = L_{0𝔓} ⊗_{L_{0}} L)

given by

α_{𝔭} ⊗ β ↦ (ι_{1}α_{𝔭} ⊗ β, …, ι_{q}α_{𝔭} ⊗ β)

where ι_{j} : K_{0𝔭} → L_{0𝔓_{j}} is the inclusion. We identify L_{𝔭} and ⊕_{𝔓|𝔭}L_{𝔓} via this isomorphism. Similarly we identify

D_{𝔭} = ⊕_{𝔓|𝔭}D_{𝔓} and W_{𝔭} = ⊕_{𝔓|𝔭}W_{𝔓},

where W is a vector space over D, via the isomorphisms

α_{𝔭} ⊗ d ↦ (ι_{1}α_{𝔭} ⊗ d, …, ι_{q}α_{𝔭} ⊗ d) and α_{𝔭} ⊗ w ↦ (ι_{1}α_{𝔭} ⊗ w, …, ι_{q}α_{𝔭} ⊗ w).

These identifications are compatible with the inclusion of D respectively W on both sides, as well as with the action of D_{𝔭} on W_{𝔭} and ⊕_{𝔓|𝔭}D_{𝔓} on ⊕_{𝔓|𝔭}W_{𝔓}.

Let f : W × W → D be a sesquilinear form.

Lemma 5.8. (a) The sesquilinear module (W_{𝔭}, f_{𝔭}) is the orthogonal direct sum of the sesquilinear modules (W_{𝔓}, f_{𝔓}). Briefly, f_{𝔭} =⊥_{𝔓|𝔭} f_{𝔓}.

(b) Let g : U × U → D be another sesquilinear form. Then f_{𝔭} ≅ g_{𝔭} if and only if f_{𝔓} ≅ g_{𝔓} for all primes 𝔓|𝔭 of L_{0}.

Proof. First of all, f_{𝔓} : W_{𝔓} × W_{𝔓} → D_{𝔓} ⊂ D_{𝔭} and so can be considered as a sesquilinear form over D_{𝔭}. Then

(⊥_{𝔓|𝔭} f_{𝔓})(α ⊗ w, α′ ⊗ w′) = (f_{𝔓_{1}}(ι_{1}α ⊗ w, ι_{1}α′ ⊗ w′), …, f_{𝔓_{q}}(ι_{q}α ⊗ w, ι_{q}α′ ⊗ w′))
= (ι_{1}αα′ ⊗ f(w, w′), …, ι_{q}αα′ ⊗ f(w, w′))
= αα′ ⊗ f(w, w′) = f_{𝔭}(α ⊗ w, α′ ⊗ w′),

which proves (a). And (b) follows since W_{𝔓} = D_{𝔓}W_{𝔭} and U_{𝔓} = D_{𝔓}U_{𝔭}.

It follows then that (5.4) holds.

Lemma 5.9. Suppose that A and B are K-involution algebras, that f : U × U → B is a sesquilinear form admitting A, and that g : V × V → A is a sesquilinear form. If 𝔭 is a prime of K_{0}, the forms (fg)_{𝔭} and f_{𝔭}g_{𝔭} over B_{𝔭} are canonically equivalent.

Proof. There is a standard K_{0𝔭}-isomorphism ϕ : (U ⊗_{A} V)_{𝔭} → U_{𝔭} ⊗_{A_{𝔭}} V_{𝔭} taking α ⊗ (u ⊗ v) to (α ⊗ u) ⊗ (1 ⊗ v) (α ∈ K_{0𝔭}, u ∈ U, v ∈ V). It is easy to check that ϕ is an isomorphism of B_{𝔭}-modules, and that it is also an equivalence between the forms (fg)_{𝔭} and f_{𝔭}g_{𝔭}. (Recall that the definition of the product fg : (U ⊗_{A} V) × (U ⊗_{A} V) → B is (fg)(u ⊗ v, u′ ⊗ v′) = f(ug(v, v′), u′)).

