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The classical spectral theorem completely describes self-adjoint operators on finite-dimensional inner product vector spaces as linear combinations of orthogonal projections onto pairwise orthogonal subspaces. We prove a similar theorem for self-adjoint operators on finite-dimensional symplectic vector spaces over perfect fields. We show that these operators decompose according to a polarization, i.e. as the product of an operator on a Lagrangian subspace and its dual on a complementary Lagrangian. Moreover, if all the eigenvalues of the operator are in the base field, then there exists a Darboux basis such that the matrix representation of the operator is [Formula: see text] blocks block-diagonal, where the first block is in Jordan normal form and the second block is the transpose of the first one.

The graph of a real symplectic linear transformation is an R-Lagrangian subspace of a complex symplectic vector space. The restriction of the complex symplectic form is thus purely imaginary and may be expressed in terms of the generating function of the transformation. We provide explicit formulas; moreover, as an application, we give an explicit general formula for the metaplectic representation of the real symplectic group.

Buildings of types $\textsf{C}_n$ and $\textsf{D}_n$ are defined by rank $n$ polar spaces. The associated building Grassmannians are polar and half-spin Grassmannians. Apartments in dual polar spaces and half-spin Grassmannians were characterized in \cite{CKS}. We characterize apartments in all polar Grassmannians consisting of non-maximal singular subspaces. This characterization is a partial case of more general results concerning embeddings of polar Johnson graphs in polar Grassmann graphs.

In this short note, completing a sequence of studies by Cooperstein, Kasikova and Shult, we consider the k-Grassmannians of a number of polar geometries of finite rank n. We classify those subspaces that are isomorphic to the j-Grassmannian of a projective m-space. In almost all cases, these are parabolic, that is, they are the residues of a flag of the polar geometry. Exceptions only occur when the subspace is isomorphic to the Grassmannian of 2-spaces in a projective m-space and we describe these in some detail. This Witt-type result implies that automorphisms of the Grassmannian are almost always induced by automorphisms of the underlying polar space.

It is shown that every automorphism of classical unitals over certain (not necessarily commutative) fields is induced by a semi-similitude of a corresponding hermitian form. In particular, this is true if the form uses an involution of the second kind. In [16], J. Tits has studied classical unitals, defined by suitable polarities. Using

It is known that every lax projective embedding e : Γ → PG(V) of a point-line geometry Γ admits a hull, namely a projective embedding ˜e : Γ → PG(V˜)uniquely determined up to isomorphisms by the following property: V and V˜are defined over the same skewfield, say K, there is morphism of embeddingsf˜ : ˜e → e and, for every embedding e ′ : Γ → PG(V ′ ) with V ′ defined overK, if there is a morphism g : e ′ → e then a morphism f : ˜e → e ′ also existssuch that f˜= gf. If e = ˜e then we say that e is dominant. Clearly, hulls aredominant. Let now Γ be a non-degenerate polar space of rank n ≥ 3. Weshall prove the following: A lax embedding e : Γ → PG(V) is dominant if andonly if, for every geometric hyperplane H of Γ, e(H) spans a hyperplane ofPG(V). We shall also give some applications of the above result. 1 Introduction In the first part of this introduction we will recall the essentials on embeddings, theirmorphisms and hulls. In the second part, we shall state our main results.1.1 Basics on embeddings and their morphimsA projective embedding of a connected point-line geometry Γ = (P,L) is an injectivemapping e from the point-set P of Γ to the point-set of a desarguesian projectivespace Σ such that(E1) the image e(P) of P spans Σ,(E2) for every line L of Γ, e(L) spans a line of Σ

We give a simple proof of the known fact that the symplectic quadrangle is self-dual if and only if the ground field is perfect of characteristic~2, and that a polarity exists exactly if there is a root of the Frobenius automorphism. Moreover, we determine all polarities, characterize the conjugacy classes of polarities, and use the results to give a simple proof that the centralizer of any polarity acts two-transitively on the ovoid of absolute points. The proofs use elementary calculations in solvable subgroups of the symplectic group.

Shult and Thas have shown in [13] that m-systems of certain finite classical polar spaces give rise to strongly regular graphs and two weight codes. The main result of this paper is to show that maximal arcs in symplectic translation planes may be obtained from certain m-systems of finite symplectic polar spaces. Many new examples of maximal arcs are then constructed. Examples of m-systems are also constructed in Q(-)(2n + 1,q) and W2n+1(q). A method different from that of Shult and Thas is used to construct strongly regular graphs using "differences" of m-systems.

A symplectic or orthogonal space admitting a hyperbolic basis over a nite eld is tensored with its Galois conjugates to obtain a symplectic or orthogonal space over a smaller eld. A mapping between these spaces is dened which takes absolute points to absolute points. It is shown that caps go to caps. Combined with a result of Dye’s one obtains a simple proof of a result due to Blokhuis and Moorehouse that ovoids do not exist on hyperbolic quadrics in dimension ten over a eld of characteristic two. Let k = GF (q), q a prime power, and K = GF (q m ) for some positive integer