The irreducible orthogonal and symplectic Galois representations of a p-adic field (the tame case)

Orthogonal, symplectic and unitary representations of finite groups lie at the crossroads of two more traditional subjects of mathematics--linear representations of finite groups, and the theory of quadratic, skew symmetric and Hermitian forms --and thus inherit some of the characteristics of both. This book is written as an introduction to the subject and not as an encyclopaedic reference text. The principal goal is an exposition of the known results on the equivalence theory, and related matters such as the Witt and Witt-Grothendieck groups, over the classical fields--algebraically closed, real closed, finite, local and global. A detailed exposition of the background material needed is given in the first chapter. It was A. Frohlich who first gave a systematic organisation of this subject, in a series of papers beginning in 1969. His paper Orthogonal and symplectic representations of groups represents the culmination of his published work on orthogonal and symplectic representations. The author has included most of the work from that paper, extending it to include unitary representations, and also providing new approaches, such as the use of the equivariant Brauer-Wall group in describing the principal invariants of orthogonal representations and their interplay with each other.

Producing geometric deformations of orthogonal and symplectic Galois representations

Let F be a p-adic field and n a positive integer. The local Langlands conjecture asserts the existence of a bijection between irreducible admissible representations of GL(n,F) and n-dimensional admissible representations of the Weil-Deligne group of F. This bijection is required to satisfy certain natural compatibilities, of which the most important is compatibility with local functional equations (preservation of L and epsilon factors of pairs). It is enough to construct a bijection with these properties between supercuspidal representations of GL(n,F) and n-dimensional irreducible representations of the Weil group of F. In a previous paper, the author constructed a canonical bijection on the etale cohomology of the rigid-analytic coverings of the p-adic upper half space constructed by Drinfeld. (That the map in the previous paper is a bijection was actually proved by Henniart.) However, the compatibility of epsilon factors of pairs was not shown. The present article uses a technique of non-Galois automorphic induction to show that the bijection previously constructed is compatible with epsilon factors of pairs of representations of GL(n,F) and GL(m,F) when n and m are prime to p (the tame case). This implies the local Langlands conjecture in degree < p. It is also shown how the local Langlands conjecture in general would follow from a generalization of Carayol's theorem on the bad reduction of Shimura curves to certain Shimura varieties of higher dimension.

This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse Galois problem for symplectic groups. For any even positive integer \(n\) and any positive integer \(d\), \(\mathrm {PSp}_n(\mathbb {F}_{\ell ^d})\) or \(\mathrm {PGSp}_n(\mathbb {F}_{\ell ^d})\) occurs as a Galois group over the rational numbers for a positive density set of primes \(\ell \). The result depends on some work of Arthur’s, which is conditional, but expected to become unconditional soon. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of \(\hbox {GL}_n({\mathbb {A}}_\mathbb {Q})\) with local types chosen so as to obtain a compatible system of Galois representations to which the results from Part II of this series apply.

Let $K$ be a field, $G$ a finite group, and $\rho : G \to \mathbf {GL}(V)$ 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 $h: V \times V \rightarrow K$ 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.

This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part is concerned with symplectic Galois representations having a huge residual image, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. A key ingredient is a classification of symplectic representations whose image contains a nontrivial transvection: these fall into three very simply describable classes, the reducible ones, the induced ones and those with huge image. Using the idea of an (n,p)-group of Khare, Larsen and Savin we give simple conditions under which a symplectic Galois representation with coefficients in a finite field has a huge image. Finally, we combine this classification result with the main result of the first part to obtain a strenghtened application to the inverse Galois problem.

We review the theory of quiver bundles over a Kähler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group [Formula: see text]. We first study the case when [Formula: see text] or [Formula: see text], interpreting them as orthogonal (respectively symplectic) bundle representations of the symmetric quivers introduced by Derksen–Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. In particular, we completely characterize the polystable forms of such representations. Finally, we discuss Hitchin–Kobayashi correspondences for these objects.

The representation of groups by automorphisms of forms

Let K be a p-adic local field where p is an odd prime and let A be the unique quaternion division algebra whose centre is K. By means of Stiefel–Whitney classes, we define an exponential homomorphism ϒK from the orthogonal representations of A*/K* to fourth roots of unity. We then evaluate this homomorphism in terms of the local root numbers of two-dimensional symplectic Galois representations of K, using the Langlands correspondence relating Galois representations to continuous representations of A*.