We now prove that the Hasse Principle holds, under the assumption that AV = 0 for every simple direct summand A of KG which satisfies (*). Any such summand is certainly one of the A_{i}, 1 ≤ i ≤ k, defined in §1.3; let A_{i}, 1 ≤ i ≤ m be the remaining A_{i}. From (5.4) and Lemma 5.9, we get

(5.5) ρ ≅ ρ′ if and only if H_{0i𝔭}ĥ_{i𝔭} ≅ H_{0i𝔭}ĥ′_{i𝔭} for all 𝔭 and 1 ≤ i ≤ m.

Now consider ρ_{𝔭} : G → I(V_{𝔭}, h_{𝔭}), h_{𝔭} : V_{𝔭} × V_{𝔭} → K_{𝔭}. Then ĥ_{𝔭} : V_{𝔭} × V_{𝔭} → K_{𝔭}G = ⊕_{i=1}^{t}A_{i𝔭} and (ĥ_{𝔭})_{i} : A_{i𝔭}V_{𝔭} × A_{i𝔭}V_{𝔭} → A_{i𝔭}. Furthermore it is easy to check that ρ_{𝔭} ≅ ρ′_{𝔭} if and only if (ĥ_{𝔭})_{i} ≅ (ĥ′_{𝔭})_{i} for all i, 1 ≤ i ≤ t.

Now suppose that i > k (cf. §1.3). If A_{i} is a hyperbolic involution algebra, so is A_{i𝔭}. Otherwise A_{i} is of index 1 with a symplectic respectively orthogonal involution (which implies that the involution on K is the identity), and so A_{i𝔭} is likewise of index 1 with a symplectic respectively orthogonal involution, or is a direct sum of such involution algebras. Thus if ρ is orthogonal respectively symplectic, the forms (ĥ_{𝔭})_{i} and (ĥ′_{𝔭})_{i} arising from these algebras are determined by the K_{𝔭}G-module structure of the spaces on which they are defined (cf. §1.3).

Thus ρ_{𝔭} ≅ ρ′_{𝔭} if and only if (ĥ_{𝔭})_{i} ≅ (ĥ′_{𝔭})_{i} for all i, 1 ≤ i ≤ k.

The next step is to show that (ĥ_{𝔭})_{i} = ĥ_{i𝔭}. Now ĥ_{i𝔭} is defined on (A_{i}V)_{𝔭} = K_{0𝔭}⊗A_{i}V which we identify with a submodule of K_{0𝔭}⊗V = V_{𝔭} in the usual way. Note that A_{i𝔭}V_{𝔭} is spanned, as an Abelian group, by elements of the form (α ⊗ a_{i})(β ⊗ v) = αβ ⊗ a_{i}v ∈ (A_{i}V)_{𝔭}. It follows that A_{i𝔭}V_{𝔭} = (A_{i}V)_{𝔭}. We can now show that (ĥ_{𝔭})_{i} = ĥ_{i𝔭}. Let α, β ∈ K_{0𝔭} and u, v ∈ A_{i}V. Then (cf. (1.2))

ĥ_{𝔭}(α ⊗ u, β ⊗ v) = ∑_{s} h_{𝔭}(ρ_{𝔭}(s^{−1})(α ⊗ u), β ⊗ v)s = ∑_{s} h_{𝔭}(α ⊗ ρ(s^{−1})u, β ⊗ v)s
= ∑_{s} αβ ⊗ h(ρ(s^{−1})u, v)s = αβ ⊗ ∑_{s} h(ρ(s^{−1})u, v)s
= αβ ⊗ ĥ(u, v) = ĥ_{i𝔭}(α ⊗ u, β ⊗ v).

Thus

ρ_{𝔭} ≅ ρ′_{𝔭} if and only if ĥ_{i𝔭} ≅ ĥ′_{i𝔭} for all i, 1 ≤ i ≤ k.

The forms h_{0i} : V_{0i} × V_{0i} → A_{i} and H_{0i} : V̄_{0i} × V̄_{0i} → D_{i} are nonsingular ϵ_{i}-Hermitian forms which satisfy the associativity relationship (5.3). The forms h_{0i𝔭} and H_{0i𝔭} also satisfy the associativity relationship:

h_{0i𝔭}(α ⊗ u, β ⊗ v)(γ ⊗ w) = (αβ ⊗ h_{0i}(u, v)) (γ ⊗ w) = αβγ ⊗ h_{0i}(u, v)w
= αβγ ⊗ (uH_{0i}(v̄, w̄)) = (α ⊗ u)(βγ ⊗ H_{0i}(v̄, w̄))
= (α ⊗ u)H_{0i𝔭}(β ⊗ v̄, γ ⊗ w̄).

It follows from the Hermitian Morita theory that h_{0i𝔭} and H_{0i𝔭} induce a category equivalence between the category of ϵ-Hermitian forms over A_{i𝔭} and the category of ϵ_{i}ϵ-Hermitian forms over D_{i𝔭}. This implies that

ρ_{𝔭} ≅ ρ′_{𝔭} if and only if H_{0i𝔭}ĥ_{i𝔭} ≅ H_{0i𝔭}ĥ′_{i𝔭} for all i, 1 ≤ i ≤ k.

This together with (5.5) shows that ρ ≅ ρ′ if and only if ρ_{𝔭} ≅ ρ′_{𝔭} for all 𝔭 (since ĥ_{i} = 0 for m < i ≤ k).

Conversely suppose that A_{i}V ≠ 0 for some A_{i} satisfying (*). In particular, ρ is orthogonal (ϵ = 1, ϵ_{i} = −1) or symplectic (ϵ = −1, ϵ_{i} = 1). Then H_{0i}ĥ_{i} is a skew Hermitian form over D_{i} on V̄_{0i}⊗A_{i}V, and there is another nonsingular skew Hermitian form over D_{i} on the same space, which is locally equivalent to H_{0i}ĥ_{i} at all primes 𝔓 but not globally equivalent to it (cf. [21], 10.4.6). By Morita equivalence there is an ϵ-Hermitian form ĥ′′_{i} over A_{i} on A_{i}V which is locally but not globally equivalent to ĥ_{i}. This gives rise in an obvious way to an orthogonal or symplectic representation ρ′′ : G → U(V, h′′) which is locally equivalent to ρ at all 𝔭 but not globally equivalent. Namely one defines ĥ′′ =⊥_{j} ĥ′′_{j} where ĥ′′_{j} = ĥ_{j} for all j ≠ i, and then h′′ = Tr_{KG/K} ∘ ĥ′ (cf. (1.3)). This finishes the proof of Theorem 5.6.

Corollary 5.10. The Hasse Principle always holds in the following cases:

(a) The representations are unitary.

(b) The characteristic of K is ≠ 0.

(c) The representations are of odd type.

(d) G is nilpotent and K contains either a cubic or quartic root of unity.

Proof. If the representation is unitary, (*) does not apply. If the characteristic is ≠ 0, the A_{i} all have index 1 and so again (*) does not apply. Suppose that the characteristic is 0. If the degree of every absolutely irreducible character is odd, no A_{i} is similar to a quaternion algebra. In (d) the A_{i} also have index 1 – see [7], 14.5, p. 77.

6. Equivariant representations of the symmetric group.

As with linear and projective representations, the equivariant representation theory of S_{r} is especially explicit – the essential points are that all representations are split and that one can calculate explicitly the invariant symmetric form h_{0} for an irreducible representation ρ : S_{r} → GL(V). This is of course the same matrix involved in the involution ā = h_{0}^{−1}a^{∗}h_{0} on the direct summand of KG corresponding to ρ, induced by the canonical involution of KG. The procedure for the calculation of h_{0} is described in [10]; we illustrate it by an example. This method applies if char K = 0 or p > r.