This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem.

We investigate the case of deformations of even Galois representations. Our methods are the group theoretic ones mainly developed by Nigel Boston to study odd representations. We present conditions for Borel and tame cases under which the universal deformation ring is isomorphic to ℤp[[T]] and where we compute the universal deformation explicitly. Furthermore we produce a family of examples of totally real S3 extensions which satisfy the above conditions in the tame case and we give examples in the Borel case. Finally we study the change of the deformation space under enlarging the ramification and thus give an example of an even representation that is not twist‐finite.

We associate one or two posets (which we call "semistandard posets") to any given irreducible representation of a rank two semisimple Lie algebra over ${\Bbb C}$. Elsewhere we have shown how the distributive lattices of order ideals taken from semistandard posets (we call these "semistandard lattices") can be used to obtain certain information about these irreducible representations. Here we show that some of these semistandard lattices can be used to present explicit actions of Lie algebra generators on weight bases (Theorem 5.1), which implies these particular semistandard lattices are supporting graphs. Our descriptions of these actions are explicit in the sense that relative to the bases obtained, the entries for the representing matrices of certain Lie algebra generators are rational coefficients we assign in pairs to the lattice edges. In Theorem 4.4 we show that if such coefficients can be assigned to the edges, then the assignment is unique up to products; we conclude that the associated weight bases enjoy certain uniqueness and extremal properties (the "solitary" and "edge-minimal" properties respectively). Our proof of this result is uniform and combinatorial in that it depends only on certain properties possessed by all semistandard posets. For certain families of semistandard lattices some of these results were obtained in previous papers; in Proposition 5.6 we explicitly construct new weight bases for a certain family of rank two symplectic representations. These results are used to help obtain in Theorem 5.1 the classification of those semistandard lattices which are supporting graphs.

It is shown that the finite- and infinite-dimensional irreducible representations ( j0, c) of the proper Lorentz group SO(3,1) may be classified into the two categories, namely, the complex-orthogonal and the symplectic representations; while all the integral-j0 representations are equivalent to complex-orthogonal ones, the remaining representations for which j0 is a half-odd integer are symplectic in nature. This implies in particular that all the representations belonging to the complementary series and the subclass of integral-j0 representations belonging to the principal series are equivalent to real-orthogonal representations. The rest of the principal series of representations for which j0 is a half-odd integer are symplectic in addition to being unitary and this in turn implies that the D j representation of SO(3) with half-odd integral j is a subgroup of the unitary symplectic group USp(2 j+1). The infinitesimal operators for the integral-j0 representations are constructed in a suitable basis wherein these are seen to be complex skew-symmetric in general and real skew-symmetric in particular for the unitary representations, exhibiting explicitly the aforementioned properties of the integral-j0 representations. Also, by introducing a suitable real basis, the finite-dimensional ( j0=0, c=n) representations, where n is an integer, are shown to be real-pseudo-orthogonal with the signature (n(n+1)/2, n(n−1)/2). In any general complex basis, these representations (0, n) are also shown to be pseudo-unitary with the same signature (n(n+1)/2, n(n−1)/2). Further it is shown that no other finite-dimensional irreducible representation of SO(3,1) possesses either of these two special properties.

Serre's conjecture relates two-dimensional odd irreducible Galois representations over 𝔽̄ p to modular forms. We discuss a generalization of this conjecture to higher-dimensional Galois representations. In particular, for n-dimensional Galois representations that are irreducible when restricted to the decomposition group at p, we strengthen a conjecture of Ash, Doud, and Pollack. We then give computational evidence for this conjecture in the case of three-dimensional representations.

Explicit Brauer Induction formulae with certain natural behaviour have been developed for complex representations, for example by work of Boltje, Snaith and Symonds. In this paper we present induction formulae for symplectic and orthogonal representations of finite groups. The problems are motivated by number theoretical and topological questions. We will prove naturality with respect to restriction and inflation. Also we investigate complexification maps and use them to compare the orthogonal and symplectic induction formulae with Boltje's complex induction formula.