We note that when G = S_{r}, every simple involution algebra component A of KG is orthogonal. In characteristic 0, this follows from the existence of a nonsingular invariant symmetric form on a simple A-module – in fact it can be assumed to be a positive definite form defined over ℚ – and from the fact that such a form is unique up to scalar multiples since A is split. In characteristic p, there is also a nonsingular invariant symmetric form on any simple module, which is described explicitly later in this section via the example.

If K has a nontrivial involution and KS_{r} has the corresponding canonical involution, then every simple involution component A of KS_{r} is again a simple algebra. This follows from the fact that KS_{r} is split over the prime subfield, so A = K ⊗_{K_{0}} A_{0} where A_{0} is a simple involution component of K_{0}S_{r}, and the involution on A is the tensor product of the involutions on K and A_{0}. This implies that the Hermitian matrix h_{0} for the unitary involution on A is identical with the symmetric matrix for the involution of the first kind on A_{0}.

All of the facts cited in the following about representations of S_{r} can be found in [10].

We recall that each irreducible representation of S_{r} arises from a partition α = [α_{1}, α_{2}, …, α_{k}] of r; thus the α_{i} are positive integers satisfying α_{1} ≥ α_{2} ≥ ⋯ ≥ α_{k} and ∑_{i} α_{i} = r (cf. [10], pp. 350 and 376). We shall describe the calculation of h_{0} through an example, namely the partition

α = [3, 2, 2]

of 7, which gives rise to an irreducible representation of S_{7}. The standard α-tableaus are the arrays

1 2 3 4 5 6 7 1 2 3 4 6 5 7 1 2 4 3 5 6 7 1 2 4 3 6 5 7 1 2 5 3 4 6 7 1 2 5 3 6 4 7

1 2 6 3 4 5 7 1 2 6 3 5 4 7 1 2 7 3 4 5 6 1 2 7 3 5 4 6 1 3 4 2 5 6 7 1 3 4 2 6 5 7

1 3 5 2 4 6 7 1 3 5 2 6 4 7 1 3 6 2 4 5 7 1 3 6 2 5 4 7 1 3 7 2 4 5 6 1 3 7 2 5 4 6

1 4 5 2 6 3 7 1 4 6 2 5 3 7 1 4 7 2 5 3 6

They are characterized by the fact that the row lengths constitute the given partition of 7, and the integers in each row and in each column are increasing. The number of standard α-tableaus, 21, is the degree of the corresponding representation ( [10], 3.1.13, p. 107).

There is an orthogonal basis u_{1}, ⋯, u_{21} (with respect to h_{0}) of the representation space for which one can calculate h_{0}(u_{i}, u_{i}). One starts with the i^{th} standard tableau; then

h_{0}(u_{i}, u_{i}) = ∏_{x=1}^{7} ∏_{k} \frac{h(x, k)}{h(x, k) − 1}

where h(x, k) is a “hook length” defined as follows. k = 1, 2, 3 is the number of a row. h(x, k) is not defined (and hence does not appear in the above product) unless the integer x is in a row lower than row k; thus h(1, k) and h(x, 3) are not defined for any k and x.

Suppose x is in a row lower than row k. Then h(x, k) is the number of integers in the “hook” consisting of

(i) the integer x,

(ii) the integers above x up to row k (but not in rows 1, …, k − 1) in the same column as x, and (iii) the integers ≤ x in row k to the right of the column of x.

For example in the 8^{th} standard tableau in the above display

1 2 6 3 5 4 7 ,

h(4, 1) = 4, h(4, 2) = 2 (and of course h(4, 3) is not defined). It is easy to check that

h_{0}(u_{8}, u_{8}) = ∏_{x=2}^{7} ∏_{k=1}^{2} \frac{h(x, k)}{h(x, k) − 1} = (\frac{3}{2})(\frac{4}{3} · \frac{2}{1})(\frac{2}{1})(\frac{4}{3} · \frac{2}{1}) = \frac{2^{6}}{3}

where the parenthesized factors correspond respectively to x = 3, 4, 5, 7. In this way one gets for h_{0}(u_{i}, u_{i}), 1 ≤ i ≤ 21,

10, 40/3, 45/4, 15, 15, 16,
16, 64/3, 18, 24, 15, 20,
20, 64/3, 64/3, 256/9, 24, 32,
24, 32, 36.

After adjusting by squares,

h_{0} = ⟨10, 30, 5, 15, 15, 1, 1, 3, 2, 6, 15, 5, 5, 3, 3, 1, 6, 2, 6, 2, 1⟩.

Thus one can compute the determinant and Hasse symbol of h_{0}; they are 1 and (3, 3)_{K} respectively. This means that over a local or global field, h_{0} ≅ ⟨1, 1, …, 1, 3, 3⟩.

References

1. N. Bourbaki, Algèbre (modules et anneaux semi-simples), Actualités Scientifiques et Industrielles, no. 1261, Hermann, 1958.

2. J. W. S. Cassels and A. Fröhlich, Algebraic number theory, Conference Proceedings, Academic Press Inc., London, 1967.

3. Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, vol. XI, Interscience Publishers, John Wiley and Sons, New York, 1962.

4. Andreas W. M. Dress, Induction and structure theorems for orthogonal representations of finite groups, Annals of Mathematics 102 (1975), 291–325.

5. Martin Epkenhans, Spurformen über Lokalen Körpern, Schriftenreihe der Mathematischen Instituts der Universität Münster, 1987.

6. , Trace forms of normal extensions over local fields, Linear and Multilinear Algebra 24 (1989), 103–116.

7. Walter Feit, Characters of Finite Groups, Mathematics Lecture Notes, W.A.Benjamin Inc., New York, 1967.

8. A. Fröhlich and A. M. McEvett, Forms over rings with involution, Journal of Algebra 12 (1969), 79–104.

9. , The representations of groups by automorphisms of forms, Journal of Algebra 12 (1969), 114–133.

10. Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and Its Applications, vol. 16, Addison-Wesley, Reading, MA, 1981.

11. Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, Colloquium Publications, vol. 44, American Mathematical Society, Providence, R.I., 1998.

12. H. Koch, Number theory II, Encyclopaedia of Mathematical Sciences, vol. 62, Springer-Verlag, 1992.

13. A. I. Malcev, On semisimple subgroups of Lie groups, Translations of the A.M.S. (1950), no. 33, 172–213.

14. A. M. McEvett, Forms over semisimple algebras with involution, Journal of Algebra 12 (1969), 105–113.

15. John Milnor, On isometries of inner product spaces, Invent. Math. 8 (1969), 83–97.

16. Gabriele Nebe, Determining invariants of orthogonal G-modules using the character table of G, to appear.

17. , Orthogonal Frobenius reciprocity, to appear in Journal of Algebra.

18. Jürgen Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, Heidelberg, 1992.

19. O. T. O’Meara, Introduction to Quadratic Forms, Grund. d. math. Wiss., vol. 117, Springer-Verlag, New York, Heidelberg, Berlin, 1963.

20. I. Reiner, Maximal orders, L.M.S. Monographs, Academic Press, Inc., London, New York, San Francisco, 1975.

21. W. Scharlau, Quadratic and hermitian forms, Grund. d. math. Wiss., vol. 270, Springer-Verlag, New York, Heidelberg, Berlin, 1985.

22. J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics, vol. 42, Springer-Verlag, New York, Heidelberg, Berlin, 1977.

23. C. T. C. Wall, Classification of Hermitian Forms. VI. Group rings, Ann. of Math. (2) 103 (1976), no. 1, 1–80.

Dept. of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1

   E-mail address: riehm@mcmaster.ca